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# Wilson Loops in string duals of Walking and Flavored Systems.
Carlos Núñez, Maurizio Piai and Antonio Rago Swansea University, School of
Physical Sciences, Singleton Park, Swansea, Wales, UK
###### Abstract
We consider the VEV of Wilson loop operators by studying the behavior of
string probes in solutions of Type IIB string theory generated by $N_{c}$ $D5$
branes wrapped on an $S^{2}$ internal manifold. In particular, we focus on
solutions to the background equations that are dual to field theories with a
walking gauge coupling as well as for flavored systems. We present in detail
our walking solution and emphasize various general aspects of the procedure to
study Wilson loops using string duals. We discuss the special features that
the strings show when probing the region associated with the walking of the
field theory coupling.
###### pacs:
11.25.Tq,11.15.Tk
###### Contents
1. I Introduction
1. I.1 AdS/CFT and Wilson loops
2. I.2 Confinement and Screening
3. I.3 Walking technicolor
4. I.4 General idea: string-theory as a laboratory for walking dynamics.
5. I.5 Outline
2. II General theory
1. II.1 Equations of motion
1. II.1.1 Boundary conditions
2. II.1.2 Turning points
2. II.2 Energy and Separation of the $Q\bar{Q}$ pair.
1. II.2.1 Some exact results.
3. II.3 Leading and subleading behaviors. Inversion points
3. III Some well-understood examples
1. III.1 The case of $AdS_{5}\times S^{5}$
2. III.2 Witten-Sakai-Sugimoto Model
3. III.3 D5 branes on $S^{2}$
4. III.4 Klebanov-Strassler Model
4. IV Walking solutions in the $D5$ system, unflavored.
1. IV.1 General set-up.
2. IV.2 Walking solutions.
3. IV.3 Probes: numerical study.
4. IV.4 Comments on this section.
5. V Wilson Loop in a Field Theory with Flavors
1. V.1 The case $N_{f}=2N_{c}$.
6. VI Summary and Conclusions
7. A UV asymptotic solutions.
8. B Van der Waals gas.
## I Introduction
In this paper we want to study the behavior of non-local operators of gauge
theories, making use of the gauge-string correspondence. We are in particular
interested in a specific class of supergravity solutions that are closely
related to what goes under the name of walking in the field theory language.
In the introduction we summarize the basic notions and ideas that will feature
prominently in the paper: the treatment of Wilson loops in the gauge-string
correspondence, the concepts of confinement and screening in gauge theories
and the meaning of walking dynamics (with a view on its role within dynamical
electro-weak symmetry breaking).
It must be stressed that we do not know the precise nature of the field-theory
dual of some of the examples we are going to consider in the body of the
paper. The Wilson loop studied here is a very important quantity, that may
help identifying such dual theory.
### I.1 AdS/CFT and Wilson loops
According to general ideas of holography and more concretely to the Maldacena
conjecture Maldacena:1997re , a quantum conformal field theory in dimension
$d$ is equivalent to a quantum theory in $AdS_{d+1}$ space. In general, the
idea is that local operators in the CFT couple to fields in the $AdS$ side, in
such a way that correlators of conformal fields are related to amplitudes in
the quantum theory in $AdS$, as explained in Gubser:1998bc . For instance,
since the CFT-side of the equivalence must contain the energy-momentum tensor
$T_{\mu\nu}$ among its operators, there must be a field on the $AdS$-side that
couples to it. Since this field is the graviton, then the theory on
$AdS_{d+1}$ space must be a quantum theory of gravity.
One can use the correspondence to study non-local operators on the CFT-side.
In particular, if the field theory is a pure Yang-Mills theory, an example of
such operator is the Wilson loop Wilson:1974sk . These objects couple to
extended objects, excited in the $AdS$ side of the correspondence. The Wilson
loops (path ordered exponentials of the holonomy of the gauge field along a
curve ${\cal C}$) are one of the most interesting observables of such a gauge
theory,
$W({\cal C})\equiv\frac{1}{N_{c}}TrPe^{i\oint_{\cal C}A_{\mu}dx^{\mu}}.$ (1)
The loop itself and products of them provide a basis of gluonic gauge
invariant operators.
The Wilson loop along a curve ${\cal C}$ is computed in the dual string theory
by calculating the action of a string bounded by ${\cal C}$ at the boundary of
the $AdS$ space. More concretely111The fundamental string is actually dual to
the generalized Wilson loop, containing a term that couples the coordinates of
the internal space with the scalars of the brane. This is due to the fact that
the string ending on the brane is source of the electric field, generating
$A_{\mu}$ but also of the scalars on the brane, from which is pulling. See
Maldacena:1998im for a clear discussion of this.,
$\left\langle\frac{}{}W({\cal C})\right\rangle=\int_{\partial F({\cal
C})}{\cal D}Fe^{-S[F]}$ (2)
where $F$ denotes all the fields of the string theory and $\partial F$ their
boundary values. A good approximation to this path integral is by steepest
descent. The Wilson loop is then related to the area of the minimal surface
bounded by the curve ${\cal C}$, spanned by classical string configurations
(with Nambu-Goto action $S_{NG}$) that explore the bulk of $AdS$.
All of this was first proposed ten years ago in Maldacena:1998im . In the
meantime, this proposal motivated lots of developments, see bunchwilson for
beautiful and influential papers on this line. See also Sonnenschein:1999if
for a review. In particular, the ideas and techniques of the original
$AdS/CFT$ correspondence have been assumed to generalize to a large class of
systems, and have been used to relate field theories that are not conformal
(and hence more closely related to phenomenological applications) with
backgrounds that are not $AdS$, extending the framework to what is more
appropriately referred to as gauge-string duality.
### I.2 Confinement and Screening
In its original definition Wilson:1974sk , the Wilson loop computes the phase
factor associated to a closed trajectory for a very massive quark in the
fundamental representation (it can also be generalized to other
representations). The quark-antiquark (static) potential can be read from the
VEV of the Wilson loop. Choosing a rectangular loop of sides $L_{QQ},T$, in
the first approximation for large times $T\rightarrow\infty$,
$\left\langle\frac{}{}W({\cal C})\right\rangle\sim e^{-E_{QQ}T}\,,$ (3)
where $E_{QQ}$ is the quark-pair energy. In the limit of large ’t Hooft
coupling, the steepest descent approximation mentioned above yields the
identification
$\left\langle\frac{}{}W({\cal C})\right\rangle\sim e^{-E_{QQ}T}\sim
e^{-S_{NG}}\,.$ (4)
The description of the Wilson loop in QCD in terms of a string partition
function is not new. Well before its modern formulation, the ideas behind
gauge/string correspondence have been used to show that the potential of a
quark-antiquark pair separated by a distance $L_{QQ}$ gets a correction of the
form $\frac{c}{L_{QQ}}$ due to quantum fluctuations of the Nambu-Goto action
Luscher:1980fr .
The definition of confinement we will adopt is the following. Consider a
$SU(N_{c})$ gauge theory with matter fields in generic representations. We
decouple (make infinitely massive) all fields with non-zero N-ality, and then
introduce a single particle-antiparticle pair of non-dynamical fields with
non-zero N-ality as a test probe for the system (effectively, the pair
dynamics is quenched). We then compute the work needed to separate the
particle-antiparticle test pair up to a distance $L_{QQ}$. If the work
approaches $E_{QQ}\simeq\sigma L_{QQ}$ for large separations, the theory is
confining222Another equivalent way of defining confinement is by computing the
VEV of the Polyakov loop, that if vanishing indicates a confining theory.
Also, the perimeter law of a ’t Hooft loop indicates confinement.. The
quantity $\sigma$ is a representation dependent constant, the string tension.
### I.3 Walking technicolor
Walking technicolor WTC is a framework within which the phenomenological
difficulties of dynamical electro-weak symmetry breaking might find a very
natural and elegant dynamical solution, thanks to the fact that, in contrast
to QCD-like theories, the guidelines provided by naive dimensional analysis
are violated. This is because of large anomalous dimensions controlling the
dynamics over a large regime of energies. The word walking refers to the fact
that the new dynamics is strongly coupled over a large range of energies,
where its fundamental coupling exhibits a $\beta$-function which is
anomalously small in respect to the coupling itself. A behavior of this type
is expected in theories which flow onto strongly-coupled IR fixed-points, and
it is reasonable to assume that it persists also when such IR fixed points are
only approximate, though in this case a degree of ambiguity as to the meaning
of approximate invites some caution.
Besides being affected by the usual calculability limitations due to the
strong coupling (as in QCD and in QCD-like technicolor), the walking dynamics
itself makes this framework very hard to work with. New, non-perturbative
instrument are needed in order to understand the (effective) field theory
properties of a walking theory. Very recent years saw a lot of progress
towards a better understanding of walking dynamics both from the lattice
lattice and thanks to ideas borrowed from the gauge-string duality. See for
instance AdSTC for a list of references focused on the precision electro-weak
parameters.
In this paper, we will use the word walking to refer to backgrounds in which
there exists a interval in the $\rho$ radial direction over which the geometry
is determined by some coupling that shows an evolution with the radial
coordinate that is anomalously slow. This agrees with the standard definition
of walking, where the beta-function of the fundamental gauge-coupling is, over
some range of energy, anomalously small in comparison with what expected from
the strength of the actual coupling itself. Not all couplings show this
behavior: this also agrees with standard definitions, in the sense that
relevant operators must be present in order to drive the RG flow away from a
possible IR fixed point, and scale invariance is present only in the sense
that in the walking region the relevant couplings are so small that their
effect can be neglected. However, our definition is significantly less
restrictive that the usual one: we do not require the presence of an actual
fixed point in the flow, and correspondingly we do not have an approximately
AdS background, nor can we recover an AdS background by dialing some parameter
to some special value.
### I.4 General idea: string-theory as a laboratory for walking dynamics.
Motivated by the difficulties described in the previous subsection, in ref.
Nunez:2008wi a more general program is proposed, based upon gauge/string
correspondence in order to go beyond the low-energy effective field theory
description. The proposal is to study the dynamics of theories that yield
walking behavior, but that are not necessarily related to the electro-weak
symmetry. In short, one would like to use string theory as a laboratory in
which to study the general properties of walking dynamics by itself, in
isolation from its complicated realization within an explicit model of
dynamical electro-weak symmetry breaking. In ref. Nunez:2008wi , it is shown
that in the context of Type IIB string theory on a background generated by a
stack of $N_{c}$ $D5$ branes, there exists a very large class of solutions to
the background equations for which a suitably defined gauge coupling exhibits
the basic properties of a putative walking theory. The running of the gauge
coupling flattens over a large range of intermediate energies, but restarts at
low energies, until the space ends into a (good) singularity in the deep IR,
so that no exact IR fixed point exists. The fact that this is not a walking
technicolor theory (there is no electro-weak symmetry in the set-up, and hence
no mass generation in the usual sense), together with the large-$N_{c}$
expansion, yields the advantage that we avoid the complications due to mixing
of weakly-coupled and strongly-coupled properties of the theory. For instance,
the spectrum of the spin-0 sector of the theory can be studied, and has been
studied Elander:2009pk , yielding remarkable surprises.
In this paper we take a further step in this direction, by studying the
behavior of the Wilson loop in backgrounds of this class. As we will explain,
we can use the techniques developed in the context of gauge-string duality, by
studying the background with a probe string. In particular, we will study
Wilson loops in the dual walking QFT. For technical reasons that will be
explained in the body of the paper, in order for this program to be carried
out we will also need to generalize further the class of backgrounds in
Nunez:2008wi . These new solutions have been already introduced in
Elander:2009pk . We explain here in deeper detail how to generate them,
characterize them, and relate them to the literature.
### I.5 Outline
The paper is organized as follows: we set up notation and introduce a set of
important ideas in section II revising the bibliography and adding important
new ingredients and derivations. Then we apply these to well-established
examples of gauge-string duality in section III, providing a simple and
compact set of exercises that are intended to yield some guidance in the
following sections, in which the dynamics is far from well-understood. Section
IV presents our new walking solution and a study of the dynamics of the Wilson
loop as a function of the length of the walking region. Section V studies the
results derived in section II when applied to background that encode the
dynamics of fundamental fields. We then conclude in section VI.
## II General theory
In this section we present general results for Wilson loops, computed using
the ideas of Maldacena:1998im . Some of the results here have been derived
long ago (see for example Kinar:1998vq ), but our approach will be different
and some new and useful points will be specially emphasized.
We study the action for a string in a background of the generic form
$ds^{2}=-g_{tt}dt^{2}+g_{xx}d\vec{x}^{2}+g_{\rho\rho}d\rho^{2}+g_{ij}d\theta^{i}d\theta^{j}.$
(5)
We assume that the functions $(g_{tt},g_{xx},g_{\rho\rho})$ depend only on the
radial coordinate $\rho$. By contrast, $g_{ij}$ for the internal (typically
compact) space can also depend on other coordinates. Whatever are the internal
coordinates, they will play no role in what follows. This is because we will
choose a configuration for a probe string that is not excited on the
$\theta^{i}$ directions, hence in what follows, we will ignore the internal
space 333Strings or other objects that extend in the internal space filling
part of it can be treated as an effective string, analogous to the one we are
studying. If these objects are allowed to vibrate in the internal space, then
a generalization of the present treatment should be done..
### II.1 Equations of motion
The configuration we choose is,
$t=\tau,\;\;\;\;x=x(\sigma),\;\;\;\;\rho=\rho(\sigma).$ (6)
and compute the Nambu-Goto action
$S=\frac{1}{2\pi\alpha^{\prime}}\int_{[0,T]}d\tau\int_{[0,2\pi]}d\sigma\sqrt{-\det
G_{\alpha\beta}}.$ (7)
The induced metric on the 2-d world-volume is
$G_{\alpha\beta}=g_{\mu\nu}\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}$,
where
$G_{\tau\tau}=-g_{tt},\;\;\;G_{\sigma\sigma}=g_{xx}(\frac{dx}{d\sigma})^{2}+g_{\rho\rho}(\frac{d\rho}{d\sigma})^{2}\,.$
(8)
Defining for convenience $f(\rho)^{2}\equiv
g_{tt}g_{xx},\;g(\rho)^{2}=g_{tt}g_{\rho\rho}$, the Nambu-Goto action is
$S=\frac{T}{2\pi\alpha^{\prime}}\int_{0}^{2\pi}d\sigma\sqrt{f^{2}x^{\prime}(\sigma)^{2}+g^{2}\rho^{\prime}(\sigma)^{2}}\,\equiv\,\frac{T}{2\pi\alpha^{\prime}}\int_{0}^{2\pi}d\sigma
L\,.$ (9)
Notice that we consider the situation in which the string does not couple to a
NS $B$-field.
We first compute the Euler-Lagrange equations from Eq. (7) and then we specify
them for the ansatz in Eq. (6). We get that for the $(t,x,\rho)$ coordinates
the eqs. of motion read, respectively,
$\displaystyle\partial_{\tau}\Big{[}\frac{1}{L}(f^{2}x^{\prime
2}+g^{2}\rho^{\prime 2})\Big{]}=0\,,$
$\displaystyle\partial_{\sigma}\Big{[}\frac{1}{L}f^{2}x^{\prime}\Big{]}=0\,,$
(10)
$\displaystyle\partial_{\sigma}\Big{[}\frac{1}{L}g^{2}\rho^{\prime}\Big{]}=\frac{1}{L}(x^{\prime
2}ff^{\prime}+\rho^{\prime 2}gg^{\prime})\,,$
where $X^{\prime}=\frac{dX(\sigma)}{d\sigma}$ for any function $X$.
The first equation in (10) is solved because we assume a background metric
independent of time (we consider the system at the equilibrium). The second
equation in (10) is solved if the quantity inside brackets is a constant (that
we call $C$), which implies that,
$\frac{d\rho}{d\sigma}=\pm\Big{(}\frac{dx}{d\sigma}\Big{)}\Big{(}\frac{f(\rho)}{Cg(\rho)}\Big{)}\sqrt{f^{2}(\rho)-C^{2}}$
(11)
One can check that the third equation in (10) is solved if Eq. (11) is
imposed. Then, we need to work with just one equation. Defining
$\displaystyle V_{eff}(\rho)$ $\displaystyle\equiv$
$\displaystyle\frac{f(\rho)}{Cg(\rho)}\sqrt{f^{2}(\rho)-C^{2}}\,,$ (12)
we write it as
$\frac{d\rho}{d\sigma}=\pm\frac{dx}{d\sigma}V_{eff}(\rho)\leftrightarrow\frac{d\rho}{dx}=\pm
V_{eff}(\rho)\,.$ (13)
Another way to arrive to the last version of Eq. (13) is to consider a
restricted ansatz for the string configuration
$[t=\tau,\;x=\sigma,\;\;\rho=\rho(\sigma)]$ and use the conserved Hamiltonian
derived from Eq. (9) to get an expression for $\rho(\sigma)$ that is precisely
Eq. (13).
The kind of solution we are interested in can be depicted as follows: a string
that hangs from infinite radial position at $x=0$ and drops down towards
smaller $\rho$ as $x$ increases. Once it arrives at the smallest $\rho$
compatible with the solution, namely $\rho_{0}$, it starts growing in the
radial direction up to infinite $\rho$ where $x=L_{QQ}$, see Fig. (1)444For
future reference we refer to the lowest end of the radial coordinate as
$\hat{\rho}_{0}$..
Figure 1: Setting of the string.
This means that in the two distinct regions $x<L_{QQ}/2$ and $x>L_{QQ}/2$ the
equations of motion will differ only in a sign
$\displaystyle x<\frac{L_{QQ}}{2}$
$\displaystyle\frac{d\rho}{dx}=-V_{eff}(\rho)$ $\displaystyle
x>\frac{L_{QQ}}{2}$ $\displaystyle\frac{d\rho}{dx}=V_{eff}(\rho)$ (14)
We can now formally integrate the equations of motion
$\displaystyle
x(\rho)=\begin{cases}\int_{\rho}^{\infty}\frac{d\rho}{V_{eff}(r)}&\quad
x<\frac{L_{QQ}}{2}\\\
L_{QQ}-\int_{\rho}^{\infty}\frac{d\rho}{V_{eff}(r)}&\quad
x>\frac{L_{QQ}}{2}\end{cases}$ (15)
or more compactly
$\left|x(\rho)-\frac{L_{QQ}}{2}\right|=\int_{\rho_{0}}^{\rho}\frac{dr}{V_{eff}(r)}\,.$
(16)
In what follows we will use only one of the solutions in Eq. (II.1) unless
explicitly noted.
#### II.1.1 Boundary conditions
We need to specify the boundary conditions for the string in Eq. (6). This is
an open string, vibrating in the bulk of a closed string background. Following
the ideas of Maldacena:1998im , we add a D-brane at a very large radial
distance where the open string will end. This string will then satisfy a
Dirichlet boundary condition at $\rho\rightarrow\infty$. This means that for
large values of the radial coordinate $\frac{dx}{d\sigma}$ must vanish. The
only way of satisfying the equation of motion in Eq. (11) for
$\rho\rightarrow\infty$, given that the left hand side has to be non
vanishing, is to have a divergent $V_{eff}(\rho)$:
$\displaystyle\lim_{\rho\rightarrow\infty}V_{eff}(\rho)$ $\displaystyle=$
$\displaystyle\infty\,.$ (17)
This implies that there are restrictions on the asymptotic behavior of the
background functions [$f(\rho),g(\rho)$] in order for the string proposed in
Eq. (6) to exist. We will come back to this in the following sections. Before
proceeding, a brief digression is needed. When studying Eqs. (13)-(17) the
reader may find unsatisfactory that the restriction we have proposed above,
namely
$\left.\frac{d\rho}{dx}\right|_{\rho\rightarrow\infty}=\left.V_{eff}\right|_{\rho\rightarrow\infty}\rightarrow\infty$
(18)
looks dependent of our choice of the radial coordinate. This is actually not
the case, because we could rewrite this restriction in a more covariant form
in the following way. We define a couple of vectors 555We thank Johannes
Schmude for the discussions that lead to this. in the $x,\rho$ directions,
$\vec{v}^{x}\equiv\frac{dx}{d\sigma}\partial_{x},\;\;\;\vec{v}^{\rho}\equiv\frac{d\rho}{d\sigma}\partial_{\rho}\,,$
(19)
compute the ratio $\mu$ of their norms and impose that this is divergent on
the surface ${\cal D}$ on which the string has to satisfy the Dirichlet
condition:
$\mu=\left.\frac{g_{\rho\rho}(\frac{d\rho}{d\sigma})^{2}}{g_{xx}(\frac{dx}{d\sigma})^{2}}\right|_{{\cal
D}}\rightarrow\infty.$ (20)
Combining this with Eq. (13), and evaluating at the boundary
$\mu=\left.\frac{g_{\rho\rho}}{g_{xx}}V_{eff}^{2}\right|_{{\cal
D}}=\frac{f^{2}({\cal D})-C^{2}}{C^{2}}\rightarrow\infty.$ (21)
This last expression is free of coordinate ambiguities, as it comes from
operating with invariants (norms of vectors). It is however easier to work
within a specific choice of coordinates, which we will do in the body of the
paper. As we will see explicitly, this occurs in the examples we will study
below.
#### II.1.2 Turning points
Once Eq. (17) is satisfied the string will move to smaller values of the
radial coordinate down to a turning point $\rho_{0}$ where the quantity
$\frac{d\rho}{dx}(\rho_{0})=0$, i. e. $V_{eff}=0$. In principle, there could
be points where $V_{eff}$ vanishes because of either isolated zeros of
$f(\rho)$, or diverging point of $g(\rho)$. However, we will not consider this
kind of inversion points, since we are interested in solutions of the
equations of motion that allow the string to probe the entire radial
direction. Hence the turning point can be placed in any possible $\rho_{0}$,
with $\hat{\rho}_{0}<\rho_{0}<\infty$ (where $\hat{\rho}_{0}$ is the end of
the space). Thus we will restrict ourselves to forms of $V_{eff}$ where the
inversion point is given by imposing $C=f(\rho_{0})$. Furthermore, in the next
section we will also impose that the envelop of the string is convex in a
neighbourhood of $\rho_{0}$, hence insuring that gauge theory quantities like
the separation between the pair of quarks and its energy will be continuous
functions of $\rho_{0}$. It is then clear from Eq. (13) that $V_{eff}(\rho)$
controls not only the boundary condition at infinity, but also the possibility
for the string to turn around and come back to the brane at infinity.
### II.2 Energy and Separation of the $Q\bar{Q}$ pair.
We now follow the standard treatment for Wilson loops summarized in
Sonnenschein:1999if . If our probe-string hangs from infinity, turns around at
a point $\rho_{0}$ as described above and goes back to the $D$-brane at
infinity, we can then compute gauge theory quantities, like the separation
between the two ends of the string, which can be thought of as the separation
between a quark-antiquark pair living on the $D$-brane and coupled to the end-
points of the string. And we can compute the Energy of the pair of quarks,
that we associate with the length of the string (computed along its path in
the bulk). Both of these quantities will be functions of the turning point
$\rho_{0}$.
The standard expressions that we will use can be derived easily. Indeed, for
the $Q\bar{Q}$ separation we only need to compute $\int dx$. To calculate the
energy of the $Q\bar{Q}$ pair, we compute the action of the string and
substract the action of two ‘rods’ that would fall from infinity to the end of
the space 666Notice that these ‘rods’ need not be strings, it is just a way of
renormalizing the infinite mass of the quarks.. The results are
Sonnenschein:1999if ,
$\displaystyle
L_{QQ}(\rho_{0})=2f(\rho_{0})\int_{\rho_{0}}^{\infty}\frac{g(z)}{f(z)}\frac{dz}{\sqrt{f^{2}(z)-f^{2}(\rho_{0})}},$
$\displaystyle
E_{QQ}(\rho_{0})=f(\rho_{0})L_{QQ}(\rho_{0})+2\int_{\rho_{0}}^{\infty}\frac{g(z)}{f(z)}\sqrt{f^{2}(z)-f^{2}(\rho_{0})}dz-2\int_{\hat{\rho}_{0}}^{\infty}g(z)dz\,.$
(22)
As discussed above, the constant $C$ defined around Eq. (11) must be taken to
be $C=f(\rho_{0})$. Using Eq. (12) we can rewrite
$L_{QQ}(\rho_{0})=2\int_{\rho_{0}}^{\infty}\frac{dz}{V_{eff}(z)}$ (23)
As discussed in section II.1.1, we must impose that for large values of the
radial coordinate $V_{eff}$ diverges. This, however, is not enough to ensure
that the integral above receives a finite contribution from the upper end of
the integral, we then have to require that for large radial coordinate
$V_{eff}$ diverges at least as
$V_{eff}\underset{\rho\rightarrow\infty}{\sim}\rho^{\beta}\,,$ (24)
with $\beta>1$. Then, the quantity $L_{QQ}$ can be finite or infinite
depending on the IR ($\rho\rightarrow\rho_{0}\equiv\hat{\rho}_{0}$)
asymptotics of the background, meaning that we consider the turning point at
the end of the space at the lower end of the integral. If, expanding around
the turning point $\rho_{0}$, we have
$V_{eff}\underset{\rho\rightarrow\rho_{0}}{\sim}(\rho-\rho_{0})^{\gamma}\,,$
(25)
then the separation of the quark pair is infinite for $\gamma\geq 1$ and
finite for $\gamma<1$. We will see examples of both behaviors in the following
sections.
Another reasonable condition that we could impose is that $\rho_{0}$ is the
first zero of $V_{eff}$ for which the string has positive convexity (a
minimum). This can be easily expressed as a condition on the background
functions. In fact, let us assume that near $\rho_{0}$ the effective potential
in Eq. (12) behaves as $V_{eff}=\kappa(\rho-\rho_{0})^{\gamma}$. Then, in a
neighbourhood of $\rho_{0}$, we have
$\displaystyle\frac{d\rho}{dx}=\pm
V_{eff}=\pm\left(\kappa(\rho-\rho_{0})^{\gamma}\right)+\mathcal{O}(\rho-\rho_{0})^{\gamma+1}$
$\displaystyle\frac{d^{2}\rho}{dx^{2}}=\frac{d}{dx}(\frac{d\rho}{dx})=\pm\frac{dV_{eff}(\rho)}{d\rho}\frac{d\rho}{dx}=V_{eff}(\rho)\frac{dV_{eff}(\rho)}{d\rho}=\kappa^{2}\gamma(\rho-\rho_{0})^{2\gamma-1}+\mathcal{O}(\rho-\rho_{0})^{2\gamma},$
(26)
by iteration and induction we obtain (we choose only one of the branches of
Eq. (II.1)),
$\frac{d^{n}\rho}{dx^{n}}=(V_{eff}(\rho)\frac{d}{d\rho})^{n}\rho=\kappa^{n}\Pi_{j=0}^{n-1}\Big{[}j(\gamma-1)+1\Big{]}(\rho-\rho_{0})^{(\gamma-1)n+1}+\mathcal{O}(\rho-\rho_{0})^{(\gamma-1)n+2}.$
(27)
In order to have a minimum we impose that the first non vanishing derivative
at $\rho=\rho_{0}$ is an even derivative (its value has to be positive). So,
for even $n$ we need (this result will not depend on the choice of branches
above),
$\gamma=1-\frac{1}{n},$ (28)
from this it follows that $\gamma$ cannot be bigger than one. This means that
the only possibility of obtaining a string that can be stretched up to
infinite distance will come from the case $\gamma=1$ (contrast this with the
result of Kinar:1998vq ). The solutions with infinite length will have all
even derivatives vanishing at $\rho_{0}$. Had we considered non-integer
corrections to the leading term in Eq. (26), so that near $\rho_{0}$ the
function
$V_{eff}=\kappa_{1}(\rho-\rho_{0})^{\gamma}+\kappa_{2}(\rho-\rho_{0})^{\gamma+\epsilon}$
(29)
for very small values of $\epsilon$ we would have had a formula like the one
in Eq. (27) where the exponent is the same, but the coefficient changes. For
large values of $\epsilon$, we have, to leading order, the same result as in
Eq. (27). The reasoning given above applies.
These constraints ensure that there exists a unique trajectory for the string
as function of $\rho_{0}$. It should be also noticed that the profile of the
string is not analytic function and in particular in $\rho_{0}$ it turns out
to be a $\mathcal{C}^{2}$ function.
#### II.2.1 Some exact results.
Next, let us derive an expression of the energy $E_{QQ}$ as function of the
inter-quark separation $L_{QQ}$. For this purpose, it is useful to introduce
the function $K[x]=\frac{1}{\sqrt{x^{2}-1}}$. The expression for
$L_{QQ}(\rho_{0})$ in Eq. (22) reads777Some of these expressions have been
derived in past collaborations with Angel Paredes.
$L_{QQ}(\rho_{0})=\lim_{\rho_{1}\rightarrow\infty}2\int_{\rho_{0}}^{\rho_{1}}\frac{g(z)}{f(z)}K\left[\frac{f(z)}{f(\rho_{0})}\right]{dz}\,.$
(30)
Performing the derivative
$\frac{dL_{QQ}}{d\rho_{0}}=-2\frac{g(\rho_{0})}{f(\rho_{0})}K[1]+\lim_{\rho_{1}\rightarrow\infty}2\int_{\rho_{0}}^{\rho_{1}}\frac{g(z)}{f(z)}\partial_{\rho_{0}}K\left[\frac{f(z)}{f(\rho_{0})}\right]$
(31)
Now, we use the following identity,
$\partial_{\rho_{0}}K\left[\frac{f(z)}{f(\rho_{0})}\right]=-\partial_{z}K\left[\frac{f(z)}{f(\rho_{0})}\right]\,\left(\frac{f(z)f^{\prime}(\rho_{0})}{f^{\prime}(z)f(\rho_{0})}\right)$
(32)
integrating by parts and after some algebra, we get
$\frac{dL_{QQ}}{d\rho_{0}}=-\lim_{\rho_{1}\rightarrow\infty}2\partial_{\rho_{0}}\log
f(\rho_{0})\left\\{\frac{g(\rho_{1})}{f^{\prime}(\rho_{1})}K\left[\frac{f(\rho_{1})}{f(\rho_{0})}\right]-\int_{\rho_{0}}^{\rho_{1}}dz~{}K\left[\frac{f(z)}{f(\rho_{0})}\right]\partial_{z}\left(\frac{g(z)}{f^{\prime}(z)}\right)\right\\}\,,$
(33)
or
$\frac{dL_{QQ}}{d\rho_{0}}=2\partial_{\rho_{0}}\log
f(\rho_{0})\lim_{\rho_{1}\rightarrow\infty}\left\\{\int_{\rho_{0}}^{\rho_{1}}dz~{}K\left[\frac{f(z)}{f(\rho_{0})}\right]\partial_{z}\left(\frac{g(z)}{f^{\prime}(z)}\right)-\frac{f(\rho_{1})}{f^{\prime}(\rho_{1})V_{eff}(\rho_{1})}\right\\}\,.$
(34)
Similarly, we compute the derivative for the energy function finding that
$\frac{dE_{QQ}}{d\rho_{0}}=f(\rho_{0})\frac{dL_{QQ}}{d\rho_{0}}\,,$ (35)
and we get
$\frac{dE_{QQ}}{dL_{QQ}}=f(\rho_{0})\,.$ (36)
Notice that (after integration) Eq. (36) is an exact expression for $E_{QQ}$
in terms of $L_{QQ}$, the non triviality residing in the function
$\rho_{0}(L_{QQ})$, this expression was also derived in Brandhuber:1999jr .
Also, Eq. (36) yields the force between the quark pair. It should be stressed
that we are assuming that we will be able to invert $\rho_{0}$ as function of
$L_{QQ}$. Whenever this cannot be done, the solution would be valid only in
the regions of monotonicity of $L_{QQ}(\rho_{0})$.
Let us comment briefly about the existence of cusps in our strings. If near
the lower point of the string (typically situated in the end of the geometry)
the quantity $V_{eff}(\rho)$ diverges, then there will be a cusp in the shape
of the string. This can be easily understood using Eq. (13) and an expansion
of $V_{eff}(\rho)$ of the form discussed in Eq. (16). If
$V_{eff}(\rho)\propto(\rho-\rho_{0})^{\gamma}$ with $\gamma<0$, one can
integrate the equation
$\rho-\rho_{0}\propto\left[(1-\gamma)\left|x-\frac{L_{QQ}}{2}\right|\right]^{\frac{1}{1-\gamma}},$
(37)
which implies that the shape of the string $\rho(x)$ is not analytic at
$\frac{L_{QQ}}{2}$.
### II.3 Leading and subleading behaviors. Inversion points
Let us focus on the case that, near the end of the space (in the IR),
$V_{eff}\sim(\rho-\rho_{0})$ (the string stretches up to infinite length
$L_{QQ}(\hat{\rho}_{0})\rightarrow\infty$). The subleading corrections to the
Energy of the quark pair Kinar:1998vq read
$E_{QQ}=f(\hat{\rho}_{0})L_{QQ}+\kappa+O(e^{-|a|L_{QQ}}(\log L_{QQ})^{b})\,.$
(38)
This formula can be obtained as an expansion around
$\rho_{0}\rightarrow\hat{\rho}_{0}$, or equivalently expanding Eq. (36) around
$L_{QQ}\rightarrow\infty$. Notice that no power-law corrections or Luscher-
terms appear in these corrections: they would appear if $\gamma>1$ in Eq.
(28), something that we ruled out. Contrast this with the result of
Kinar:1998vq . Because Eq. (28) and the discussion around it rely only on the
generic properties of $V_{eff}$, such power-law corrections can only descend
from higher-derivative corrections to Eq. (7), which cannot be repackaged into
the form of Eq. (13).
In the following sections, we will discuss these subleading behaviors in
various examples and precisely find the coefficients that apply to each
particular case. Before proceeding, a few comments are due in order to avoid
ambiguities.
The partition function of the string is studied by using two different
expansions, in $\alpha^{\prime}$ and $g_{s}$. First, the Nambu-Goto action is
characterized by the string tension, and one can think of expanding any
physical quantity (correlation function) in powers of $\alpha^{\prime}$. This
procedure is effectively equivalent to quantizing the (particle-like)
excitations of the string, and in this sense $\alpha^{\prime}$ corrections can
be associated with loops of the dynamics of the string modes. One can also
rephrase the results in terms of a classical effective theory, in which the
$\alpha^{\prime}$ corrections are encoded in higher derivative corrections to
the Nambu-Goto classical action. By doing so, one can associate the resulting
$E_{QQ}$ to the expectation value of a Wilson loop in the dual theory, and
hence interpret the generalization of Eq. (38) in terms of the actual quantum
corrections to the quark-antiquark potential at large distance, for a
confining string, as done for example in Aharony:2009gg . As explained above,
this procedure yields power-law corrections to the leading-order result
$E_{QQ}\propto L_{QQ}$. This is not what we are doing in this paper. All the
results we obtain are at the leading-order in $\alpha^{\prime}$, the
calculations being completely classical and based on the Nambu-Goto action. In
particular, this explains why we do not find a Luscher term.
We are going to truncate at the leading-order also the expansion in $g_{s}$,
which controls processes where the string breaks. This is also done in
Aharony:2009gg , where the quantum corrections that are computed are the ones
in $\alpha^{\prime}$, but the whole analysis treats the string itself as
effectively free. Not only quantum corrections related to the $g_{s}$
expansion, but the whole many-body nature of string-theory is neglected in
this way. This is a very important limitation: strings that can stretch up to
infinite separation for the quark pair can be treated in this way, while
theories in which fragmentation and hadronization take place via string-
breaking are accessible to our approach only up to a scale smaller than the
scale of breaking itself (real world QCD, if it admits a string dual
description, should fall into this class in which $g_{s}$ corrections must be
included and actually dominate the dynamics from the scale of breaking and
below).
We will see in the following examples in which Eq. (25) holds with $\gamma<1$.
This will yield a finite value for the maximal displacement in the Minkowski
directions between the quark-antiquark pair. Studying the large-$L_{QQ}$ limit
would necessarily require the inclusion both of quantum effects in the
$\alpha^{\prime}$ expansion, but also the effect of string breaking, which
might be decoupled in the large quark-mass limit.
Another important comment is the following: if the function $L_{QQ}(\rho_{0})$
is not monotonic, then it is not invertible, and $\rho_{0}(L_{QQ})$ is
multivalued. If this is the case, it will happen that for some given $L_{QQ}$
we can find that also $E_{QQ}(L_{QQ})$ is multivalued. Among all the possible
values of $E_{QQ}$ for a given $L_{QQ}$ (all of which satisfy the equations of
motion) we will refer to the lowest one as the stable solution, while the
others will represent either excited or unstable states. The Van der Waals
liquid-gas system provides a nice realization of these excited and unstable
states, and we report its treatment in Appendix B.
The existence of inversion points (or extrema of $L_{QQ}(\rho_{0})$) implies
that the first derivative $\frac{dL_{QQ}}{d\rho_{0}}$ vanishes. Using the
expression in Eq. (34),
$\lim_{\rho_{1}\rightarrow\infty}\left\\{\int_{\rho_{0}}^{\rho_{1}}dz~{}K\left[\frac{f(z)}{f(\rho_{0})}\right]\partial_{z}\left(\frac{g(z)}{f^{\prime}(z)}\right)-\frac{f(\rho_{1})}{f^{\prime}(\rho_{1})V_{eff}(\rho_{1})}\right\\}=0.$
(39)
If it happens that
$\frac{f(\rho_{1})}{f^{\prime}(\rho_{1})V_{eff}(\rho_{1})}\rightarrow 0$ for
large values of $\rho_{1}$, then a necessary condition for the existence of
turning points is that the integral in Eq. (34) vanishes, or more simply that
the integrand changes sign. Notice, nevertheless, that this is a not a
criterium of practical application except for some particular easy backgrounds
as we will see in section V.
To close this section let us stress that given the constraints we have
specified along this section, the shape of the string that we are proposing as
solution is the only allowed shape, since there can be only one minimun of
$\rho(x)$.
## III Some well-understood examples
In this section we illustrate the points of the previous section in a set of
very well known examples. All of them are string-theory backgrounds that are
simple enough that the dynamics of the probe string can be treated
analytically. We will write the 10-dimensional background in string frame.
### III.1 The case of $AdS_{5}\times S^{5}$
This is certainly the main example, where the conjecture was originally
proposed Maldacena:1997re and the ideas for Wilson loops described in the
introduction first developed Maldacena:1998im . After a rescaling of the
radial coordinate, the metric reads,
$ds^{2}=\alpha^{\prime}\Big{[}\frac{\rho^{2}}{R^{2}}dx_{1,3}^{2}+\frac{R^{2}}{\rho^{2}}d\rho^{2}+R^{2}d\Omega_{5}^{2}\Big{]}\,,$
(40)
where $R^{2}=\sqrt{4\pi g_{s}N_{c}}$ is dimensionless, $\rho$ has dimensions
of inverse-length, and the constant $\alpha^{\prime}$ in front of the metric
ensures that when we compute the functions $f,g$ defined below Eq. (8) and
plug into Eq. (7) the factors of $\alpha^{\prime}$ will cancel (this will
happen in the examples discussed below also). Hence we have,
$\begin{cases}g(\rho)^{2}=\alpha^{\prime\,2}&\\\
f(\rho)^{2}=\alpha^{\prime\,2}\frac{\rho^{4}}{R^{4}}&\\\
C^{2}=f(\rho_{0})^{2}=\alpha^{\prime\,2}\frac{\rho_{0}^{4}}{R^{4}}&\end{cases}V_{eff}=\frac{\rho^{2}}{\rho_{0}^{2}R^{2}}\sqrt{\rho^{4}-\rho_{0}^{4}}$
(41)
One can check that the functions of the background respect all the constraints
we imposed in section II regarding the boundary conditions and convexity. We
can exactly integrate $L_{QQ}(\rho_{0})$
$L_{QQ}(\rho_{0})=2\int_{\rho_{0}}^{\infty}d\rho\frac{R^{2}\rho_{0}^{2}}{\rho^{2}\sqrt{\rho^{4}-\rho_{0}^{4}}}=(2\pi)^{\frac{3}{2}}\frac{R^{2}}{\rho_{0}\Gamma(\frac{1}{4})^{2}}$
(42)
Since it is possible to invert the relation we can then write
$\rho_{0}(L_{QQ})$, and using Eq. (36)
$\frac{dE_{QQ}}{dL_{QQ}}=\frac{(2\pi)^{3}R^{2}}{L_{QQ}^{2}\Gamma(\frac{1}{4})^{4}}\Rightarrow
E_{QQ}(L_{QQ})=-\frac{(2\pi)^{3}R^{2}}{L_{QQ}\Gamma(\frac{1}{4})^{4}}$ (43)
in agreement with the result of Maldacena:1998im . Of course, the separation
for the quark pair diverges for $\rho_{0}\rightarrow 0$ (the end of the
space). Also notice that in this case, the expression of $L_{QQ}(\rho_{0})$ is
invertible. There are no corrections to Eq. (43).
A few words of comment might be useful to a reader who is not familiar with
this set-up. The Yang-Mills coupling of the dual ${\cal N}=4$ field theory is
defined as $g_{YM}^{2}\equiv 2\pi g_{s}$, so that the (dimensionless)
curvature $R^{4}$ is proportional to the ’t Hooft coupling in the CFT. The
supergravity approximation holds when $R^{4}\gg 1$.
### III.2 Witten-Sakai-Sugimoto Model
This model, based on $D4$ branes wrapped on a circle with SUSY breaking
periodicity conditions Witten:1998zw received a great deal of attention
thanks to the observation by Sakai and Sugimoto Sakai:2004cn that the
introduction of $N_{f}\ll N_{c}$ flavor $D8$ branes as probes allows to
construct a model where a peculiar realization of chiral symmetry is
spontaneously broken. The metric reads
$\frac{ds^{2}}{\alpha^{\prime}}=\left(\frac{\rho}{R}\right)^{3/2}\left[\frac{}{}dx_{1,3}^{2}+\hat{F}(\rho)dx_{4}^{2}\right]+\left(\frac{R}{\rho}\right)^{3/2}\left[\frac{}{}\frac{d\rho^{2}}{\hat{F}(\rho)}+\rho^{2}d\Omega_{4}^{2}\right]\,,$
(44)
where $x_{4}$ is the coordinate on the circle, $R^{3}=\pi
g_{s}\sqrt{\alpha^{\prime}}N_{c}$, and
$\hat{F}(\rho)=\frac{\rho^{3}-\Lambda^{3}}{\rho^{3}}$. Notice that in this
case the gauge coupling
$g_{YM}^{2}\equiv(2\pi)^{2}g_{s}\sqrt{\alpha^{\prime}}$ is dimensionful, so
that $\rho$ has dimensions of inverse-length, as in the previous subsection,
and so does $\Lambda$. The relevant functions are,
$\begin{cases}f^{2}(\rho)=\alpha^{\prime\,2}\frac{\rho^{3}}{R^{3}}&\\\
g^{2}(\rho)=\alpha^{\prime\,2}\frac{\rho^{3}}{\rho^{3}-\Lambda^{3}}&\\\
C^{2}=f(\rho_{0})^{2}=\alpha^{\prime\,2}\frac{\rho_{0}^{3}}{R^{3}}&\end{cases}V_{eff}=\sqrt{\frac{(\rho^{3}-\Lambda^{3})(\rho^{3}-\rho_{0}^{3})}{(\rho_{0}R)^{3}}}$
(45)
hence,
$L_{QQ}=2(\rho_{0}R)^{3/2}\int_{\rho_{0}}^{\infty}\frac{d\rho}{\sqrt{(\rho^{3}-\rho_{0}^{3})(\rho^{3}-\Lambda^{3})}}$
(46)
One can explicitly perform the integral above, but the result, in terms of
special functions, is not very illuminating. To get a better handle on the
underlying dynamics, we consider the case in which $\rho_{0}=\Lambda$. In this
case, we get
$L_{QQ}(\rho_{0})=-\frac{2\sqrt{R^{3}\rho_{0}^{3}}}{6\rho_{0}^{2}}\left.\left[\sqrt{12}\arctan\left(\frac{2\rho+\rho_{0}}{\sqrt{3}\rho_{0}}\right)+\log\left(\frac{\rho^{2}+\rho\rho_{0}+\rho_{0}^{2}}{(\rho-\rho_{0})^{2}}\right)\right]\right|_{\rho_{0}}^{\infty}$
(47)
that we can see diverges logarithmically for $\rho=\rho_{0}$. So, the string
here again, has infinite length as is expected in a dual of a confining field
theory. If $\rho_{0}>\Lambda$ then the separation of the pair is computed from
the integral of Eq. (46) and turns out to be finite.
On the other hand, it is possible to iteratively invert the relation between
$L_{QQ}$ and $\rho_{0}$ for $\rho_{0}\sim\Lambda$ and hence for
$L_{QQ}\rightarrow\infty$
$\rho_{0}(L_{QQ})=\Lambda+4\sqrt{3}e^{-\frac{\pi}{2\sqrt{3}}-\frac{3L_{QQ}\sqrt{\Lambda}}{4R^{\frac{3}{2}}}}\Lambda+\dots$
(48)
and from Eq. (36) we get for the Energy of the pair $E_{QQ}$ in terms of the
separation $L_{QQ}$,
$E_{QQ}(L_{QQ})\underset{L_{QQ}\rightarrow\infty}{=}L_{QQ}\left(\frac{\Lambda}{R}\right)^{\frac{3}{2}}-O(e^{-\frac{\pi}{2\sqrt{3}}-\frac{3L_{QQ}\sqrt{\Lambda}}{4R^{\frac{3}{2}}}}\Lambda)+\dots$
(49)
### III.3 D5 branes on $S^{2}$
In this example the system consists of $N_{c}$ $D5$ branes that wrap a two
cycle inside the resolved conifold preserving four supercharges (${\cal N}=1$
in four dimensions) Maldacena:2000yy . After a geometrical transition takes
place, a background dual to this field theory contains a metric, dilaton and
RR three form. In this case $g_{YM}^{2}\equiv(2\pi)^{3}g_{s}\alpha^{\prime}$.
We quote only the part of the metric relevant to this computation (after a
rescaling of the Minkowski coordinates is done)
$\frac{ds^{2}}{g_{s}\alpha^{\prime}N_{c}}=e^{\phi}\Big{[}dx_{1,3}^{2}+d\rho^{2}+ds_{int}^{2}\Big{]}\,.$
(50)
The relevant functions read,
$\begin{cases}g(\rho)=f(\rho)=e^{\phi}\,g_{s}\alpha^{\prime}N_{c}&\\\
e^{2\phi}=\frac{e^{2\phi_{0}}\sinh(2\rho)}{2\sqrt{\rho\coth(2\rho)-\frac{\rho^{2}}{\sinh(2\rho)^{2}}-\frac{1}{4}}}\end{cases}V_{eff}=e^{-\phi(\rho_{0})}\sqrt{e^{2\phi(\rho)}-e^{2\phi(\rho_{0})}}$
(51)
The integral defining the separation of the quark pair cannot be evaluated
explicitly, but we can check that the upper limit of the integral gives a
finite contribution (it goes as
$\int^{\infty}\rho^{\frac{1}{4}}e^{-\rho}d\rho$), while from the lower
extremun of the string reaches the end of the space ($\rho_{0}\rightarrow 0$)
we get
$L_{QQ}\sim\int_{\rho_{0}=0}\frac{d\rho}{\sqrt{e^{2\phi_{0}}\rho^{2}+...}}\sim\lim_{\rho_{0}\rightarrow
0}\log(\rho_{0})\rightarrow\infty.$ (52)
indicating that (like in the Witten-Sakai-Sugimoto example discussed above )
strings that reach the end of the space correspond to an infinite separation
between the quark pair, which in turn indicates the absence of screening
(expected from the QFT that contains only fields with zero N-ality).
We compute the subleading terms in the Energy of the pair in terms of the
separation to obtain,
$E_{QQ}=e^{\phi(0)}L_{QQ}+O(e^{-\frac{2\sqrt{2}}{3}L_{QQ}})$ (53)
as above a clear sign of a confining dual QFT.
### III.4 Klebanov-Strassler Model
Certainly, the Klebanov-Strassler model is the cleanest example for a dual to
a four-dimensional field theory that confines in the IR and approaches a
conformal point in the UV (modulo important subtleties) Klebanov:2000hb . Here
again, we will quote only the relevant part of the metric,
$ds^{2}=h(\rho)^{-1/2}dx_{1,3}^{2}+h(\rho)^{1/2}\epsilon^{4/3}\frac{d\rho^{2}}{6K(\rho)^{2}}+ds_{int}^{2}$
(54)
with
$K^{3}(\rho)=\frac{\sinh
2\rho-2\rho}{2\sinh^{3}(\rho)},\;\;h(\rho)=2^{2/3}(g_{s}\alpha^{\prime}N_{c})^{2}\epsilon^{-8/3}\int_{\rho}^{\infty}\frac{x\coth
x-1}{\sinh^{2}x}(\sinh 2x-2x)^{1/3}.$ (55)
The functions $f,g,V_{eff}$ read,
$f^{2}=h(\rho)^{-1},\;\;g^{2}=\frac{\epsilon^{4/3}}{6K(\rho)^{2}},\;\;V_{eff}=\frac{\sqrt{6h(\rho_{0})}K(\rho)}{\sqrt{h(\rho)\epsilon^{4/3}}}\sqrt{h(\rho)^{-1}-h(\rho_{0})^{-1}}$
(56)
In this example again, using the asymptotics for the various functions, it can
be checked that when the string approaches the end of the space $\rho_{0}=0$
the separation of the quark pair diverges logarithmically, as in the two
previous examples. Also, the energy of the pair reads
$E_{QQ}=f(0)L_{QQ}+O(e^{-\frac{2\epsilon^{2/3}}{\sqrt{3}a_{0}}L_{QQ}})$ (57)
with
$a_{0}=\frac{h(0)}{(g_{s}\alpha^{\prime}N_{c}\epsilon^{-4/3})^{2}}=1.1398..$.
As above, it is clear that the model shows confining behavior.
## IV Walking solutions in the $D5$ system, unflavored.
This section is devoted to the specific case of a class of solutions to the
$D5$ system that exhibit walking behavior in the IR, in the sense that a
suitably defined gauge coupling becomes almost constant in a finite range of
energies Nunez:2008wi . We remind the reader about the set-up, based on the
geometry produced by stacking on top of each other $N_{c}$ $D5$-branes that
wrap on a $S^{2}$ inside a CY3-fold and then taking the strongly-coupled limit
of the gauge theory on this stack (that leaves us in the supergravity
approximation for this system). We introduce the class of solutions of
interest and apply the formalism of the previous sections to these solutions.
As we will see, these solutions do confine, in the sense defined earlier on,
but also show a remarkable behavior in the walking energy region, leading to a
phenomenology resemblant a first order phase transition. As a result, the
leading-order, long-distance behavior of the quark-antiquark potential is
linear, but the coefficient is very different from what computed in section
III.3.
### IV.1 General set-up.
We start by recalling the basic definitions that yield the general class of
backgrounds obtained from the $D5$ system, which includes Maldacena:2000yy as
a very special case. We start from the action of type-IIB truncated to include
only gravity, dilaton and the RR 3-form $F$:
$S_{IIB}=\frac{1}{G_{10}}\int
d^{10}x\sqrt{-g}\Big{[}R-\frac{1}{2}(\partial\phi)^{2}-\frac{e^{\phi}}{12}F_{3}^{2}\Big{]},$
(58)
We define the $SU(2)$ left-invariant one forms as,
$\displaystyle\tilde{\omega}_{1}\,=\,\cos\psi
d\tilde{\theta}\,+\,\sin\psi\sin\tilde{\theta}d\tilde{\varphi}\,\,,\,\tilde{\omega}_{2}\,=\,-\sin\psi
d\tilde{\theta}\,+\,\cos\psi\sin\tilde{\theta}d\tilde{\varphi}\,\,,\,\tilde{\omega}_{3}\,=\,d\psi\,+\,\cos\tilde{\theta}d\tilde{\varphi}\,\,.$
(59)
and write an ansatz for the solution Papadopoulos:2000gj assuming that the
functions appearing in the background depend only the radial coordinate
$\rho$, but not on $x$ nor the 5 angles
$\theta,\tilde{\theta},\phi,\tilde{\phi},\psi$ (in string frame):
$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\alpha^{\prime}g_{s}e^{\phi(\rho)}\Big{[}\frac{dx_{1,3}^{2}}{\alpha^{\prime}g_{s}}+e^{2k(\rho)}d\rho^{2}+e^{2h(\rho)}(d\theta^{2}+\sin^{2}\theta
d\varphi^{2})+$ $\displaystyle+$
$\displaystyle\frac{e^{2g(\rho)}}{4}\left((\tilde{\omega}_{1}+a(\rho)d\theta)^{2}+(\tilde{\omega}_{2}-a(\rho)\sin\theta
d\varphi)^{2}\right)+\frac{e^{2k(\rho)}}{4}(\tilde{\omega}_{3}+\cos\theta
d\varphi)^{2}\Big{]},$ $\displaystyle F_{3}$ $\displaystyle=$
$\displaystyle\frac{N_{c}}{4}\Bigg{[}-(\tilde{\omega}_{1}+b(\rho)d\theta)\wedge(\tilde{\omega}_{2}-b(\rho)\sin\theta
d\varphi)\wedge(\tilde{\omega}_{3}+\cos\theta d\varphi)+$ (60) $\displaystyle
b^{\prime}d\rho\wedge(-d\theta\wedge\tilde{\omega}_{1}+\sin\theta
d\varphi\wedge\tilde{\omega}_{2})+(1-b(\rho)^{2})\sin\theta d\theta\wedge
d\varphi\wedge\tilde{\omega}_{3}\Bigg{]}.$
The system of BPS equations can be rearranged in a convenient form, by
rewriting the functions of the background in terms of a set of functions
$P(\rho),Q(\rho),Y(\rho),\tau(\rho),\sigma(\rho)$ as HoyosBadajoz:2008fw
$4e^{2h}=\frac{P^{2}-Q^{2}}{P\cosh\tau-Q},\;\;e^{2g}=P\cosh\tau-Q,\;\;e^{2k}=4Y,\;\;a=\frac{P\sinh\tau}{P\cosh\tau-Q},\;\;N_{c}b=\sigma.$
(61)
Using these new variables, one can manipulate the BPS equations to obtain a
single decoupled second order equation for $P(\rho)$, while all other
functions are simply obtained from $P(\rho)$ as follows:
$\displaystyle Q(\rho)=(Q_{0}+N_{c})\cosh\tau+N_{c}(2\rho\cosh\tau-1),$
$\displaystyle\sinh\tau(\rho)=\frac{1}{\sinh(2\rho-2\hat{\rho_{0}})},\quad\cosh\tau(\rho)=\coth(2\rho-2\hat{\rho_{0}}),$
$\displaystyle Y(\rho)=\frac{P^{\prime}}{8},$ $\displaystyle
e^{4\phi}=\frac{e^{4\phi_{o}}\cosh(2\hat{\rho_{0}})^{2}}{(P^{2}-Q^{2})Y\sinh^{2}\tau},$
$\displaystyle\sigma=\tanh\tau(Q+N_{c})=\frac{(2N_{c}\rho+Q_{o}+N_{c})}{\sinh(2\rho-2\hat{\rho_{0}})}.$
(62)
The second order equation mentioned above reads,
$P^{\prime\prime}+P^{\prime}\Big{(}\frac{P^{\prime}+Q^{\prime}}{P-Q}+\frac{P^{\prime}-Q^{\prime}}{P+Q}-4\coth(2\rho-2\hat{\rho}_{0})\Big{)}=0.$
(63)
In the following we will fix the integration constant $Q_{0}=-N_{c}$, so that
no singularity appears in the function $Q(\rho)$. We also choose
$\hat{\rho}_{o}=0$ for notational convenience, together with
$\alpha^{\prime}g_{s}=1$. For our purposes, it is also convenient to fix
$8e^{4\phi_{0}}=1$. With all of this, the functions we need for the probe
string are:
$\displaystyle f^{2}(\rho)$ $\displaystyle=$ $\displaystyle
e^{2\phi}\,=\,\sqrt{\frac{\sinh^{2}(2\rho)}{(P^{2}-Q^{2})P^{\prime}}}\,,$ (64)
$\displaystyle g^{2}(\rho)$ $\displaystyle=$
$\displaystyle\frac{1}{2}P^{\prime}f^{2}(\rho)\,,$ (65) $\displaystyle
V_{eff}^{2}(\rho)$ $\displaystyle=$
$\displaystyle\frac{2}{C^{2}P^{\prime}}\left(\sqrt{\frac{\sinh^{2}(2\rho)}{(P^{2}-Q^{2})P^{\prime}}}\,-\,C^{2}\right)\,.$
(66)
### IV.2 Walking solutions.
The simplest and better understood solution of the system described in the
previous subsection is defined by
$\displaystyle\hat{P}$ $\displaystyle=$ $\displaystyle 2N_{c}\rho\,.$ (67)
It belongs to class I, and it has already been presented in section III.3,
where is shown the study of the behaviour of the Wilson loop888see Appendix A
for the definition of class I and II and for more details about the solutions
to the equation for P..
For this background it is possible to show that
$\displaystyle f^{2}(0)$ $\displaystyle=$
$\displaystyle\frac{1}{N_{c}\sqrt{2N_{c}}}\,,$ (68)
and that for $C^{2}=f^{2}(0)$, expanding around $\rho\sim 0$
$\displaystyle V_{eff}^{2}(\rho)$ $\displaystyle=$
$\displaystyle\frac{8\rho^{2}}{9N_{c}}\,+\cdots\,.$ (69)
In particular, on the basis of what we already discussed around Eq. (25) this
means that $L_{QQ}$ and $E_{QQ}$ diverge for $C^{2}\rightarrow f^{2}(0)$.
In Nunez:2008wi , it was observed that there exists a class of well-behaved
solutions for which a suitably defined (four-dimensional) gauge coupling
exhibits a walking regime, meaning that for a long range in the radial
coordinate $\rho_{IR}<\rho<\rho_{\ast}$ the running becomes very slow, and the
gauge coupling effectively is constant. These solutions depend on two free
parameters $c$ and $\alpha$, and can be obtained recursively, by assuming $c$
large and expanding in powers of $N_{c}/c$, so that
$\displaystyle P(\rho)$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}c^{1-n}P_{1-n}.$ (70)
with
$P_{1}(\rho)\equiv\left(\cos^{3}\alpha+\sin^{3}\alpha(\sinh(4\rho)-4\rho)\right)^{1/3}.$
(71)
In particular, for $c/N_{c}\rightarrow+\infty$, the solution is very well
approximated by $P\simeq cP_{1}$.
The solutions of the form in Eq. (70) belong to class II in the language
introduced in Appendix A. This type of solution is not suited for the present
study, because of the exponential behavior of $P$ and $P^{\prime}$ at
large-$\rho$, which is not compatible with the boundary conditions for the
string in the UV as required in Eq. (17). (Equivalently, the presence of high-
dimensional operators dominating the dynamics in the far UV renders the study
of the probes problematic. Analogous problems arise when studying the spectrum
of excitations of the background Elander:2009pk and/or of probe fields ENP .)
However, we are mostly interested in what happens in the IR. We are hence
going to construct and study a different class of solutions, which can be
thought of as a generalization of Eq. (70), with UV-asymptotics in class I.
Such solutions can be expanded for small-$\rho$, yielding,
$\displaystyle
P(\rho)=c_{0}+k_{3}c_{0}\rho^{3}+\frac{4k_{3}c_{0}\rho^{5}}{5}-k_{3}^{2}c_{0}\rho^{6}+\frac{16\left(2c_{0}^{2}k_{3}-5k_{3}N_{c}^{2}\right)\rho^{7}}{105c_{0}}\,+\cdots\,,$
(72)
with $c_{0}$ and $k_{3}$ the two integration constants. Notice how this
expansion does not contain a term linear in $\rho$. This means that in the
$c_{0}\rightarrow 0$ limit one does not trivially recover Eq. (67).
In order to build numerically the solution, we start by expanding Eq. (63), by
assuming that the solution can be written as
$\displaystyle P(\rho)$ $\displaystyle=$
$\displaystyle\hat{P}(\rho)\,+\,\varepsilon f(\rho)\,,$ (73)
and replacing in Eq. (63):
$\displaystyle 0$ $\displaystyle=$ $\displaystyle F_{0}\,+\,\varepsilon
F_{1}\,+\,{\cal O}(\varepsilon^{2})\,.$ (74)
Because $\hat{P}$ is an exact solution, $F_{0}=0$. Hence one finds a new
equation, that now is linear:
$\displaystyle 0$ $\displaystyle=$ $\displaystyle
F_{1}\,=\,\frac{8\left(f(\rho)(-\cosh(4\rho)+4\rho\sinh(4\rho)+1)-2\rho\sinh^{2}(2\rho)f^{\prime}(\rho)\right)}{8\rho^{2}-4\sinh(4\rho)\rho+\cosh(4\rho)-1}+f^{\prime\prime}(\rho)\,$
(75) $\displaystyle\simeq$ $\displaystyle\frac{8\left((1-4\rho)f(\rho)+\rho
f^{\prime}(\rho)\right)}{4\rho-1}+f^{\prime\prime}(\rho)$ (76)
In the last step, we approximated the equation by assuming that $\rho\gg 0$.
The resulting equation can be solved exactly, yielding, in terms of
hypergeometric functions:
$\displaystyle f(\rho)$ $\displaystyle=$ $\displaystyle
e^{-4\rho}\sqrt{4\rho-1}c_{1}U\left(\frac{5}{6},\frac{3}{2},6\rho-\frac{3}{2}\right)+e^{-4\rho}\sqrt{4\rho-1}c_{2}L\left(-\frac{5}{6},\frac{1}{2},6\rho-\frac{3}{2}\right)\,.$
(77)
Asymptotically, neglecting power-law corrections, this means
$\displaystyle f(\rho)$ $\displaystyle\simeq$ $\displaystyle
c_{1}e^{-4\rho}\,+c_{2}e^{2\rho}\,,$ (78)
implying that consistency of the perturbative expansion in Eq. (74) enforces
the choice $c_{2}=0$. Indeed, there are no asymptotic (in the UV) solutions
that behave as $e^{2\rho}$. Allowing for $c_{2}\neq 0$ would imply that the
solution is not a deformation of $\hat{P}$, but rather that the expansion in
Eq. (73) breaks down, and the solution (if regular) falls in class II. The
procedure of allowing for a small component $c_{1}\neq 0$ can be interpreted
as the insertion of a small, relevant deformation, which does not affect the
UV-asymptotics, but has very important physical effects in the IR.
We have now obtained an important result: at least up to large values of
$\rho$, there exists a class of solutions that approach asymptotically the
$\hat{P}$ solution. We cannot prove that such solutions are well behaved all
the way to $\rho\rightarrow 0$. However, we can use this result in setting up
the boundary conditions (at large-$\rho$) and numerically solve Eq. (63)
towards the IR. By inspection, these solutions are precisely the ones we were
looking for. They start deviating significantly from $\hat{P}$ below some
$\rho_{\ast}>0$, below which $P$ is approximately constant. We plot in Fig. 2
two such solutions, with $\rho_{\ast}\simeq 4$ and $\rho_{\ast}\simeq 9$,
together with the $\hat{P}$ solution for the same value of $N_{c}$. We also
plot in Fig. 3 the functions appearing in metric $(e^{2g},e^{2h},e^{2k},\phi)$
for the same solutions. Notice the behavior of $e^{2g}$ for $\rho\rightarrow
0$, but also the fact that the dilaton $\phi$ is finite for $\rho\rightarrow
0$.
A short digression is due at this point. The reader should be aware that in
the presented case we are likely working with a singular background, as the
function $e^{2g}$, a warp factor inside the internal space, is divergent at
$\rho=0$.
The problem of singularities in gauge-string duality has a long history.
Surely it is better to work with non singular backgrounds, but it is possible
to obtain interesting information also from singular manifolds. The procedure,
developed along the years, to decide whether one is a dealing with a “good
singularity” or not, consists in analyzing all the physical observables and
showing that no track of the singularity can be found in them. One specific
suggestion is to look at the behavior of the $g_{tt}$ component of the metric
(in the present case, the dilaton) and make sure that it is finite. Many are
in literature the examples of such a class of backgrounds. To corroborate the
idea that our background falls in this class, we decide to investigate not
only the value of the Wilson loop but also two invariants of the metric,
namely the Ricci scalar and a suitable contraction of the Ricci tensor. As can
be seen in Fig. 4 no trace of singularities can be found in these two
observables for our background, in support of the idea that the backgrounds we
are considering are indeed acceptable.
Figure 2: The numerical solutions for $P(\rho)/N_{c}$ used in the analysis as
an example, for $N_{c}=100$. The three solutions correspond to the $\hat{P}$
case with $\rho_{\ast}=0$, and to two new numerical solutions with,
respectively, $\rho_{\ast}\simeq 4$ and $\rho_{\ast}\simeq 9$. The numerical
solutions can be plotted up to $\rho\simeq 150$, but in the following we will
truncate them at $\rho_{1}=30$.
Figure 3: The functions $(e^{2g},e^{2h},e^{2k},\phi)$ appearing in the metric
for the same solutions as in Fig. 2, computed rescaling $P\rightarrow
P/N_{c}$, and $Q\rightarrow Q/N_{c}$.
Figure 4: The (10-dimensional) Ricci scalar $R$ and the scalar $R^{2}\equiv
R_{MN}R_{PQ}g^{MP}g^{NQ}$, plotted as a function of the radial coordinate
$\rho$, for several numerical solutions in the class discussed in the body of
the paper. Each curve corresponds to a different value of $\rho_{\ast}$.
Notice that both scalars are finite in the $\rho\rightarrow 0$ limit.
### IV.3 Probes: numerical study.
Figure 5: Upper panel, the radial coordinate $\rho_{0}$ of the middle point of
the string as a function of $L_{QQ}$. Middle and lower panel, the energy
$E_{QQ}$ as a function of the quark-antiquark separation $E_{QQ}(L_{QQ})$. The
three solutions in Fig. 2 are used, with the same color-coding.
We set-up the configuration of the string by assuming that its extremes are
attached to the brane at $\rho=\rho_{1}\gg 1$, and treat this as a UV cut-off.
In the numerical study and in the resulting plots, we used $\rho_{1}=30$. The
string is stretched in the Minkowski direction $x=x(\rho)$, with
$x(\rho_{1})=0$ for convenience. We vary the integration constant
$C^{2}>f^{2}(0)$. For each choice of $C$ we define $\rho_{0}$ as
$V^{2}_{eff}(\rho_{0})=0$. In this way, the coordinates of the string are
$(x(\rho),\rho)$, where
$\displaystyle x(\rho)$ $\displaystyle=$
$\displaystyle\int_{\rho}^{\rho_{1}}\frac{\mbox{d}r}{{V_{eff}(r)}}\,.$ (79)
The Minkowski distance between the the end-points of the string is hence
$L_{QQ}=2x(\rho_{0})$. For the energy, because in the numerical study we do
not remove the UV cut-off, rather that Eq. (22), we use the unsubtracted
action (setting $T/(2\pi\alpha^{\prime})=1$) evaluated up to the cut-off:
$\displaystyle E_{QQ}$ $\displaystyle=$ $\displaystyle
2\int_{\rho_{0}}^{\rho_{1}}\mbox{d}r\sqrt{\frac{f^{2}(r)g^{2}(r)}{f^{2}(r)-C^{2}}}\,.$
(80)
The numerical results obtained for the three different solutions $P$ are shown
in Fig. 5 and Fig. 6.
Let us first focus our attention on the $\hat{P}$ case of Eq. (67). The deeper
the string probes the radial coordinate (smaller values of $\rho_{0}$), the
longer the separation $L_{QQ}$ between the end-points on the UV brane, in
agreement with our criteria and results of sections II and III.
Two regimes can be identified: as long as $\rho_{0}>\rho_{IR}$, then $L_{QQ}$
varies very little with $\rho_{0}$, while for small $\rho_{0}$, further
reductions of $\rho_{0}$ imply much longer $L_{QQ}$. The scale
$\rho_{IR}\sim{\cal O}(1)$ is the scale in which the function $Q$ changes from
linear to approximately quadratic in $\rho$, and is also the scale below which
the gaugino condensate is appearing (the function $b(\rho)$ in the background
is non-zero). This result is better visible in the upper panel of Fig. 5. The
dependence of $L_{QQ}$ on $\rho_{0}$ is monotonic, but shows two very
different behaviors for $\rho_{0}<\rho_{IR}$ and $\rho_{0}>\rho_{IR}$,
respectively. The transition between the two is completely smooth.
The physical meaning of this behavior is well illustrated by studying the
total energy $E_{QQ}$ of the classical configurations, as a function of
$L_{QQ}$ and of $\rho_{0}$ (see the middle and lower panel of Fig. 5). One
sees that for small $L_{QQ}$, the energy grows very fast with $L_{QQ}$, until
a critical value beyond which the dependence becomes linear. We have already
studied analytically this behavior, which can be interpreted in terms of the
linear behavior of the quark-antiquark potential obtained from the Wilson loop
in agreement with the discussion of sections II and III. The energy is also a
monotonic function of $\rho_{0}$.
Figure 6: The strings in $(x,\rho)$-plane, obtained with various choices of
$C^{2}>f^{2}(0)$. Top to bottom, the three numerical solutions corresponding
to increasing values of $\rho_{\ast}$
By comparing with the solutions that walk in the IR, one sees that a very
different behavior appears. Starting from the upper panel in Fig. 5, one sees
that as long as $\rho_{0}>\rho_{\ast}$, the dependence of $L_{QQ}$ from
$\rho_{0}$ reproduces the $\hat{P}$ case. Beginning from such large
$\rho_{0}$, we start pulling the string down to smaller values of $\rho_{0}$,
and follow the classical evolution. Provided we do this adiabatically, we can
describe the motion of the string as the set of classical equilibrium
solutions we obtained in the previous section. Going to smaller $\rho_{0}$,
$L_{QQ}$ increses, and nothing special happens until the tip of the string
touches $\rho_{0}\simeq\rho_{\ast}$. At this point $L_{QQ}=L_{max}$. From here
on, the string can keep probing smaller values of $\rho_{0}$ only at the price
of becoming shorter in the Minkowsi direction (smaller $L_{QQ}$). Another
change happens when $L_{QQ}=L_{min}$, at which point the tip of the string
entered the bottom section of the space, $\rho_{0}<\rho_{IR}$. From here on,
further reducing $\rho_{0}$ requires larger values of $L_{QQ}$. Asymptotically
for $\rho_{0}\rightarrow 0$, the separation between the end-points of the
string is diverging, $L_{QQ}\rightarrow\infty$.
Even more interesting is the behavior of the energy (lower panel of Fig. 5):
for very short $L_{QQ}$, and again for very large-$L_{QQ}$, it is just a
monotonic function, but for a range $L_{min}<L_{QQ}<L_{max}$ there are three
different configurations allowed by the classical equations for the string we
are studying 999 Notice that the qualitative behavior of the solutions with
$\rho_{\ast}\simeq 4$ and $\rho_{\ast}\simeq 9$ are identical. However, we
were not able to follow numerically the solution with larger $\rho_{\ast}$,
and hence the plots show only two such solution. The existence of the third is
assured by the fact that this background must yield a confining potential..
One of the three solutions (smoothly connected to the small-$L_{QQ}$
configurations) is just the Coulombic potential already seen with $\hat{P}$.
The highest energy one is an unstable configuration, with much higher energy.
The third solution (smoothly connected to the unique solution with
$L_{QQ}>L_{max}$) reproduces the linear potential typical of confinement.
Notice (from the lower panel of Fig. 5) that the solution at large-$L_{QQ}$ is
linear, but has a slope much larger than what seen in the $\hat{P}$ case. This
co-existence of several disjoint classical solutions is expected in systems
leading to phase transitions (see Appendix B). The
instability/metastability/instability of the solutions can be illustrated by
comparing Fig. 5 with Fig. 10 in Appendix B. This is just an analogy, and one
should not push it too far. However, identifying the pressure $P$, volume $V$
and Gibbs free energy $G$ as $L_{QQ}\leftrightarrow P$,
$\rho_{0}\leftrightarrow V$ and $E_{QQ}\leftrightarrow G$, one sees obvious
similarities. In particular, there is a critical distance
$L_{min}<L_{c}<L_{max}$ at which the minimum of $E_{QQ}$ is not
differentiable.
In order to better understand and characterize the solutions we find, it is
useful to look more in details at the shape of the string configurations,
focusing in particular on the middle panel in Fig. 6, in which we plot the
string configuration that solves the equations of motion for various values of
$\rho_{0}$, on the background with $\rho_{\ast}\simeq 4$. Consider those
strings that penetrate below $\rho_{\ast}$. Besides having a shorter $L_{QQ}$,
and higher $E_{QQ}$ than those for which $\rho_{0}>\rho_{\ast}$, this strings
show another interesting feature. They start developing a non trivial
structure around their middle point, that becomes progressively more curved
the further the string falls at small $\rho$. Ultimately, this degenerates
into a cusp-like configuration, which disappears once $\rho_{0}$ approaches
the end of the space. Notice that, as a result, the three different solutions
for $L_{min}<L_{QQ}<L_{max}$ have three very different geometric
configurations. One (stable or metastable) configuration is completely
featureless, and practically indistinguishable from the solutions in the
background generated by $\hat{P}$. The second (stable or metastable)
configuration shows a funnel-like structure below $\rho_{\ast}$, and then the
string lies very close to the end of the space. The third (unstable) solution
presents a highly curved configuration around its middle point 101010 This
should not be interpreted literally as a cusp. The classical solutions we
found are always continuous and differentiable, and this structure disappears
once the string approaches the end of the space.. All of this seems to be
consistent with the fact that in the region $\rho_{IR}<\rho<\rho_{\ast}$ the
background has a higher curvature, and as a result the classical
configurations prefer to lie either in the far-UV or deep-IR. It must be noted
here that the Ricci curvature is indeed bigger in this region, but that it
converges to a constant for $\rho\rightarrow 0$.
### IV.4 Comments on this section.
We conclude this section by summarizing here three important lessons we
learned.
First, we are able to derive the linear $E_{QQ}(L_{QQ})$ expected from
confinement. This linear behavior emerges for $L_{QQ}\gg L_{c}$, $L_{c}$ being
the critical distance between the end-points of the string, of the order of
the distance $L_{QQ}$ computed for the string the tip of which reaches
$\rho_{IR}$. This fact provides a physical meaning for the scale $\rho_{IR}$,
which clearly shows in the (scheme-dependent) background functions and in the
gauge coupling 111111A change of scheme in the string picture corresponds to a
redefinition of the radial coordinate..
In particular, we can conclude that the walking solution we looked at is dual
to a confining theory, but a very different one from the one of the non-
walking solution $\hat{P}$. This is signaled by the fact that the leading
order behavior of the walking solution is related to the value of $V_{eff}$ in
$\rho_{0}\gtrsim\rho_{\ast}$ while the non-walking is related to the zero of
$V_{eff}$ in $\rho_{0}\gtrsim 0$, and hence the presence of this two different
solutions has to be related to the existence of two different scales in the
background.
A limitation of this formalism is that, as we explained at length, it does not
allow to study the sub-leading corrections to the linear behavior, which would
require a treatment in which $\alpha^{\prime}$-suppressed quantum corrections
are included. It would be very interesting to calculate these corrections,
which might provide useful information as to the nature of the dual theory.
The second important lesson we learn is related to the second scale
$\rho_{\ast}$ appearing in solutions with walking behavior. Again, this scale
appears in $P$ and as a consequence in all the functions in the background,
including the (scheme-dependent) gauge coupling. The main point is that,
provided $\rho_{\ast}>\rho_{IR}$, $\rho_{\ast}$ has a very important physical
meaning: it separates the small-$L_{QQ}$ regime, where the background and all
physical quantities are the same as in the original $\hat{P}$ solution, from
the large-$L_{QQ}$ regime, in which the dual theory is completely different
from the dual to the $\hat{P}$ solution. The scale $\rho_{\ast}$ is not just a
scheme-dependent fluke effect: its value is somehow related to the coefficient
of the linear leading behavior of the quark-antiquark potential.
The third lesson we learn is that in the region $\rho_{IR}<\rho<\rho_{\ast}$,
the dynamics being more strongly coupled than elsewhere, classical solutions
are unstable. The string configurations that are stable are those that either
do not reach $\rho_{\ast}$, or those that go through this region only in order
to reach the region near the end of the space, where the string can lie up to
indefinitely large separations (in the Minkowski directions). This is a very
interesting result, that might be related with what found in Elander:2009pk ,
where it is shown that for large values of $\rho_{\ast}$ the spectrum of
scalar glueballs on these backgrounds splits into a set of towers of heavy
states and some light state, separated by a large gap. This is going to be
studied elsewhere ENP .
## V Wilson Loop in a Field Theory with Flavors
In this section we consider the effects of fundamental (flavor) degrees of
freedom on the Wilson loop, using backgrounds that encode the dynamics of
fields charged under a flavor group. As discussed in the Introduction, we
expect that confinement does not take place, instead the theory will screen.
We will observe the existence of a maximal length, requiring the string to
break. As we explained, we are not including $g_{s}$ effects hence the pair-
creation will not be accessible to the description we are giving here. In
Karch:2002xe these effects are taken into account by constructing the
screened solution explicitly.
The construction of string backgrounds where the effects of flavors is
included was considered in a large variety of models, see bunch . We will
concentrate on the models developed in Casero:2006pt . We follow the treatment
and notation of HoyosBadajoz:2008fw , as done in the previous sections. Other
authors have studied effects similar to the ones described in this section by
using different string models, see Bigazzi:2008gd -Ramallo:2008ew .
At this point, let us stress that, what we will do in the following is to use
as probe a closed string in a classical theory with no $g_{s}$ correction.
Such an observable cannot describe the physics of the broken $Q\bar{Q}$ pair
in the QFT sector; It will have a good overlap with the state describing the
$Q\bar{Q}$ pair only for small separation distance among the quarks. If the
distance among the quark is increased another configuration, that will mimic
the breaking of the pair, will become more energetically favorable. On the
other hand if we decide to measure only the Wilson loop we will see no signal
of such a breaking. Conversely if a cusps appears in the profile of the string
used to measure the wilson loop, this signal has to be read as the loss of
validity of the approximation imposed on our theory. While this signal cannot
be interpreted as corresponding physical quantity (the distance at which the
cusp appears has nothing to do with the scale of breaking of the $Q\bar{Q}$
pair), the observation can give a suggestion about the range of validity of
our approximation. The results of this section should be understood as the
application of the formalism of section II to the case of flavored background,
with the proviso of the validity of the Nambu-Goto approximation and the
comment made above about the screening phenomena. Aside form this, the
numerical solution Fig. 7 below is new material included here.
In order to find the background solutions, we will need to solve a
differential equation that is a generalization of the one discussed in the
previous section, Eq. (63):
$\displaystyle
P^{\prime\prime}+(P^{\prime}+N_{f})\Big{[}\frac{P^{\prime}+Q^{\prime}+2N_{f}}{P-Q}+\frac{P^{\prime}-Q^{\prime}+2N_{f}}{P+Q}-4\coth(2\rho)\Big{]}=0,$
$\displaystyle Q(\rho)=\frac{2N_{c}-N_{f}}{2}(2\rho\coth(2\rho)-1).$ (81)
The relation to the functions that appear explicitly in the background is
given in Eqs. (3.18) and (3.19) of HoyosBadajoz:2008fw .
There are various known solutions to Eq. (81). We focus on the so called
type-I solutions that are known only as a series expansion. For small values
of the radial coordinate ($\rho\rightarrow 0$), the expansion is given in Eq.
(4.24) of HoyosBadajoz:2008fw , while for large values of the radial
coordinate ($\rho\rightarrow\infty$) is given in Eqs. (4.9)-(4.11) of
HoyosBadajoz:2008fw :
$\displaystyle P(\rho\sim
0)=P_{0}-N_{f}\rho+\frac{4}{3}c^{3}P_{0}^{2}\rho^{3}+\cdots\,,$ $\displaystyle
P(\rho\sim\infty)=Q+N_{c}(1+\frac{N_{f}}{4Q}+\frac{N_{f}(N_{f}-2N_{c})}{8Q^{2}}+\cdots)\,.$
(82)
In the $N_{f}\rightarrow 0$ limit this can be matched with the expansion Eq.
(72). Below, we find it useful to have the IR ($\rho\rightarrow 0$)
asymptotics for the functions
$e^{4\phi}\sim\frac{8e^{4\phi_{0}}}{c^{3}P_{0}^{4}}\Big{[}1+\frac{4N_{f}}{P_{0}}\rho+\frac{10N_{f}^{2}}{P_{0}^{2}}\rho^{2}\Big{]}+....,\;\;\;e^{2k}\sim
2c^{3}\Big{[}P_{0}^{2}\rho^{2}-2P_{0}N_{f}\rho^{3}\Big{]}+....$ (83)
and in the UV ($\rho\rightarrow\infty$),
$e^{4\phi}\sim\frac{e^{4\rho}}{\rho},\;\;\;\;\;\;\;e^{2k}\sim 1.$ (84)
Figure 7: Numerical solutions of $P(\rho)$ for various values of
$N_{f}/N_{c}$.
We plot in Fig. 7 a set of numerical solutions, which reproduce the asymptotic
behaviors in Eq. (82), for several different values of $N_{f}/N_{c}$. The
existence of these solutions smoothly joining the asymptotic behaviors has
been assumed in HoyosBadajoz:2008fw . Here we are explicitly showing that this
assumption is correct. Numerically, we learn that (provided $P_{0}$ is not too
small) the function $P$ is convex, as shown in Fig. 7. Let us move to the
study of string probes in these backgrounds.
In the light of the discussion of section II.1, we form the combinations
$\displaystyle
f^{2}=g_{tt}g_{xx}=e^{2\phi(\rho)}=\frac{\sqrt{8}e^{2\phi_{0}}\sinh(2\rho)}{\sqrt{(P^{2}-Q^{2})(P^{\prime}+N_{f})}},\;\;\;C^{2}=e^{2\phi(\rho_{0})}$
$\displaystyle
g^{2}=g_{tt}g_{\rho\rho}=e^{2\phi(\rho)+2k(\rho)}=\frac{\sqrt{2}e^{2\phi_{0}}\sinh(2\rho)\sqrt{P^{\prime}+N_{f}}}{\sqrt{(P^{2}-Q^{2})}},$
$\displaystyle
V_{eff}=\frac{1}{e^{k(\rho)}C}\sqrt{e^{2\phi(\rho)}-C^{2}}=\frac{1}{e^{k(\rho)}}\sqrt{e^{2\phi(\rho)-2\phi(\rho_{0})}-1}.$
(85)
We will choose for convenience the integration constant $\phi_{0}$ such that
$8e^{4\phi_{0}}=c^{3}P_{0}^{4}$, hence we will have $\phi(\rho=0)=0$.
Following the discussion below Eq. (23) and using eq.(84), the contribution to
$L_{QQ}$ of the integral for large values of the radial coordinate goes like
$L_{QQ}(\rho_{0}\rightarrow\infty)\sim\int^{\infty}\frac{d\rho}{e^{2\rho}}$
(86)
which is finite. Similarly, using Eq. (83), the lower-end of the integral
contributes with
$L_{QQ}(\rho_{0}\rightarrow 0)\sim\int_{\rho_{0}\rightarrow
0}d\rho\sqrt{\rho},$ (87)
which is itself finite. In this case we can see that in Eq. (25) the quantity
$\gamma=-\frac{1}{2}$, according to the discussion around Eq. (25) implies a
finite $L_{QQ}$. Contrast this with the examples studied in section III and V.
The fact that near $\rho=0$ the quantity $V_{eff}\sim\rho^{-\frac{1}{2}}$
implies, using Eq. (13), that near the IR there must be a cusp-like behavior.
A plot of the shape of the string makes this clear, see Fig. 8 where we string
probing a background with $N_{f}/N_{c}=1.2$.
Figure 8: Shapes of the string for various choice of $\rho_{0}$
In contrast with the example studied in section IV these backgrounds have a
divergent Ricci scalar at $\rho=0$, in agreement with the fact that only for
$\rho_{0}=0$ the string presents a cusp. This is the only example presented in
this paper with a true cusp in the string profile.
### V.1 The case $N_{f}=2N_{c}$.
Finally, we turn our attention to the very peculiar, limiting case with
$N_{f}=2N_{c}$. The solution is described in Eq. (4.22) of the first paper in
Casero:2006pt . It corresponds to
$\displaystyle
Q=4N_{c}\frac{(2-\xi)}{\xi(4-\xi)}\,,P=N_{c}+\sqrt{N_{c}^{2}+Q^{2}}\,,$ (88)
and the functions $f(\rho),g(\rho)$ read in this case
$f^{2}=g^{2}=e^{2\phi_{0}+2\rho}$ (89)
where now the radial coordinate is defined in the interval
$-\infty<\rho<\infty$. The explicit computation of Eqs. (22), now for the
lower end of the integral $\rho_{0}\rightarrow-\infty$, gives
$L_{QQ}=\sqrt{g_{s}\alpha^{\prime}N_{c}}\pi,\;\;\;\;E_{QQ}=0$ (90)
We illustrate in Fig. 9 the behavior of the string on this background. Notice
how the length of the string is fixed $L_{QQ}=\pi$, in units where
$\alpha^{\prime}g_{s}N_{c}=1$, irrespectively of how deep into the radial
direction the string probes the background. Another interesting fact is that
the quantity of Eq. (39) vanishes identically in this background, meaning that
$dL_{QQ}/d\rho_{0}=0$ for any $\rho_{0}$, implying that it is always possible
to find a solution with any given $\rho_{0}$ and with the same $L_{QQ}$, as
Fig. 9 clearly indicates. The field theory dual to this background is quite
peculiar. On the one hand, the numerology $N_{f}=2N_{c}$ brings to mind the
situation in which the theory becomes conformal. Nevertheless, we should not
forget that this is a background constructed with wrapped branes, so, clearly
a scale is introduced (the scale set by the inverse size of the cycle
wrapped). It may be the case that once the singularity is resolved one finds
in the deep IR an $AdS_{5}$ space. But the theory should remind that in the UV
conformality is broken, hence developing the scale
$\Lambda^{-2}=\alpha^{\prime}g_{s}N_{c}$, that is typical of six dimensional
theories. Notice that the only allowed separation, according to the
calculations above is precisely $\pi$ in units of this scale and the energy is
zero. These results do not need to be generic for all solutions with
$N_{f}=2N_{c}$, but for this particular one discussed above. Note also that in
this case, even when the background is singular in the IR, the string does not
become cuspy, as the case studied in the previous section.
Figure 9: The string hanging on the special background in Eq. (88).
## VI Summary and Conclusions
Let us summarize what we learnt in this paper and emphasize the reasons that
motivated our study. In section II we recovered many well-known results about
holographic Wilson loops, but at the same time introduced a certain amount of
formalism and some new results that allow us to make a systematic study of
many qualitative features of string probes dual to Wilson loops in field
theory. For example, we introduced the quantity $V_{eff}$ in Eq. (12) which
dictates the behavior of the string near the boundary of the space
($\rho\rightarrow\infty$) and tells us if the Dirichlet boundary condition is
attained. At the same time, $V_{eff}$ near the IR tells us about the
possibility (or not) of stretching the probe string indefinitely. Assuming
certain characteristic power-law behavior of the function $V_{eff}$ near the
IR, we were able to derive a condition on the power ensuring the presence of a
minimum (or turning point). In that same section, we derived an exact
expression of the relation between the energy and the separation of the quark
pair. We also found an integral expression giving a necessary condition for
the existence of turning points in the string probing the bulk and the
condition for the existence of cusps.
In section III, we applied the formalism described above to some well-
understood examples and made some comments about the absence of Luscher terms
in the $E_{QQ}-L_{QQ}$ relation.
The material in section IV was derived with the following motivation: in the
paper Nunez:2008wi a new background was proposed to be dual to a QFT with a
walking regime (for a particular coupling). Nevertheless, aside from an
argument based on symmetries given in that paper, it could have been the case
that this walking region is just a fluke of the coordinates choice that
dissapears under a simple diffeomorphism (in the dual field theory, gauge
couplings and beta functions are scheme dependent). To clarify if there is any
real physical effect in the walking region we computed Wilson loops in the
(putative) walking field theory. In order to do this, we needed to find a new
walking solution that allows the string to satisfy the boundary conditions
discussed in section II. The new solution was found numerically as a
perturbation of the solution in Maldacena:2000yy . We gave an interpretation
of this new background in the dual field theory as the appearance of a VEV for
a quasi-marginal operator. When studying the dynamics of strings in this new
background we observed that there are various physical effects associated with
the presence and length of the walking regime: the value of the Ricci scalar,
the separation between the quark pair, the relation between the energy and the
separation of the pair, etc. In conclusion, the walking regime has physically
observable effects, hence it can not erased by a change of coordinates
(conversely, it is not an effect of a choice of scheme in the field theory).
Recent experience seems to suggest that whenever we deal with a system with
two independent scales, the phenomenology of the Wilson loop will be similar
to what we described in section IV.
Finally we moved to the study of the Wilson loops in field theories with
flavors. Here we benefitted from some work done in the past, where the
asymptotic behavior of the background functions was given. We constructed
numerically the full solution in terms of a convenient formalization of the
problem described in HoyosBadajoz:2008fw . We applied the material of section
II to this case and checked that the probe string was behaving as expected
(screening).
We believe that this paper clarifies a considerable number of new and
interesting points. Certainly, it pushes forward the idea of using string
theory methods to study models of walking dynamics. This may become important
at the moment of modelling the mechanism electroweak symmetry breaking. On the
phenomenological side we provide a set of tools that can be applied to other
backgrounds and may be even used in bottom-up approaches to the dual of QCD.
It would be nice to find new models of walking dynamics and apply the ideas in
this paper to study and compare features. It would also be interesting to
apply the formalism developed here to some of the examples of backgrounds dual
to field theories with flavors, for which certain aspects of Wilson loops were
studied Bigazzi:2008gd \- Ramallo:2008ew .
## A UV asymptotic solutions.
In this appendix we present a short digression about the asymptotic behavior
of the solutions to Eq. (63), in order to make contact with a possible field
theory interpretation of the results that have been presented in the section
IV. Generically, one can expand all the functions appearing in the background,
and associate the integration constants appearing in this expansion with the
insertion in the UV of the dual field theory of operators with scaling
behavior determined by the $\rho$-dependence of the corresponding term in the
expansion. This is reminiscent of what is done in the case of backgrounds that
are asymptotically AdS, though in the present case the procedure is less
solid, and the results should be taken with caution.
We can start from the function $Q$.
$\displaystyle Q$ $\displaystyle\sim$ $\displaystyle
N_{c}\left(2\rho-1\,+\,{\cal O}(e^{-4\rho})\right)\,,$ (91)
where we dropped factors as powers of $\rho$ in the exponential corrections.
We will do so in rest of this analysis, the logic of which will not be
affected.
If one thinks that the two behaviors correspond to the insertion of an
operator of dimension $d$ and of its $6-d$ dimensional coupling in the (six-
dimensional) theory living on the $D5$ branes, this means that this expression
should be written as
$\displaystyle Q$ $\displaystyle\sim$ $\displaystyle z^{d}\,+\,z^{6-d}\,,$
(92)
with $z$ proportional to a length scale. By comparison with the expansion of
$Q$, one is lead to the identification $\rho=-\frac{3}{2}\log z$. This result
agrees with what was done by comparing an appropriately defined beta-function
to the NSVZ beta function in DiVecchia:2002ks . One can interpret the two
terms in this expansion as the deformation of the theory by the insertion of a
marginally relevant operator of dimension $d\sim 6-\epsilon$, with coupling of
dimension $6-d\sim\epsilon$. Notice that the coefficient of the ${\cal
O}(e^{-4\rho})$ correction is very small, and has a sizable effect only at
very small values of $\rho$. This is not due to the fact that we chose a
particular value of the integration constant $Q_{0}$, in order to avoid the
arising of a pathology in the IR. Allowing for $Q_{0}$ would yield the same
expansion, and the modification of the coefficients would be such that only
for $\rho<\rho_{IR}$ would the sub-leading correction have an effect. The
scale $\rho_{IR}\sim{\cal O}(1)$ has an important role in the present study.
Another important quantity in the background is
$\displaystyle b(\rho)$ $\displaystyle=$
$\displaystyle\frac{2\rho}{\sinh(2\rho)}\,\sim\,{\cal O}(e^{-2\rho})\,,$ (93)
which by means of the previous identification is equivalent to $b\sim z^{3}$.
This can be interpreted as the VEV of a dimension-3 operator (customarily
identified with the gaugino condensate in the dimensionally-reduced
4-dimensional low-energy theory). This is a relevant deformation. Because this
is a more relevant operator than the VEV mentioned above, it always dominates
in the IR over the sub-leading correction to $Q$, which can be ignored. Notice
also that $b(\rho)$ is practically vanishing for $\rho>\rho_{IR}$. This
provides an intuitive explanation for this scale: it is the scale at which the
gaugino condensate emerges dynamically. It also suggests that the expansion of
$Q$, rather than identifying an independent operator, might be interpreted in
terms of the insertion of an operator loosely corresponding to the square of
the gaugino condensate.
There exist two different classes of UV-asymptotic solutions for $P$
HoyosBadajoz:2008fw :
$\displaystyle P$ $\displaystyle\sim$ $\displaystyle 2N_{c}\rho\,+\,{\cal
O}(e^{-4\rho})\,\,\,\,{\rm(class\,I)}\,,$ (94) $\displaystyle P$
$\displaystyle\sim$ $\displaystyle{\cal O}(e^{4/3\rho})\,+\,{\cal
O}(e^{-4/3\rho})\,+\,{\cal O}(e^{-8/3\rho})\,\,\,\,{\rm(class\,II)}\,.$ (95)
In class I, there is only one integration constant, in the ${\cal
O}(e^{-4\rho})$ term. Notice that $P$ has the same leading and sub-leading
components as $Q$. However, the sub-leading correction depends on a free
parameter: the corresponding VEV can be enhanced in such a way that there be a
range of $\rho$ over which the dynamics is dominated by this deformation, over
the deformation present in $b(\rho)$, while the latter will become important
only at very small values of $\rho$.
Solutions in the class II are rather different. The independent coefficients
appear in the ${\cal O}(e^{4/3\rho})$ and ${\cal O}(e^{-8/3\rho})$ terms, and
the former cannot be dialed to zero independently of the latter (see
HoyosBadajoz:2008fw for details). Using the same identification between
$\rho$ and $z$, the leading order component of $P$ scales as $z^{-2}$, and can
be interpreted as the insertion of a dimension-8 operator in the six-
dimensional theory. It is somewhat natural to think that it is related to a
gauge coupling is six-dimensions. The presence of sub-leading corrections that
scale as $z^{2}$, and $z^{4}$ suggest that the gravity field $P$ should not be
interpreted as a simple operator in the underlying dual dynamics, and that
some caution should be used. One might want to interpret the $z^{4}$ as the
insertion of the VEV of a four dimensional operator. But it could as well
arise from the combination of the coupling scaling as $z^{-2}$ and the same
VEV of marginal operator scaling as $z^{6}$ that is present in class I. All of
this should not be taken too literally, but provides some guidance in what we
do in the body of the paper, when constructing the class of walking solutions
we are interested in.
## B Van der Waals gas.
Here we summarize some aspect of first order phase transitions that plays an
important conceptual role in the body of the paper. By way of example, we
remind the reader about the classical treatment of the van der Waals gas, in
terms of its pressure $P$, temperature $T$ and volume $V$ of $N$ moles of
particles, by means of the equation of state
$P=\frac{NRT}{V-bN}-\frac{N^{2}a}{V^{2}}\,,$ (96)
where $R,b,a$ are constants.
In Fig. 10, we plot one isotherm. The condition for stability of the
equilibrium $\left(\partial^{2}F/\partial
V^{2}\right)_{T}=-\left(\frac{\partial P}{\partial V}\right)_{T}>0$ is not
satisfied in some region. This implies that a phase transition is taking
place.
Figure 10: The pressure $P$ as a function of the volume $V$ (left panel) and
the Gibbs free energy $G$ as a function of the pressure $P$(right panel) for
the same isotherm curve.
In order to understand what the physical trajectory followed by the system at
equilibrium is, we plot in Fig. 10 the Gibbs free energy $G=G(T,P)=G(P)$ for
the same isotherm. From this plot, one sees that the system evolves on the
path $ABCOQ$, where $C=N$, in such a way that for every choice of $P$ the
Gibbs free energy $G$ is at its minimum. The evolution is smooth along $ABC$
(gas phase: $|\partial P/\partial V|$ is small), but at $C=N$ the free energy
is not differentiable, signaling that a first-order phase transition is taking
place. In the $(P,V)$ plane the system runs along the horizontal line
(constant $P$) joining $C$ and $N$. This explains the Maxwell rule introducing
a curve of constant pressure that separates two regions of equal areas above
and below it, delimited by the original isotherm. Afterwards, the evolution
follows smoothly the curve $NOQ$ (liquid phase: $|\partial P/\partial V|$ is
large).
While the trajectory $ABCNOQ$ follows the stable equilibrium configurations,
it is possible to have the system evolving along the path $CDE$ or $LMN$. Both
these paths represent metastable configurations, because $\left(\partial
P/\partial V\right)_{T}<0$. Indeed, these metastable states can be realized in
laboratory experiments. For example, in a bubble chamber or in supercooled
water, the metastability is exploited as a detector device, because small
perturbations induced by passing-by charged particles are sufficient to drive
the system out of the state and into the stable minimum. The evolution along
the path $EFHIL$ is completely unstable and not realized physically (it is a
local maximum, as clear from the right panel of Fig. 10, and by the fact that
$\left(\partial P/\partial V\right)_{T}>0$).
The analogy with the examples in the main body of the paper is apparent.
Notice for instance that the pressure $P$ in this system as a function of the
volume $V$ behaves as a non monotonic function, hence there are inversion
points in the curve (the points $E,L$ in figure 10), in the same sense in
which we discuss the presence of inversion points in the body of the paper.
## Acknowledgments
We wish to thank A. Cotrone, F. Bigazzi, D. Elander, S. P. Kumar, J. Schmude,
A. Paredes for discussions and correspondence. The work of MP is supported in
part by the Wales Institute of Mathematical and Computational Sciences.
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|
arxiv-papers
| 2009-09-04T14:14:08 |
2024-09-04T02:49:05.048765
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Carlos Nunez, Maurizio Piai and Antonio Rago",
"submitter": "Antonio Rago Dr",
"url": "https://arxiv.org/abs/0909.0748"
}
|
0909.0995
|
# Coefficients of cyclotomic polynomials
Pingzhi Yuan
School of Mathematics, South China Normal University , Guangzhou 510631,
P.R.CHINA
e-mail mcsypz@mail.sysu.edu.cn
## Abstract
Let $a(n,k)$ be the $k$-th coefficient of the $n$-th cyclotomic polynomial.
Recently, Ji, Li and Moree [12] proved that for any integer $m\geq 1$,
$\\{a(mn,k)|n,k\in\mathbb{N}\\}=\mathbb{Z}$. In this paper, we improve this
result and prove that for any integers $s>t\geq 0$,
$\\{a(ns+t,k)|n,k\in\mathbb{N}\\}=\mathbb{Z}.$
2000 Mathematics Subject Classification:11B83; 11C08
Keywords: Cyclotomic polynomials; Dirichlet’s theorem; Squarefree integers
## 1 Introduction
Let $\Phi_{n}(x)=\sum_{k=0}^{\varphi(n)}a(n,k)x^{k}$ be the $n$th cyclotomic
polynomial. The Taylor series of $1/\Phi_{n}(x)$ around $x=0$ is given by
$1/\Phi_{n}(x)=\sum_{k=0}^{\varphi(n)}c(n,k)x^{k}$. It is not difficult to
show that $a(n,k)$ and $c(n,k)$ are all integers. The coefficients $a(n,k)$
and $c(n,k)$ are quite small in absolute value, for example for $n<105$ it is
well-known that $|a(n,k)|\leq 1$ and for $n<561$ we have $|c(n,k)|\leq 1$(see
[13]). Migotti [8] showed that all $a(pq,i)\in\\{0,\pm 1\\}$, where $p$ and
$q$ are distinct primes. Beiter [3] and [4] gave a criterion on $i$ for
$a(pq,i)$ to be $0,1$ or -1, see also Lam and Leung [6]. Also Carlitz [5]
computed the number of non-zero $a(pq,i)$’s. For more information on this
topic, we refer to the beautiful survey paper of Thangadurai [14]. Bachman [1,
2] proved the existence of an infinite family of $n=pqr$ with all
$a(pqr,i)\in\\{0,\pm 1\\}$, where $p,q,r$ are distinct odd primes.
Let $m\geq 1$ be a integer. Put
$S(m)=\\{a(mn,k)|n\geq 1,k\geq 0\\}\quad\mbox{ and}\quad R(m)=\\{c(mn,k)|n\geq
1,k\geq 0\\}.$
Schur poved in 1931 (in a letter to E. Landau) that $S(1)$ is not a finite
set, see Lenstra [7]. In 1987 Suzuki [10] proved that $S(1)=\mathbb{Z}$.
Recently, Ji, Li and Moree [12], [11] proved that with $S(m)=R(m)=\mathbb{Z}$
for any integer $m\geq 1$.
Let $m\geq 1,s>t\geq 0$ be positive integers with $\gcd(s,t)=1$. Put
$S(m;s,t)=\\{a(m(sn+t),k)|n\geq 1,k\geq 0\\}\quad\mbox{ and}\quad
R(m;s,t)=\\{c(m(sn+t),k)|n\geq 1,k\geq 0\\}.$
In this note, by a slight modification of the proof in [12], we prove the
following generalization of the result in [12].
###### Theorem 1.1.
Let $m\geq 1,s>t\geq 0$ be positive integers with $\gcd(s,t)=1$. Then
$S(m;r,t)=R(m;s,t)=\mathbb{Z}$.
An equivalent statement of Theorem 1.1 is the following result, which is the
motivation to write this paper.
###### Theorem 1.2.
Let $s>t\geq 0$ be integers, then
$\\{a(ns+t,k)|n,k\in\mathbb{N}\\}=\\{c(ns+t,k)|n,k\in\mathbb{N}\\}=\mathbb{Z}.$
## 2 Some Lemmas
###### Lemma 2.1.
([12] Lemma 1) The coefficient $c(n,k)$ is an integer whose value only depends
on the congruence class of $k$ modulo $n$.
Let $\kappa(m)=\prod_{p|m}p$ denote the squarefree kernel of $m$.
###### Lemma 2.2.
([12] Corollary 1) We have $S(m)=S(\kappa(m))$ and $R(m)=R(\kappa(m))$.
###### Lemma 2.3.
(Quantitative Form of Dirichlet s Theorem) Let $a$ and $m$ be coprime natural
numbers and let $\pi(x;m,a)$ denote the number of primes $p\leq x$ that
satisfy $p\equiv a\pmod{m}$. Then, as $x$ tends to infinity,
$\pi(x;m,a)\sim\frac{x}{\varphi(m)\log x},$
where $\phi$ is Euler’s toitent function.
###### Lemma 2.4.
([12] Corollary 2) Given $m,t\geq 1$ and any real number $r>1$ , there exists
a constant $N_{0}(t,m,r)$ such that for every $n>N_{0}(t,m,r)$ the interval
$(n,rn)$ contains at least $t$ primes $p\equiv 1\pmod{m}$.
## 3 The proof of Theorem 1
###### Proof.
We first prove that $S(m;s,t)=\mathbb{Z}$. Since $S(m;s,t)=S(\kappa(m);s,t)$
and $S(m;s,t)\supseteq S(mp;s,t)$, where $p\equiv 1\pmod{s}$ is an odd prime,
we may assume that $m$ is square-free, $m>1$ and $\mu(m)=1$. Suppose that
$n>N_{0}(t,ms,\frac{15}{8})$, then, by Lemma 2.4, there exist primes
$p_{1},p_{2},\ldots,p_{t}$ such that
$N<p_{1}<p_{2}<\cdots<p_{t}<\frac{15}{8}n\quad\mbox{and}\quad p_{j}\equiv
1\pmod{ms},\quad j=1,2,\ldots,t.$
Let $q_{1},q_{2}$ be primes such that $q_{2}>q_{1}>2p_{1}$, $q_{1}\equiv
t\pmod{s}$ and $q_{2}\equiv 1\pmod{s}$ and put
$m_{1}=\left\\{\begin{aligned} p_{1}p_{2}\cdots
p_{t}q_{1}&\quad\mbox{if}\,t\,\mbox{is even};\\\ p_{1}p_{2}\cdots
p_{t}q_{1}q_{2}&\quad{\rm otherwise}.\end{aligned}\right.$ (1)
Note that $m$ and $m_{1}$ are coprime, $m_{1}\equiv t\pmod{s}$ and that
$\mu(m_{1})=-1$, where $\mu$ denotes the Möbius function. Using these
observations we conclude that
$\begin{split}\Phi_{mm_{1}}(x)&\equiv\prod_{d|mm_{1},d<2p_{1}}(1-x^{d})^{\mu(\frac{mm_{1}}{d})}\pmod{x^{2p_{1}}}\\\
&\equiv\prod_{d|m}(1-x^{d})^{\mu(\frac{m}{d})\mu(m_{1})}\prod_{j=1}^{t}(1-x^{p_{j}})^{\mu(\frac{mm_{1}}{p_{j}})}\pmod{x^{2p_{1}}}\\\
&\equiv\Phi_{m}(x)^{\mu(m_{1})}\prod_{j=1}^{t}(1-x^{p_{j}})^{-\mu(mm_{1})}\pmod{x^{2p_{1}}}\\\
&\equiv\frac{1}{\Phi_{m}(x)}\prod_{j=1}^{t}(1-x^{p_{j}})^{\mu(m)}\pmod{x^{2p_{1}}}\\\
&\equiv\frac{1}{\Phi_{m}(x)}(1-\mu(m)(x^{p_{1}}+\cdots+x^{p_{t}}))\pmod{x^{2p_{1}}}.\end{split}$
(2)
From (2) it follows that, if $p_{t}\leq k<2p_{1}$, then
$a(mm_{1},k)=c(m,k)-\mu(m)\sum_{j=1}^{t}c(m,k-p_{j}).$
By Lemma 2.1 we have $c(m,k-p_{j})=c(m,k-1)$, and therefore
$a(mm_{1},k)=c(m,k)-\mu(m)tc(m,k-1)\,\,\mbox{with}\,\,p_{t}\leq k<2p_{1}.$ (3)
Since $\mu(m)=1$, we let $q_{3}<q_{4}$ be the smallest two prime divisors of
$m$. Here we also required that $n\geq 8q_{4}$, which ensures that
$p_{t}+q_{4}<2p_{1}$. Note that
$\begin{split}\frac{1}{\Phi_{m}(x)}&\equiv\frac{(1-x^{q_{3}})(1-x^{q_{4}})}{1-x}\pmod{x^{q_{4}+2}}\\\
&\equiv
1+x+x^{2}+\cdots+x^{q_{3}-1}-x^{q_{4}}-x^{q_{4}+1}\pmod{x^{q_{4}+2}}.\end{split}$
(4)
Thus $c(m,k)=1$ if $k\equiv\beta\pmod{m}$ with $\beta\in\\{1,2\\}$ and
$c(m,k)=-1$ if $k\equiv\beta\pmod{m}$ with $\beta\in\\{q_{4},q_{4}+1\\}$. This
in combination with (3) shows that $a(m_{1}m,p_{t}+1)=1-t$ and
$a(m_{1}m,p_{t}+q_{4})=t-1$. Since $\\{1-t,t-1|t\geq 1\\}=\mathbb{Z}$, then
$S(m;s,t)=\mathbb{Z}$ and the first result follows.
To prove $R(m;s,t)=\mathbb{Z}$. As before we may assume that $m>1$ is square-
free and $\mu(m)=1$.
Let $q_{1},q_{2}$ be primes such that $q_{2}>q_{1}>2p_{1}$, $q_{1}\equiv
t\pmod{s}$ and $q_{2}\equiv 1\pmod{s}$ and put
$\bar{m_{1}}=\left\\{\begin{aligned} p_{1}p_{2}\cdots
p_{t}q_{1}q_{2}&\quad\mbox{if}\,t\,\mbox{is even};\\\ p_{1}p_{2}\cdots
p_{t}q_{1}&\quad{\rm otherwise}.\end{aligned}\right.$ (5)
Note that $m$ and $m_{1}$ are coprime and that $\mu(\bar{m_{1}})=1$. Reasoning
as in the derivation of (2) we obtain
$\frac{1}{\Phi_{mm_{1}}(x)}\equiv\frac{1}{\Phi_{m}(x)}(1-\mu(m)(x^{p_{1}}+\cdots+x^{p_{t}}))\pmod{x^{2p_{1}}}$
(6)
and from this $c(\bar{m_{1}}m,k)=a(m_{1}m,k)$ for $k\leq 2p_{1}$. Reasoning as
in the proof $S(m;s,t)=\mathbb{Z}$, we obtain $R(m;s,t)=\mathbb{Z}$. This
completes the proof. ∎
Remark: Since we do not need to consider the case $\mu(m)=-1$, so a proof a
little easier than that given in [12] is obtained.
Acknowledgments: The author is supported by NSF of China (No. 10971072) and by
the Guangdong Provincial Natural Science Foundation (No. 8151027501000114).
## References
* [1] G. Bachman, _Flat cyclotomic polynomials of order three_ , Bull. London Math. Soc. 38 (2006), pp. 53-60.
* [2] G. Bachman, _Ternary cyclotomic polynomials with an optimally large set of coefficients_ , Proc. Amer. Math. Soc. 132 (2004), pp. 1943-1950.
* [3] M. Beiter, _The midterm coefficient of the cyclotomic polynomials_ , Amer. Math. Monthly, 71(1964), 769-770.
* [4] M. Beiter, _Coefficients in the cyclotomic polynomials for numbers with at most three distinct odd primes in their factorization_ , The Catholic University of American Press, Washington 1960.
* [5] L. Carlitz, _The number of terms in the cyclotomic polynomial $\Phi_{pq}(X)$_, Amer. Math. Monthly, 73(1966), 979-981.
* [6] T.Y. Lam and K.H. Leung, _On the cyclotomic polynomial $\Phi_{pq}(X)$_, Amer. Math. Monthly, 103(1996), 562-564.
* [7] H.W. Lenstra Jr., _Vanishing sums of roots of unity_ , Proc. Bicentennial Cong. Wiskundig Genootschap, Vrije Univ., Amsterdam (1978), pp. 249-268.
* [8] A. Migotti, _Zur Theorie der Kreisteilungsgleichung_ , Sitzber. Math.-Naturwiss. Classe der Kaiser. Akad. der Wiss. 87 (1883), pp. 7-14.
* [9] P. Moree, H. Hommersom, _Value distribution of Ramanujan sums and of cyclotomic polynomial coefficients_. arXiv: math.NT/0307352.
* [10] J. Suzuki, _On coefficients of cyclotomic polynomials_ , Proc. Japan Acad. Ser. A Math. Sci. 63 (1987), pp. 279-280.
* [11] Chun-Gang Ji, Wei-Ping Li, _Values of coefficients of cyclotomic polynomials_ , Discrete Mathematics, 308(2008), 5860-5863.
* [12] Chun-Gang Ji, Wei-Ping Li, Pieter Moree, _Values of coefficients of cyclotomic polynomials II_ , Discrete Mathematics, 309(2009), 1720-1723.
* [13] Pieter Moree, _Reciprocal cyclotomic polynomials_ , Journal of Number Theory 129(2009), 667-680.
* [14] Ravindranathan Thangadurai, _On the coefficients of cyclotomic polynomials_ , in: Cyclotomic Fields and Related Topics (Pune, 1999), Bhaskaracharya Pratishthana, Pune, 2000, pp. 311 C322.
|
arxiv-papers
| 2009-09-05T06:22:45 |
2024-09-04T02:49:05.063844
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Pingzhi Yuan",
"submitter": "Pingzhi Yuan",
"url": "https://arxiv.org/abs/0909.0995"
}
|
0909.1089
|
# Fusion dynamics of symmetric systems near barrier energies
Zhao-Qing Fenga,b111Corresponding author. Tel. +86 931 4969215.
_E-mail address:_ fengzhq@impcas.ac.cn, Gen-Ming Jina,b
a _Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000,
China_
Abstract
The enhancement of the sub-barrier fusion cross sections was explained as the
lowering of the dynamical fusion barriers within the framework of the improved
isospin-dependent quantum molecular dynamics (ImIQMD) model. The numbers of
nucleon transfer in the neck region are appreciably dependent on the incident
energies, but strongly on the reaction systems. A comparison of the neck
dynamics is performed for the symmetric reactions 58Ni+58Ni and 64Ni+64Ni at
energies in the vicinity of the Coulomb barrier. An increase of the ratios of
neutron to proton in the neck region at initial collision stage is observed
and obvious for neutron-rich systems, which can reduce the interaction
potential of two colliding nuclei. The distribution of the dynamical fusion
barriers and the fusion excitation functions are calculated and compared them
with the available experimental data.
_PACS_ : 25.60.Pj, 25.70.Jj, 24.10.-i
_Keywords:_ ImIQMD model; dynamical fusion barrier; nucleon transfer; fusion
excitation functions
Heavy-ion fusion reactions at energies in the vicinity of the Coulomb barrier
has been an important subject in nuclear physics for more than 20 years, which
is involved in not only exploring several fundamental problems such as quantum
tunneling in the multidimensional potential barrier etc, also investigating
nuclear physics itself associated with nuclear structure, synthesis of
superheavy nuclei etc [1]. The experimental fusion cross sections can be well
reproduced by the various coupled channel methods, which include the couplings
of the relative motion to the nuclear shape deformations, vibrations,
rotations, and nucleon-transfer, such as CCFULL code [2]. However, the coupled
channel models still have some difficulties in describing the fusion reactions
for symmetric systems, especially for heavy combinations, in which the neck
dynamics in the fusion process of two colliding nuclei plays an important role
on the interaction potential, and consequently on the fusion cross section.
Microscopic mechanism of the neck dynamics is significant for properly
understanding the capture and fusion process in the formation of superheavy
nuclei in massive fusion reactions [3]. The ImIQMD model has been successfully
applied to treat heavy-ion fusion reactions near barrier energies in our
previous works [4], in which the interaction potential energy is
microscopically derived from the Skyrme energy-density functional besides the
spin-orbit term and the shell correction is considered properly. In this
letter, we will concentrate on exploring the influence of the dynamical
mechanism in heavy-ion collisions near barrier energies on the fusion cross
sections.
In the ImIQMD model, the time evolutions of the nucleons under the self-
consistently generated mean-field are governed by Hamiltonian equations of
motion, which are derived from the time dependent variational principle and
read as
$\displaystyle\dot{\mathbf{p}}_{i}=-\frac{\partial
H}{\partial\mathbf{r}_{i}},\quad\dot{\mathbf{r}}_{i}=\frac{\partial
H}{\partial\mathbf{p}_{i}}.$ (1)
The total Hamiltonian $H$ consists of the kinetic energy, the effective
interaction potential and the shell correction part as
$H=T+U_{int}+U_{sh}.$ (2)
The details of the three terms can be found in details in Ref. [4]. The shell
correction term is important for magic nuclei induced fusion reactions, which
constrains the fusion cross section in the sub-barrier region.
For the lighter reaction systems, the compound nucleus is formed after the two
colliding nuclei is captured by the interaction potential. The quasi-fission
reactions after passing over the barrier take place when the product
$Z_{p}Z_{t}$ of the charges of the projectile and target nuclei is larger than
about 1600. In the ImIQMD model, the interaction potential $V(R)$ of two
colliding nuclei as a function of the distance $R$ between their centers is
defined as [5]
$V(R)=E_{pt}(R)-E_{p}-E_{t}.$ (3)
Here the $E_{pt}$, $E_{p}$ and $E_{t}$ are the total energies of the whole
system, projectile and target, respectively. The total energy is the sum of
the kinetic energy, the effective potential energy and the shell correction
energy. In the calculation, the Thomas-Fermi approximation is adopted for
evaluating the kinetic energy. Shown in in Fig. 1 is a comparison of the
various static interaction potentials, such as Bass potential [6], double-
folding potential used in dinuclear system model [3], proximity potential of
Myers and Swiatecki [7], the adiabatic barrier as mentioned in Ref. [8] and
ImIQMD static and dynamical interaction potentials for head on collisions of
the reaction system 58Ni+58Ni. It should be noted that the potentials
calculated by the ImIQMD model have included the shell effects that evolve
from the projectile and target nuclei into the composite system. The
contribution of the shell correction energy to the interaction potential is
shown separately in the right panel of the figure at frozen densities and
different incident energies. The static interaction potential means that the
density distribution of projectile and target is always assumed to be the same
as that at initial time, which is a diabatic process and depends on the
collision orientations and the mass asymmetry of the reaction systems. The
corresponding barrier heights are indicated for the various cases. However,
for a realistic heavy-ion collision, the density distribution of the whole
system will evolve with the reaction time, which is dependent on the incident
energy and impact parameter of the reaction system [9]. In the calculation of
the dynamical potentials, we only pay attention to the fusion events, which
give the dynamical fusion barrier. At the same time, a stochastic rotation is
performed for each simulation event. One can see that the heights of the
dynamical barriers are reduced gradually with decreasing the incident energy,
which result from the reorganization of the density distribution of two
colliding nuclei due to the influence of the effective interaction potential
on each nucleon. The dynamical barrier with incident energy $E_{c.m.}$=105 MeV
approaches the static one. The lowering of the dynamical fusion barrier is in
favor of the enhancement of the sub-barrier fusion cross sections, which can
give a little information that the cold fusion reactions are also suitable to
produce superheavy nuclei although an extra-push energy is needed for heavy
reaction systems [10]. The energy dependence of the nucleus-nucleus
interaction potential in heavy-ion fusion reactions was also investigated by
the time dependent Hartree-Fock theory and the lowering of dynamical barrier
near Coulomb energies was also observed [11].
The influence of the structure quantities such as excitation energies,
deformation parameters of the collective motion can be embodied by comparing
the fusion barrier distributions calculated from the coupled channel models
and the measured fusion excitation functions. In the ImIQMD model, the
dynamical fusion barrier is calculated by averaging the fusion events at a
given incident energy and a fixed impact parameter. To explore more
information on the fusion dynamics, we also investigate the distribution of
the dynamical fusion barrier, which counts the dynamical barrier per fusion
event and satisfies the condition $\int f(B_{fus})dB_{fus}=1$. Fig. 2 shows
the barrier distribution for head on collisions of the reaction 58Ni+58Ni at
the center of mass incident energies 96 MeV and 100 MeV, respectively, which
correspond to below and above the static barrier $V_{b}=97.32$ MeV as labeled
in Fig. 1, and a comparison with the neutron-rich system 64Ni+64Ni. The
distribution trend moves towards the low-barrier region with decreasing the
incident energy, which can be explained from the slow evolution of the
colliding system. The system has enough time to exchange and reorganize
nucleons of the reaction partners at lower incident energies. A number of
fusion events are located at the sub-barrier region, which is favorable to
enhance sub-barrier fusion cross sections. There is a little distribution
probability that the fusion barrier is higher than the incident energy 96 MeV
owing to dynamical evolution of two touching nuclei. We should note that the
fusion events decrease dramatically with incident energy in the sub-barrier
region. Neutron-rich system has the distribution towards the low-barrier
region owing to the lower dynamical fusion barrier, which favors the
enhancement of the fusion cross section.
The neck formation in heavy-ion collisions close to the Coulomb barrier is of
importance for understanding the enhancement of the sub-barrier cross
sections. A phenomenological approach (neck formation fusion model) was
proposed by Vorkapić [12] to fit experimental data that can not be reproduced
properly by the coupled channel models. Using a classical dynamical model
Aguiar, Canto, and Donangelo have pointed out that the neck formation in
heavy-ion fusion reactions may explain the lowering of the barrier [13]. Using
the ImIQMD model, we carefully investigate the dynamics of the formation of
the neck in heavy-ion fusion reactions. The neck region is defined as a
cylindrical shape along the collision orientation with the high 4 fm when the
density at the touching point reaches 0.02$\rho_{0}$. Shown in Fig. 3 is the
numbers of nucleon transfer from projectile to target in the neck region at
incident energies 95 MeV and 100 MeV in the left panel and a comparison of the
system 58Ni+58Ni and 64Ni+64Ni in the right panel. The evolution time starts
at the stage of the neck formation. A slight peak appears for both cases
because the dynamical fluctuation takes place in the formation process of the
neck. Larger numbers of neutron transfer are obvious especially for neutron-
rich system, which can be easily understood because the neutron transfer does
not affected by the repulsive Coulomb force. The transfer of protons reduces
the interaction potential of two colliding nuclei. The time evolution of the
ratio of neutron to proton in the neck region and the radius of the neck at
incident energy 100 MeV are also calculated as shown in Fig. 4 for the
reactions 58Ni+58Ni and 64Ni+64Ni. It is clear that the neutron-rich system
has the larger values of the N/Z ratio and the neck radius. An obvious bump in
the evolution of the N/Z ratio appears at the initial stage of the formation
of the neck for both systems due to the Coulomb repulsion for protons.
In the ImIQMD model, the fusion cross section is calculated by the formula [4]
$\sigma_{fus}(E)=2\pi\int_{0}^{b_{max}}bp_{fus}(E,b)db=2\pi\sum_{b=\Delta
b}^{b_{max}}bp_{fus}(E,b)\Delta b,$ (4)
where $p_{fus}(E,b)$ stands for the fusion probability and is given by the
ratio of the fusion events $N_{fus}$ to the total events $N_{tot}$. In the
calculation, the step of the impact parameter is set to be $\Delta b=0.5$ fm.
In Fig. 5 we show a comparison of the calculated fusion excitation functions
and the well-known one dimensional Hill-Wheeler formula [14] as well as the
experimental data for the reactions 58Ni+58Ni [15] and 64Ni+64Ni [16]. One can
see that a strong enhancement of the fusion cross sections for the neutron-
rich combination 64Ni+64Ni is obvious, especially in the sub-barrier region.
The Hill-Wheeler formula reproduces rather well the fusion cross sections at
above barrier energies, but underestimate obviously the sub-barrier cross
sections. The ImIQMD model reproduces the experimental data rather well over
the whole range. In the piont of view from dynamical calculations, the
reorganization of the density distribution of the colliding system results in
the lowering of the dynamical fusion barrier, which consequently enhances the
sub-barrier fusion cross sections. The phenomenon is more clearly for neutron-
rich combinations.
In conclusion, using the ImIQMD model, the fusion dynamics in heavy-ion
collisions in the vicinity of the Coulomb barrier is investigated
systematically. The dynamical fusion barrier is reduced with decreasing the
incident energies, which results in the enhancement of the sub-barrier fusion
cross sections. The distribution forms of the dynamical fusion barrier are
dependent on the incident energies and the N/Z ratios in the neck region of
the reaction systems. The nucleon transfer in the neck region reduces the
interaction potential of two colliding nuclei. The lower fusion barrier is in
favor of the enhancement of the fusion cross sections of the neutron-rich
systems.
Acknowledgements
This work was supported by the National Natural Science Foundation of China
under Grant No. 10805061, the special foundation of the president fellowship,
the west doctoral project of Chinese Academy of Sciences, and major state
basic research development program under Grant No. 2007CB815000.
## References
* [1] A.B. Balantekin, N. Takigawa, Rev. Mod. Phys. 79 (1998) 77\.
* [2] K. Hagino, N. Rowley, A.T. Kruppa, Comput. Phys. Commun. 123 (1999) 143.
* [3] Z.Q. Feng, G.M. Jin, J.Q. Li, W. Scheid, Phys. Rev. C 76 (2007) 044606; Z.Q. Feng, G.M. Jin, J.Q. Li, W. Scheid, Nucl. Phys. A 816 (2009) 33.
* [4] Z.Q. Feng, F.S. Zhang, G.M. Jin, X. Huang, Nucl. Phys. A 750 (2005) 232; Z.Q. Feng, G.M. Jin, F.S. Zhang, Nucl. Phys. A 802 (2008) 91; Z.Q. Feng, G.M. Jin, F.S. Zhang, et al., Chin. Phys. Lett. 22 (2005) 3040.
* [5] K.A. Brueckner, J.R. Buchler, M.M. Kelly, Phys. Rev. 173 (1968) 944.
* [6] R. Bass, Phys. Rev. Lett. 39 (1977) 265.
* [7] W.D. Myers, W.J. Swiatecki, Phys. Rev. C 62 (2000) 044610.
* [8] K. Siwek-Wilczynska, J. Wilczynski, Phys. Rev. C 64 (2001) 024611.
* [9] Z.Q. Feng, G.M. Jin, F.S. Zhang, et al., Chin. Phys. Lett. 22 (2005) 3040.
* [10] S. Bjornholm and W.J. Swiatecki, Nucl. Phys. A 391 (1982) 471.
* [11] K. Washiyama, D. Lacroix, Phys. Rev. C 78 (2008) 024610.
* [12] D. Vorkapić, Phys. Rev. C 49 (1994) 2812.
* [13] C.E. Aguiar, L.F. Canto, R. Donangelo, Phys. Rev. C 31 (1985) 1969.
* [14] D.L. Hill, J.A. Wheeler, Phys. Rev. 89 (1953) 1102.
* [15] M. Beckerman, J. Ball, H. Enge, et al., Phys. Rev. C 23 (1981) 1581.
* [16] C.L. Jiang, K.E. Rehm, R.V.F. Janssens, et al., Phys. Rev. Lett. 93 (2004) 102701.
Figure 1: Comparisons of the reaction 58Ni+58Ni for various static interaction
potentials (Bass, double-folding, proximity and ImIQMD potential at frozen
density), the dynamical fusion potentials at different incident energies and
the adiabatic potential in Ref. [8] (left panel), and the contributions of the
shell corrections calculated at the frozen densities and at incident energies
95 MeV, 100 MeV and 105 MeV, respectively. Figure 2: Distribution of the
dynamical fusion barriers at incident energies 96 MeV and 100 MeV in the
center of mass frame (left panel) and comparison of the systems 58Ni+58Ni and
64Ni+64Ni (right panel). Figure 3: Nucleon transfer from projectile to target
nucleus in the neck region at different incident energies (left panel) and for
systems 58Ni+58Ni and 64Ni+64Ni (right panel). Figure 4: The ratio of neutron
to proton in the neck region (left panel) and the radius of the neck (right
panel) as functions of the evolution time at incident energy 100 MeV. Figure
5: The calculated fusion excitation functions for the reactions 58Ni+58Ni and
64Ni+64Ni, and compared them with the Hill-Wheeler formula [14] and the
experimental data [15, 16].
|
arxiv-papers
| 2009-09-06T15:48:27 |
2024-09-04T02:49:05.069131
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zhao-Qing Feng, Gen-Ming Jin",
"submitter": "Zhaoqing Feng",
"url": "https://arxiv.org/abs/0909.1089"
}
|
0909.1131
|
# On the vanishing and finiteness properties of generalized local cohomology
modules
Moharram Aghapournahr Moharram Aghapournahr
Arak University, Beheshti St, P.O. Box:879, Arak, Iran m-aghapour@araku.ac.ir
###### Abstract.
Let $R$ be a commutative noetherian ring, $\mathfrak{a}$ an ideal of $R$ and
$M,N$ finite $R$–modules. We prove that the following statements are
equivalent.
1. (i)
$\operatorname{H}^{i}_{\mathfrak{a}}(M,N)$ is finite for all $i<n$.
2. (ii)
$\operatorname{Coass}_{R}(\operatorname{H}^{i}_{\mathfrak{a}}(M,N))\subset\operatorname{V}{(\mathfrak{a})}$
for all $i<n$.
3. (iii)
$\operatorname{H}^{i}_{\mathfrak{a}}(M,N)$ is coatomic for all $i<n$.
If $\operatorname{pd}M$ is finite and $r$ be a non-negative integer such that
$r>\operatorname{pd}M$ and $\operatorname{H}^{i}_{\mathfrak{a}}(M,N)$ is
finite (resp. minimax) for all $i\geq r$, then
$\operatorname{H}^{i}_{\mathfrak{a}}(M,N)$ is zero (resp. artinian) for all
$i\geq r$.
###### Key words and phrases:
Generalized local cohomology, Minimax module, coatomic module, Projective
dimension.
###### 2000 Mathematics Subject Classification:
13D45, 13D07
## 1\. Introduction
Throughout $R$ is a commutative noetherian ring. Generalized local cohomology
was given in the local case by J. Herzog [5] and in the more general case by
M.H Bijan-Zadeh [2]. Let $\mathfrak{a}$ denote an ideal of a ring $R$. The
generalized local cohomology defined by
$\operatorname{H}^{i}_{\mathfrak{a}}(M,N)\cong\underset{n}{\varinjlim}\operatorname{Ext}^{i}_{R}(M/{\mathfrak{a}}^{n}M,N).$
This concept was studied in the articles [8], [5] and [9]. Note that this is
in fact a generalization of the usual local cohomology, because if $M=R$, then
$\operatorname{H}^{i}_{\mathfrak{a}}(R,N)=\operatorname{H}^{i}_{\mathfrak{a}}(N)$.
Important problems concerning local cohomology are vanishing, finiteness and
artinianness results (see [6]).
In Section 2 we show in 2.1 that if $M$ is finite and all generalized local
cohomology modules $\operatorname{H}^{i}_{\mathfrak{a}}(M,N)$ are coatomic for
all $i<n$, then they are finite for all $i<n$. In fact this is another
condition equivalent to Falting’s Local-global Principle for the finiteness of
generalized local cohomology modules (see [1, Theorem 2.9]). In Theorem 2.2 we
generalize Yoshida’s theorem ( [10, Theorem 3.1]).
In Section 3, We prove in 3.2, that when $M$ is a finite $R$–module of finite
projective dimension such that the generalized local cohomology modules
$\operatorname{H}^{i}_{\mathfrak{a}}(M,N)$ are minimax modules for all $i\geq
r$, (where $r>\operatorname{pd}M$) then they must be artinian.
For unexplained terminology we refer to [3] and [4].
## 2\. Finiteness and vanishing
An $R$–module $M$ is called coatomic when each proper submodule $N$ of $M$ is
contained in a maximal submodule $N^{\prime}$ of $M$ (i.e. such that
$M/N^{\prime}\cong R/\mathfrak{m}$ for some
$\mathfrak{m}\in\operatorname{Max}{R}$). This property can also be expressed
by $\operatorname{Coass}_{R}(M)\subset\operatorname{Max}{R}$ or equivalently
that any artinian homomorphic image of $M$ must have finite length. In
particular all finite modules are coatomic. Coatomic modules have been studied
by Zöschinger [12].
###### Theorem 2.1.
Let $R$ be a noetherian ring, $\mathfrak{a}$ an ideal of $R$ and $M,N$ finite
$R$–modules. The following statements are equivalent:
1. (i)
$H^{i}_{\mathfrak{a}}(M,N)$ is coatomic for all $i<n$.
2. (ii)
$\operatorname{Coass}_{R}(\operatorname{H}^{i}_{\mathfrak{a}}(M,N))\subset\operatorname{V}{(\mathfrak{a})}$
for all $i<n$.
3. (iii)
$H^{i}_{\mathfrak{a}}(M,N)$ is finite for all $i<n$.
###### Proof.
By [1, Theorem 2.9] and[12, 1.1, Folgerung] we may assume that
$(R,\mathfrak{m})$ is a local ring.
$\Rightarrow$ (ii) It is trivial by the definition of coatomic modules.
$\Rightarrow$ (iii) By [15, Satz 1.2] there is $t\geq 1$ such that
$\mathfrak{a}^{t}\operatorname{H}^{i}_{\mathfrak{a}}(M,N)$ is finite for all
$i<n$. Therefore there is $s\geq t$ such that
$\mathfrak{a}^{s}\operatorname{H}^{i}_{\mathfrak{a}}(M,N)=0$ for all $i<n$,
and apply [1, Theorem 2.9].
$\Rightarrow$ (i) Any finite $R$–module is coatomic. ∎
The following results are generalizations of [10, Proposition 3.1].
###### Theorem 2.2.
Let $(R,\mathfrak{m})$ be a local ring, $\mathfrak{a}$ be an ideal of $R$ and
$M$ be a finite module of finite projective dimension. Let $N$ be a finite
module and $r>\operatorname{pd}{M}$. If
$\operatorname{H}^{i}_{\mathfrak{a}}(M,N)$ is finite for all $i\geq r$, then
$\operatorname{H}^{i}_{\mathfrak{a}}(M,N)=0$ for all $i\geq r$.
###### Proof.
We prove by induction on $d=\dim N$. If $d=0$, By [9, Theorem 3.7], it follows
that $\operatorname{H}^{i}_{\mathfrak{a}}(M,N)=0$ for all
$i>\operatorname{pd}M+\dim(M\otimes_{R}N)$ and so the claim clearly holds for
$n=0$. Now suppose $d>0$ and $\operatorname{H}^{i}_{\mathfrak{a}}(M,N)=0$ for
all $i>r$. It is enough to show $\operatorname{H}^{r}_{\mathfrak{a}}(M,N)=0$.
First suppose $\operatorname{depth}_{R}{N}>0$. Take $x\in\mathfrak{m}$ which
is $N$–regular. Then $\dim{N/{x}N}=d-1$. The exact sequence
$0\longrightarrow N\overset{x}{\longrightarrow}N\longrightarrow
N/{x}N\longrightarrow 0$
induces the exact sequence
$\operatorname{H}^{r}_{\mathfrak{a}}(M,N)\overset{x}{\longrightarrow}\operatorname{H}^{r}_{\mathfrak{a}}(M,N)\longrightarrow\operatorname{H}^{r}_{\mathfrak{a}}(M,N/{x}N)\longrightarrow\operatorname{H}^{r+1}_{\mathfrak{a}}(M,N)=0$
It yields that $\operatorname{H}^{i}_{\mathfrak{a}}(M,N/{x}N)=0$ for all
$i>r$. Hence by induction hypothesis we get
$\operatorname{H}^{r}_{\mathfrak{a}}(M,N/{x}N)=0$. Thus we have
$\operatorname{H}^{r}_{\mathfrak{a}}(M,N)=0$ by Nakayama’s lemma. Next suppose
$\operatorname{depth}_{R}{N}=0$. Put
$L=\operatorname{\Gamma}_{\mathfrak{m}}(N)$. Since $L$ have finite length, so
we have $\dim L=0$ and therefore $\operatorname{H}^{i}_{\mathfrak{a}}(M,L)=0$
for all $i>\operatorname{pd}M$ . But from the exact sequence
$0\longrightarrow L\longrightarrow N\longrightarrow N/L\longrightarrow 0$
we get the exact sequence
$...\rightarrow\operatorname{H}^{i}_{\mathfrak{a}}(M,L)\rightarrow\operatorname{H}^{i}_{\mathfrak{a}}(M,N)\rightarrow\operatorname{H}^{i}_{\mathfrak{a}}(M,N/L)\rightarrow\operatorname{H}^{i+1}_{\mathfrak{a}}(M,L)\rightarrow...$
hence we have
$\operatorname{H}^{i}_{\mathfrak{a}}(M,N)\cong\operatorname{H}^{i}_{\mathfrak{a}}(M,N/L)$
for all $i>\operatorname{pd}M$, and we get the required assertion from the
first step. ∎
###### Theorem 2.3.
Let $\mathfrak{a}$ be an ideal of $R$ and $M$ a finite $R$–module of finite
projective dimension. Let $N$ be a finite $R$–module and
$r>\operatorname{pd}M$. The following statements are equivalent:
1. (i)
$\operatorname{H}^{i}_{\mathfrak{a}}(M,N)=0$ for all $i\geq r$.
2. (ii)
$\operatorname{H}^{i}_{\mathfrak{a}}(M,N)$ is finite for all $i\geq r$.
3. (iii)
$\operatorname{H}^{i}_{\mathfrak{a}}(M,N)$ is coatomic for all $i\geq r$.
###### Proof.
$(i)\Rightarrow(ii)\Rightarrow(iii)$ Trivial. $(iii)\Rightarrow(i)$ By use of
theorem 2.2 and [12, 1.1, Folgerung] we may assume that $(R,\mathfrak{m})$ is
a local ring. Note that coatomic modules satisfy Nakayama’s lemma. So the
proof is the same as in theorem 2.2. ∎
In the following corollary $\operatorname{cd}_{\mathfrak{a}}(M,N)$ denote the
supremum of $i$’s such that $\operatorname{H}^{i}_{\mathfrak{a}}(M,N)\neq 0$.
###### Corollary 2.4.
Let $\mathfrak{a}$ an ideal of $R$, $M$ a finite $R$–module of finite
projective dimention and $N$ a finite $R$–module. If
$c:=\operatorname{cd}_{\mathfrak{a}}(M,N)>\operatorname{pd}M$, then
$\operatorname{H}^{c}_{\mathfrak{a}}(M,N)$ is not coatomic in particular is
not finite.
## 3\. Artinianness
Recall that a module $M$ is a minimax module if there is a finite (i.e.
finitely generated) submodule $N$ of $M$ such that the quotient module $M/N$
is artinian. Thus the class of minimax modules includes all finite and all
artinian modules. Moreover, it is closed under taking submodules, quotients
and extensions, i.e., it is a Serre subcategory of the category of
$R$–modules. Minimax modules have been studied by Zink in [11] and Zöschinger
in [13, 14]. See also [7].
###### Lemma 3.1.
Let $M$ and $N$ be two $R$–module. If $f:R\longrightarrow S$ is a flat ring
homomorphism, then
$\operatorname{H}^{i}_{\mathfrak{a}}(M,N)\otimes_{R}{S}\cong\operatorname{H}^{i}_{{\mathfrak{a}}}S(M\otimes_{R}{S},N\otimes_{R}{S}).$
###### Proof.
It is easy and we lift it to the reader.
∎
###### Theorem 3.2.
Let $\mathfrak{a}$ an ideal of $R$ and $M$ a finite $R$–module of finite
projective dimension. Let $N$ be a finite $R$–module and
$r>\operatorname{pd}M$. If $\operatorname{H}^{i}_{\mathfrak{a}}(M,N)$ is a
minimax module for all $i\geq r$, then
$\operatorname{H}^{i}_{\mathfrak{a}}(M,N)$ is an artinian module for all
$i\geq r$.
###### Proof.
Let $\mathfrak{p}$ be a non-maximal prime ideal of $R$. Then by the definition
of minimax module and lemma 3.1
$\operatorname{H}^{i}_{\mathfrak{a}}(M,N)_{\mathfrak{p}}\cong\operatorname{H}^{i}_{{\mathfrak{a}}R_{\mathfrak{p}}}(M_{\mathfrak{p}},N_{\mathfrak{p}})$
is a finite $R_{\mathfrak{p}}$–module for all $i\geq r$. By theorem 2.2,
$\operatorname{H}^{i}_{\mathfrak{a}}(M,N)_{\mathfrak{p}}=0$ for all $i\geq r$,
thus
$\operatorname{Supp}_{R}(\operatorname{H}^{i}_{\mathfrak{a}}(M,N))\subset{\operatorname{Max}{R}}$
for all $i\geq r$. By [7, Theorem 2.1],
$\operatorname{H}^{i}_{\mathfrak{a}}(M,N)$ is artinian for all $i\geq r$. ∎
Let $q_{\mathfrak{a}}(M,N)$ denote the supremum of the $i$’s such that
$\operatorname{H}^{i}_{\mathfrak{a}}(M,N)$ is not artinian with the usual
convention that the supremum of the empty set of integers is interpreted as
$-\infty$.
###### Corollary 3.3.
Let $\mathfrak{a}$ an ideal of $R$, $M$ a finite $R$–module of finite
projective dimension and $N$ a finite $R$–module. If
$q:=q_{\mathfrak{a}}(M,N)>\operatorname{pd}M$, then
$\operatorname{H}^{q}_{\mathfrak{a}}(M,N)$ is not minimax in particular is not
finite.
## References
* [1] A. Abbasi, K. Khashyarmanesh, A new version of Local-global Principal for annihilations of local cohomology modules, Colloq. Math. 100(2004), 213-219.
* [2] M. H. Bijan-Zadeh, A commen generalization of local cohomology theories, Glasgow Math. J. 21(1980), 173-181.
* [3] M.P. Brodmann, R.Y. Sharp, Local cohomology: an algebraic introduction with geometric applications, Cambridge University Press, 1998.
* [4] W. Bruns, J. Herzog, Cohen-Macaulay rings, Cambridge University Press, revised ed., 1998.
* [5] J. Herzog, _Komplexe, Auflösungen und Dualität in der lokalen Algebra_ , Habilitationsschrift, Universitat Regensburg 1970. Invent. Math. 9 (1970), 145–164.
* [6] C. Huneke, Problems on local cohomology :Free resolutions in commutative algebra and algebraic geometry, (Sundance, UT, 1990), 93-108, Jones and Bartlett, 1992.
* [7] P. Rudlof, _On minimax and related modules_ , Can. J. Math. 44 (1992), 154–166.
* [8] N. Suzuki, _On the generalized local cohomology and its duality_ , J. Math. Kyoto. Univ. 18 (1978), 71–85.
* [9] S. Yassemi, _Generalized section functors_ , J. Pure Appl. Algebra 95 (1994), 103–119.
* [10] K. I. Yoshida, Cofiniteness of local cohomology modules for ideals of dimension one, Nagoya Math. J. 147(1997), 179-191.
* [11] T. Zink, _Endlichkeitsbedingungen für Moduln über einem Noetherschen Ring_ , Math. Nachr. 164 (1974), 239–252.
* [12] H. Zöschinger, Koatomare Moduln, Math. Z. 170(1980) 221-232.
* [13] H. Zöschinger, Minimax Moduln, J. Algebra. 102(1986), 1-32.
* [14] H. Zöschinger, _Über die Maximalbedingung für radikalvolle Untermoduln_ , Hokkaido Math. J. 17 (1988), 101–116.
* [15] H. Zöschinger, _Über koassoziierte Primideale_ , Math Scand. 63(1988), 196-211.
|
arxiv-papers
| 2009-09-07T02:11:54 |
2024-09-04T02:49:05.074002
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Moharram Aghapournahr",
"submitter": "Moharram Aghapournahr",
"url": "https://arxiv.org/abs/0909.1131"
}
|
0909.1182
|
# Results from PAMELA, ATIC and FERMI : Pulsars or Dark Matter ?
Debtosh Chowdhury debtosh@cts.iisc.ernet.in Centre for High Energy Physics,
Indian Institute of Science, Bangalore, India Chanda J. Jog
cjjog@physics.iisc.ernet.in Department of Physics, Indian Institute of
Science, Bangalore, India Sudhir K. Vempati vempati@cts.iisc.ernet.in Centre
for High Energy Physics, Indian Institute of Science, Bangalore, India
###### Abstract
It is well known that the dark matter dominates the dynamics of galaxies and
clusters of galaxies. Its constituents remain a mystery despite an assiduous
search for them over the past three decades. Recent results from the
satellite-based PAMELA experiment detect an excess in the positron fraction at
energies between $10-100$ GeV in the secondary cosmic ray spectrum. Other
experiments namely ATIC, HESS and FERMI show an excess in the total electron
($e^{+}$\+ $e^{-}$) spectrum for energies greater 100 GeV. These excesses in
the positron fraction as well as the electron spectrum could arise in local
astrophysical processes like pulsars, or can be attributed to the annihilation
of the dark matter particles. The second possibility gives clues to the
possible candidates for the dark matter in galaxies and other astrophysical
systems. In this article, we give a report of these exciting developments.
## I Introduction
The evidence for the existence of dark matter in various astrophysical systems
has been gathering over the past three decades. It is now well-recognized that
the presence of dark matter is required in order to explain the observations
of galaxies and other astrophysical systems on larger scales. The clearest
support for the existence of dark matter comes from the now well-known
observation of nearly flat rotation curves or constant rotation velocity in
the outer parts of galaxies rubin ; Sofue:2000jx . Surprisingly the rotation
velocity is observed to remain nearly constant till the last point at which it
can be measured111In the absence of dark matter, one would expect that the
curves to fall off as we move towards the outer parts of the galaxy.. The
simple principle of rotational equilibrium then tells one that the amount of
dark to visible mass must increase at larger radii. Thus the existence of the
dark matter is deduced from its dynamical effect on the visible matter, namely
the stars and the interstellar gas in galaxies.
The presence of dark matter in the elliptical galaxies is more problematic to
ascertain since these do not contain much interstellar hydrogen gas which
could be used as a tracer of their dynamics, and also because these galaxies
are not rotationally supported. These galaxies are instead supported by
pressure or random motion of stars (see Binney Binney:1987 for details of
physical properties of the spiral and elliptical galaxies). As a result, the
total mass cannot be deduced using the rotation curve for elliptical galaxies.
Instead, here the motions of planetary nebulae which arise from old, evolved
stars, as well as lensing, have been used to trace the dark matter dekel05 .
The fraction of dark matter at four effective radii is still uncertain with
values ranging from 20% to 60% given in the literature, for the extensively
studied elliptical galaxy NGC 3379 mamon .
Historically the first evidence for the unseen or dark matter was found in
clusters of galaxies. Assuming the cluster to be in a virial equilibrium, the
total or the virial mass can be deduced from the observed kinematics. Zwicky
zwicky noted that there is a discrepancy of a factor of $\sim$ 10 between the
observed mass in clusters of galaxies and the virial mass deduced from the
kinematics. In other words, the random motions are too large for the cluster
to be bound and a substantial amount of dark matter ($\sim$ 10 times the
visible matter in galaxies) is needed for the clusters of galaxies to remain
bound. This discrepancy remained a puzzle for over four decades, and was only
realized to be a part of the general trend after the galactic-scale dark
matter was discovered in the late 1970’s.
On the much larger cosmological scale, there has been some evidence for non-
baryonic dark matter from theoretical estimates of primordial elements during
Big Bang Nucleosynthesis and measurements of them, particularly, primordial
deuterium. Accurate measurements of the Cosmic Microwave Background Radiation
(CMBR) could as well give information about the total dark matter relic
density of the Universe. The satellite based COBE experiment was one of the
first experiments to provide accurate “ mapping” of the CMBRcobe . The recent
high precision determination of the cosmological parameters using Type I
supernova data sndata as well as precise measurements of the cosmic
background radiation by the WMAP collaboration wmap ; kom08 has pinpointed
the total relic dark matter density in the early universe with an accuracy of
a few percent. Accordingly, dark matter forms almost 26% of all the matter
density of the universe, with visible matter about 4% and the dark energy
roughly about 70% of the total energy density. This goes under the name of
$\Lambda$CDM model with $\Lambda$ standing for dark energy and denoted by the
Einstein’s constant, and CDM standing for Cold Dark Matter cosmoreviews .
Numerical simulations for the currently popular scenario of galaxy formation,
based on the ${\Lambda}$CDM model, predicts a universal profile for the dark
matter in halos of spherical galaxies NFW97 . While this model was initially
successful, over the years many discrepancies between the predictions from it
and the observations have been pointed out. The strongest one has been the
‘cusp-core’ issue of the central mass distribution. While Navarro et al. NFW97
predict a cuspy222Sharp increase in the density at the centre. central mass
distribution, the observations of rotation curves of central regions of
galaxies, especially the low surface brightness galaxies, when modeled show a
flat or cored density distribution BMBR01 .
A significantly different alternative to the dark matter, which can be used to
explain the rotation curves of the galaxies and clusters was proposed early on
by Milgrom. He claimed that Milgrom:1983ca for low accelerations, Newtonian
law has to be modified by addition of a small repulsive term. This idea is
known as ‘MOND’ or the MOdified Newtonian Dynamics. While initially this idea
was not taken seriously by the majority of astrophysics community, it has
gained more acceptance in the recent years. For example some of the standard
features seen in galaxies such as the frequency of bars can be better
explained under the MOND paradigm, see Tiret et al. Tiret:2007dd . For a
summary of the predictions and comparisons of these two alternatives (dark
matter and MOND), see Combes et al. Combes:2009ab .
So far the most direct empirical proof for the existence of dark matter, and
hence the evidence against MOND, comes from the study of the so-called Bullet
clusterClowe:2006eq . This is a pair of galaxies undergoing a supersonic
collision at a redshift of $\sim 0.3$. The main visible baryonic component in
clusters is hot, X-ray emitting gas. In a supersonic collision, this hot gas
would collide and be left at the center of mass of the colliding system while
the stars will just pass through since they occupy a small volume333This is
exactly analogous to the reason why the atomic hydrogen gas from two colliding
galaxies is left at the center of mass while the stars and the molecular gas
pass through each other unaffected, as proposed and studied by Valluri et
al.jog:1990 to explain the observed HI deficiency but normal molecular gas
content of galaxies in clusters..
In the Bullet cluster, the gravitational potential as traced by the weak-
lensing shows peaks that are separated from the central region traced by the
hot gas. In MOND, these two would be expected to coincide444The relativistic
MOND theory Bek1 proposed by Bekenstein could be used to explain the Bullet
Cluster Bek2 ., since the gravitational potential would trace the dominant
visible component namely the hot gas, while if there is dark matter it would
be expected to peak at the location of the stellar component in the galaxies.
The latter case is what has been observed as can been seen in Fig.1 of
Clowe:2006eq . For the rest of the article, we will not consider the MOND
explanation, but instead take the view point that the flat rotation curves of
galaxies and clusters at large radii as an evidence for the existence of dark
matter. Furthermore, we believe that the dark matter explanation is much
simpler and more natural compared to the MOND explanation.
Despite the fact that the existence of dark matter has been postulated for
over three decades, there is still no consensus of what its constituents are.
This has been summarized well in many review articles. Refs trimble87 ; AKS09
are couple of examples that span from the early to recent times on this topic.
Over the years, both astrophysicists as well as particle physicists have
speculated on the nature of dark matter.
The baryonic dark matter in the form of low-mass stars, binary stars, or
Jupiter-like massive planets were ruled out early on (see trimble87 for a
summary). From the amount of dark matter required to explain the flat rotation
curves, it can be shown that the number densities required of these possible
constituents would be large, and hence it would be hard to hide these massive
objects. Because, if present in these forms, they should have been detected
either from their absorption or from their emission signals. It has also been
proposed that the galactic dark matter could be in the form of dense, cold
molecular clumps PCM94 , though this has not yet been detected. This
alternative cannot be expected to explain the dark matter necessary to “fit”
the observations of clusters, or indeed the elliptical galaxies since the
latter have very little interstellar gas.
There is also a more interesting possibility of the dark matter being
essentially of baryonic nature, but due to the dynamics of the QCD phase
transition in the early universe which left behind a form of cold quark-gluon-
plasma, the baryon number content of the dark matter is hidden from us. This
idea was first proposed by Witten in 1984 witten , who called these quantities
as quark nuggets. An upper limit on the total number of baryons in a quark
nugget is determined by the baryon to photon ratio in the early universe (See
for example raha1 ). Taking in to consideration these constraints, it is
possible to fit the observed relic density with a mass (density) distribution
of the quark nuggets raha2 . For the observational possibilities of such quark
nuggets, see for example, Ref.nuggetstudy .
From a more fundamental point of view, it is not clear what kind of elementary
particle could form dark matter. The standard model of particle physics
describes all matter to be made up of quarks and leptons of which neutrinos
are the only ones which can play the role of dark matter as they are
electrically neutral. However with the present indications from various
neutrino oscillation experiments putting the standard model neutrino masses in
the range $\lesssim 1$ eV valleneutrinoreview they will not form significant
amount of dark matter. There could however, be non-standard sterile neutrinos
with masses of the order of keV-MeV which could form warm555Depending on the
mass of the particle which sets its thermal and relativistic properties, dark
matter can be classified as hot, warm and cold Peacock:1999ye .dark matter
(for reviews, see Refs. strumiavissani ; julien1 ). Cold Dark Matter (CDM), on
the other hand, is favored over the warm dark matter by the hierarchical
clustering observed in numerical simulations for large scale structure
formation, see for example Ref.Peacock:1999ye . Recent analysis including
X-ray flux observations from Coma Cluster and Andromeda galaxy have shown that
the room for sterile neutrino warm dark matter is highly constrained julien2 .
However, if one does not insist that the total relic dark matter density is
due to sterile neutrinos then, it is still possible that they form a sub-
dominant warm component of the total dark matter silk2 relic density666On the
other hand, if the neutrinos are not thermally produced and their production
is suppressed like in models with low reheating temperature gelmini1 , it is
possible to weaken the cosmological bounds, especially from extra galactic
radiation and distortion of CMBR spectra gelmini2 . See also julupdate ..
The Standard Model thus, needs to be extended to incorporate a dark matter
candidate. The simplest extensions would be to just include a new particle
which is a singlet under the SM gauge group (i.e., does not carry the Standard
Model interactions). Further, we might have to impose an additional symmetry
under which the Dark Matter particle transforms non-trivially to keep it
stable or at least sufficiently long lived with a life time typically larger
than the age of the universe. Some of the simplest models would just involve
adding additional light ($\sim$ GeV) scalar particles to the SM and with an
additional $U(1)$ symmetry (see for example, Boehm et al. fayet1 ). Similar
extensions of SM can be constructed with fermions too fayet2 ; wells . An
interesting aspect of these set of models is that they can be tested at
existing $e^{+}e^{-}$ colliders like for the example, the one at present at
Frascati, Italy dreeschoudhury . A heavier set of dark matter candidates can
be achived by extending the Higgs sector by adding additional Higgs scalar
doublets. These go by the name of inert Higgs models Barbieri:2005kf ;
marajaji . In this extension, there is a additional neutral higgs boson which
does not have SM gauge interactions (hence inert), which can be a dark matter
candidate. With the inclusion of this extra inert higgs doublet, the SM
particle spectrum has some added features, like it can evade the “naturalness
problem” up to 1.5 TeV while preserving the perturbativity of Higgs couplings
up to high scales and further it is consistent with the electroweak precision
tests Barbieri:2006dq .
On the other hand, there exist extensions of the Standard Model (generally
labeled Beyond Standard Model (BSM) physics) which have been constructed to
address a completely different problem called the hierarchy problem. The
hierarchy problem addresses the lack the symmetry for the mass of the Higgs
boson in the Standard Model and the consequences of this in the light of the
large difference of energy scales between the weak interaction scale ($\sim
10^{2}$ GeV) and the quantum gravity or grand unification scale ($\sim
10^{16}$ GeV). Such a huge difference in the energy scales could destabilize
the Higgs mass due to quantum corrections. To protect the Higgs mass from
these dangerous radiative corrections, new theories such as supersymmetry,
large extra dimensions and little Higgs have been proposed. It turns out that
most of these BSM physics models contain a particle which can be the dark
matter. A few examples of these theories and the corresponding candidates for
dark matter are as follows. (i) Axions are pseudo-scalar particles which
appear in theories with Peccei-Quinn symmetry Peccei:1977hh ; Peccei:2006as
proposed as solution to the strong CP problem of the standard model. They also
appear in Superstring theories which are theories of quantum gravity. The
present limits on axions are Bertone:2004pz extremely strong from
astrophysical data. In spite of this, there is still room for axions to form a
significant part of the dark matter relic density.
(ii) Supersymmetric theories Martin:1997ns ; Drees:2004jm which incorporate
fermion-boson interchange symmetry are proposed as extensions of Standard
Model to protect the Higgs mass from large radiative corrections. The dark
matter candidate is the lightest supersymmetric particle (LSP) which is stable
or sufficiently long lived as mentioned before777The corresponding symmetry
here is called $R$-parity. If this symmetry is exact, the particle is stable.
If is broken very mildly, the LSP could be sufficiently long lived, close to
the age of the universe.. Depending on how supersymmetry is broken jun96 ,
there are several possible dark matter candidates in these models. In some
models, the lightest supersymmetric particle and hence the dark matter
candidate is a neutralino. The neutralino is a linear combination of super-
partners of $Z,\gamma$ as well as the neutral Higgs bosons888The neutralino
could be either gaugino dominated or higgsino dominated depending on the
composition. It turns out that neutralino composition should be sufficiently
well-temperedArkaniHamed:2006mb to explain the observed relic density. While
one might debate the some what philosophical requirement of ‘fine-tuning’, it
is now known that in simplest models of supersymmetry breaking, like mSUGRA,
only special regions in the parameter space, corresponding to the special
conditions in the neutralino-neutralino annihilation channels satisfy the
relic density constraint dreesdjouadi . . The other possible candidates are
the super-partner of the graviton, called the gravitino and the super-partners
of the axinos, the scalar saxion and the fermionic axino. These particles also
can explain the observed relic density covireview .
(iii) Other classic extensions of the Standard Model either based on
additional space dimensions or larger symmetries also have dark matter
candidates. In both versions of the extra dimensional models, i.e., the
Arkani-Hamed, Dimopoulos, Dvali (ADD) ArkaniHamed:1998rs ; ArkaniHamed:1998nn
and Randall-Sundrum (RS) Randall:1999ee ; Randall:1999vf , models, the
lightest Kaluza-Klein particle999The extra space dimensions are compactified.
The compact extra dimension manifests it selves in ordinary four dimensional
space-time as an infinite tower of massive particles called Kaluza-Klein (KK)
particles. can be considered as the dark matter candidate Servant:2002aq ;
Hooper:2007qk ; che02 ; Bertone:2002ms . Similarly, in the little-Higgs models
where the Higgs boson is a pseudo-Goldstone boson of a much larger symmetry, a
symmetry called T-parity Hubisz:2004ft assures us a stable and neutral
particle which can form the dark matter. Very heavy neutrinos with masses of
$\mathcal{O}(100~{}\text{GeV}-1~{}\text{TeV})$ can also naturally appear
within some classes of Randall-Sundrum and Little Higgs models. Under suitable
conditions, these neutrinos can act like cold dark matter. (For a recent
study, please see geraldine ). In addition to these particles, more exotic
candidates like simpzillas simpzillas and wimpzillas kolb2 with masses close
to the GUT scale ($\sim 10^{15}$ GeV) have also been proposed in the
literature. Indirect searches like ICECUBE icecube (discussed below) already
have strong constraints on simpzillas.
## II Dark Matter Experiments
If the dark matter candidate is indeed a new particle and it has interactions
other than gravitational interactions101010It cannot have electromagnetic
interactions as this would mean it is charged, and it cannot have strong
interactions as this would most likely mean it would be baryonic in form -
both these prospects are already ruled out by experiments., then the most
probable interactions it could have are the weak interactions111111In spite of
being electrically neutral, dark-matter particle can have a nonzero electric
and/or magnetic dipole moment, if it has a nonzero spin. In such a case the
strongest constraint comes from Big Bang Nucleosynthesis. Interested readers
are referred to the paper by Kamoinkowski et al. Sigurdson:2004zp and
particularly Fig. 1 therein.. This weakly interacting particle, dubbed as WIMP
(Weakly Interacting Massive Particle) could interact with ordinary matter and
leave traces of its nature. There are two ways in which the WIMP could be
detected (a) Direct Detection: here one looks for the interaction of the WIMP
on a target, the target being typically nuclei in a scintillator. It is
expected that the WIMPs present all over the galaxy scatter off the target
nuclei once in a while. Measuring the recoil of the nuclei in these rarely
occurring events would give us information about the properties of the WIMP.
The scattering cross section would depend on whether it was elastic or
inelastic and is a function of the spin of the WIMP 121212More generally, the
WIMP-Nucleon cross section can be divided as (i) elastic spin-dependent (eSD),
(ii) elastic spin-independent (eSI) , (iii) in-elastic spin-dependent ( iSD)
and (iv) in-elastic spin-independent (iSI).. There are more than 20
experiments located all over the world, which are currently looking for WIMP
through this technique. Some of them are DAMA, CDMS, CRESST, CUORICINO, DRIFT
etc. (b) Indirect detection : when WIMPs cluster together in the galatic halo,
they can annihilate with themselves giving rise to electron-positron pairs,
gamma rays, proton-anti-proton pairs, neutrinos etc. The flux of such
radiation is directly proportional to the annihilation rate and the the WIMP
matter density. Observation of this radiation could lead to information about
the mass and the cross section strength of the WIMPs. Currently, there are
several experiments which are looking for this radiation131313These are
typically the same experiments which measure the cosmic ray spectrum. For a
comprehensive list of all these experiments and other useful information like
propagation packages, please have a look at:
http://www.mpi-hd.mpg.de/hfm/CosmicRay/CosmicRaySites.html. (i) MAGIC, HESS,
CANGAROO, FERMI/GLAST, EGRET etc. look for the gamma ray photons. (ii) HEAT,
CAPRICE, BESS, PAMELA, AMS can observe anti-protons and positron flux.
(iii)Very highly energetic neutrinos/cosmic rays $\sim$ a few TeV to multi-TeV
can be observed by large detectors like AMANDA, ANTARES, ICECUBE etc. (for a
more detailed discussion see Bertone:2004pz ; pijush1 ).
Over the years, there have been indications of presence of the dark matter
through both direct and indirect experiments. The most popular of these
signals are INTEGRAL and DAMA results (for a nice discussion on these topics
please see, Hooper:2009zm ). INTEGRAL (International Gamma -Ray Astrophysics
Laboratory) is a satellite based experiment looking for gamma rays in outer
space. In 2003, it has observed a very bright emission of the 511 keV photons
from the Galactic Bulge integral1 at the centre. The 511 KeV line is special
as it is dominated by $e^{+}e^{-}$ annihilations via the positronium. The
observed rate of (3-15) $\times 10^{42}$ positrons/sec in the inner galaxy was
much larger than the expected rate from pair creation via cosmic ray
interactions with the interstellar medium in the galactic bulge by orders of
magnitude141414It should be noted that the Integral spectrometer has a very
good resolution of about 2 KeV over a range of energies 20 keV to 8 MeV..
Further, the signal is approximately spherically symmetric with very little
positrons from galactic bulge contributing to the signal integral2 . Several
explanations have been put forward to explain this excess. Astrophysical
entities like hypernovae, gamma ray burts and X-ray binaries have been
proposed as the likely objects contributing to this excess. On the other hand,
this signal can also be attributed to the presence of dark matter which could
annihilate itself giving rise to electron-positron pairs. To explain the
INTEGRAL signal in terms of dark matter, extensions of Standard Model
involving light $\sim(\text{MeV}-\text{GeV})$ particles and light gauge bosons
($\sim\text{GeV}$) are ideally suited. These models which have been already
reviewed in the previous section, can be probed directly at the existing and
future $e^{+}e^{-}$ colliders and hence could be tested. Until further
confirmation from either future astrophysical experiments or through ground
based colliders comes about, the INTEGRAL remains an ‘anomaly’ as of now.
While the INTEGRAL is an indirect detection experiment, the DAMA (DArk MAtter
) is a direct detection experiment located in the Gran Sasso mountains of
Italy. The target material consists of highly radio pure NaI crystal
scintillators; the scintillating light from WIMP-Nucleon scattering and recoil
is measured. The experiment looks for an annual modulation of the signal as
the earth revolves around the sun dama1 . Such modulation of the signal is due
to the gravitational effects of the Sun as well as rotatory motion of the
earth151515Looking for such modulations further limit any systematics present
in the experiment.. DAMA and its upgraded version DAMA/LIBRA have collected
data for seven annual cycles and four annual cycles respectively161616These
results have been recently updated with six annual cycles for DAMA/LIBRA; the
CL has now moved up to $8.9\sigma$ dama2 .. Together they have reported an
annual modulation at 8.2$\sigma$ confidence level. If confirmed, the DAMA
results would be the first direct experimental evidence for the existence of
WIMP dark matter particle. However, the DAMA results became controversial as
this positive signal has not been confirmed by other experiments like XENON
and CDMS, which have all reported null results in the spin independent WIMP-
Nucleon scattering signal region.
The Xenon 10 detector also at Gran Sasso laboratories uses a Xenon target
while measuring simultaneously the scintillation and ionization produced by
the scattering of the dark matter particle. The simultaneous measurement
reduces the background significantly down to $4.5~{}\text{KeV}$. With a
fiducial mass of 5.4 Kg, they set an upper limit of WIMP-Nucleon spin
independent cross section to be 8.8 $\times~{}10^{-44}\text{cm}^{2}$ for a
WIMP mass of 100 GeVxenon1 . An upgraded version Xenon 100 has roughly double
the fiducial mass has started taking data from Oct 2009. In the first results,
they present null results, with upper limits of about $3.4\times
10^{-44}\text{cm}^{2}$ for 55 GeV WIMPs xenon2 . These results severely
constraint interpretation of the DAMA results in terms of an elastic spin
independent WIMP-nucleon scattering.
The CDMS (Cryogenic Dark Matter Search ) experiment has 19 Germanium detectors
located in the underground Soudan Mine, USA. It is maintained at temperatures
$\sim 40\text{mK}$ (milli-Kelvin). Nuclear recoils can be “seen” by measuring
the ionisation energy in the detector. Efficient separation between electron
recoils and nuclear recoils is possible by employing various techniques like
signal timing and measuring the ratios of the ionization energies. Similar to
Xenon, this experiment cdms1 too reported null results in the signal
region171717The final results have a non-zero probability of two events in the
signal region, we comment on it in the next section. and puts an upper limit
$\sim 4.6\times 10^{-44}\text{cm}^{2}$ on the WIMP-Nucleon cross-section for a
WIMP mass of around 60 GeV.
The CoGeNT (Cryogenic Germanium Neutrino Technology) collaboration runs
another recent experiment which uses ultra low noise Germanium detectors. It
is also located in the Soudan Man, USA. The experiment has one of the lowest
backgrounds below 3 KeVee ( KeV electron equivalent (ee) ionisation energy).
It could further go down to 0.4 KeVee, the electron noise threshold. The first
initial runs have again reported null results cogent1 consistent with the
observed background. At this point, the experiment did not have the
sensitivity to confirm/rule out the DAMA results. However, later runs have
shown some excess events over the expected background in the low energy
regions cogent2 . While, the collaboration could not find a suitable
explanation for this excess ( as of now) there is a possibility of these
excess events having their origins in a very light WIMP dark matter particle.
However, care should be taken before proceeding with this interpretation as
the CoGeNT collaboration does not distinguish between electron recoils and
nucleon recoilsweinercogent .
In the light of these experimental results, the DAMA results are hard to
explain. One of the ways out to make the DAMA results consistent with other
experiments is to include an effect called “channelling” which could be
present only in the NaI crystals which DAMA uses. However, even the inclusion
of this effect does not improve the situation significantly. To summarize, the
situation is as follows for various interpretations of the WIMP-Nucleon cross
section. For eSI (elastic Spin Independent) interpretation, the DAMA regions
are excluded by both CDMS as well as Xenon 10. This is irrespective of whether
one considers the channeling effect or not. It is also hard to reconcile DAMA
results with CoGeNT in this case. For elastic Spin Dependent (eSD)
interpretation, the DAMA and CoGeNT results though consistent with each other
are in conflict with other experiments. For an interpretation in terms of
WIMP-proton scattering, the results are in conflict with several experiments
like SIMPLE , PICASSO etc. On the other hand, an interpretation in terms of
WIMP-neutron scattering is ruled out by XENON and CDMS data. For the inelastic
dark matter interpretations, spin -independent cross section with a medium
mass ($\sim 50~{}\text{GeV}$) WIMP is disfavored by CRESST as well as CDMS
data. For a low mass (close to 10 GeV) WIMP, with the help of channeling in
the NaI crystals, it is possible to explain the DAMA results, in terms of
spin- independent inelastic dark matter - nucleon scattering. However, the
relevant parameters (dark matter mass and mass splittings) should be fine
tuned and further, the WIMP velocity distribution in the galaxy should be
close to the escape velocity. Inelastic Spin dependent interpretation of the
DAMA results is a possibility (because it can change relative signals at
different experiments koppzupan ) which does not have significant constraints
from other experiments. However, it has been shownweinercogent that inelastic
dark matter either with spin dependent or spin independent interpretation of
the DAMA results is difficult to reconcile with the CoGeNT results, unless one
introduces substantial exponential background in the CoGeNT data.
## III The Data
The focus of the present topical review is a set of new experimental results
which have appeared over the past year. In terms of the discussion in the
previous section, these experiments follow “indirect” methods to detect dark
matter. The data from these experiments seems to be pointing to either
“discovery” of the dark matter or some yet non-understood new astrophysics
being operative within the vicinity of our Galaxy. The four main experiments
which have led to this excitement are (i) PAMELAAdriani:2008zr (ii)
ATIC:2008zzr (iii) HESSCollaboration:2008aaa and (iv) FERMIAbdo:2009zk . All
of these experiments involve international collaborations spanning several
nations. While PAMELA and FERMI are satellite based experiments, ATIC is a
balloon borne experiment and HESS is a ground based telescope. All these
experiments contain significant improvements in technology over previous
generation experiments of similar type. The H.E.S.S experiment has a factor
$\sim 10$ improvement in $\gamma$-ray flux sensitivity over previous
experiments largely due to its superior rejection of the hadronic background.
Similarly, ATIC is the next generation balloon based experiment equipped to
have higher resolution as well as larger statistics. Similar statements also
hold for the satellite based experiments, PAMELA and FERMI. It should be noted
that the satellite based experiments have some inherent advantages over the
balloon based ones. Firstly, they have enhanced data taking period, unlike the
balloon based ones which can take data only for small periods. And
furthermore, these experiments also do not have problems with the residual
atmosphere on the top of the instrument which plagues the balloon based
experiments.
Figure 1: Results from PAMELA and ATIC with theoretical models. The left panel
shows PAMELAAdriani:2008zr positron fraction along with theoretical model.
The solid black line shows a calculation by Moskalenko & Strongmos98 for pure
secondary production of positrons during the propagation of cosmic-rays in the
Galaxy. The right panel shows the differential electron energy spectrum
measured by ATIC:2008zzr (red filled circles) compared with other experiments
and also with theoretical prediction using the GALPROPStrong:2001fu code
(solid line). The other data points are from AMSAguilar:2002ad (green stars),
HEATBarwick:1997kh (open black triangles), BETSTorii:2001aw (open blue
circles), PPB-BETSTorii:2008xu (blue crosses) and emulsion chambers (black
open diamonds) and the dashed curve at the beginning is the spectrum of solar
modulated electron. All the data points have uncertainties of one standard
deviation. The ATIC spectrum is scaled by $E_{e}^{3.0}$. The figures of PAMELA
and ATIC are reproduced from their original papers cited above.
The satellite-based Payload for Antimatter Matter Exploration and Light-nuclei
Astrophysics or PAMELA collects cosmic ray protons, anti-protons, electrons,
positrons and also light nuclei like Helium and anti-Helium. One of the main
strengths of PAMELA is that it could distinguish between electrons and anti-
electrons, protons and anti-protons and measure their energies accurately. The
sensitivity of the experiment in the positron channel is up to approximately
300 GeV and in the anti-proton channel up to approximately 200 GeV. Since it
was launched in June 2006, it was placed in an elliptical orbit at an altitude
ranging between $350-610$ km with an inclination of 70.0°. About 500 days of
data was analyzed and recently presented. The present data is from 1.5 GeV to
100 GeV has been published in the journal Nature Adriani:2008zr . In this
paper, PAMELA reported an excess of positron flux compared to earlier
experiments. In the left panel of the Fig. 1, we see PAMELA results along with
the other existing results. The y-axis is given by
$\phi(e^{+})/(\phi(e^{-})+\phi(e^{+}))$, which $\phi$ represents the flux of
the corresponding particle. According to the analysis presented by PAMELA, the
results of PAMELA are consistent with the earlier experiments up to 20 GeV,
taking into consideration the solar modulations between the times of PAMELA
and previous experiments. Particles with energies up to 20 GeV are strongly
effected by solar wind activity which varies with the solar cycle. On the
other hand, PAMELA has data from 10 GeV to 100 GeV, which sees an increase in
the positron flux (Fig. 1). The only other experimental data in this energy
regime (up to 40 GeV) are the AMS and HEAT, which while having large errors
are consistent with the excess seen by PAMELA. In the low energy regime most
other experiments are in accordance with each other but have large error bars.
Figure 2: The Fermi LAT CR electron spectrum. The red filled circles shows the
data from Fermi along with the gray bands showing systematic errors. The
dashed line correspond to a theoretical model by Moskalenko et al.
Strong:2004de . The figure of FERMI is reproduced from their original paper
cited above.
Cosmic ray positrons at these energies are expected to be from secondary
sources i.e. as result of interactions of primary cosmic rays (mainly protons
and electrons) with interstellar medium. The flux of this secondary sources
can be estimated by numerical simulations. There are several numerical codes
available to compute the secondary flux, the most popular publicly available
codes being GALPROP galprop ; galprop2 and CRPropa crpropa . These codes
compute the effects of interactions and energy loses during cosmic ray
propagation within galactic medium taking also in to account the galactic
magnetic fields. GALPROP solves the differential equations of motion either
using a 2D grid or a 3D grid while CRPropa does the same using a 1D or 3D
grids. While GALPROP contains a detailed exponential model of the galactic
magnetic fields, CRPropa implements only extragalactic turbulent magnetic
fields. In particular CRPropa is not optimised for convoluted galactic
magnetic fields. For this reason, GALPROP is best suited for solving diffusion
equations involving low energy (GeV-TeV) cosmic rays in galactic magnetic
fields.
The main input parameters of the GALPROP code are the primary cosmic ray
injection spectra, the spatial distribution of cosmic ray sources, the size of
the propagation region, the spatial and momentum diffusion coefficients and
their dependencies on particle rigidity. These inputs are mostly fixed by
observations, like the interstellar gas distribution is based on observations
of neutral atomic and molecular gas, ionized gas ; cross sections and energy
fitting functions are build from Nuclear Data sheets (based on Las Almos
Nuclear compilation of nuclear crosssections and modern nuclear codes) and
other phenomenological estimates. Interstellar radiation fields and galactic
magnetic fields are based on various models present in literature. The
uncertainties in these inputs would constitute the main uncertainties in the
flux computation from GALPROP181818Some codes are constructed to fix the
various parameters of their own cosmic ray propagation model. See for example,
DRAGON dragon . Here one can fix the diffusion coefficients from PAMELA and
other experimental data.. Recently, a new code called CRT which emphasizes
more on the minimization of the computation time was introduced. Here most of
the input parameters are user defined crt . Finally, using the popular Monte
Carlo routine GEANT geant one can construct cosmic ray propagation code as
has been done by desorgher1 ; strumia . On the other hand, dark matter relic
density calculators like DARKSUSY Gondolo:2004sc also compute cosmic ray
propagation in the galaxies required for indirect searches of dark matter. It
is further interfaced with GALPROP.
In summary, GALPROP is most suited for the present purposes i.e, understanding
of PAMELA and ATIC data which is mostly in the GeV-TeV range. It has been
shown the results from these experiments do not vary much if one instead
chooses to use a GEANT simulation. In fact, most of the experimental
collaborations use GALPROP for their predictions of secondary cosmic ray
spectrum. In the left panel of Fig. 1, the expectations based on GALPROP are
given as a solid line running across the figure. From the figure it is obvious
that PAMELA results show that the positron fraction increases with energy
compared to what GALPROP expects. The excess in the positron fraction as
measured by PAMELA with respect to GALPROP indicates that this could be a
result due to new primary sources rather than secondary sources 191919For an
independent analysis which confirms the PAMELA excess, please see, delahaye2 .
This new primary source could be either dark matter decay/annihilation or a
nearby astrophysical object like a pulsar. Before going to the details of the
interpretations, let us summarize the results from ATIC and FERMI also.
Advanced Thin Ionization Calorimeter or in short ATIC is a balloon-borne
experiment to measure energy spectrum of individual cosmic ray elements within
the region of GeV up to almost a TeV (thousand GeV) with high precision. As
mentioned, this experiment was designed to be a high-resolution and high
statistics experiment in this energy regime compared to the earlier ones. ATIC
measures all the components of the cosmic rays such as electrons, protons (and
their anti-particles) with high energy resolution, while distinguishing well
between electrons and protons. ATIC (right panel in Fig. 1) presented its
primary cosmic ray electron ($e^{-}$\+ $e^{+}$) spectrum between the energies
3 GeV to about 2.5 TeV202020The cosmic ray electrons follow a power law
spectrum, the index being $\sim$ $-3.0$. Thus it is normalized by a factor
$E^{3.0}$ .. The results show that the spectrum while agreeing with the
GALPROP expectations up to 100 GeV, show a sharp increase above 100 GeV. The
total flux increases till about 600 GeV where it peaks and then sharply falls
till about 800 GeV. Thus, ATIC sees an excess of the primary cosmic ray
($e^{-}$\+ $e^{+}$) spectrum between the energy range $300-800$ GeV. The rest
of the spectrum is consistent with the expectations within the errors. What is
interesting about such peaks in the spectrum is that, if they are confirmed
they could point towards a Breit-Wigner resonance in dark matter annihilation
cross section with a life time as given by its width. As we will discuss in
the next section, this possibility is severely constrained by the data from
the FERMI experiment.
Another ground based experiment sensitive to cosmic rays within this energy
range is H.E.S.S which can measure gamma rays from few hundred GeV to few TeV.
This large reflecting array telescope operating from Namibia has presented
data (shown in figure 2) from $600$ GeV to about 5 TeV. It could confirm
neither the ‘peaking’ like behavior at 600 GeV nor the sharp cut-off at 800
GeV of the ATIC data. The ATIC results can be made consistent with those of
HESS. This would require a $15\%$ overall normalisation of the HESS data. Such
a normalisation is well within the uncertainty of the energy resolution of
HESS. However notice that HESS data does not have a sharp fall about and after
800 GeV.
The Large Area Telescope (LAT) is one of the main components of the Fermi
Gamma Ray Space Telescope, which was launched in June 2008. Due to its high
resolution and high statistical capabilities, it has been one of the most
anticipated experiments in the recent times. Fermi can measure Gamma rays
between 20 MeV and 300 GeV with high accuracy and primary cosmic ray electron
($e^{-}$\+ $e^{+}$) flux between 20 GeV and 1 TeV. The energy resolution
averaged over the LAT acceptance is 11% FWHM (Full-Width-at-Half-Maximum) for
20-100 GeV, increasing to 13% FWHM for 150-200 GeV. The photon angular
resolution is less than 0.1° over the energy range of interest (68%
containment). The FERMI-LAT collaboration has recently published its six month
data on the primary cosmic ray electron flux. More than 4 million electron
events above 20 GeV were selected in survey (sky scanning) mode from 4 August
2008 to 31 January 2009. The systematic error on the absolute energy of the
LAT was determined to be ${}^{-10\%}_{+5\%}~{}$ for 20-300 GeV. Please see
Table I for more details on the errors in Abdo:2009zk . In Fig. 2 we reproduce
the result produced by the FERMI collaboration. They find that the primary
cosmic ray electron spectrum more or less goes along the expected lines up to
100 GeV (its slightly below the expected flux between 10 and 50 GeV), however
above 100 GeV, there is strong signal for an excess of the flux ranging up to
1 TeV. The FERMI data thus confirms the excess in the electron spectrum which
was seen by ATIC, the excess however has a much flatter profile with respect
to the peak seen by ATIC. Thus, ATIC could in principle signify a ‘resonance’
in the spectrum, whereas FERMI cannot. However, in comparing both the spectra
from the figures presented above, one should keep in mind that the FERMI
excess is in the total electron spectrum ($e^{+}+e^{-}$ ) whereas the ATIC
data is presented in terms of positron excess only. If the excess in FERMI is
caused by the excess only through excess positrons, one should expect that the
FERMI spectra to also have similar ‘peak’ like behavior at 600 GeV. From
Fig.(2), where both FERMI and ATIC data are presented, we see that the ATIC
data points are far above that of FERMI’s.
## IV The Interpretations
Lets now summarise the experimental observations strumia which would require
an interpretation :
* •
The excess in the flux of positron fraction
$\left(\tfrac{\phi(e^{+})}{\phi(e^{-})+\phi(e^{+})}\right)$ measured by PAMELA
up to 100 GeV.
* •
The lack of excess in the anti-proton fraction measured by PAMELA up to 100
GeV.
* •
The excess in the total flux $\left(\phi(e^{-})+\phi(e^{+})\right)$ in the
spectrum above 100 GeV seen by FERMI, HESS etc. While below 100 GeV, the
measurements have been consistent with GALPROP expectations.
* •
The absence of ‘peaking’ like behavior as seen by ATIC, which indicates a long
lived particle, in the total electron spectrum measured by FERMI.
Two main interpretations have been put forward: (a) A nearby astrophysical
source which has a mechanism to accelerate particles to high energies and (b)
A dark matter particle which decays or annihilates leading to excess of
electron and positron flux. Which of the interpretations is valid will be
known within the coming years with enhanced data from both PAMELA and FERMI.
Let us now turn to both the interpretations:
Pulsars and supernova shocks have been proposed as likely astrophysical local
sources of energetic particles that could explain the observed excess of the
positron fraction Hooper:2008kg ; Blasi:2009hv . In the high magnetic fields
present in the pulsar magnetosphere, electrons can be accelerated and induce
an electromagnetic cascade through the emission of curvature
radiation212121The curvature radiation arises due to relativistic, charged
particles moving around curved magnetic field lines, see for details Gil et
al.Gil:2003ug .. This can lead to a production of high energy photons above
the threshold for pair production; and on combining with the number density of
pulsars in the Galaxy, the resulting emission can explain the observed
positron excess Hooper:2008kg . The energy of the positrons tell us about the
site of their origin and their propagation history coutu99 . The cosmic ray
positrons above 1 TeV could be primary and arise due to a source like a young
plusar within a distance of 100 pc ato95 . This would also naturally explain
the observed anisotropy, as argued for two of the nearest pulsars, namely
$B0656+14$ and the $Geminga$ bue08 ; Yuksel:2008rf . On a similar note,
diffusive shocks as in a supernova remnant hardens the spectrum, hence this
process can explain the observed positron excess above 10 GeV as seen from
PAMELA Ahlers09 .
Another possible astrophysical source that has been proposed is the pion
production during acceleration of hadronic cosmic rays in the local sources
Blasi:2009hv . It has been argued Mertsch:2009ph that the measurement of
secondary nuclei produced by cosmic ray spallation can confirm whether this
process or pulsars are more important as the production mechanism. It has been
show that the present data from ATIC-II supports the hadronic model and can
account for the entire positron excess observed.
If the excess observed by PAMELA, HESS and FERMI is not due to some yet not
fully-understood astrophysics but is a signature of the dark matter, then
there are two main processes through which such an excess can occur:
1. (i)
The annihilation of dark matter particles into Standard Model (SM) particles
and
2. (ii)
The decay of the dark matter particle into SM particles.
Interpretation in terms of annihilating dark matter, however, leads to
conflicts with cosmology. The observed excesses in the PAMELA/FERMI data would
set a limit on the product of annihilation cross section and the velocity of
the dark matter particle in the galaxy (for a known dark matter density
profile). Annihilation of the dark matter particles also happens in the early
universe with the same cross section but at much larger velocities for
particles (about 1000 times the particle velocities in galaxies). The
resultant relic density is not compatible with observations. The factor $\sim
1000$ difference in the velocities should some how be compensated in the cross
sections. This can be compensated by considering “boost” factors for the
particles in the galaxy which can enhance the cross section by several orders
of magnitude. The boost factors essentially emanate from assuming local
substructures for the dark matter particles, like clumps of dark matter and
are typically free parameters of the model (see however, delahaye1 ). Another
mechanism which goes by the name Sommerfeld mechanism can also enhance the
annihilation cross sections. For very heavy dark matter ( with masses much
greater than the relevant gauge boson masses) trapped in the galactic
potential, non-perturbative effects could push the annihilation cross-sections
to much larger values. For SU(2) charged dark matter, the masses of dark
matter particles should be $\gg M_{W}$ hisano . The Sommerfeld mechanism is
more general and applicable to other (new) interactions alsohannestead .
Another way of avoiding conflict with cosmology would be to consider non-
thermal production of dark matter in the early universe222222Non-thermal
production typically refers to production mechanisms through decays of very
heavy particles like inflaton. See for example riotto1 .. Before the release
of FERMI data, the annihilating dark matter model with a very heavy dark
matter $\sim\mathcal{O}(2-3)\ $TeV was much in favour to explain the
“resonance peak” of the ATIC and the excess in PAMELA data. Post FERMI, whose
data does not have sharp raise and fall associated with a resonance, the
annihilating dark matter interpretation has been rendered incompatible.
However, considering possible variations in the local astro physical
background profile due to presence of local cosmic ray accelerator, it has
been shown that it is still possible to explain the observed excess, along
with FERMI data with annihilating dark matter. The typical mass of the dark
matter particle could lie even within sub-TeV region Dodelson:2009ih ;
Belikov:2009cx ; Cholis:2009gv ; Hooper:2009cs and as low as $30-40$ GeV
Goodenough:2009gk . Some more detailed analysis can be found in Pato:2009fn .
Several existing BSM physics models of annihilating dark matter become highly
constrained or ruled out if one requires to explain PAMELA/ATIC and FERMI
data. The popular supersymmetric DM candidate neutralino with its annihilating
partners such as chargino, stop, stau etc., can explain the cosmological relic
density but not the excess observed by PAMELA/ATIC. Novel models involving a
new ‘dark force’, with a gauge boson having mass of about 1 GeV
ArkaniHamed:2008qn , which predominantly decays to leptons, together with the
so-called Sommerfeld enhancement seem to fit the data well. The above class of
models, which are extensions of standard model with an additional $U(1)$ gauge
group, caught the imagination of the theorists Katz:2009qq ; Cholis:2008vb ;
Cholis:2008hb ; Cholis:2008qq . A similar supersymmetric version of this
mechanism where the neutralinos in the MSSM can annihilate to a scalar
particle, which can then decay the observed excess in the cosmic ray data
Hooper:2009gm . Models involving Type II seesaw mechanism Gogoladze:2009gi
have also been considered recently where neutrino mass generation is linked
with the positron excess. In addition to the above it has been shown that
extra dimensional models with KK gravitions can also produce the excess
Hooper:2009fj 232323Some of the first simulations using PYTHIA and DARK SUSY
for the KK gravition has been done in hooperkribs . Similar study for SUSY can
be found in susy01 . These have been done when HEAT results have shown an
excess though in a less statistically significant way.. Models with Nambu-
Goldstone bosons as dark matter have been studied in murayama .
In the case of decaying dark matter, the relic density constraint of the early
universe is not applicable, however, the lifetime of the dark matter particle
(typically of a mass of $\mathcal{O}(1)$ TeV) should be much much larger than
($\sim 10^{9}$ times) the age of the universestrumia . Such a particle can fit
the data well. A crucial difference in this picture with respect to the
annihilation picture is that the decay rate is directly proportional to the
density of the dark matter ($\rho$), whereas the annihilation rate is
proportional to its square, ($\rho^{2}$). The most promising candidates in the
decaying dark matter seem to be a fermion (scalar) particle decaying in to
$W^{\pm}l^{\pm}$ etc. ($W^{+}W^{-}$ etc.) Ibarra:2009dr ; Ibarra:2009nw ;
Ibarra:2008jk ; Nardi:2008ix . In terms of the BSM physics, supersymmetric
models with a heavy gravitino and small R-parity violation have been proposed
as candidates for decaying dark matter Buchmuller:2007ui . A heavy neutralino
with $R$-parity violation can also play a similar role Gogoladze:2009kv
stated above. A recent more general model independent analysis has shown that,
assuming the GALPROP background, gravitino decays cannot simultaneously
explain both PAMELA and FERMI excess. However, the presence of additional
astrophysical sources can change the situation Buchmuller:2009xv . Independent
of the gravitino model, it has been pointed out that, the decays of the Dark
Matter particle could be new signals for unification where the Dark Matter
candidate decays through dimension six operators suppressed by two powers of
GUT scale Arvanitaki:2008hq ; Arvanitaki:2009yb ; Buckley:2009kw . Finally,
there has also been some discussion about the possibilities of dark matter
consisting of not one particle but two particles, of which one is the decaying
partner. This goes under the name of ‘two-component dark matter’ and analysis
of this scenario has been presented by Fairbairn:2008fb .
We have so far mentioned just a sample of the theoretical ideas proposed in
the literature. Several other equally interesting and exciting ideas have been
put forward, which have not been presented to avoid the article becoming too
expansive.
## V Outlook and Remarks
An interesting aspect about the present situation is that, future data from
PAMELA and FERMI could distinguish whether the astrophysical interpretation
i.e. in terms of pulsars or the particle physics interpretation in terms of
dark matter is valid Malyshev:2009tw . PAMELA is sensitive to up to 300 GeV in
its positron fraction and this together with the measurement of the total
electron spectrum can strongly effect the dark matter interpretations. FERMI
with its improved statistics, can on the other hand look for anisotropies
within its data Grasso:2009ma which can exist if the pulsars are the origin
of this excess. Further measurements of the anti-Deuteron could possibly gives
us a hint why there is no excess in the anti-Proton channel Kadastik:2009ts .
Similarly neutrino physics experiments could give us valuable information on
the possible modelshisano2 . Finally, the Large Hadron collider could also
give strong hints on the nature of dark matter through direct production LHCdm
.
As we have been preparing this note, there has been news from one the
experiments called CDMS-II (Cryogenic Dark Matter Search Experiment)cdms2 . As
mentioned before this experiment conducts direct searches for WIMP dark matter
by looking at collisions of WIMPs on super-cooled nuclear target material. The
present and final analysis of this experiment have shown two events in the
signal region, with the probability of observing two or more background events
in that region being close to $23\%$. Thus, while these results are positive
and encouraging, they are not conclusive. However these results already set a
stringent upper bound on the WIMP-nucleus cross section for a WIMP mass of
around 70 GeV. The exclusion plots in the parameter space of WIMP cross
section and WIMP mass are presented in the paper cdms2 . The interpretations
of this positive signal are quite different compared to the signal of PAMELA
and FERMI. While PAMELA and FERMI as we have seen would require severe
modifications for the existing beyond standard model (BSM) models of Dark
Matter, CDMS results if confirmed would prefer the existing BSM dark matter
candidates like neutralino of the supersymmetry. There are ways of making both
PAMELA/FERMI and CDMS-II consistent through dark matter interpretations,
however, we will not discuss it further here. Finally, it has been shown that
it is possible to make CDMS-II results consistent with DAMA annual modulation
results by assuming a spin-dependent inelastic scattering of WIMP on Nuclei
koppzupan .
In the present note, we have tried to convey exciting developments which have
been happening recently within the interface of astrophysics and particle
physics, especially on the one of the most intriguing subjects of our time,
namely, the Dark Matter. Though it has been proposed about sixty years ago, so
far we have not have any conclusive evidence of its existence other than
through gravitational interactions, or we do not of its fundamental
composition. Experimental searches which have been going on for decades have
not bore fruit in answering either of these questions. For these reasons, the
present indications from PAMELA and FERMI have presented us with a unique
opportunity of unraveling at least some of mystery surrounding the dark
matter. These experimental results, if they hold and get confirmed as due to
dark matter, would strongly modify the way dark matter was perceived in the
scientific community. As a closing point, let us note that there are several
new experiments being planned to explore the dark matter either directly or
indirectly and thus some information about the nature of the dark matter might
just around the corner.
###### Acknowledgements.
We thank PAMELA collaboration, ATIC collaboration and FERMI-LAT collaboration
for giving us permission to reproduce their figures. We thank Diptiman Sen for
a careful reading of this article and useful comments. C. J. would like to
thank Gary Mamon for illuminating discussions regarding the search for dark
matter in elliptical galaxies and clusters. We thank A. Iyer for bringing to
our notice a reference. Finally, we thank the anonymous referee for
suggestions and comments which have contributed in improving the article.
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|
arxiv-papers
| 2009-09-07T09:19:12 |
2024-09-04T02:49:05.079642
|
{
"license": "Public Domain",
"authors": "Debtosh Chowdhury, Chanda J. Jog, Sudhir K Vempati",
"submitter": "Sudhir Vempati",
"url": "https://arxiv.org/abs/0909.1182"
}
|
0909.1188
|
Persistent current and low-field magnetic susceptibility in one-dimensional
mesoscopic rings: Effect of long-range hopping
Santanu K. Maiti1,2,∗
1Theoretical Condensed Matter Physics Division, Saha Institute of Nuclear
Physics,
1/AF, Bidhannagar, Kolkata-700 064, India
2Department of Physics, Narasinha Dutt College, 129, Belilious Road,
Howrah-711 101, India
Abstract
Persistent current and low-field magnetic susceptibility in single-channel
normal metal rings threaded by a magnetic flux $\phi$ are investigated within
the tight-binding framework considering long-range hopping of electrons in the
shortest path. The higher order hopping integrals try to reduce the effect of
disorder by delocalizing the energy eigenstates, and accordingly, current
amplitude in disordered rings becomes comparable to that of an ordered ring.
Our study of low-field magnetic susceptibility predicts that the sign of
persistent currents can be mentioned precisely in mesoscopic rings with fixed
number of electrons, even in the presence of impurity in the rings. For
perfect rings, low-field current shows only the diamagnetic sign irrespective
of the total number of electrons $N_{e}$. On the other hand, in disordered
rings it exhibits the diamagnetic and paramagnetic natures for the rings with
odd and even $N_{e}$ respectively.
PACS No.: 73.23.Ra; 73.23.-b; 71.23.-k; 73.20.Jc; 75.20.-g
Keywords: Persistent current; Magnetic susceptibility; Long-range hopping;
Finite temperature; Disorder.
∗Corresponding Author: Santanu K. Maiti
Electronic mail: santanu.maiti@saha.ac.in
## 1 Introduction
The phenomenon of persistent current in mesoscopic normal metal rings has
generated a lot of excitement as well as controversy over the past years. In a
pioneering work, Büttiker, Imry and Landauer [1] predicted that, even in the
presence of disorder, an isolated one-dimensional metallic ring threaded by a
magnetic flux $\phi$ can support an equilibrium persistent current with
periodicity $\phi_{0}=ch/e$, the elementary flux quantum. Later, the existence
of persistent current was further confirmed by several experiments [2, 3, 4,
5, 6, 7, 8]. However, these experiments yield many results those are not well-
understood theoretically even today [9, 10, 11, 12, 13, 14, 15, 16, 17, 18,
19, 20, 21, 22, 23, 24, 25]. The results of the single loop experiments are
significantly different from those for the ensemble of isolated loops.
Persistent currents with expected $\phi_{0}$ periodicity have been observed in
isolated single Au rings [2] and in a GaAs-AlGaAs ring [3]. Levy et al. [4]
found oscillations with period $\phi_{0}/2$ rather than $\phi_{0}$ in an
ensemble of $10^{7}$ independent Cu rings. Similar $\phi_{0}/2$ oscillations
were also reported for an ensemble of disconnected $10^{5}$ Ag rings [5] as
well as for an array of $10^{5}$ isolated GaAs-AlGaAs rings [6]. In an
experiment, Jariwala et al. [7] obtained both $\phi_{0}$ and $\phi_{0}/2$
periodic persistent currents for an array of thirty diffusive mesoscopic Au
rings. Except for the case of nearly ballistic GaAs-AlGaAs ring [3], all the
measured currents are in general one or two orders of magnitude larger than
those expected from theory. The diamagnetic response of the measured
$\phi_{0}/2$ oscillations of ensemble-averaged persistent currents near zero
magnetic field also contrasts with most predictions [15, 16].
Free electron theory predicts that, at zero temperature, an ordered one-
dimensional metallic ring threaded by a magnetic flux $\phi$ supports
persistent current with maximum amplitude $I_{0}=ev_{F}/L$, where $v_{F}$ is
the Fermi velocity of an electron and $L$ is the circumference of the ring.
Metals are intrinsically disordered which tends to decrease persistent
current, and calculations show that the magnitude of the currents reduces to
$I_{0}l/L$, where $l$ is the elastic mean free path of the electrons. This
expression remains valid even if one takes into account the finite width of
the ring by adding contributions from the transverse channels, since disorder
leads to a compensation between the channels [10, 11]. However, measurements
on single isolated mesoscopic rings [2, 3] detected $\phi_{0}$-periodic
persistent currents with amplitudes of the order of $I_{0}\sim ev_{F}/L$,
(close to the value for an ordered ring). Though theory seems to agree with
experiment [3] only when disorder is weak, the amplitudes of the currents in
single-isolated-diffusive gold rings [2] are two orders of magnitude larger
than the theoretical estimates. This discrepancy initiated intense theoretical
activity, and it is generally believed that the electron-electron correlation
plays an important role in the disordered rings [17, 18, 19], though the
physical origin behind this enhancement of persistent current is still
unclear.
In this article we investigate a detailed study of persistent current and low-
field magnetic susceptibility in single channel rings in the tight-binding
framework considering long-range hopping of electrons in the shortest path.
Our calculations show that in a disordered ring with higher order hopping
integrals, current amplitude is comparable to that of an ordered ring. This is
due to the fact that higher order hopping integrals try to delocalize the
energy eigenstates and thus compensate the effect of disorder. In the rest
part of this article, we describe the dependences of the sign of low-field
currents as a function of the total number of electrons $N_{e}$, and also
discuss the effect of temperature on these low-field currents.
The plan of the paper is as follow. Section $2$ relates the behavior of
persistent current in the presence of long-range hopping integrals and clearly
describes how current amplitude in disordered rings becomes comparable to that
of an ordered ring. In Section $3$, we investigate the behavior of low-field
magnetic susceptibility at absolute zero temperature both for ordered and
disordered rings as a function of $N_{e}$. Section $4$ focuses the effect of
temperature on the low-field currents and determines the critical value of
magnetic flux $\phi_{c}(T)$ where current changes its sign from the
paramagnetic to diamagnetic phase for the rings with even number of electrons.
Finally, the conclusions of our study can be found in Section $5$.
## 2 Persistent current
We describe a $N$-site ring (see Fig. 1) enclosing a magnetic flux $\phi$ (in
units of the elementary flux quantum $\phi_{0}=ch/e$) by the following tight-
binding Hamiltonian in the Wannier basis,
$H=\sum_{i}\epsilon_{i}c_{i}^{\dagger}c_{i}+\sum_{i\neq
j}v_{ij}\left[e^{-i\theta}c_{i}^{\dagger}c_{j}+h.c.\right]$ (1)
where $\epsilon_{i}$’s are the site energies, $v_{ij}$’s are the hopping
integrals between the sites $i$ and $j$, and $\theta={2\pi\phi}(|i-j|)/N$. The
long-range hopping (LRH) integrals are taken as,
$v_{ij}=\frac{v}{\left|\frac{N}{\pi}\sin\left[\frac{\pi}{N}(i-j)\right]\right|^{\alpha}}$
(2)
where, $v$ being a constant representing the nearest-neighbor hopping (NNH)
integral. In the present work, electron-electron interaction is not included,
and therefore, we do not consider the spin of the electrons since it will not
change any qualitative behavior of persistent currents. Throughout our study
we set $v=-1$, and use the units where $c=e=h=1$.
An electron in an eigenstate with energy $E_{n}$ carries a persistent
Figure 1: One-dimensional mesoscopic normal metal ring threaded by a magnetic
flux $\phi$. A persistent current $I$ is established in the ring.
current $I_{n}(\phi)=-\partial E_{n}/\partial\phi$, and at zero temperature
total persistent current is given by $I(\phi)=\sum_{n}I_{n}(\phi)$, where the
summation is over all the states below the Fermi level.
For an ordered ring, setting $\epsilon_{i}=0$ for all $i$, the energy of the
$n$-th eigenstate can be expressed as,
$E_{n}(\phi)=\sum_{m}\frac{2v}{m^{\alpha}}\cos\left[\frac{2\pi
m}{N}(n+\phi)\right]$ (3)
where $m$ is an integer and it runs from $1$ to $N/2$ for the rings with even
$N$, while it goes from $1$ to $(N-1)/2$ for those rings described by odd $N$.
Now the persistent current carried by this $n$-th eigenstate becomes,
$I_{n}(\phi)=\left(\frac{4\pi
v}{N}\right)\sum_{m}m^{1-\alpha}\sin\left[\frac{2\pi m}{N}(n+\phi)\right]$ (4)
For very large value of $\alpha$, Eqs. (3) and (4) essentially reduce to the
expressions for the energy spectrum and persistent current of an ordered ring
described by the nearest-neighbor tight-binding Hamiltonian. As we decrease
$\alpha$, contributions from the higher order hopping integrals become
appreciable which modify the energy spectrum and persistent current, and we
will see that the modifications are quite significant in the presence of
disorder. Figure 2 shows the variation of persistent current as a function of
magnetic flux $\phi$ for some perfect rings ($N=120$). The solid and dotted
curves represent the results for the rings described by LRH ($\alpha=1.3$) and
NNH integrals respectively, where the curves plotted in (a) give the variation
of the currents for the rings with odd $N_{e}$
Figure 2: Current-flux characteristics for some perfect rings with size
$N=120$. The solid and dotted curves are respectively for the rings with LRH
($\alpha=1.3$) and NNH integrals, where (a) $N_{e}=35$ (odd) and (b)
$N_{e}=40$ (even).
($N_{e}=35$) and the same are plotted for the rings with even $N_{e}$
($N_{e}$=40) in (b). The results predict that the current amplitude increases
in the presence of LRH integrals, compared to the NNH integral. In this
context it has been examined that, in the presence of LRH integrals, the
amplitude of the current initially increases (not shown here in the figure) as
we increase the ring size, but eventually it falls when the ring becomes
larger. This is due to the fact that as we increase the number of sites, the
Hamiltonian Eq. (1) includes additional higher order hopping integrals which
cause an increase in the net velocity of the electrons, but after certain ring
size the increment in velocity drops to zero since the additional hopping
integrals are then between far enough sites giving negligible contributions.
Now we address the problem of persistent current in the presence of disorder.
In order to introduce the disorder in the ring, we choose the site energies
$\epsilon_{i}$’s randomly from a “Box” distribution function of width $W$,
which reveal that the ring is subjected to the diagonal disorder. As
representative examples of persistent current in disordered rings, we plot
Figure 3: Current-flux characteristics for some typical disordered rings
($W=1$) with size $N=120$. The solid and dotted curves correspond to the rings
with LRH ($\alpha=1.3$) and NNH integrals respectively, where (a) $N_{e}=35$
(odd) and (b) $N_{e}=40$ (even).
the results in Fig. 3 for some $120$-site rings taking $W=1$. All the curves
shown in Fig. 3 are performed for the distinct disordered configurations of
the rings and no averaging is done here since in the averaging process several
mesoscopic phenomena disappear. The solid and dotted curves correspond to the
rings with LRH and NNH integrals respectively, and our results predict that
current amplitude gets an order of magnitude enhancement in the rings
described by the LRH integrals compared to those rings described by the NNH
integral. Figure 3(a) shows the variation of the persistent current for the
rings with odd $N_{e}$ ($N_{e}=35$) and Fig. 3(b) gives the variation of the
currents for the rings with even $N_{e}$ ($N_{e}=40$). It is apparent from
Figs. 2 and 3 that the current amplitudes in disordered rings with LRH
integrals are of the same order of magnitude as observed in ordered rings. We
have also seen that the decrease in amplitude of the current is quite small
even if we increase the strength of disorder. In the NNH models current
amplitudes are suppressed due to the localization of the energy eigenstates
[26]. On the other hand, the present tight-binding model with LRH integrals
supports extended electronic eigenstates even in the presence of disorder and
for this reason persistent currents are not reduced by the impurities.
## 3 Magnetic susceptibility at $T=0$ K
The sign of persistent currents can be determined exactly by calculating
magnetic susceptibility, and here we investigate the properties of low-field
Figure 4: $\chi(\phi)$ versus $N_{e}$ curves for some perfect rings with
$N=150$. The solid and dotted curves represent the rings with LRH
($\alpha=1.4$) and NNH integrals respectively.
currents for the rings with fixed number of electrons $N_{e}$. The general
expression of magnetic susceptibility is expressed in the form,
$\chi(\phi)=\frac{N^{3}}{16\pi^{2}}\left(\frac{\partial
I(\phi)}{\partial\phi}\right)$ (5)
Thus by calculating the sign of $\chi(\phi)$ we can predict whether the
current is diamagnetic or paramagnetic.
Let us first discuss the properties of low-field currents in perfect rings at
absolute zero temperature ($T=0$ K). In Fig. 4, we plot $\chi(\phi)$ as a
function of $N_{e}$ in the limit $\phi\rightarrow 0$ for some perfect rings
with $N=150$. The solid and dotted curves represent the variation of $\chi$
for the rings described by LRH and NNH integrals respectively. The results
show that, for perfect rings, low-field currents exhibit only the diamagnetic
sign irrespective of the total number of electrons $N_{e}$ in the ring. This
variation can be clearly understood if we consider the slope of the curves as
plotted in Figs. 2(a) and (b)
Figure 5: $\chi(\phi)$ versus $N_{e}$ curves of some disordered rings ($W=1$)
with size $N=150$. The solid and dotted lines are respectively for the rings
with even and odd $N_{e}$, where (a) NNH and (b) LRH ($\alpha=1.4$) integrals.
near $\phi=0$. Thus, both for the perfect rings with odd and even $N_{e}$,
current has only negative slope which predicts the diamagnetic persistent
current.
The effect of impurities on the sign of low-field currents is quite
interesting. As representative examples, in Fig. 5 we display the variation of
$\chi$ as a function of $N_{e}$, where (a) and (b) represent the rings with
NNH and LNH integrals respectively. The solid and dotted lines correspond to
the results for the rings containing even and odd number of electrons
respectively. These results emphasize that, in the disordered rings the low-
field currents exhibit the diamagnetic sign for odd $N_{e}$, while we get the
paramagnetic response for the rings with even $N_{e}$. The diamagnetic and the
paramagnetic natures of the low-field currents in the presence of impurity in
the rings can be understood easily if we take the slope of the curves as given
in Figs. 3(a) and (b). Such an effect of disorder on the low-field currents is
true for any disordered configuration. Accordingly, in the presence of
impurity one can easily predict the sign of the low-field currents both for
the rings with odd and even $N_{e}$, irrespective of the specific realization
of disordered configuration of the rings.
## 4 Magnetic susceptibility at finite temperature
This section focuses the effect of temperature on the low-field currents.
Figure 6: Variation of $\phi_{c}(T)$ with $N_{e}$ (even $N_{e}$ only) for some
disordered rings ($W=1$) taking $N=60$, where (a) rings with NNH integrals and
(b) rings with LRH ($\alpha=1.6$) integrals. The upper and lower curves both
in (a) and (b) are respectively for the rings with $T/T^{\star}=1.0$ and
$T/T^{\star}=0.5$.
As temperature increases the probability that electrons occupy higher energy
levels, those may carry larger currents, increases. But if we increase the
temperature in such a way that crosses the energy gap between two successive
energy levels, which carry currents in opposite directions, then mutual
cancellations of the positive and negative currents decrease the net current
amplitude. Therefore, it is necessary to specify a characteristic temperature
$T^{\star}$, which is determined by the energy level spacing $\Delta$. At
finite temperature, thermal excitations, such as phonons, are present which
interact with the electrons inelastically and thus randomizes the phase of the
electronic wave functions. These interactions try to destroy the phase
coherence of the electrons which can remove the quantum effects. Hence, it is
necessary to do the calculations at sufficiently low temperatures such that
the phase coherence length of an electron exceeds the circumference $L$ of the
ring.
In our calculations of low-field magnetic susceptibility at absolute zero
temperature ($T=0$ K), we see that the current has only diamagnetic sign for
perfect rings irrespective of the total number of electrons, while in the
presence of impurity, it exhibits respectively the diamagnetic and
paramagnetic sign for the rings with odd and even $N_{e}$. It is well known
that at any finite temperature ($T\neq 0$ K), low-field current has a
paramagnetic sign for the rings with even $N_{e}$ and in this section we
determine that critical value of magnetic flux, $\phi_{c}(T)$, where the low-
field current changes its sign from the paramagnetic to diamagnetic nature. In
Fig. 6 we display the variation of $\phi_{c}(T)$ as a function of $N_{e}$
(here $N_{e}$ is even only) for $60$-site disordered rings ($W=1$) at two
different temperatures. The upper and lower curves in Figs. 6(a) and (b) are
respectively for the rings with $T/T^{\star}=1.0$ and $0.5$. Figure 6(a) shows
the results for the rings with NNH integral, while the same are plotted for
the rings with LRH ($\alpha=1.6$) integrals in Fig. 6(b). From these results
we can emphasize that, as the temperature increases the critical value of
magnetic flux $\phi_{c}(T)$, where the low-field current changes its sign from
the paramagnetic phase to the diamagnetic one, increases.
## 5 Concluding remarks
In conclusion, we have investigated the behavior of persistent current in
single-isolated mesoscopic rings subjected to both NNH and LRH integrals
within the tight-binding framework. Our exact numerical calculations have
shown that the current amplitude in disordered rings are comparable to that of
ordered rings if we consider the model with LRH integrals instead of usual NNH
integral models. This is due to the fact that higher order hopping integrals
try to delocalize the energy eigenstates and thus prevents the reduction of
current due to disorder in the rings. Later, we have studied the low-field
magnetic response at $T=0$ K both for the perfect and disordered rings and our
results have predicted that the sign of the currents can be mentioned
precisely even in the presence of impurity in the rings. At the end, we have
calculated the magnetic response at finite temperatures ($T\neq 0$ K) and
estimated the critical value of magnetic flux $\phi_{c}(T)$ where the low-
field current changes its sign from the paramagnetic to the diamagnetic
nature.
## References
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* [13] B. L. Altshuler, Y. Gefen, and Y. Imry, Phys. Rev. Lett. 66, 88 (1991).
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* [15] A. Schmid, Phys. Rev. Lett. 66, 80 (1991).
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* [17] M. Abraham and R. Berkovits, Phys. Rev. Lett. 70, 1509 (1993).
* [18] G. Bouzerar, D. Poilblanc, and G. Montambaux, Phys. Rev. B 49, 8258 (1994).
* [19] T. Giamarchi and B. S. Shastry, Phys. Rev. B 51, 10915 (1995).
* [20] X. Waintal, G. Fleury, K. Kazymyrenko, M. Houzet, P. Schmitteckert and D. Weinmann, Phys. Rev. Lett. 101, 106804 (2008).
* [21] K. Yakubo, Y. Avishai and D. Cohen, Phys. Rev. B 67, 125319 (2003).
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|
arxiv-papers
| 2009-09-07T09:36:36 |
2024-09-04T02:49:05.089015
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Santanu K. Maiti",
"submitter": "Santanu Maiti Kumar",
"url": "https://arxiv.org/abs/0909.1188"
}
|
0909.1255
|
# T-Zamfirescu and T-weak contraction mappings on cone metric spaces
José R. Morales and Edixon Rojas Department of Mathematics, Faculty of
Science, University of Los Andes, Mérida-5101, Venezuela. moralesj@ula.ve
edixonr@ula.ve
###### Abstract.
The purpose of this paper is to obtain sufficient conditions for the existence
of a unique fixed point of T-Zamfirescu and T-weak contraction mappings in the
framework of complete cone metric spaces.
###### Key words and phrases:
Fixed point, complete cone metric space, $T-$zamficescu mapping, $T-$weak
contraction, subsequentially convergent.
###### 1991 Mathematics Subject Classification:
47H10, 46J10.
## 1\. Introduction
In 2007, Guang and Xiang [11] generalized the concept of metric space,
replacing the set of real numbers by an ordered Banach space and defined a
cone metric space. The authors there described the convergence of sequences in
cone metric spaces and introduced the completeness. Also, they proved some
fixed point theorems of contractive mappings on complete cone metric spaces.
Since then, fixed point theorems for different (classic) classes of mappings
on these spaces have been appeared, see for instance [1], [7], [8], [10],
[15], [16] and [17].
On the other hand, recently A. Beiranvand S. Moradi, M. Omid and H. Pazandeh
[5] introduced the $T-$contraction and $T-$contractive mappings and then they
extended the Banach contraction principle and the Edelstein’s fixed point
Theorem. S. Moradi [12] introduced the $T-$Kannan contractive mappings,
extending in this way the Kannan’s fixed point theorem [9]. The corresponding
version of $T$-contractive, $T$-Kannan mappings and $T-$Chalterjea
contractions on cone metric spaces was studied in [13] and [14] respectively.
In view of these facts, thereby the purpose of this paper is to study the
existence of fixed points of $T-$Zamficescu and $T-$weak contraction mappings
defined on a complete cone metric space $(M,d)$, generalizing consequently the
results given in [11] and [18].
## 2\. General framework
In this section we recall the definition of cone metric space and some of
their properties (see, [11]). The following notions will be useful for us in
order to prove the main results.
###### Definition 2.1.
Let $E$ be a real Banach space. A subset $P$ of $E$ is called a cone if and
only if:
(P1):
$P$ is closed, nonempty and $P\neq\\{0\\}$;
(P2):
$a,b\in\mathbb{R},\,\,a,b\geq 0,\,\,x,y\in P$ imply $ax+by\in P$;
(P3):
$x\in P$ and $-x\in P\Rightarrow x=0$. I.e., $P\cap(-P)=\\{0\\}$.
Given a cone $P\subset E,$ we define a partial ordering $\leq$ with respect to
$P$ by $x\leq y$ if and only if $y-x\in P$. We write $x<y$ to indicate that
$x\leq y$ but $x\neq y$, while $x\ll y$ will stand for
$y-x\in\operatorname{Int}P$. (interior of $P$.)
###### Definition 2.2.
Let $E$ be a Banach space and $P\subset E$ a cone. The cone $P$ is called
normal if there is a number $K>0$ such that for all $x,y\in E,\,\,0\leq x\leq
y$ implies $\|x\|\leq K\|y\|.$ The least positive number satisfying the above
is called the normal constant of $P.$
In the following, we always suppose that $E$ is a Banach space, $P$ is a cone
in $E$ with $\operatorname{Int}P\neq\emptyset$ and $\leq$ is partial ordering
with respect to $P$.
###### Definition 2.3 ([11]).
Let $M$ be a nonempty set. Suppose that the mapping $d:M\times
M\longrightarrow E$ satisfies:
(d1):
$0<d(x,y)$ for all $x,y\in M$, and $d(x,y)=0$ if and only if $x=y$;
(d2):
$d(x,y)=d(y,x)$ for all $x,y\in M$;
(d3):
$d(x,y)\leq d(x,z)+d(z,y)$ for all $x,y,z\in M$.
Then, $d$ is called a cone metric on $M$ and $(M,d)$ is called a cone metric
space.
Note that the notion of cone metric space is more general that the concept of
metric space.
###### Definition 2.4.
Let $(M,d)$ be a cone metric space. Let $(x_{n})$ be a sequence in $M$ and
$x\in M$.
* (i)
$(x_{n})$ converges to $x$ if for every $c\in E$ with $0\ll c$ there is an
$n_{0}$ such that for all $n>n_{0},\,\,d(x_{n},x)\ll c.$ We denote this by
$\displaystyle\lim_{n\rightarrow\infty}x_{n}=x$ or $x_{n}\rightarrow
x,\,\,(n\rightarrow\infty)$.
* (ii)
If for any $c\in E$ with $0\ll c$ there is an $n_{0}$ such that for all
$n,m\geq n_{0}$, $\;d(x_{n},x_{m})\ll c$, then $(x_{n})$ is called a Cauchy
sequence in $M$.
Let $(M,d)$ be a cone metric space. If every Cauchy sequence is convergent in
$M,$ then $M$ is called a complete cone metric space.
###### Lemma 2.1 ([11]).
Let $(M,d)$ be a cone metric space, $P\subset E$ a normal cone with normal
constant $K.$ Let $(x_{n}),\,\,(y_{n})$ be sequences in $M$ and $x,y\in M$.
* (i)
$(x_{n})$ converges to $x$ if and only if
$\displaystyle\lim_{n\rightarrow\infty}d(x_{n},x)=0$.
* (ii)
If $(x_{n})$ converges to $x$ and $(x_{n})$ converges to $y$, then $x=y$.
* (iii)
If $(x_{n})$ converges to $x$, then $(x_{n})$ is a Cauchy sequence.
* (iv)
$(x_{n})$ is a Cauchy sequence if and only if
$\displaystyle\lim_{n,m\rightarrow\infty}d(x_{n},x_{m})=0$.
* (v)
If $x_{n}\longrightarrow x$ and $y_{n}\longrightarrow
y,\,\,(n\rightarrow\infty)$, then $d(x_{n},y_{n})\longrightarrow d(x,y)$.
###### Definition 2.5.
Let $(M,d)$ be a cone metric space, $P$ a normal cone with normal constant $K$
and $T:M\longrightarrow M$. Then
* (i)
$T$ is said to be continuous, if
$\displaystyle\lim_{n\rightarrow\infty}x_{n}=x$ implies that
$\displaystyle\lim_{n\rightarrow\infty}T(x_{n})=T(x)$ for all $(x_{n})$ and
$x$ in $M$.
* (ii)
$T$ is said to be subsequentially convergent if we have, for every sequence
$(y_{n}),$ if $T(y_{n})$ is convergent, then $(y_{n})$ has a convergent
subsequence.
* (iii)
$T$ is said to be sequentially convergent if we have, for every sequence
$(y_{n}),$ if $T(y_{n})$ is convergent then $(y_{n})$ also is convergent.
Examples of cone metric spaces can be found for instance in [11], [17] and
references therein.
## 3\. Main Results
This section is devoted to give fixed point results for $T$-Zamfirescu and
$T$-weak contraction mappings on complete (normal) cone metric spaces, as well
as, their asymptotic behavior. First, we recall the following classes of
contraction type mappings:
###### Definition 3.1.
Let $(M,d)$ be a cone metric space and $T,S:M\longrightarrow M$ two mappings
* (i)
The mapping $S$ is called a $T-$Banach contraction, (TB - Contraction) if
there is $a\in[0,1)$ such that
$d(TSx,TSy)\leq ad(Tx,Ty)$
for all $x,y\in M$.
* (ii)
The mapping $S$ is called a $T-$Kannan contraction, (TK - Contraction) if
there is $b\in[0,1/2)$ such that
$d(TSx,TSy)\leq b[d(Tx,TSx)+d(y,TSy)]$
for all $x,y\in M$.
* (iii)
A mapping $S$ is said to be a Chatterjea contraction, (TC - Contraction) if
there is $c\in[0,1/2)$ such that
$d(TSx,TSy)\leq c[d(Tx,TSy)+d(Ty,TSx)]$
for all $x,y\in M.$
It is clear that if we take $T=I_{d}$ (the identity map) in the Definition 3.1
we obtain the definitions of Banach contraction, Kannan mapping ([9]) and
Chatterjea mapping ([6]).
Now, following the ideas of T. Zamfirescu [18] we introduce the notion of
$T-$Zamfirescu mappings.
###### Definition 3.2.
Let $(M,d)$ be a cone metric space and $T,S:M\longrightarrow M$ two mappings.
$S$ is called a $T-$Zamfirescu mapping, (TZ -mapping), if and only if, there
are real numbers, $0\leq a<1,\,\,0\leq b,c<1/2$ such that for all $x,y\in M,$
at least one of the next conditions are true:
($TZ_{1}$):
$d(TSx,TSy)\leq ad(Tx,Ty)$.
($TZ_{2}$):
$d(TSx,TSy)\leq b[d(Tx,TSx)+d(Ty,TSy)]$.
($TZ_{3}$):
$d(TSx,TSy)\leq c[d(Tx,TSy)+d(Ty,TSx)]$.
If in Definition 3.2 we take $T=I_{d}$ and $E=\mathbb{R}_{+}$ we obtain the
definition of T. Zamfirescu [18].
###### Lemma 3.1.
Let $(M,d)$ be a cone metric space and $T,S:M\longrightarrow M$ two mappings.
If $S$ is a $TZ-$mapping, then there is $0\leq\delta<1$ such that
(3.1) $d(TSx,TSy)\leq\delta d(Tx,Ty)+2\delta d(Tx,TSx)$
for all $x,y\in M$.
###### Proof.
If $S$ is a $TZ-$mapping, then at least one of $(TZ_{1})$, $(TZ_{2})$ o
$(TZ_{3})$ condition is true.
If $(TZ_{2})$ holds, then:
$\begin{array}[]{ccl}d(TSx,TSy)&\leq&b[d(Tx,TSx)+d(Ty,TSy)]\\\ \\\
&\leq&b[d(Tx,TSx)+d(Ty,Tx)+d(Tx,TSx)+d(TSx,TSy)]\end{array}$
thus,
$(1-b)d(TSx,TSy)\leq bd(Tx,Ty)+2bd(Tx,TSx).$
From the fact that $0\leq b<1/2$ we get:
$d(TSx,TSy)\leq\displaystyle\frac{b}{1-b}d(Tx,Ty)+\displaystyle\frac{2b}{1-b}d(Tx,TSx).$
with $\frac{b}{1-b}<1$. If $(TZ_{3})$ holds, then similarly we get
$d(TSx,TSy)\leq\displaystyle\frac{c}{1-c}d(Tx,Ty)+\displaystyle\frac{2c}{1-c}d(Tx,TSx).$
Therefore, denoting by
$\delta:=\max\left\\{a,\,\displaystyle\frac{b}{1-b},\,\displaystyle\frac{c}{1-c}\right\\}$
we have that $0\leq\delta<1$. Hence, for all $x,y\in M,$ the following
inequality holds:
$d(TSx,TSy)\leq\delta d(Tx,Ty)+2\delta d(Tx,TSx).$
∎
###### Remark 1.
Notice that inequality (3.1) in Lemma 3.1 can be replace by
$d(TSx,TSy)\leq\delta d(Tx,Ty)+2\delta d(Tx,TSy)$
for all $x,y\in M$.
###### Theorem 3.2.
Let $(M,d)$ be a complete cone metric space, $P$ be a normal cone with normal
constant $K$. Moreover, let $T:M\longrightarrow M$ be a continuous and one to
one mapping and $S:M\longrightarrow M$ a $T-$Zamfirescu continuous mapping.
Then
* (i)
For every $x_{0}\in M$,
$\displaystyle\lim_{n\rightarrow\infty}d(TS^{n}x_{0},TS^{n+1}x_{0})=0.$
* (ii)
There is $y_{0}\in M$ such that
$\displaystyle\lim_{n\rightarrow\infty}TS^{n}x_{0}=y_{0}.$
* (iii)
If $T$ is subsequentially convergent, then $(S^{n}x_{0})$ has a convergent
subsequence.
* (iv)
There is a unique $z_{0}\in M$ such that $Sz_{0}=z_{0}$.
* (v)
If $T$ is sequentially convergent, then for each $x_{0}\in M$ the iterate
sequence $(S^{n}x_{0})$ converges to $z_{0}$.
###### Proof.
* (i)
Since $S$ is a $T-$Zamfirescu mapping, then by Lemma 3.1, there exists
$0<\delta<1$ such that
$d(TSx,TSy)\leq\delta d(Tx,Ty)+2\delta d(Tx,TSx)$
for all $x,y\in M$.
Suppose $x_{0}\in M$ is an arbitrary point and the Picard iteration associated
to $S,$ $\;(x_{n})$ is defined by
$x_{n+1}=Sx_{n}=S^{n}x_{0},\qquad n=0,1,2,\ldots.$
Thus,
$d(TS^{n+1}x_{0},TS^{n}x_{0})\leq hd(TS^{n}x_{0},TS^{n-1}x_{0})$
where $h=\displaystyle\frac{\delta}{1-2\delta}<1$. Therefore, for all $n$ we
have
$d(TS^{n+1}x_{0},TS^{n}x_{0})\leq h^{n}d(TSx_{0},Tx_{0}).$
From the above, and the fact the cone $P$ is a normal cone we obtain that
$\|d(TS^{n+1}x_{0},TS^{n}x_{0})\|\leq Kh^{n}\|d(TSx_{0},Tx_{0})\|,$
taking limit $n\longrightarrow\infty$ in the above inequality we can conclude
that
$\displaystyle\lim_{n\rightarrow\infty}d(TS^{n+1}x_{0},TS^{n}x_{0})=0.$
* (ii)
Now, for $m,n\in\mathbb{N}$ with $m>n$ we get
$\begin{array}[]{ccl}d(TS^{m}x_{0},TS^{n}x_{0})&\leq&(h^{n}+\ldots+h^{m-1})d(TSx_{0},Tx_{0})\\\
\\\ &\leq&\displaystyle\frac{h^{n}}{1-h}d(TSx_{0},Tx_{0}).\end{array}$
Again; as above, since $P$ is a normal cone we obtain
$\displaystyle\lim_{n,m\rightarrow\infty}d(TS^{m}x_{0},TS^{n}x_{0})=0.$
Hence, the fact that $(M,d)$ is a complete cone metric space, imply that
$(TS^{n}x_{0})$ is a Cauchy sequence in $M$, therefore there is $y_{0}\in M$
such that
$\displaystyle\lim_{n\rightarrow\infty}TS^{n}x_{0}=y_{0}.$
* (iii)
If $T$ is subsequentially convergent, $(S^{n}x_{0})$ has a convergent
subsequence, so there is $z_{0}\in M$ and $(n_{k})_{k=1}^{\infty}$ such that
$\displaystyle\lim_{k\rightarrow\infty}S^{n_{k}}x_{0}=z_{0}.$
* (iv)
Since $T$ and $S$ are continuous mappings we obtain:
$\displaystyle\lim_{k\rightarrow\infty}TS^{n_{k}}x_{0}=Tz_{0},\qquad\displaystyle\lim_{k\rightarrow\infty}TS^{n_{k}+1}x_{0}=TSz_{0}$
therefore, $Tz_{0}=y_{0}=TSz_{0},$ and since $T$ is one to one, then
$Sz_{0}=z_{0}.$ So $S$ has a fixed point.
Now, suppose that $Sz_{0}=z_{0}$ and $Sz_{1}=z_{1}$.
$\begin{array}[]{ccl}d(TSz_{0},TSz_{1})&\leq&\delta d(Tz_{0},Tz_{1})+2\delta
d(Tz_{0},TSz_{0})\\\ \\\ d(Tz_{0},Tz_{1})&\leq&\delta
d(Tz_{0},Tz_{1})\end{array}$
from the fact that $0\leq\delta<1$ and that $T$ is one to one we obtain that
$z_{0}=z_{1}$.
* (v)
It is clear that if $T$ is sequentially convergent, then for each $x_{0}\in
M$, the iterate sequence $(S^{n}x_{0})$ converges to $z_{0}$.
∎
In 2003, V. Berinde (see, [2], [3]) introduced a new class of contraction
mappings on metric spaces, which are called weak contractions. We will extend
these kind of mappings by introducing a new function $T$ and we define it in
the framework of cone metric spaces.
###### Definition 3.3.
Let $(M,d)$ be a cone metric space and $T,S:M\longrightarrow M$ two mappings.
$S$ is called a $T-$weak contraction, (TW- Contraction,
$T_{(S,L)}-$Contraction), if there exist a constant $\delta\in(0,1)$ and some
$L\geq 0$ such that
$d(TSx,TSy)\leq\delta d(Tx,Ty)+Ld(Ty,TSx)$
for all $x,y\in M$.
It is clear that if we take $T=I_{d}$ and $E=\mathbb{R}_{+}$ then we obtain
the notion of Berinde [2].
Due to the symmetry of the metric, the $T-$weak contractive condition
implicitly include the following dual one:
$d(TSx,TSy)\leq\delta d(Tx,Ty)+Ld(Tx,TSy)$
for all $x,y\in M$.
The next proposition gives examples of $T-$weak contraction and it proof is
similar to the proof of Lemma 3.1.
###### Proposition 3.3.
Let $(M,d)$ be a cone metric space and $T,S:M\longrightarrow M$ two mappings.
* (i)
If $S$ is a TB - contraction, then $S$ is a $T-$weak contraction.
* (ii)
If $S$ is a TK - contraction, then $S$ is a $T-$weak contraction.
* (iii)
If $S$ is a TC - contraction, then $S$ is a $T-$weak contraction.
* (iv)
If $S$ is TZ - mapping, then $S$ is a $T-$weak contraction.
Now we have the following result:
###### Theorem 3.4.
Let $(M,d)$ be a complete cone metric space, $P$ a normal cone with normal
constant $K$. Let furthermore $T:M\longrightarrow M$ a continuous and one to
one mapping and $S:M\longrightarrow M$ a continuous $T-$weak contraction. Then
* (i)
For every $x_{0}\in M$,
$\displaystyle\lim_{n\rightarrow\infty}d(TS^{n}x_{0},TS^{n+1}x_{0})=0.$
* (ii)
There is $y_{0}\in M$ such that
$\displaystyle\lim_{n\rightarrow\infty}TS^{n}x_{0}=y_{0}.$
* (iii)
If $T$ is subsequentially convergent, then $(S^{n}x_{0})$ has a convergent
subsequence.
* (iv)
There is $z_{0}\in M$ such that
$Sz_{0}=z_{0}.$
* (v)
If $T$ is sequentially convergent, then for each $x_{0}\in M$ the iterate
sequence $(S^{n}x_{0})$ converges to $z_{0}$.
###### Proof.
Similar to the proof of Theorem 3.2. ∎
As we see in Theorem 3.2, a $T-$Zamfirescu mapping has a unique fixed point.
The next example shows that a $T-$weak contraction may has infinitely fixed
points.
###### Example 1 ([4]).
Let $M=[0,1]$ be the unit interval with the usual metric and
$T,S:M\longrightarrow M$ the identity maps, that is, $Tx=Sx=x$ for all $x\in
M$. Then, taking $0\leq a<1$ and $L\geq 1-a$ we obtain
$\begin{array}[]{ccl}d(TSx,TSy)&=&|TSx-TSy|\\\ \\\
|x-y|&\leq&a|x-y|+L|y-x|\end{array}$
which is valid for all $x,y\in[0,1]$. Thus the set of the fixed points $F_{S}$
of the map $S$ is the interval $[0,1]$. I.e.,
$F_{S}=\\{x\in[0,1]\,/\,Sx=x\\}=[0,1].$
It is possible to force the uniqueness of the fixed point of a $T-$weak
contraction by imposing an additional contractive condition, as is shown in
the next theorem.
###### Theorem 3.5.
Let $(M,d)$ be a complete cone metric space, $P$ be a normal cone with normal
constant $K$. Let furthermore $T:M\longrightarrow M$ a continuous and one to
one mapping and $S:M\longrightarrow M$ a $T-$weak contraction for which there
is $\theta\in(0,1)$ and some $L_{1}\geq 0$ such that
$d(TSx,TSy)\leq\theta d(Tx,Ty)+L_{1}d(Tx,TSx)$
for all $x,y\in M$. Then:
* (i)
For every $x_{0}\in M$
$\displaystyle\lim_{n\rightarrow\infty}d(TS^{n}x_{0},TS^{n+1}x_{0})=0.$
* (ii)
There is $y_{0}\in M$ such that
$\displaystyle\lim_{n\rightarrow\infty}TS^{n}x_{0}=y_{0}.$
* (iii)
It $T$ is subsequentially convergent, then $(S^{n}x_{0})$ has a convergent
subsequence.
* (iv)
There is a unique $z_{0}\in M$ such that
$Sz_{0}=z_{0}.$
* (v)
If $T$ is sequentially convergent, then for each $x_{0}\in M$ the iterate
sequence $(S^{n}x_{0})$ converges to $z_{0}.$
###### Proof.
Assume $S$ has two distinct fixed points $x^{*},y^{*}\in M.$ Then
$d(Tx^{*},Ty^{*})=d(TSx^{*},TSy^{*})\leq\theta
d(Tx^{*},Ty^{*})+L_{1}d(Tx^{*},TSx^{*})$
thus, we get
$d(Tx^{*},Ty^{*})\leq\theta
d(Tx^{*},Ty^{*})\Leftrightarrow(1-\theta)d(Tx^{*},Ty^{*})\leq 0.$
Therefore, $d(Tx^{*},Ty^{*})=0$. Since $T$ is one to one, then $x^{*}=y^{*}$.
The rest of the proof follows as the the proof of Theorem 3.2. ∎
## References
* [1] M. Abbas and B.E. Rhoades, Fixed and periodic results in cone metric space, Appl. Math. Lett., 22, (4), (2009), 511–515.
* [2] V. Berinde, Iterate Approximation of fixed points, lect. Notes Math., Vol 1912, (2nd ed.), Springer Verlag, Berlin, 2007.
* [3] V. Berinde, On the approximation of fixed points of weak contractive mappings, Carpathian J. Math, 19, (1), (2003), 7–22.
* [4] V. Berinde, On the convergence of the Ishikawa Iteration in the class of quasi contractive operators, Acta Math. Univ. Comenianae, 73, (1), (2004), 119–126.
* [5] A. Beiranvand, S. Moradi, M. Omid and H. Pazandeh, Two fixed point theorem for special mapping, arXiv:0903.1504v1 [math.FA].
* [6] S. K. Chatterjea, Fixed point theorems, C. R. Acad. Bulgare Sci., 25, (1972), 727–730.
* [7] D. Ilić and V. Rakočević, Quasi-contraction on a cone metric space, Appl. Math. Lett., 22, (5), (2009), 728–731.
* [8] Z. Kadelburg, S. Radenović and V. Rakočević, Remarks on “Quasi-contraction on a cone metric space”, Appl. Math. Lett., (2009), doi:10.1016/j.aml.2009.06.003.
* [9] R. Kannan, Some results on fixed points, Bull. Calcutta Math. Soc., 60, (1968), 71–76.
* [10] M.S. Khan and M. Samanipour, Fixed point theorems for some discontinuous operators in cone metric space, Mathematica Moravica, Vol 12-2, (2008), 29–34.
* [11] Huan Long - Guang and Zhan Xian, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl., 332, (2007), 1468–1476.
* [12] S. Moradi, Kannan fixed point theorem on complete metric spaces and on generalized metric spaces depended on another function, arXiv:0903.1577v1 [math.FA].
* [13] J. Morales and E. Rojas, Cone metric spaces and fixed point theorems of $T-$contractive mappings, preprint, 2009.
* [14] J. Morales and E. Rojas, Cone metric spaces and fixed point theorems of $T-$Kannan contractive mappings, arXiv:0907.3949v1 [math.FA].
* [15] H.K. Pathak and N. Shahzad, Fixed points results for generalized quasicontraction mappings in abstract metric spaces, Nonlinear Analysis, (2009), doi:10.1016/j.na.2009.05.052.
* [16] V. Raja and S.M. Vaezpour, Some extension of Banach’s contraction principle in complete cone metric spaces, Fixed Point Theory and Applications, (2008), 11 p.
* [17] Sh. Rezapour and R. Hamlbarani, Some notes on the paper “Cone metric spaces and fixed point theorems of contractive mappings”, J. Math. Anal. Appl., 345, (2), (2008), 719–724.
* [18] T. Zamfirescu, Fixed points theorems in metric spaces, Arch. Math., 23, (1972), 292–298.
|
arxiv-papers
| 2009-09-07T14:40:22 |
2024-09-04T02:49:05.094802
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jos\\'e R. Morales and Edixon Rojas",
"submitter": "Edixon Rojas",
"url": "https://arxiv.org/abs/0909.1255"
}
|
0909.1303
|
Key principle of the efficient running, swimming, and flying
.
by Valery B. Kokshenev
Submitted to EPL 11 June 2009, resubmitted 31 August 2009
.
Abstract. Empirical observations indicate striking similarities among
locomotion in terrestrial animals, birds, and fish, but unifying physical
grounds are lacking. When applied to efficient locomotion, the analytical
mechanics principle of minimum action yields two patterns of mechanical
similarity via two explicit spatiotemporal coherent states. In steady
locomotory modes, the slow muscles determining maximal optimum speeds maintain
universal intrinsic muscular pressure. Otherwise, maximal speeds are due to
constant mass-dependent stiffness of fast muscles generating a uniform force
field, exceeding gravitation. Being coherent in displacements, velocities and
forces, the body appendages of animals are tuned to natural propagation
frequency through the state-dependent elastic muscle moduli.
Key words: variational principle of minimum action (04.20.Fy), locomotion
(87.19.ru), biomechanics (87.85.G-).
## I Introduction
Although evolutionary biologists and comparative zoologists make wonderful
generalizations about the movements of terrestrial animals, birds, and fish of
different size [1-12], the fundamental physical principles underlying striking
similarities in distinct types of movement for organisms remain a challenge
[13]. Within the scope of the simplest pendulum model (stiff-legged
approximation), it has been demonstrated [14] that humans and other animals,
in contrast to human-made engines, accomplish efficient propulsion (maximum
power output at minimum power consumption) by tuning musculoskeletal system to
the resonant propagation frequency. Storing mechanical energy in elastic
oscillations of body parts and in pendulum oscillations of legs or other
appendages, animals thereby reduce the energy consumption [1,3], which is
minimal at the resonance conditions [14]. In this study, instead of searching
for uncovered principles of body mass effects in biology [5], or doing in-
depth analysis of equations of motion in pendulum [14], spring [7,8], or
vortex [15] approximations and other engineer constructive approaches [9], I
address the key principle of mechanics.
In analytical mechanics, the requirement of __ minimum action between two
fixed points of the conceivable trajectory of an arbitrary isolated mechanical
system determines Lagrangian $\mathcal{L}(q,v)$, the function of time-
dependent coordinates $q(t)$ and instant velocities $v(t)=dq/dt$. The most
general property of a freely __ moving system is spatiotemporal homogeneity
implying that the multiplication of $\mathcal{L}$ on an arbitrary constant
does not affect the equations of motion, arising from $\mathcal{L}$. This
property, designated as a _mechanical similarity_ [16], permits one to
establish the major mechanical constraints without consideration of equations
of motion. Indeed, following Landau and Lifshitz [16], let us consider the
uniform transformation of mechanical trajectories due to linear changing of
all coordinates $q\rightarrow aq$ and times $t\rightarrow bt$, and hence
velocities $v\rightarrow(a/b)v$, via arbitrary coefficients $a$ and $b$. Let
the potential energy change consequently through a certain exponent $s$, i.e.,
$\mathcal{U}(aq)=a^{s}\mathcal{U}(q)$. Being a quadratic function of
velocities, the kinetic energy scales as
$\mathcal{K}(av/b)=\left(a/b\right)^{2}\mathcal{K}(v)$. The requirement of
homogeneity of $\mathcal{L}(q,v)=\mathcal{K}(v)-\mathcal{U}(q)$ is self-
consistent when both the energies change similar, i.e., $(a/b)^{2}=a^{s}$ or
$b=a^{1-s/2}$ . Thereby, the frictionless propagation of a classical system
obeys the scaling relationships imposed on all principal mechanical
characteristics: _period_ $T$, overall-system _speed_ $V$, and _force_
_amplitude_ $F$, namely [16]
$T\backsim t\varpropto L^{1-s/2}\text{, }V\backsim v\varpropto L^{s/2}\text{,
and }F\varpropto L^{s-1}\text{.}$ (1)
The seminal case $s=-1$ introduces Newtonian’s intertrajectory coupling force
$F\backsim M^{2}L^{-2}$, where mass $M$ emerges as the dimensional coefficient
of proportionality.
It will be demonstrated how the mechanical principle of minimum action applied
to musculoskeletal system of animals involved in efficient locomotion may
provide basic patterns of biomechanical similarity.
## II Minimum action in biomechanics
During locomotion, chemical energy released by muscles and mechanical elastic
energy stored in body system is transformed into external and internal work
and partially lost as a heat. In the case of the off resonance human walking
[17], the small velocity-dependent frictional effects were accounted for in
the second order of perturbation theory, thereby generalizing the Lagrangian
formalism over weakly open systems.
During the muscle forced resonance walking and running (or flying) minimizing
energy consumption, the small damping effects restrict only the amplitude of
motion, i.e., _stride length_ $\Delta L$ (or stroke amplitude) and _muscle
length change_ $\Delta L_{m}$, but not the propagation _speed_ $V=\Delta L/T$
and _period_ $T$, constrained geometrically [14]. Likewise [17], frictional
effects can be therefore neglected in the equations of motion [14], on the
first approximation. With the same precision, the principle of mechanical
similarity (1) provides
$\displaystyle T^{-1}$
$\displaystyle=1/T_{ms}\varpropto\sqrt{E_{ms}}L_{m}^{-1}\text{, }V\backsim
V_{ms}\varpropto\sqrt{E_{ms}}\text{,}$ $\displaystyle F$
$\displaystyle\backsim\Delta F\backsim F_{ms}\backsim\Delta
F_{ms}=\varepsilon_{m}A_{m}E_{ms}\text{, with
}E_{ms}\varpropto(L_{m})^{s}\text{ and }g_{ms}\varpropto(L_{m})^{s-1}\text{,}$
(2)
when presented in the linear-displacement body ($\Delta L\backsim L$) and
muscle ($\Delta L_{m}\backsim L_{m}$) approximation. Introducing in eq. (2)
the force change $\Delta F$ for the body _force output_ $F$, driving a given
animal (of _characteristic length_ $L$, _cross-sectional area_ $A$, and _body
mass_ $M$) through the environment, and the effective body rigidness, or
longitudinal _stiffness_ $K=\Delta F/\Delta L$, one also determines the
natural (resonant) _cyclic frequency_ $T^{-1}\backsim\sqrt{K/M}$
[1,7,8,17,18]. Since the animal locomotion is substantially muscular [1,3,18],
the _muscle stiffness_ $K_{m}=E_{m}A_{m}/L_{m}$ (of a muscle of length $L_{m}$
and cross-sectional area $A_{m}$), controlled by the geometry-independent
muscle rigidity or _elastic modulus_ $E_{m}$ (ratio of _stress_ $\sigma_{m}$
to _strain_ $\varepsilon_{m}$, i.e., $(\Delta F_{m}/A_{m})/(\Delta
L_{m}/L_{m})$) [7, 18], is also under our consideration. To improve the
integrative approach to animal locomotion [1-18] via mechanical [19] and
elastic strain [19, 20] similarities, let us determine a muscle-force _field_
$g_{m}\equiv F_{m}/m$, where the _muscle mass_ $m$ (or _motor mass_[6]) is a
source of the _active force output_ $F_{m}$. Furthermore, the scaling
relations for physical quantities (shown in eq. (2) by symbol $\varpropto$)
result from provided relations and constraints imposed by the invariable
_body_ _density_ $\rho$ ($=M/AL$) and _muscle density_ $\rho_{m}$
($m/A_{m}L_{m}$), all common in scaling biomechanics [1,7,9,18].
In this study, the intrinsic muscle modulus $E_{ms}$, substituting $E_{m}$ in
eq. (2), describes a new dynamic degree of freedom characterizing muscle
ability of tuning to the resonance [15] in different locomotory gaits
distinguished by the single _dynamic-state exponent_ $s$.
## III Results and discussion
The _steady-speed locomotion_ for flight mode was first recognized by Hill:
”the frequencies of hovering birds are in inverse proportionality to the cube
roots of the weights, i.e., to the linear size” [2]. This dynamic regime is
pronounced in eq. (2), taken with $s=0$, by the propagation frequency
$T^{-1}\backsim\sqrt{E_{m0}/\rho}L^{-1}$, contrasting with the rigid-pendulum
estimate $T_{pend}^{-1}\backsim\sqrt{g}L^{-1/2}$ ($g$ is gravitation field)
[7,14]. Broadly speaking, Hill’s observation plays the role similar to
Kepler’s observation of third law for planets $T^{2}\varpropto L^{3}$,
following from eq. (1) with $s=-1$.
Hence, when the animal’s body travels or cruises slowly for long distances [4]
with the constant optimum speed $V_{body}^{(\max)}$
$\backsim\sqrt{E_{m0}^{(\max)}/\rho}$, invariant with body weight and
frequency, or moves throughout the terrestrial, air, or water environment
resisting drag forces, the legs, wings, and tails suggest to maintain constant
elastic modulus $E_{m0}^{(\max)}$ in _slow muscles_ responsible for the steady
locomotion [21]. Consequently, a constant functional intrinsic muscle stress
$\varepsilon_{m}E_{m0}$ is also predicted in eq. (2) with $s=0$, providing in
turn constant safety factor (ratio of muscle strength to peak functional
stress), also expected by Hill [2]. These and other relevant constraints of
steady-speed locomotion are displayed in table 1.
.
$s=0$ | Frequency | Length | Speed | Force | Mass
---|---|---|---|---|---
$\ \ T^{-1}$ | $T^{-1}$ | $\rho^{{\small-}\frac{1}{2}}E_{{\small 0}}^{\frac{1}{2}}\cdot L^{{\small-1}}$ | $\rho^{{\small-}\frac{1}{4}}E_{{\small 0}}^{\frac{1}{4}}\cdot V^{-\frac{1}{2}}$ | $F^{0}$ | $\rho^{{\small-}\frac{1}{6}}E_{{\small 0}}^{{\small-}\frac{1}{2}}\cdot M^{{\small-}\frac{1}{3}}$
$\Delta L,L$ | $\rho^{{\small-}\frac{1}{2}}E_{{\small 0}}^{\frac{1}{2}}\cdot T$ | $L$ | $\rho^{{\small-}\frac{1}{4}}E_{{\small 0}}^{\frac{1}{4}}\cdot V^{\frac{1}{2}}$ | $F^{0}$ | $\rho^{{\small-}\frac{1}{3}}\cdot M^{\frac{1}{3}}$
$V^{(\max)}$ | $\rho^{{\small-}\frac{1}{2}}E_{{\small 0}}^{\frac{1}{2}}\cdot T^{0}$ | $\rho^{{\small-}\frac{1}{2}}E_{{\small 0}}^{\frac{1}{2}}\cdot L^{0}$ | $\rho^{{\small-}\frac{1}{2}}E_{{\small 0}}^{\frac{1}{2}}$ | $\rho^{{\small-}\frac{1}{2}}E_{{\small 0}}^{\frac{1}{2}}\cdot F^{0}$ | $\rho^{{\small-}\frac{1}{2}}E_{{\small 0}}^{\frac{1}{2}}\cdot M^{{\small 0}}$
$K_{body}^{(\max)}$ | $\rho^{\frac{1}{2}}E_{{\small 0}}^{\frac{1}{2}}A\cdot T^{-1}$ | $E_{{\small 0}}A\cdot L^{-1}$ | $\rho^{\frac{1}{4}}E_{{\small 0}}^{-\frac{1}{4}}\cdot V^{-\frac{1}{2}}$ | $L^{-1}\cdot F$ | $\rho^{{\small-}\frac{1}{3}}E_{0}\cdot M^{\frac{1}{3}}$
$\sigma_{slow}^{(\max)}$ | $\varepsilon_{m}E_{0}\cdot T^{0}$ | $\varepsilon_{m}E_{0}\cdot L^{0}$ | $\varepsilon_{m}E_{0}\cdot V^{0}$ | $\varepsilon_{m}E_{0}\cdot F^{0}$ | $\varepsilon_{m}E_{0}\cdot m^{{\small 0}}$
$F_{slow}^{(\max)}$ | $\varepsilon_{m}E_{0}A_{m}\cdot T^{0}$ | $\varepsilon_{m}E_{0}A_{m}\cdot L^{0}$ | $\varepsilon_{m}E_{0}A_{m}\cdot V^{0}$ | $\varepsilon_{m}^{(\max)}E_{0}A_{m}$ | $\rho_{{\small m}}^{{\small-}\frac{2}{3}}\varepsilon_{m}E_{0}\cdot m^{\frac{2}{3}}$
Table 1. Mechanical characteristics of body system and slow individual muscles
in the steady-motion dynamic states $s=0$ prescribed by the principle of
minimum muscular action in eq. (2). _Abbreviation_ : $E_{0}=E_{m0}^{(\max)}$.
.
The constant maximum propulsive force $F_{body}^{(\max)}\backsim
E_{m0}^{(\max)}A$, equilibrating all drag forces via slow muscles, i.e.,
$F_{drag}^{(\max)}\backsim F_{slow}^{(\max)}$ shown in table 1, was first
documented by Alexander as the peak body force $F_{body}^{(\exp)}\varpropto
M^{2/3}$ [10] exerted on the environment by running, flying, and swimming
animals ranged over nine orders of body mass. More recently, the slow-fiber
force output $F_{slow}^{(\max)}\varpropto m^{2/3}$ (table 1) was revealed [6]
by statistical regression method in both biological and human-made _slow
motors_. The underlying muscle longitudinal field ”caused by intrinsic muscle
quantity (here associated with $E_{m0}$), equally stimulated electrically and
by the nervous system” [2] decreases linearly with the distance $r$:
$g_{slow}^{(\max)}(r)\thickapprox
E_{m0}^{(\max)}\varepsilon_{m}^{(\max)}/\rho_{m}r$, where
$\varepsilon_{m}^{(\max)}$ is nearly isometric strain, as follows from eq. (2)
with $s=0$.
The resonant efficient locomotion broadly prescribes a concerted behavior
synchronized in time and coordinated in displacements and forces of the body’s
appendages. Consequently, the muscle _duty factor_ $\beta_{m}=\Delta t_{m}/T$,
where $\Delta t_{m}$ is timing of the muscle lengthening/shortening $\Delta
L_{m}$, is constant, besides the body-mass invariable _Strouhal number_
$St=\Delta L/VT$, explaining the tail and wing oscillations in swimmers and
flyers [22]. At maximum propulsive efficiency of cruising dolphins, birds, and
bats, it was observed as $St_{cruis}\thickapprox 0.3$ [4].
The steady-speed locomotion state also was remarkably established in hovering
flying motors via the wing frequencies $1/T^{(\exp)}\varpropto M^{-1/3}$ [23],
as predicted in table 1. However, departures from Hill’s findings rationalized
here by the dynamic-state exponent $s=0$ were also debated [24]. For example,
it was claimed [7] that Hill’s maximal optimum speeds are in sharp
disagreement with the peak trot-gallop _crossover speeds_ $V_{cross}^{(\exp)}$
measured in quadrupeds [12]. The same could refer to the bipeds [11]. However,
as can be seen from the proper empirical data $1/T^{(\exp)}\varpropto
M^{-0.178}$ [11] and $1/T^{(\exp)}\varpropto V_{cross}^{-1}$ $\varpropto
M^{-0.145}$ [7,12], the measured stride frequencies indicate observations of
another kind of mechanical similarity attributed to the _non-steady dynamic
state_ $s=1$, prescribed in eq. (2) through the mass-dependent muscle modulus
$E_{m1}\varpropto L_{m}\varpropto M^{1/3}$.
The minimum muscle action of legs in fast running rats, wallaby, dog, goat,
horse, and human was indirectly revealed through the mechanical similarity
derived with the help of _leg spring model_ [8], providing the stride
frequency $T^{-1}\backsim\Delta t_{leg}^{-1}\varpropto M^{-0.19}$, stride
length $\Delta L\varpropto M^{0.30}$, model-body length $L_{leg}\varpropto
M^{0.34}$, body stiffness $K_{leg}^{(\max)}\varpropto M^{0.67}$, and body
force output $F_{leg}^{(\max)}\varpropto M^{0.97}$. Relations between the
quantities underlying these findings are discussed below and summarized in
table 2.
.
$s=1$ | Frequency | Length | Speed | Force | Mass
---|---|---|---|---|---
$T^{-1}$ | $T^{-1}$ | $g_{1}^{\frac{1}{2}}\cdot L^{-\frac{1}{2}}$ | $g_{1}\cdot V^{-1}$ | $(\rho g_{1}^{2}A)^{{\small-}\frac{1}{2}}\cdot F^{\frac{1}{2}}$ | $\rho^{\frac{1}{6}}g_{1}^{\frac{1}{2}}\cdot M^{{\small-}\frac{1}{6}}$
$\Delta L,L$ | $g_{1}\cdot T^{2}$ | $L$ | $g_{1}^{-1}\cdot V^{2}$ | $(\rho g_{1}A)^{{\small-1}}\cdot F$ | $\ \rho^{{\small-}\frac{1}{3}}\cdot M^{\frac{1}{3}}$
$V_{cross}^{(\max)}$ | $g_{1}\cdot T$ | $g_{1}^{\frac{1}{2}}\cdot L^{\frac{1}{2}}$ | $V$ | $(\rho A)^{{\small-}\frac{1}{2}}\cdot F^{\frac{1}{2}}$ | $\rho^{{\small-}\frac{1}{6}}g_{1}^{\frac{1}{2}}\cdot M^{\frac{1}{6}}$
$K_{body}^{(\max)}$ | $\rho g_{1}A\cdot T^{0}$ | $\rho g_{1}A\cdot L^{0}$ | $\rho g_{1}A\cdot V^{0}$ | $\rho g_{1}A\cdot F^{0}$ | $\rho^{{\small-}\frac{1}{3}}g_{1}\cdot M^{\frac{2}{3}}$
$\sigma_{fast}^{(\max)}$ | $\rho_{m}g_{1}^{2}\cdot T^{2}$ | $\rho_{m}g_{1}\cdot L_{m}$ | $\rho_{m}\cdot V^{2}$ | $A_{m}^{-1}\cdot F_{m}$ | $\rho_{m}^{\frac{2}{3}}g_{1}\cdot m^{\frac{1}{3}}$
$F_{fast}^{(\max)}$ | $\rho_{m}g_{1}^{2}A_{m}\cdot T^{2}$ | $\rho_{m}g_{1}A_{m}\cdot L$ | $\rho_{m}A_{m}\cdot V^{2}$ | $F_{m}$ | $g_{1}\cdot m$
Table 2. Mechanical characterization of body of animals and fast muscles in
physiologically equivalent non-steady states $s=1$ prescribed by eq. (2).
_Abbreviation_ : $g_{1}=g_{m1}$.
.
In accord with table 2, the equilibration of the air drag by wings of flapping
birds is manifested by the observed wing frequencies $1/T^{(\exp)}\varpropto
M^{-1/6}$ [23]. Moreover, the mechanical similarity between animals resisting
air, ground, and water friction forces was demonstrated via the energy cost
minimization [9], where the spatiotemporal correlations $V\varpropto L^{1/2}$
($\varpropto M^{1/6}$) were critically explored on ad hoc basis.
When the non-steady locomotion conditions associated with the physiologically
equivalent (or transient-equilibrium [19]) states $s=1$ are applied to
individual fast-twitch-fiber muscles controlling fast gaits [21], the muscle
field is apparently uniform and likely universal [6]. Indeed, the body force
field $F_{body}^{(\max)}/M\thickapprox 3g$ was first observed via the maximum
force output in fast trotting and hopping quadrupeds [8]. Later, mass-specific
force output $g_{m1}^{(\exp)}$ was empirically established [6] for locomotory
individual muscles associated with _fast motors_ in running, flying, and
swimming animals. One therefore infers that the gravitation field $g$ is not
crucial in fast running modes, as proposed in [9]. Moreover, the principle of
minimum muscular action suggests that fast muscles may generate force into the
whole muscle bulk [25] maintaining constant body stiffness (table 2), unlike
the constant pressure characteristic of steady gaits (table 1). In other
words, the fast muscles are not simple passive springs [3,26], attributed to
$s=2$ and having length-independent period, but are complex systems being able
to activate fibres in both parallel and series. Maintaining the uniform muscle
force field $g_{m1}$, the _Froude number_($Fr=V/\sqrt{gL}$ [1]) must be mass-
invariable, for both muscle system ($Fr_{fast}\backsim\sqrt{g_{m1}/g}$) and
body system, apart from the corresponding Strouhal number. For fast running
gaits in mammals, $Fr_{run}^{(\exp)}\thickapprox 1.5$ and
$St_{run}^{(\exp)}\thickapprox 0.4$ [8].
## IV Conclusion
The main goal of this letter is to demonstrate how the complex biological
phenomenon of mechanical similarity in animal locomotion allows to be
rationalized and formulated as a predictive, quantitative framework. It has
been shown how the fundamental physical principle of minimum action applied to
locomotory muscles via intrinsic elastic moduli quantifies amazing
similarities established empirically between maximal speeds, frequencies,
forces, and other relevant mechanical characteristics of animals locomoting in
a certain gait. Naturally operating the softness of legs, wings, and tails,
the efficient runners, flyers, and swimmers are shown to maintain constant
Strouhal number via the universal constant muscle pressure, when traveling or
cruising at steady speeds. When acting quickly at higher speeds, escaping from
predators, or when hunting, the successful runners, flyers, and swimmers
appear to maintain the universal field in the whole bulk of fast muscles, at
least at crossover speeds. This uniform field eventually results in the
bodyweight depending, fixed muscle stiffness and universal Froude and Strouhal
numbers. The provided from first principles study illuminates and supplements
a wide spectrum of reliable empirical findings in walking and running bipeds
[3,11], trotting and galloping quadrupeds [6-9,12]; hovering and flapping
birds [2-4,10,11], bats, and insects [3,4,9]; undulating and tail-beating fish
[2-4,9,10], dolphins [2,4], sharks [4], and whales [2].
On the other hand, the study of muscle characteristics, including obtained
scaling relations to muscle and body mass, is limited by the linear-
displacement muscle approximation. It can been shown however that the top
speeds attributed to limiting animal performance [19,24] cannot be achieved by
the linear-strain elastic muscle fields. The consequences of application of
the minimum action to specific fast locomotory muscles structurally adapted to
a certain mechanical activity, such as motor, brake, or strut functions [3]
prescribed by non-linear elastic effects [25] will be discussed elsewhere.
.
Acknowledgments. Financial support by the national agency CNPq is
acknowledged.
.
References
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Press, Princeton and Oxford) 2002 pp.53-67
2\. Hill A. V. 1950 The dimensions of animals and their muscular dynamics Sci.
Progr. 38 209-230
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Lehman S. 2000 How animals move: an integrative view Science 288 100-106
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animals cruise at a Strouhal number tuned for high power efficiency Nature 425
707-710
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universal performance characteristics of motors Proc. Natl. Acad. Sci. USA 99
4161-4166
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size and limb posture in birds and animals J. Zool. Lond. 224 127-147
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and gait to animal size: mice to horses Science 186 1112-1113
13\. Cressey D. 2008 Moving forward together Nature doi:10.1038/news.2008.1268
14\. Ahlborn B. K. and Blake R.W. 2002 Walking and running at resonance
Zoology 105 165–174
15\. Ahlborn B. K., Blake R.W. and Megill W. M. 2006 Frequency tuning in
animal locomotion Zoology 109 43–53
16\. Landau L. D. and Lifshitz E. M. Mechanics (Pergamon Press, 3rd ed.,
Oxford) 1976 section 10
17\. Kokshenev V. B. 2004 Dynamics of human walking at steady speeds Phys.
Rev. Lett. 93 208101–208105
18\. McMahon T. A. Muscles, reflexes, and locomotion (Princeton University
Press, Princeton and New Jersey) 1984
19\. Kokshenev V. B. 2007 New insights into long-bone biomechanics: Are limb
safety factors invariable across mammalian species? J. Biomech. 40 2911-2918
20\. Rubin C. T. and Lanyon L. E. 1984 Dynamic strain similarity in
vertebrates; an alternative to allometric limb bone scaling J. Theor. Biol.
107 321–327
21\. Rome L .C., Funke R. P., Alexander R. McN., Lutz G., Aldridge H., Scott
F. and Freadman M. 1988 Why animals have different muscle fibre types Nature
335 824-829
22\. Whitfield J.2003 One number explains animal flightNature
doi:10.1038/news031013-9
23\. Ellington C. P. 1991 Limitations on animal flight performance J. Exp.
Biol. 160 71-91
24\. Jones J. H. and Lindstedt S. 1993 Limits of maximal performance Annu.
Rev. Physiol. 55 547-569
25\. Kokshenev V. B. 2008 A force-similarity model of the activated muscle is
able to predict primary locomotor functions J. Biomech. 41 912–915
26\. Lindstedt S. L., Reich T. E., Keim P. and LaStayo P. C. 2002 Do muscle
functions as adaptable locomotor springs? J. Exp. Biol. 205 2211-2116
|
arxiv-papers
| 2009-09-07T18:44:52 |
2024-09-04T02:49:05.100402
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Valery B. Kokshenev",
"submitter": "Valery B. Kokshenev",
"url": "https://arxiv.org/abs/0909.1303"
}
|
0909.1331
|
# FLUCTUATIONS OF MULTI-DIMENSIONAL
KINGMAN-LÉVY PROCESSES
Thu Nguyen Department of Mathematics; International University, HCM City;
No.6 Linh Trung ward, Thu Duc District, HCM City; Email: nvthu@hcmiu.edu.vn
(Date: August 10, 2009)
###### Abstract.
In the recent paper [15] we have introduced a method of studying the multi-
dimensional Kingman convolutions and their associated stochastic processes by
embedding them into some multi-dimensional ordinary convolutions which allows
to study multi-dimensional Bessel processes in terms of the cooresponding
Brownian motions. Our further aim in this paper is to introduce k-dimensional
Kingman-Lévy (KL) processes and prove some of their fluctuation properties
which are analoguous to that of k-symmetric Lévy processes. In particular, the
Lévy-Itô decomposition and the series representation of Rosiński type for
k-dimensional KL-processes are obtained.
Keywords and phrases: Cartesian products of Kingman convolutions; Rayleigh
distributions
## 1\. Introduction. notations and prelimilaries
The purpose of this paper is to introduce and study the multivariate KL
processes defined in terms of multicariate Kingman convolutions. To begin with
we review the following information of the Kingman convolutions and their
Cartesian products.
Let $\mathcal{P}:=\mathcal{P}(\mathbb{R}^{+})$ denote the set of all
probability measures (p.m.’s) on the positive half-line $\mathbb{R}^{+}$. Put,
for each continuous bounded function f on $\mathbb{R}^{+}$,
(1)
$\int_{0}^{\infty}f(x)\mu\ast_{1,\delta}\nu(dx)=\frac{\Gamma(s+1)}{\sqrt{\pi}\Gamma(s+\frac{1}{2})}\\\
\int_{0}^{\infty}\int_{0}^{\infty}\int_{-1}^{1}f((x^{2}+2uxy+y^{2})^{1/2})(1-u^{2})^{s-1/2}\mu(dx)\nu(dy)du,$
where $\mu\mbox{ and }\nu\in\mathcal{P}\mbox{ and }\delta=2(s+1)\geq 1$ (cf.
Kingman[7] and Urbanik[19]). The convolution algebra
$(\mathcal{P},\ast_{1,\delta})$ is the most important example of Urbanik
convolution algebras (cf. Urbanik[19]). In language of the Urbanik convolution
algebras, the characteristic measure, say $\sigma_{s}$, of the Kingman
convolution has the Rayleigh density
(2)
$d\sigma_{s}(y)=\frac{2{(s+1)^{s+1}}}{\Gamma(s+1)}y^{2s+1}\exp{(-(s+1)y^{2})}dy$
with the characteristic exponent $\varkappa=2$ and the kernel $\Lambda_{s}$
(3) $\Lambda_{s}(x)=\Gamma(s+1)J_{s}(x)/(1/2x)^{s},$
where $J_{s}(x)$ denotes the Bessel function of the first kind,
(4)
$J_{s}(x):=\Sigma_{k=0}^{\infty}\frac{(-1)^{k}(x/2)^{\nu+2k}}{k!\Gamma(\nu+k+1)}.$
It is known (cf. Kingman [7], Theorem 1), that the kernel $\Lambda_{s}$ itself
is an ordinary characteristic function (ch.f.) of a symmetric p.m., say
$F_{s}$, defined on the interval [-1,1]. Thus, if $\theta_{s}$ denotes a
random variable (r.v.) with distribution $F_{s}$ then for each
$t\in\mathbb{R}^{+}$,
(5) $\Lambda_{s}(t)=E\exp{(it\theta_{s})}=\int_{-1}^{1}\cos{(tx)}dF_{s}(x).$
Suppose that $X$ is a nonnegative r.v. with distribution $\mu\in\mathcal{P}$
and $X$ is independent of $\theta_{s}$. The radial characteristic function
(rad.ch.f.) of $\mu$, denoted by $\hat{\mu}(t),$ is defined by
(6)
$\hat{\mu}(t)=E\exp{(itX\theta_{s})}=\int_{0}^{\infty}\Lambda_{s}(tx)\mu(dx),$
for every $t\in\mathbb{R}^{+}$. The characteristic measure of the Kingman
convolution $\ast_{1,\delta}$, denoted by $\sigma_{s}$, has the Maxwell
density function
(7)
$\frac{d\sigma_{s}(x)}{dx}=\frac{2(s+1)^{s+1}}{\Gamma(s+1)}x^{2s+1}exp\\{-(s+1)x^{2}\\},\quad(0<x<\infty).$
and the rad.ch.f.
(8) $\hat{\sigma}_{s}(t)=exp\\{-t^{2}/4(s+1)\\}.$
Let $\tilde{P}:=\tilde{\mathcal{P}}(\mathbb{R})$ denote the class of symmetric
p.m.’s on $\mathbb{R}.$ Putting, for every $G\in\mathcal{P}$,
$F_{s}(G)=\int_{0}^{\infty}F_{cs}G(dc),$
we get a continuous homeomorphism from the Kingman convolution algebra
$(\mathcal{P},\ast_{1,\delta})$ onto the ordinary convolution algebra
$(\tilde{\mathcal{P}},\ast)$ such that
(9) $\displaystyle F_{s}\\{G_{1}\ast_{1,\delta}G_{2}\\}$ $\displaystyle=$
$\displaystyle(F_{s}G_{1})\ast(F_{s}G_{2})\qquad(G_{1},G_{2}\in\mathcal{P})$
(10) $\displaystyle F_{s}\sigma_{s}$ $\displaystyle=$ $\displaystyle
N(0,2s+1)$
which shows that every Kingman convolution can be embedded into the ordinary
convolution $\ast$.
Denote by $\mathbb{R}^{+k},k=1,2,...$ the k-dimensional nonnegative cone of
$\mathbb{R}^{k}$ and $\mathcal{P}(\mathbb{\mathbb{R}}^{+k})$ the class of all
p.m.’s on $\mathbb{\mathbb{R}}^{+k}$ equipped with the weak convergence. In
the sequel, we will denote the multidimensional vectors and random vectors
(r.vec.’s) and their distributions by bold face letters.
For each point z of any set $A$ let $\delta_{z}$ denote the Dirac measure (the
unit mass) at the point z. In particular, if
$\mathbf{x}=(x_{1},x_{2},\cdots,x_{k})\in\mathbb{R}^{k+}$, then
(11)
$\delta_{\mathbf{x}}=\delta_{x_{1}}\times\delta_{x_{2}}\times\ldots\times\delta_{x_{k}},\quad(k\;times),$
where the sign $"\times"$ denotes the Cartesian product of measures. We put,
for $\mathbf{x}=(x_{1},\cdots,x_{k})\mbox{ and
}\mathbf{y}=(y_{1},y_{2},\cdots,y_{k})\in\mathbb{R}^{+k},$
(12)
$\delta_{\mathbf{x}}\bigcirc_{s,k}\delta_{\mathbf{y}}=\\{\delta_{x_{1}}\circ_{s}\delta_{y_{1}}\\}\times\\{\delta_{x_{2}}\circ_{s}\delta_{y_{2}}\\}\times\cdots\
\times\\{\delta_{x_{k}}\circ_{s}\delta_{y_{k}}\\},\quad(k\;times),$
here and somewhere below for the sake of simplicity we denote the Kingman
convolution operation $\ast_{1,\delta},\delta=2(s+1)\geq 1$ simply by
$\circ_{s},s\geq\frac{!}{2}.$ Since convex combinations of p.m.’s of the form
(11) are dense in $\mathcal{P}(\mathbb{R}^{+k})$ the relation (12) can be
extended to arbitrary p.m.’s $\mathbf{G}_{1}\mbox{ and
}\mathbf{G}_{2}\in\mathcal{P}(\mathbb{R}^{+k})$. Namely, we put
(13)
$\mathbf{G}_{1}\bigcirc_{s,k}\mathbf{G}_{2}=\iint\limits_{\mathbb{R}^{+k}}\delta_{\mathbf{x}}\bigcirc_{s,k}\delta_{\mathbf{y}}{\mathbf{G}}_{1}(d\mathbf{x}){\mathbf{G}}_{2}(d\mathbf{y})$
which means that for each continuous bounded function $\phi$ defined on
$\mathbb{R}^{+k}$
(14)
$\int\limits_{\mathbb{R}^{+k}}\phi({\mathbf{z}}){\mathbf{G}}_{1}\bigcirc_{s,k}{\mathbf{G}}_{2}(d{\mathbf{z}})=\iint\limits_{\mathbb{R}^{+k}}\big{\\{}\int\limits_{\mathbb{R}^{+k}}\phi({\mathbf{z}})\delta_{{\mathbf{x}}}\bigcirc_{s,k}\delta_{{\mathbf{y}}}(d{\mathbf{z}})\big{\\}}{\mathbf{G}}_{1}(d{\mathbf{x}}){\mathbf{G}}_{2}(d{\mathbf{y}}).$
In the sequel, the binary operation $\bigcirc_{s,k}$ will be called the
k-times Cartesian product of Kingman convolutions and the pair
$(\mathcal{P}(\mathbb{R}^{+k}),\bigcirc_{s,k})$ will be called the
k-dimensional Kingman convolution algebra. It is easy to show that the binary
operation $\bigcirc_{s,k}$ is continuous in the weak topology which together
with (1) and (13) implies the following theorem.
###### Theorem 1.
The pair $(\mathcal{P}{(\mathbb{R}^{+k})},\bigcirc_{s,k})$ is a commutative
topological semigroup with $\delta_{\mathbf{0}}$ as the unit element.
Moreover, the operation $\bigcirc_{s,k}$ is distributive w.r.t. convex
combinations of p.m.’s in $\mathcal{P}(\mathbb{R}^{+k})$.
For every ${\mathbf{G}}\in\mathcal{P}(\mathbb{R}^{+k})$ the k-dimensional
rad.ch.f. $\hat{{\mathbf{G}}}({\mathbf{t}}),{\mathbf{t}}=(t_{1},t_{2},\cdots
t_{k})\in\mathbb{R}^{+k},$ is defined by
(15)
$\hat{\mathbf{G}}(\mathbf{t})=\int\limits_{\mathbb{R}^{+k}}\prod_{j=1}^{k}\Lambda_{s}(t_{j}x_{j}){\mathbf{G}}(\mathbf{dx}),$
where $\mathbf{x}=(x_{1},x_{2},\cdots x_{k})\in\mathbb{R}^{+k}.$ Let
$\mathbf{\Theta_{s}}=\\{\theta_{s,1},\theta_{s,2},\cdots,\theta_{s,k}\\}$,
where $\theta_{s,j}$ are independent r.v.’s with the same distribution
$F_{s}$. Next, assume that ${\mathbf{X}}=\\{X_{1},X_{2},...,X_{k}\\}$ is a
k-dimensional r.vec. with distribution $\mathbf{G}$ and $\mathbf{X}$ is
independent of $\mathbf{\Theta}_{s}$. We put
(16)
$[{\mathbf{\Theta}}_{s},{\mathbf{X}}]=\\{{\theta_{s,1}X_{1},\theta_{s,2}X_{2},...,\theta_{s,k}X_{k}}\\}.$
Then, the following formula is equivalent to (15) (cf. [14])
(17)
$\widehat{\mathbf{G}}({\mathbf{t}})=Ee^{i<{\mathbf{t}},[{\mathbf{\Theta}_{s},\mathbf{X}}]>},\qquad({\mathbf{t}}\in\mathbb{R}^{+k}).$
The Reader is referred to Corollary 2.1, Theorems 2.3 & 2.4 [14] for the
principal properties of the above rad.ch.f. Given $s\geq-1/2$ define a map
$F_{s,k}:\mathcal{P}(\mathbb{R}^{+k})\rightarrow\mathcal{P}(\mathbb{R}^{k})$
by
(18)
$F_{s,k}({\mathbf{G}})=\int\limits_{\mathbb{R}^{+k}}(T_{c_{1}}F_{s})\times(T_{c_{2}}F_{s})\times\ldots\times(T_{c_{k}}F_{s}){\mathbf{G}}(d{\mathbf{c}}),$
here and in the sequel, for a distribution $\mathbf{G}$ of a r.vec.
$\mathbf{X}$ and a real number c we denote by $T_{c}{\mathbf{G}}$ the
distribution of $c{\mathbf{X}}$. Let us denote by
$\tilde{\mathcal{P}}_{s,k}(\mathbb{R}^{+k})$ the sub-class of
$\mathcal{P}(\mathbb{R}^{k})$ consisted of all p.m.’s defined by the right-
hand side of (18). By virtue of (15)-(18) one can prove the following theorem.
###### Theorem 2.
The set $\tilde{\mathcal{P}}_{s,k}(\mathbb{R}^{+k})$ is closed w.r.t. the weak
convergence and the ordinary convolution $\big{.}\ast$ and the following
equation holds
(19)
$\hat{\mathbf{G}}({\mathbf{t}})=\mathcal{F}(F_{s,k}({\mathbf{G}}))({\mathbf{t}})\qquad({\mathbf{t}}\in{\mathbb{R}^{+k}})$
where $\mathcal{F}({\mathbf{K}})$ denotes the ordinary characteristic function
(Fourier transform) of a p.m. ${\mathbf{K}}$. Therefore, for any
${\mathbf{G}}_{1}\mbox{ and }{\mathbf{G}}_{2}\in\mathbb{R}^{+k}$
(20) $F_{s,k}({\mathbf{G}}_{1})\big{.}\ast
F_{s,k}({\mathbf{G}}_{2})=F_{s,k}({\mathbf{G}}_{1}\bigcirc_{s,k}{\mathbf{G}}_{2})$
and the map $F_{s,k}$ commutes with convex combinations of distributions and
scale changes $T_{c},c>0$. Moreover,
(21) $F_{s,k}({\Sigma_{s,k}})=N({\mathbf{0}},2(s+1){\mathbf{I}})$
where $\Sigma_{s,k}$ denotes the k-dimensional Rayleigh distribution and
$N({\mathbf{0}},2(s+1){\mathbf{I}})$ is the symmetric normal distribution on
$\mathbb{R}^{k}\mbox{ with variance operator
}{\mathbf{R}}=2(s+1){\mathbf{I}},{\mathbf{I}}$ being the identity operator.
Consequently, Every Kingman convolution algebra
$\big{(}\mathcal{P}(\mathbb{R}^{+k}),\bigcirc_{s,k}\big{)}$ is embedded in the
ordinary convolution algebra
$\big{(}\mathcal{P}_{s,k}(\mathbb{R}^{+k}),\big{.}\star\big{)}$ and the map
$F_{s,k}$ stands for a homeomorphism.
Let us denote by
$\mathcal{E}=\\{{\mathbf{e}}=(e_{1},e_{2},\ldots,e_{k}),e_{j}=\pm,j=1,2,\ldots,k\\}$.
It is convenient to regard the elements of $\mathcal{E}$ as sign vectors.
Denote
$\mathbb{R}^{+k}_{\mathbf{e}}=\\{[{\mathbf{e}},{\mathbf{x}}]:{\mathbf{x}}\in\mathbb{R}^{+k}\\},\mbox{
where
}[{\mathbf{e}},{\mathbf{x}}]:=(e_{1}x_{1},e_{2}x_{2},\ldots,e_{k}x_{k}).$ Then
the family $\\{\mathbb{R}^{+k}_{\mathbf{e}},{\mathbf{e}}\in\mathcal{E}\\}$ is
a partition of the space $\mathbb{R}^{k}.$ If $\mathbf{X}$ is a k-dimensional
r.vec. with distribution $\mathbf{G},$ the k-symmetrization of $\mathbf{G}$,
denoted by $\tilde{\mathbf{G}},$ is defined by
(22)
$\tilde{\mathbf{G}}=\frac{1}{2^{k}}\sum_{\mathbf{e}\in\mathcal{E}}S_{\mathbf{e}}{\mathbf{G}},$
where the operator $S_{\mathbf{e}}$ is defined by
(23)
$S_{\mathbf{e}}({\mathbf{x}})=[{\mathbf{e}},{\mathbf{x}}]\qquad{\mathbf{x}}\in{\mathbb{R}^{k}}$
and the symbol $S_{\mathbf{e}}\tilde{\mathbf{G}}$ denotes the image of
$\mathbf{G}$ under $S_{\mathbf{e}}$.
###### Definition 1.
We say that a distribution $\mathbf{G}\in\mathcal{P}(\mathbb{R}^{k})$ is
k-symmetric, if the equation $\mathbf{G}=\tilde{\mathbf{G}}$ holds.
###### Definition 2.
A p.m. ${\mathbf{F}}\in\mathcal{P}(\mathbb{R}^{+k})$ is called
$\bigcirc_{s,k}-$infinitely divisible ($\bigcirc_{s,k}-$ID), if for every m=1,
2, …there exists $\mathbf{F}_{m}\in\mathbf{P}(\mathbb{R}^{+k})$ such that
(24)
${\mathbf{F}}={\mathbf{F}}_{m}\bigcirc_{s,k}{\mathbf{F}}_{m}\bigcirc_{s,k}\ldots\bigcirc_{s,k}{\mathbf{F}}_{m}\quad(m\;times).$
###### Definition 3.
$\mathbf{F}$ is called stable, if for any positive numbers a and b there
exists a positive number c such that
(25)
$T_{a}{\mathbf{F}}\;{\bigcirc_{s,k}}\;T_{b}{\mathbf{F}}=T_{c}{\mathbf{F}}$
By virtue of Theorem 2 it follows that the following theorem holds.
###### Theorem 3.
A p.m. $\mathbf{G}\mbox{ is }\bigcirc_{s,k}-ID$, resp. stable if and only if
$H_{s,k}({\mathbf{G}})$ is ID, resp. stable, in the usual sense.
The following theorem gives a representation of rad.ch.f.’s of
$\bigcirc_{s,k}-$ID distributions. The proof is a verbatim reprint of that for
([14], Theorem 2.6).
###### Theorem 4.
A p.m. $\mu\in ID(\bigcirc_{s,k})$ if and only if there exist a
$\sigma$-finite measure M (a Lévy’s measure) on $\mathbb{R}^{+k}$ with the
property that $M({\mathbf{0}})=0,{\mathbf{M}}$ is finite outside every
neighborhood of ${\mathbf{0}}$ and
(26)
$\int_{\mathbb{R}^{+k}}\frac{\|{\mathbf{x}}\|^{2}}{1+\|{\mathbf{x}}\|^{2}}{\mathbf{M}}(d{\mathbf{x}})<\infty$
and for each ${\mathbf{t}}=(t_{1},...,t_{k})\in\mathbb{R}^{+k}$
(27)
$-\log{\hat{\mu}({\mathbf{t}})}=\int_{\mathbb{R}^{+k}}\\{1-\prod_{j=1}^{k}\Lambda_{s}(t_{j}x_{j})\\}\frac{1+\|{\mathbf{x}}\|^{2}}{\|{\mathbf{x}}\|^{2}}M({\mathbf{dx}}),$
where, at the origin $\mathbf{0}$, the integrand on the right-hand side of
(27) is assumed to be
(28) $lim_{\|\mathbf{x}\|\rightarrow
0}\\{1-\prod_{j=1}^{k}\Lambda_{s}(t_{j}x_{j})\\}\frac{1+\|\mathbf{x}\|^{2}}{\|\mathbf{x}\|^{2}}=\Sigma_{j=1}^{k}\lambda^{2}_{j}t_{j}^{2}$
for nonnegative $\lambda_{j},j=1,2,...,k.$ In particular, if $M=0,\mbox{ then
}\mu$ becomes a Rayleighian distribution with the rad.ch.f (see definition 4)
(29)
$-\log{\hat{\mu}({\mathbf{t}})}=\frac{1}{2}\sum_{j=1}^{k}\lambda^{2}_{j}t_{j}^{2},\quad{\mathbf{t}}\in\mathbb{R}^{+k},$
for some nonnegative $\lambda_{j},j=1,...,k.$ Moreover, the representation
(27) is unique.
An immediate consequence of the above theorem is the following:
###### Corollary 1.
Each distribution $\mu\in ID(\bigcirc_{s,k})$ is uniquely determined by the
pair $[\mathbf{M},\boldsymbol{\lambda}]$, where $\mathbf{M}$ is a Levy’s
measure on $\mathbb{R}^{+k}$ such that $\mathbf{M}(\mathbf{0})=0,$
$\mathbf{M}$ is finite outsite every neighbourhood of $\mathbf{0}$ and the
condition (26) is satisfied and
$\boldsymbol{\lambda}:=\\{\lambda_{1},\lambda_{2},\cdots\lambda_{k}\\}\in\mathbb{R}^{+k}$
is a vector of nonnegative numbers appearing in (29). Consequently, one can
write $\mu\equiv[\mathbf{M},\boldsymbol{\lambda}].$
In particular, if $\mathbf{M}$ is zero measure then
$\mu=[\boldsymbol{\lambda}]$ becomes a Rayleighian p.m. on $\mathbb{R}^{+k}$
as defined as follows:
###### Definition 4.
A k-dimensional distribution, say $\boldsymbol{\mathbf{\Sigma}}_{s,k}$, is
called a Rayleigh distribution, if
(30)
$\boldsymbol{\mathbf{\Sigma}}_{s,k}=\sigma_{s}\times\sigma_{s}\times\cdots\times\sigma_{s}\quad(k\;times).$
Further, a distribution ${\mathbf{F}}\in\mathcal{P}(\mathbb{R}^{+k})$ is
called a Rayleighian distribution if there exist nonnegative numbers
$\lambda_{r},r=1,2\cdots k$ such that
(31)
${\mathbf{F}}=\\{T_{\lambda_{1}}\sigma_{s}\\}\times\\{T_{\lambda_{2}}\sigma_{s}\\}\times\ldots\times\\{T_{\lambda_{k}}\sigma_{s}\\}.$
It is evident that every Rayleigh distribution is a Rayleighian distribution.
Moreover, every Rayleighian distribution is $\bigcirc_{s,k}-$ID. By virtue of
(7 ) and (30) it follows that the k-dimensional Rayleigh density is given by
(32)
$g({\mathbf{x}})=\Pi_{j=1}^{k}\frac{2^{k}(s+1)^{k(s+1)}}{\Gamma^{k}(s+1)}x_{j}^{2s+1}exp\\{-(s+1)||{\mathbf{x}}||^{2}\\},$
where ${\mathbf{x}}=(x_{1},x_{2},\ldots,x_{k})\in\mathbb{R}^{+k}$ and the
corresponding rad.ch.f. is given by
(33)
$\hat{\Sigma}_{s,k}({\mathbf{t}})=Exp(-|{\mathbf{t}}|^{2}/4(s+1)),\quad{\mathbf{t}}\in\mathbb{R}^{+k}.$
Finally, the rad.ch.f. of a Rayleighian distribution $\mathbf{F}\mbox{ on
}\mathbb{R}^{+k}$ is given by
(34)
$\hat{\mathbf{F}}({\mathbf{t}})=Exp(-\frac{1}{2}\sum_{j=1}^{k}\lambda_{j}^{2}t_{j}^{2})$
where $\lambda_{j},j=1,2,\ldots,k$ are some nonnegative numbers.
## 2\. Multivariate Bessel processes
## 3\. Multivrariate Kingman-Lévy processes and their Lévy-Itô decomposition
Suppose that $\mu_{t},t\geq 0$ is continuous semigroup in
$ID(\bigcirc_{s,k})$, that is for any $t,s\geq 0$
(35) $\mu_{t}\bigcirc_{s,k}\mu_{s}=\mu_{t+s}$
and $\\{\mu_{t}\\}$ is continuous at 0 i.e.
$lim_{t\rightarrow 0}\mu_{t}=\delta_{\mbox{0}}.$
By virtue of Theorem 2 it follows that $\\{\mathcal{F}_{s,k}(\mu_{t})\\}$ is
an ordinary continuous convolution semigroup on $\mathbb{R}^{k}.$ Putting, for
each $\mathbf{x}\in\mathbb{R}^{k+}$ and for every Borel subset
$\mathcal{E}\mbox{ of }\mathbb{R}^{k+},$
(36)
$\mathbf{P}(t,\mathcal{E},\mathbf{x})=\mu_{t}\bigcirc_{s,k}\delta_{\mathbf{x}}(\mathcal{E})$
and using the rad.ch.f. it follows that the family
$\\{\mathbf{P}(t,\mathcal{E},\mathbf{x}),t\geq 0\\}$ satisfies the Chapman-
Kolmogorov equation and, consequently, the formula (36) defines transition
probabilities of a $\mathbb{R}^{k+}-$valued homogeneous strong Markov Feller
process $\\{\mathbf{X}^{\mathbf{x}}_{t},t\geq 0\\}$, say, such that it is
stochastically continuous and has a cadlag version (compare [11], Theorem
2.6).
###### Definition 5.
A $\mathbb{R}^{k+}$-valued stochastic process $\\{\mathbf{X}_{t},t\geq 0\\}$
is called a Kingman-Lévy process, if $\mathbf{X}_{t}=$
(i) $\mathbf{X}_{0}=\mathbf{0}\qquad(P.1);$
(ii) There exists a $\mathbb{R}^{k+}-$valued homogeneous strong Markov Feller
process having a cadlag version $\\{\mathbf{X}^{\mathbf{x}}_{t},t\geq 0\\}$
with transition probabilities defined by (36) and
$\mathbf{X}_{t}=\mathbf{X}^{\mathbf{0}}_{t},t\geq 0;$
## 4\. Fluctuations of Multidimensional Bessel Processes
###### Definition 6.
Let $(W_{t},t\geq 0)$ be a d-dimensional Brownian motion (d=1, 2, …). The
Euclidean norm of $(W_{t})$, denoted by $B_{t},t\geq 0$ is called a Bessel
process.
It has been proved that Bessel processes inherit the well-known
characteristics of Brownian motions: They are independent stationary
”increments” processes with continuous sample paths. The term ’increment’ is
defined as follows:
###### Definition 7.
For any $s>u$ the random variable $|W_{s}-W_{u}|$ is called an increments of
the Bessel process.
The following theorem gives a Lévy-Khinczyn representation of the Bessel
process in the sense of the Kingman convolution.
###### Theorem 5.
The radial characteristic function $\phi(x)$ of the Bessel process $(B_{t})$
is of the form
(37) $\phi(x)=exp\\{-\frac{tx^{2}}{4(s+1)}\\}\qquad x,t\geq 0$
where d=2(s+1).
Since for any $s>u$ the ’increment’ of the Bessel process $(B_{t})$ is
infinitely divisible in the ordinary convolution $\ast$ we have the following
representation of the Fourier transform of $B_{s-u}.$
(38) $\mathcal{F}_{B_{s-u}}(x)=exp(-(s-u)\psi(x))$
where $\psi(x)$ is a symmetric characteristic exponent
(39) $\psi(x)=\frac{1}{2}\sigma^{2}+\int_{0}^{\infty}(1-cos\,xv)\Pi(dv)$
where the measure $\Pi$ satisfies the condition
begin equation
(40) $\int_{0}^{\infty}(min(1,x^{2})\Pi(dx)<\infty.$
which implies the following Lévy-Itô decomposition.
###### Theorem 6.
(Lévy-Itô decomposition) There exists a Brownian motion $X^{(1)}_{t}$ and a
compound Poison process $X^{(2)}_{t}$ independent of $X^{(1)}_{t}$ such that
(41) $B_{t}=||W_{t}||\overset{d}{=}X^{(1)}_{t}+X^{(2)}_{t}\qquad(t\geq 0).$
Before stating the Wienner-Hopf factorization (WHf) theorem for Bessel
processes we introduce some concepts and notations. The importance of WHf is
that it gives us information of the ascending and descending ladder processes.
We begin by recalling that for $\alpha,\beta\geq 0$ the Laplace exponents
$\kappa(\alpha,\beta)\mbox{ and }\hat{\kappa}(\alpha,\beta)$ of the ascending
ladder process $(\hat{L}^{-1},\hat{H})$ and the descending ladder process
$(\hat{L}^{-1},\hat{H}).$ Further, we define
$\overset{-}{G}_{t}=sup\\{s<t:\overset{-}{X}_{s}=X_{s}\\}\mbox{ and
}\underset{-}{G_{t}}=sup\\{s<t:\underset{-}{X_{t}}=X_{s}.$
###### Theorem 7.
(Wienner-Hopf Factorization) Let $(B_{t},t\geq 0)$ be a Bessel process. Denote
by ${\mathbf{e}}_{p}$ an independent and exponentially distributed random
variable.
The pairs
$(\overset{-}{G}_{{\mathbf{e}}_{p}},\overset{-}{X}_{{\mathbf{e}}_{p}})\mbox{
and
}({\mathbf{e}}_{p}-\overset{-}{G}_{{\mathbf{e}}_{p}},\overset{-}{X}_{{\mathbf{e}}_{p}}-X_{{\mathbf{e}}_{p}})$
are independent and infinitely divisible, yielding the factorization
(42)
$\frac{p}{p-i\nu+\psi(\theta)}=\Psi^{+}(\nu,\theta).\Psi^{-}(\nu,\theta)\qquad\nu,\theta\in\mathbb{R},$
$\psi^{+},\psi^{-}$ being Fourier transforms and called the Wienner-Hopf
factors.
## 5\. Levy-Ito decomposition of Kingman-Levy processes
## References
* [1] Bingham, N.H., Random walks on spheres, Z. Wahrscheinlichkeitstheorie Verw. Geb., 22, (1973), 169-172.
* [2] Bingham, N.H., On a Theorem of Klosowska about generalized convolutions, Colloquium Math., 28 No. 1, (1984), 117-125.
* [3] Cox, J.C., Ingersoll, J.E.Jr., and Ross, S.A., A theory of the term structure of interest rates. Econometrica, 53(2), (1985).
* [4] Feller, W., An Introduction to probability Theory and Its Applications, John Wiley & Sons Inc., Vol.II, 2nd Ed., (1971).
* [5] Ito, K., Mckean H.P., Jr., Diffusion processes and their sample paths, Berlin-Heidelberg-New York. Springer (1996).
* [6] Kalenberg O., Random measures, 3rd ed. New York: Academic Press, (1983).
* [7] Kingman, J.F.C., Random walks with spherical symmetry, Acta Math., 109, (1963), 11-53.
* [8] Kyprianou, Andreas E., Introductory lectures on fluctuations of Lévy processes with applications,
* [9] Levitan B.M., Generalized translation operators and some of their applications, Israel program for Scientific Translations, Jerusalem, (1962).
* [10] Linnik Ju. V., Ostrovskii, I. V., Decomposition of random variables and vectors, Translation of Mathematical Monographs, vol. 48, American Mathematical Society, Providence R. L, 1977, ix+380 pp.,$38.80. (Translated from the Russian, 1972, by Israel Program for Scientific Translations).
* [11] Nguyen V.T, Generalized independent increments processes, Nagoya Math. J.133, (1994), 155-175.
* [12] Nguyen V.T., Generalized translation operators and Markov processes, Demonstratio Mathematica, 34 No 2, ,295-304.
* [13] Nguyen T.V., OGAWA S., Yamazato M. A convolution Approach to Mutivariate Bessel Processes, Proceedings of the 6th Ritsumeikan International Symposium on ”Stochastic Processes and Applications to Mathematical Finance”, edt. J. Akahori, S. OGAWA and S. Watanabe, World Scientific, (2006) 233-244.
* [14] Nguyen V. T., A Kingman convolution approach to Bessel processes, Probab. Math. Stat, Probab. Math. Stat. 29, fasc. 1(2009) 119-134.
* [15] Nguyen V. T., An analogue of the Cramér-Lévy theorem for multi-dimensional Rayleigh distributions, arxiv.org/abs/0907.5035.
* [16] Revuz, D. and Yor, M., Continuous martingals and Brownian motion. Springer-verlag Berlin Heidelberg, (1991).
* [17] Sato K, Lévy processes and infinitely divisible distributions, Cambridge University of Press, (1999).
* [18] Shiga T., Watanabe S., Bessel diffusions as a one-parameter family of diffusion processes, Z. Warscheinlichkeitstheorie Verw. geb. 27,(1973), 34-46.
* [19] Urbanik K., Generalized convolutions, Studia math., 23 (1964), 217-245.
* [20] Urbanik K., Cramér property of generalized convolutions,Bull. Polish Acad. Sci. Math.37 No 16 (1989), 213-218.
* [21] Vólkovich, V. E., On symmetric stochastic convolutions, J. Theor. Prob. 5, No. 3(1992), 417-430.
|
arxiv-papers
| 2009-09-07T20:13:00 |
2024-09-04T02:49:05.105477
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Thu Nguyen",
"submitter": "Thu Nguyen",
"url": "https://arxiv.org/abs/0909.1331"
}
|
0909.1378
|
Anomalous Quantum Diffusion in Order-Disorder Separated Double Quantum Ring
Santanu K. Maiti†,‡,∗
†Theoretical Condensed Matter Physics Division, Saha Institute of Nuclear
Physics,
1/AF, Bidhannagar, Kolkata-700 064, India
‡Department of Physics, Narasinha Dutt College, 129, Belilious Road,
Howrah-711 101, India
Abstract
A novel feature for control of carrier mobility is explored in an order-
disorder separated double quantum ring, where the two rings thread different
magnetic fluxes. Here we use simple tight-binding formulation to describe the
system. In our model, the two rings are connected through a single bond and
one of the rings is subjected to impurity, keeping the other ring as impurity
free. In the strong impurity regime, the electron diffusion length increases
with the increase of the impurity strength, while it decreases in the weak
impurity regime. This phenomenon is completely opposite to that of a
conventional disordered double quantum ring, where the electron diffusion
length always decreases with the increase of the disorder strength.
PACS No.: 73.23.Ra; 73.23.-b; 73.63.-b
Keywords: Persistent current; Drude weight; Double quantum ring; Impurity.
∗Corresponding Author: Santanu K. Maiti
Electronic mail: santanu.maiti@saha.ac.in
## 1 Introduction
Over the last few decades, the physics at sub-micron length scale provides
enormous evaluation both in terms of our understanding of basic physics as
well as in terms of the development of revolutionary technologies. In this
length scale, the so-called mesoscopic or nanoscopic regime, several
characteristic quantum length scales for the electrons such as system size and
phase coherence length or elastic mean free path and phase coherence length
are comparable. Due to the dominance of the quantum effects in the
mesoscopic/nanoscopic regime, intense research in this field has revolved its
richness. The most significant issue is probably the persistent currents in
small normal metal rings. In thermodynamic equilibrium, a small metallic ring
threaded by magnetic flux $\phi$ supports a current that does not decay
dissipatively even at non-zero temperature. It is the well-known phenomenon of
persistent current in mesoscopic normal metal rings which is a purely quantum
mechanical effect and gives an obvious demonstration of the Aharonov-Bohm
effect.1 The possibility of persistent current was predicted in the very early
days of quantum mechanics by Hund,2 but their experimental evidences came much
later only after realization of the mesoscopic systems. In $1983$, Büttiker et
al.3 predicted theoretically that persistent current can exist in mesoscopic
normal metal rings threaded by a magnetic flux $\phi$, even in the presence of
impurity. In a pioneering experiment, Levy et al.4 first gave the experimental
evidence of persistent current in the mesoscopic normal metal ring, and later,
the existence of the persistent current was further confirmed by several
experiments.5-8 Though the phenomenon of persistent current has been addressed
quite extensively over the last twenty years both theoretically9-27 as well as
experimentally,4-8 but yet we cannot resolve the controversy between the
theory and experiment. The main controversies come in the determinations of
(a) the current amplitude, (b) flux-quantum periodicities, (c) low-field
magnetic susceptibilities, etc. In recent works,24-26 we have pointed out that
the higher order hopping integrals, in addition to the nearest-neighbor
hopping integral, have a significant role to enhance the current amplitude
(even an order of magnitude). In other recent work,27 we have focused that the
low-field magnetic susceptibility can be predicted exactly only for the one-
channel systems with fixed number of electrons, while for all other cases it
becomes random. To grasp the experimental behavior of the persistent current,
one has to focus attention on the interplay of quantum phase coherence,
disorder and electron-electron correlation and this is a highly complex
problem.
Using the advanced molecular beam epitaxial growth technique, one can easily
fabricate a quantum system where the impurities are located only in some
particular region of the system, keeping the other region free from any
impurity. This is completely opposite from a conventional disordered system,
where the disorders are given uniformly throughout the system. Traditional
wisdom is that, the larger the disorder stronger the localization.28 However,
some recent experimental studies29-31 as well as theoretical
investigations32-35 on these special class of systems where the disorders are
not distributed uniformly, have yielded completely different behavior which
predicts that the electron diffusion length decreases in the weak disorder
regime, while it increases in the strong disorder regime. Motivated with these
results, in this article, we focus our attention in an order-disorder
separated double quantum ring system. To reveal the variation of the electron
diffusion length in such a particular system, here we study the behavior of
persistent current and Drude weight and our results may illuminate some of the
unusual experimental results for such diverse transport property. The
parameter Drude weight $D$ characterizes the conducting nature of the system
as originally introduced by Kohn.36 In our present model, two mesoscopic
rings, threaded by different magnetic fluxes, are connected by a single bond
and impurities are given in any one of these two rings, while the other ring
becomes impurity free. For this order-disorder separated double quantum ring,
we observe an anomalous behavior of electron mobility in which the electron
diffusion length increases with the increase of the impurity strength in the
strong impurity regime, while the diffusion length decreases in the weak
impurity regime. This phenomenon is completely opposite to that of a
conventional disordered double quantum ring, in which the electron diffusion
length always decreases with the increase of the disorder strength.
In what follows, we describe the model and the method in Section $2$. Section
$3$ contains the significant results and the discussion, and finally, we
summarize our results in Section $4$.
## 2 The model and the method
The schematic representation of a double quantum ring is shown in Fig. 1 where
the two rings, threaded by different magnetic fluxes, are connected by a
single bond. In the non-interacting picture, the system is usually modeled by
a single-band tight-binding Hamiltonian,
$\displaystyle H$ $\displaystyle=$
$\displaystyle\sum_{i}\epsilon_{i}^{I}c_{i}^{\dagger}c_{i}+v_{I}\sum_{<ij>}\left[e^{i\theta_{I}}c_{i}^{\dagger}c_{j}+e^{-i\theta_{I}}c_{j}^{\dagger}c_{i}\right]$
(1) $\displaystyle+$
$\displaystyle\sum_{k}\epsilon_{k}^{II}c_{k}^{\dagger}c_{k}+v_{II}\sum_{<kl>}\left[e^{i\theta_{II}}c_{k}^{\dagger}c_{l}+e^{-i\theta_{II}}c_{l}^{\dagger}c_{k}\right]$
$\displaystyle+$ $\displaystyle
v_{\alpha\beta}\left[c_{\alpha}^{\dagger}c_{\beta}+c_{\beta}^{\dagger}c_{\alpha}\right]$
Here $\epsilon_{i}^{I}$’s ($\epsilon_{i}^{II}$’s) are the site energies in the
ring $I$ (ring $II$), $c_{i}^{\dagger}$ ($c_{k}^{\dagger}$) is the creation
operator of an electron at site $i$ ($k$) of the ring $I$ (ring $II$) and
$c_{i}$ ($c_{k}$) is the annihilation operator of an electron at site $i$
($k$)
Figure 1: Schematic view of a double quantum ring in which the two rings
thread magnetic fluxes $\phi_{I}$ and $\phi_{II}$ respectively. These two
rings are connected through the lattice sites $\alpha$ and $\beta$. The filled
circles correspond to the position of the atomic sites (for color
illustration, see the web version).
of the ring $I$ (ring $II$), $v_{I}$ ($v_{II}$) is the hopping strength
between nearest-neighbor sites in the ring $I$ (ring $II$), and
$v_{\alpha\beta}$ gives the hopping strength between these two rings. In this
expression, $\theta_{I}=2\pi\phi_{I}/N_{I}$ and
$\theta_{II}=2\pi\phi_{II}/N_{II}$ are the phase factors due to the fluxes
$\phi_{I}$ and $\phi_{II}$ (measured in units of $\phi_{0}=ch/e$, the
elementary flux quantum), respectively where $N_{I}$ and $N_{II}$ correspond
to the total number of atomic sites in the ring $I$ and ring $II$,
respectively. In order to introduce the impurities in the system, we choose
the site energies ($\epsilon_{i}$’s, omitting the ring index in the
superscript) from the relation: $\epsilon_{i}=W\cos(i\lambda\pi)$, where $W$
is the strength of the disorder and $\lambda$ is an irrational number, and as
a typical example we take it as the golden mean $\left(1+\sqrt{5}\right)/2$.
Setting $\lambda=0$, we get back the pure system with identical site potential
$W$. The idea of considering such an incommensurate potential is that, for
such a correlated disorder we do not require any configuration averaging and
therefore the numerical calculations can be done in the low cost of time. Now
to achieve the order-disorder separated double quantum ring, we introduce the
correlated disorder in any one of the rings, keeping the other one as impurity
free.
At absolute zero temperature, the persistent currents in the two rings can be
calculated from the expressions,
$\displaystyle
I(\phi_{I})=-\frac{\partial{E(\phi_{I},\phi_{II})}}{\partial{\phi_{I}}}$ (2)
$\displaystyle
I(\phi_{II})=-\frac{\partial{E(\phi_{I},\phi_{II})}}{\partial{\phi_{II}}}$ (3)
where, $I(\phi_{I})$ and $I(\phi_{II})$ correspond to the currents in the ring
$I$ and ring $II$, respectively and $E(\phi_{I},\phi_{II})$ represents the
ground state energy of the complete system. We evaluate this energy exactly to
understand unambiguously the anomalous behavior of persistent current, and
this is achieved by exact diagonalization of the tight-binding Hamiltonian Eq.
(1).
Now the response of the double quantum ring system to a uniform time-dependent
electric field can be determined in terms of the Drude weight $D$,37-38 a
closely related parameter that characterizes the conducting nature of the
system as originally noted by Kohn.36 The Drude weights for the two rings can
be calculated through the relations,39
$\displaystyle
D_{I}=\left.\frac{N_{I}}{4\pi^{2}}\left(\frac{\partial{{}^{2}E(\phi_{I},\phi_{II})}}{\partial{\phi_{I}}^{2}}\right)\right|_{\phi_{I}\rightarrow
0,\phi_{II}\rightarrow 0}$ (4) $\displaystyle
D_{II}=\left.\frac{N_{II}}{4\pi^{2}}\left(\frac{\partial{{}^{2}E(\phi_{I},\phi_{II})}}{\partial{\phi_{II}}^{2}}\right)\right|_{\phi_{I}\rightarrow
0,\phi_{II}\rightarrow 0}$ (5)
where $D_{I}$ and $D_{II}$ represent the Drude weights for the ring $I$ and
ring $II$ respectively.
Our main aim in this article is the determination of the conducting properties
of an order-disorder separated double quantum ring, which can be computed
through the parameters $D_{I}$ and $D_{II}$. From these parameters we can
clearly describe the mobility of the charge carriers in the system and
accordingly, the variation of the electron diffusion length might be expected.
## 3 Results and discussion
In the order-disorder separated double quantum ring, we introduce the
correlated disorder in ring $I$ (for the sake of simplicity), keeping the ring
$II$ as impurity free. Throughout the numerical computations, we take the
values of the different parameters as: $v=-1$, $v_{\alpha\beta}=-1$ and for
the sake of simplicity, we use the units where $c=1$, $e=1$ and $h=1$. During
these calculations, we fix the chemical potential ($\mu$) for all the systems
to a constant value $0$. The main focus of this article is to describe how the
disordered states affect the ordered states in the order-disorder separated
system. Since for such a system the impurities are introduced in the ring $I$,
we evaluate the conducting properties of the double quantum ring by measuring
the Drude weight $D_{II}$. This actually provides the response of the ordered
states in presence of the disordered states.
Figure 2: Drude weight ($D_{II}$) of the ring $II$ as a function of the
disorder strength ($W$) for the systems with $N_{I}=30$, $N_{II}=30$ and the
fixed chemical potential $\mu=0$. The red and the blue lines correspond to the
order-disorder separated and the complete disordered double quantum ring
systems respectively (for color illustration, see the web version).
Otherwise, if we measure the parameter $D_{I}$, then we will get the trivial
result as obtained in a traditional disordered system since the response of
the disordered states will not be changed by coupling these states with the
ordered states.
In Fig. 2, we show the variation of the Drude weight $D_{II}$ as a function of
the disorder strength $W$ for some double quantum rings, where we choose
$N_{I}=30$ and $N_{II}=30$. The chemical potential $\mu$ is fixed to $0$. The
red and the blue curves represent the results for the order-disorder separated
and the complete disordered double quantum rings, respectively. From the
results it is observed that, in the complete disordered double quantum ring
the Drude weight sharply decreases with the increase of the disorder strength
and eventually it drops to zero. Therefore, we can say that for such a system
the electron diffusion length as well as the electron mobility decreases
sharply with the disorder strength. Such a behavior can be well understood
from the theory of Anderson localization, where we get more localization with
the increase of the disorder strength.28 The anomalous behavior is observed
when the impurities are given only in any one of the two rings, keeping the
other one as impurity free i.e., for the order-disorder separated system. Our
results predict that the Drude weight initially decreases with the increase of
the disorder strength, but after reaching to a minimum it again increases with
the strength of the disorder. Such a phenomenon is completely opposite to that
of the traditional disordered system and can be justified in the following
way. For the order-disorder separated double quantum ring, the energy spectra
of the disordered ring are gradually separated from the energy spectra of the
ordered ring with the increase of the disorder strength $W$. Therefore, the
influence of random scattering in the ordered ring due to the strong
localization in the disordered ring decreases. It has been examined that the
energy spectrum of the order-disorder separated double quantum ring with large
disorder contains localized tail states with much small and central states
with much large values of localization length, contributed approximately by
disordered and ordered rings, respectively. Hence the central states gradually
separated from the tail states and delocalized with the increase of the
strength of the disorder. Thus we see that, for the coupled order-disorder
separated double quantum ring, the coupling between the localized states with
the extended states is strongly influenced by the strength of the disorder,
and this coupling is inversely proportional to the disorder strength $W$.
Accordingly, in the limit of weak disorder the coupling effect is
significantly high, while the coupling effect becomes very weak in the strong
disorder regime. Hence, in the limit of weak disorder the electron transport
is strongly influenced by the impurities at the disordered ring such that the
electron states are scattered more and therefore the electron diffusion length
decreases which manifests the lesser electron mobility. On the other hand, for
the stronger disorder limit the extended states are weakly influenced by the
disordered ring and the coupling effect gradually decreases with the increase
of the disorder strength which provides the larger electron mobility in the
strong disorder limit. This reveals that the electron diffusion length
increases in this limit. For large enough impurity strength, the extended
states are almost unaffected by the impurities at the disordered ring and in
that case the electrons are carried only by these extended states in the
ordered ring which is the trivial limit. So the novel phenomenon will be
observed only in the intermediate limit of $W$.
In order to emphasize the dependence of the electron mobility on the system
size, here we focus our attention on the results those are plotted in Fig. 3.
In this figure, we display the Drude weight for some
Figure 3: Drude weight ($D_{II}$) of the ring $II$ as a function of the
disorder strength ($W$) for the systems with $N_{I}=50$, $N_{II}=50$ and the
chemical potential $\mu=0$. The red and the blue lines correspond to the
order-disorder separated and the complete disordered double quantum ring
systems respectively (for color illustration, see the web version).
typical double quantum rings, where we fix $N_{I}=50$ and $N_{II}=50$. Similar
to the previous systems, here we also take $\mu=0$ for these systems. The red
and the blue lines correspond to the same meaning as in Fig. 2. From this
figure (Fig. 3) it is also observed that, the Drude weight in the order-
disorder separated double quantum ring decreases with the increase of the
disorder strength $W$ in the weak disorder regime, while it increases with the
strength $W$ in the strong disorder regime. On the other hand, the Drude
weight always decreases with the strength of the disorder for the complete
disordered system, as expected. Though the results plotted in Fig. 3 seem to
be quite similar in nature with the results those are described in Fig. 2, but
the significant point is that, the typical magnitude of the Drude weight
strongly depends on the size of both these two rings which manifest the finite
quantum size effects. Now the other significant factor that raises to our mind
is the existence of the location of the minimum in the Drude weight versus
disorder curves of the order-disorder separated double quantum rings. This
minimum can be implemented as follows. The carrier mobility in the order-
disorder separated double quantum ring is controlled by the two competing
mechanisms. One is the random scattering in the ordered ring due to the
localization in the disordered ring which tends to decrease the carrier
mobility, and the other one is the vanishing influence of random scattering in
the ordered ring due to the strong localization in the disordered ring which
provides the enhancement of the carrier mobility. Depending on the ratio of
the total number of atomic sites in the disordered ring to the total number of
atomic sites in the ordered ring, the vanishing effect of random scattering
from the ordered states dominates over the non-vanishing effect of random
scattering from these states for a particular disorder strength $(W=W_{c})$,
which provides the location of the minimum in the Drude weight versus disorder
curve.
## 4 Concluding remarks
In conclusion, we have established a novel feature for control of the electron
diffusion length in an order-disorder separated double quantum ring in which
the two rings thread different magnetic fluxes. From our study it has been
observed that, in the order-disorder separated double quantum ring, the
electron diffusion length increases with the increase of the disorder strength
in the strong disorder regime, while it decreases in the weak disorder regime.
Such a peculiar behavior is completely opposite to that of the conventional
disordered systems, where the electron diffusion length always decreases with
the increase of the disorder strength. Lastly, we have noticed that, both the
electron mobility and the location of the minimum in the Drude weight versus
disorder curve strongly depend on the size of both the two rings which
manifest the finite quantum size effects. Our theoretical results in this
article might be helpful to illuminate some of the unusual experimental
phenomena which have been observed in the order-disorder separated quantum
systems.29-31
Throughout our study, we have ignored the effect of the electron-electron
(e-e) correlation since the inclusion of the e-e correlation will not provide
any new significant result in our present investigations.
Acknowledgment
I acknowledge with deep sense of gratitude the illuminating comments and
suggestions I have received from Prof. S. Sil during the calculations.
## References
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* [25] S. K. Maiti, J. Chowdhury and S. N. Karmakar, Synthetic Metals 155, 430 (2005).
* [26] S. K. Maiti, J. Chowdhury and S. N. Karmakar, J. Phys.: Condens. Matter 18, 5349 (2006).
* [27] S. K. Maiti, Physica E 31, 117 (2006).
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* [29] Y. Cui, X. F. Duan, J. T. Hu and C. M. Lieber, J. Phys. Chem. B 104, 5213 (2000).
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* [32] J. X. Zhong and G. M. Stocks, Nano. Lett. 6, 128 (2006).
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|
arxiv-papers
| 2009-09-08T03:19:07 |
2024-09-04T02:49:05.111512
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Santanu K. Maiti",
"submitter": "Santanu Maiti Kumar",
"url": "https://arxiv.org/abs/0909.1378"
}
|
0909.1427
|
# Temperature dependence of magnetic susceptibility of nuclear matter: lowest
order constrained variational calculations
M. Bigdeli1,3 111E-mail: m bigdeli@znu.ac.ir, G.H. Bordbar 2,3222E-mail:
ghbordbar@shirazu.ac.ir and Z. Rezaei 2 1Department of Physics, Zanjan
University, P.O. Box 45195-313, Zanjan, Iran333Permanent address
2Department of Physics, Shiraz University, Shiraz 71454, Iran444Permanent
address
3Research Institute for Astronomy and Astrophysics of Maragha,
P.O. Box 55134-441, Maragha, Iran
###### Abstract
In this paper we study the magnetic susceptibility and other thermodynamic
properties of the polarized nuclear matter at finite temperature using the
lowest order constrained variational (LOCV) method employing the $AV_{18}$
potential. Our results show a monotonic behavior for the magnetic
susceptibility which indicates that the spontaneous transition to the
ferromagnetic phase does not occur for this system.
###### pacs:
21.65.-f, 26.60.-c, 64.70.-p
## I INTRODUCTION
The magnetic susceptibility is one of the most important magnetic properties
of the dense matter, its behavior specifies whether the spontaneous phase
transition to a ferromagnetic state occurs. This transition in nuclear matter
could have important consequences for the physical origin of the magnetic
field of the pulsars which are believed to be rapidly rotating neutron stars
with strong surface magnetic fields in the range of $10^{12}-10^{13}$ Gauss
shap ; paci ; gold ; navarro . Considering different stages of the neutron
star formation at different temperatures, the study of the magnetic properties
of the polarized nuclear matter at finite temperature is of special interest
in the description of protoneutron stars. A protoneutron star (newborn neutron
star) is born within a short time after the supernovae collapse. In this stage
the interior temperature of the neutron star matter is of the order 20-50 MeV
burro . The magnetic susceptibility of nucleonic matter is also an useful
quantity to estimate the mean free path of the neutrino in the dense nucleonic
matter which is relevant information for the understanding of the mechanism
underlying the supernova explosion and the cooling process of the neutron
stars iwamoto .
There exists several possibilities of the generation of the magnetic field in
a neutron star. From the nuclear physics point of view, such a possibility has
been studied by several authors using different theoretical approaches [7-30],
but the results are still contradictory. In most calculations, the neutron
star matter is approximated by pure neutron matter at zero temperature. The
properties of the polarized neutron matter both at finite and zero temperature
have been studied by several authors apv ; dapv ; bprrv . Some calculations
show that the neutron matter becomes ferromagnetic for some densities brown ;
rice ; pandh ; marcos ; apv . Some others, using the modern two-body and
three-body realistic interactions, show no indication of the ferromagnetic
transition at any density for the neutron matter and the asymmetrical nuclear
matter kutsb ; fanto ; vida ; bprrv . The results of another calculation show
both behaviors; the D1P force exhibits a ferromagnetic transition whereas no
sign of such transition is found for D1 at any density and temperature dapv .
The influence of the finite temperature on the antiferromagnetic (AFM) spin
ordering in the symmetric nuclear matter with the effective Gogny interaction
within the framework of a Fermi liquid formalism has been studied by Isayev
isay1 ; isay2 . We note that in the symmetric nuclear matter corresponding to
AFM spin ordering, we have
$\Delta\rho_{\uparrow\downarrow}=(\rho_{n\uparrow}+\rho_{p\downarrow})-(\rho_{n\downarrow}+\rho_{p\uparrow})\neq
0$ and $\Delta\rho_{\uparrow\uparrow}=\rho_{\uparrow}-\rho_{\downarrow}=0$. In
this article, we use the lowest order constrained variational (LOCV) formalism
to investigate the possibility of the transition to a ferromagnetic phase for
the polarized hot symmetrical nuclear matter.
The LOCV method has been developed to study the bulk properties of the quantal
fluids OBI1 ; OBI2 ; OBI3 . This technique has been used for studying the
ground state properties of finite nuclei and treatment of isobars BHIM ; MI1 ;
MI2 . Modarres has extended the LOCV method to the finite temperature
calculations and has applied it to the neutron matter, nuclear matter and
asymmetrical nuclear matter in order to calculate the different thermodynamic
properties of these systems Mod93 ; Mod95 ; Mod97 ; MM98 . Few years ago, we
calculated the properties of nuclear matter at zero and finite temperature
using the LOCV method with the new nucleon-nucleon potentials BM97 ; BM98 ;
MB98 . The LOCV method has several advantages with respect to the other many-
body formalism. These are as follows: (i) Since the method is fully self-
consistent, it does not introduce any free parameters into the calculations.
(ii) It considers the constraint in the form of a normalization constraint
Feen to keep the higher-order terms as small as possible OBI3 ; MI1 ; MI2 ;
MM98 ; BM97 and it also assumes a particular form for the long-range behavior
of the correlation function in order to perform an exact functional
minimization of the two-body energy with respect to the short-range behavior
of the correlation function. (iii) The functional minimization procedure
represents an enormous computational simplification over the unconstrained
methods (i.e. to parameterize the short-range behavior of the correlation
functions) which attempt to go beyond the lowest order.
Recently, we have computed the properties of the polarized neutron matter
bordbig , polarized symmetrical bordbig2 and asymmetrical nuclear matters
bordbig3 and also polarized neutron star matter bordbig3 at zero temperature
using the microscopic calculations employing the LOCV method with the
realistic nucleon-nucleon potentials. We have concluded that the spontaneous
phase transition to a ferromagnetic state in these matters does not occur. We
have also calculated the thermodynamic properties of the polarized neutron
matter at finite temperature bordbig4 such as the total energy, magnetic
susceptibility, entropy and pressure using the LOCV method employing the
$AV_{18}$ potential wiring . Our calculations do not show any transition to a
ferromagnetic phase for a hot neutron matter.
In the present work, we intend to apply the LOCV calculation for the polarized
symmetrical nuclear matter at finite temperature using the $AV_{18}$
potential.
## II Finite temperature calculations for polarized nuclear matter with the
LOCV method
We consider a system of $A$ interacting nucleons with $A^{(+)}$ spin-up and
$A^{(-)}$ spin-down nucleons. For this system, the total number density
($\rho$) and spin asymmetry parameter ($\delta$) are defined as
$\displaystyle\rho$ $\displaystyle=$ $\displaystyle\rho^{(+)}+\rho^{(-)},$
$\displaystyle\delta$ $\displaystyle=$
$\displaystyle\frac{\rho^{(+)}-\rho^{(-)}}{\rho}.$ (1)
$\delta$ shows the spin ordering of the matter which can have a value in the
range of $\delta=0.0$ (unpolarized matter) to $\delta=1.0$ (fully polarized
matter). To obtain the macroscopic properties of this system, we should
calculate the total free energy per nucleon, $F$,
$\displaystyle F=E-{\cal T}S^{(+)}-{\cal T}S^{(-)}.$ (2)
$E$ is total energy per nucleon and $S^{(i)}$ is the entropy per nucleon
corresponding to spin projection $i$,
$\displaystyle S^{(i)}(\rho,T)$ $\displaystyle=$
$\displaystyle-\frac{1}{A}\sum_{k}\\{[1-n^{(i)}(k,{\cal
T},\rho^{(i)})]\textrm{ln}[1-n^{(i)}(k,{\cal T},\rho^{(i)})]$ (3)
$\displaystyle+n^{(i)}(k,{\cal T},\rho^{(i)})\textrm{ln}n^{(i)}(k,{\cal
T},\rho^{(i)})\\}.$
where $n^{(i)}(k,{\cal T},\rho^{(i)})$ is the Fermi-Dirac distribution
function,
$\displaystyle n^{(i)}(k,{\cal
T},\rho^{(i)})=\frac{1}{e^{\beta[\epsilon^{(i)}(k,{\cal
T},\rho^{(i)})-\mu^{(i)}({\cal T},\rho^{(i)})]}+1}\cdot$ (4)
In the above equation $\beta=\frac{1}{k_{B}{\cal T}}$ , $\mu^{(i)}$ being the
chemical potential which is determined at any adopted value of the temperature
$\cal T$, number density $\rho^{(i)}$ and spin polarization $\delta$, by
applying the following constraint,
$\displaystyle\sum_{k}n^{(i)}(k,{\cal T},\rho^{(i)})=A^{(i)},$ (5)
and $\epsilon^{(i)}$ is the single particle energy of a nucleon. In our
formalism, the single particle energy of a nucleon with momentum $k$ and spin
projection $i$ is approximately written in terms of the effective mass as
follows apv ; dapv ; isay2
$\displaystyle\epsilon^{(i)}(k,{\cal
T},\rho^{(i)})=\frac{\hbar^{2}{k^{2}}}{2{m^{*}}^{(i)}(\rho,{\cal
T})}+U^{(i)}({\cal T},\rho^{(i)}).$ (6)
In fact, we use a quadratic approximation for single particle potential
incorporated in the single particle energy as a momentum independent effective
mass. $U^{(i)}({\cal T},\rho^{(i)})$ is the momentum independent single
particle potential. We introduce the effective masses, $m^{{*}{(i)}}$, as
variational parameters bordbig4 ; fp . We minimize the free energy with
respect to the variations in the effective masses and then we obtain the
chemical potentials and the effective masses of the spin-up and spin-down
nucleons at the minimum point of the free energy. This minimization is done
numerically.
As it is also mentioned in the pervious section, for calculating the total
energy of the polarized symmetrical nuclear matter, we use the LOCV method. We
adopt a trial many-body wave function of the form
$\displaystyle\psi=\cal{F}\phi,$ (7)
where $\phi$ is the uncorrelated ground state wave function (simply the Slater
determinant of plane waves) of $A$ independent nucleons and ${\cal F}={\cal
F}(1\cdots A)$ is an appropriate A-body correlation operator which can be
replaced by a Jastrow form i.e.,
$\displaystyle{\cal F}={\cal S}\prod_{i>j}f(ij),$ (8)
in which ${\cal S}$ is a symmetrizing operator. Now, we consider the cluster
expansion of the energy functional up to the two-body term clark ,
$\displaystyle
E([f])=\frac{1}{A}\frac{\langle\psi|H\psi\rangle}{\langle\psi|\psi\rangle}=E_{1}+E_{2}\cdot$
(9)
For the hot nuclear matter, the one-body term $E_{1}$ is
$\displaystyle E_{1}=E_{1}^{(+)}+E_{1}^{(-)},$ (10)
where
$\displaystyle E_{1}^{(i)}=\sum_{k}\frac{\hbar^{2}{k^{2}}}{2m}n^{(i)}(k,{\cal
T},\rho^{(i)}).$ (11)
The two-body energy $E_{2}$ is
$\displaystyle E_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2A}\sum_{ij}\langle ij\left|\nu(12)\right|ij-
ji\rangle,$ (12)
where
$\displaystyle\nu(12)=-\frac{\hbar^{2}}{2m}[f(12),[\nabla_{12}^{2},f(12)]]+f(12)V(12)f(12).$
(13)
In above equation, $f(12)$ and $V(12)$ are the two-body correlation and
potential. In our calculations, we use the $AV_{18}$ two-body potential which
has the following form wiring ,
$V(12)=\sum^{18}_{p=1}V^{(p)}(r_{12})O^{(p)}_{12},$ (14)
where
$\displaystyle O_{12}^{(p=1-18)}$ $\displaystyle=$ $\displaystyle 1,\
{\bf\sigma_{1}}\cdot{\bf\sigma_{2}},\ {\bf\tau_{1}}\cdot{\bf\tau_{2}},\
({\bf\sigma_{1}}\cdot{\bf\sigma_{2}})\ ({\bf\tau_{1}}\cdot{\bf\tau_{2}}),\
S_{12},\ S_{12}({\bf\tau_{1}}\cdot{\bf\tau_{2}}),$ (15) $\displaystyle{\bf
L}\cdot{\bf S},\ {\bf L}\cdot{\bf S}({\bf\tau_{1}}\cdot{\bf\tau_{2}}),\ {\bf
L}^{2},\ {\bf L}^{2}({\bf\sigma_{1}}\cdot{\bf\sigma_{2}}),\ {\bf
L}^{2}({\bf\tau_{1}}\cdot{\bf\tau_{2}}),$ $\displaystyle{\bf
L}^{2}({\bf\sigma_{1}}\cdot{\bf\sigma_{2}})({\bf\tau_{1}}\cdot{\bf\tau_{2}}),\
({\bf L}\cdot{\bf S})^{2},\ ({\bf L}\cdot{\bf
S})^{2}({\bf\tau_{1}}\cdot{\bf\tau_{2}}),$ $\displaystyle{\bf T_{12}},\
({\bf\sigma_{1}}\cdot{\bf\sigma_{2}}){\bf T_{12}},\ S_{12}{\bf T_{12}},\
(\bf\tau_{z1}+\bf\tau_{z2}).$
In above equation,
$S_{12}=[3({\bf\sigma_{1}}\cdot\hat{r})({\bf\sigma_{2}}\cdot\hat{r})-{\bf\sigma_{1}}\cdot{\bf\sigma_{2}}]$
is the tensor operator and ${\bf
T_{12}}=[3({\bf\tau_{1}}\cdot\hat{r})({\bf\tau_{2}}\cdot\hat{r})-{\bf\tau_{1}}\cdot{\bf\tau_{2}}]$
is the isotensor operator. The above 18 components of the $AV_{18}$ two-body
potential are denoted by the labels
$c,\sigma,\tau,\sigma\tau,t,t\tau,ls,ls\tau,l2,l2\sigma,l2\tau,l2\sigma\tau,ls2,ls2\tau,T,\sigma
T,tT$ and $\tau z$, respectively wiring . In the LOCV formalism, the two-body
correlation $f(12)$ is considered as the following form OBI3 ,
$\displaystyle f(12)$ $\displaystyle=$
$\displaystyle\sum^{3}_{k=1}f^{(k)}(r_{12})P^{(k)}_{12},$ (16)
where
$\displaystyle P_{12}^{(k=1-3)}$ $\displaystyle=$
$\displaystyle\left(\frac{1}{4}-\frac{1}{4}O^{(2)}_{12}\right),\
\left(\frac{1}{2}+\frac{1}{6}O^{(2)}_{12}+\frac{1}{6}O^{(5)}_{12}\right),$
(17)
$\displaystyle\left(\frac{1}{4}+\frac{1}{12}O^{(2)}_{12}-\frac{1}{6}O^{(5)}_{12}\right).$
The operators $O^{(2)}_{12}$ and $O^{(5)}_{12}$ are given in Eq. (15). Using
the above two-body correlation and potential, after doing some algebra we find
the following equation for the two-body energy,
$\displaystyle E_{2}$ $\displaystyle=$
$\displaystyle\frac{2}{\pi^{4}\rho}\left(\frac{h^{2}}{2m}\right)\sum_{JLTSS_{z}T_{z}}\frac{(2J+1)(2T+1)}{2(2S+1)}[1-(-1)^{L+S+T}]\left|\left\langle\frac{1}{2}\sigma_{z1}\frac{1}{2}\sigma_{z2}\mid
SS_{z}\right\rangle\right|^{2}$ (18) $\displaystyle\times\int
dr\left\\{\left[{f_{\alpha}^{(1)^{{}^{\prime}}}}^{2}{a_{\alpha}^{(1)}}^{2}(k_{f}r)+\frac{2m}{h^{2}}\left(\\{V_{c}-3V_{\sigma}+(V_{\tau}-3V_{\sigma\tau})(4T-3)\right.\right.\right.$
$\displaystyle\left.\left.\left.\ \ \ \ \ \ \ \ \
+(V_{T}-3V_{\sigma\tau})(4T)\\}{a_{\alpha}^{(1)}}^{2}(k_{f}r)\
+[V_{l2}-3V_{l2\sigma}\right.\right.\right.$ $\displaystyle\left.\left.\left.\
\ \ \ \ \ \ \ \
+(V_{l2\tau}-3V_{l2\sigma\tau})(4T-3)]{c_{\alpha}^{(1)}}^{2}(k_{f}r)\right)(f_{\alpha}^{(1)})^{2}\right]\right.$
$\displaystyle\left.\ \ \ \ \ \ \ \ \ \
+\sum_{k=2,3}\left[{f_{\alpha}^{(k)^{{}^{\prime}}}}^{2}{a_{\alpha}^{(k)}}^{2}+\frac{2m}{h^{2}}\left(\left\\{V_{c}+V_{\sigma}+(-6k+14)V_{t}+-(k-1)V_{ls}\right.\right.\right.\right.$
$\displaystyle\left.\left.\left.\left.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
+[V_{\tau}+V_{\sigma\tau}+(-6k+14)V_{tz}-(k-1)V_{ls\tau}](4T-3)\right.\right.\right.\right.$
$\displaystyle\left.\left.\left.\left.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
+[V_{T}+V_{\sigma\tau}+(-6k+14)V_{tT}][T(6T_{z}^{2}-4)]+2V_{\tau
z}T_{z}\right\\}{a_{\alpha}^{(k)}}^{2}(k_{f}r)\right.\right.\right.$
$\displaystyle\left.\left.\left.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
+[V_{l2}+V_{l2\sigma}+(V_{l2\tau}+V_{l2\sigma\tau})(4T-3)]{c_{\alpha}^{(k)}}^{2}(k_{f}r)\right.\right.\right.$
$\displaystyle\left.\left.\left.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
+[(V_{ls2}+V_{ls2\tau})(4T-3)]{d_{\alpha}^{(k)}}^{2}(k_{f}r)\right){f_{\alpha}^{(k)}}^{2}\right]\right.$
$\displaystyle\left.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
+\frac{2m}{h^{2}}[[(V_{ls\tau}-2(V_{l2\sigma\tau}+V_{l2\tau})-3V_{ls2\tau})(4T-3)]\right.$
$\displaystyle\left.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
+V_{ls}-2(V_{l2}+V_{l2\sigma})-3V_{ls2}]b_{\alpha}^{2}(k_{f}r)f_{\alpha}^{(2)}f_{\alpha}^{(3)}\right.$
$\displaystyle\left.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
+\frac{1}{r^{2}}(f_{\alpha}^{(2)}-f_{\alpha}^{(3)})^{2}b_{\alpha}^{2}(k_{f}r)\right\\},$
where $\alpha=\\{J,L,S,S_{z}\\}$ and the coefficient ${a_{\alpha}^{(1)}}^{2}$,
etc., are as follows,
$\displaystyle{a_{\alpha}^{(1)}}^{2}(x)=x^{2}I_{L,S_{z}}(x),$ (19)
$\displaystyle{a_{\alpha}^{(2)}}^{2}(x)=x^{2}[\beta I_{J-1,S_{z}}(x)+\gamma
I_{J+1,S_{z}}(x)],$ (20) $\displaystyle{a_{\alpha}^{(3)}}^{2}(x)=x^{2}[\gamma
I_{J-1,S_{z}}(x)+\beta I_{J+1,S_{z}}(x)],$ (21) $\displaystyle
b_{\alpha}^{(2)}(x)=x^{2}[\beta_{23}I_{J-1,S_{z}}(x)-\beta_{23}I_{J+1,S_{z}}(x)],$
(22) $\displaystyle{c_{\alpha}^{(1)}}^{2}(x)=x^{2}\nu_{1}I_{L,S_{z}}(x),$ (23)
$\displaystyle{c_{\alpha}^{(2)}}^{2}(x)=x^{2}[\eta_{2}I_{J-1,S_{z}}(x)+\nu_{2}I_{J+1,S_{z}}(x)],$
(24)
$\displaystyle{c_{\alpha}^{(3)}}^{2}(x)=x^{2}[\eta_{3}I_{J-1,S_{z}}(x)+\nu_{3}I_{J+1,S_{z}}(x)],$
(25)
$\displaystyle{d_{\alpha}^{(2)}}^{2}(x)=x^{2}[\xi_{2}I_{J-1,S_{z}}(x)+\lambda_{2}I_{J+1,S_{z}}(x)],$
(26)
$\displaystyle{d_{\alpha}^{(3)}}^{2}(x)=x^{2}[\xi_{3}I_{J-1,S_{z}}(x)+\lambda_{3}I_{J+1,S_{z}}(x)].$
(27)
In above equations, we have
$\displaystyle\beta=\frac{J+1}{2J+1},\ \ \gamma=\frac{J}{2J+1},\ \
\beta_{23}=\frac{2J(J+1)}{2J+1},$ (28) $\displaystyle\nu_{1}=L(L+1),\ \
\nu_{2}=\frac{J^{2}(J+1)}{2J+1},\ \ \nu_{3}=\frac{J^{3}+2J^{2}+3J+2}{2J+1},$
(29) $\displaystyle\eta_{2}=\frac{J(J^{2}+2J+1)}{2J+1},\ \
\eta_{3}=\frac{J(J^{2}+J+2)}{2J+1},$ (30)
$\displaystyle\xi_{2}=\frac{J^{3}+2J^{2}+2J+1}{2J+1},\ \
\xi_{3}=\frac{J(J^{2}+J+4)}{2J+1},$ (31)
$\displaystyle\lambda_{2}=\frac{J(J^{2}+J+1)}{2J+1},\ \
\lambda_{3}=\frac{J^{3}+2J^{2}+5J+4}{2J+1},$ (32)
and
$\displaystyle I_{J,S_{z}}(r,\rho,T)=\frac{1}{2\pi^{6}\rho^{2}}\int
k_{1}^{2}dk_{1}k_{2}^{2}dk_{2}n_{i}(k_{1},T,\rho_{i})n_{j}(k_{2},T,\rho_{j})J_{J}^{2}(|k_{2}-k_{1}|r),$
(33)
where $J_{J}(x)$ is the Bessel’s function .
Now, we minimize the two-body energy Eq. (18) with respect to the variations
in the correlation functions ${f_{\alpha}}^{(k)}$, but subject to the
normalization constraint OBI3 ; BM98 ,
$\displaystyle\frac{1}{A}\sum_{ij}\langle
ij\left|h_{S_{z}}^{2}-f^{2}(12)\right|ij\rangle_{a}=0\cdot$ (34)
In the case of polarized symmetrical nuclear matter, the Pauli function
$h_{S_{z}}(r)$ is as follows
$\displaystyle
h_{S_{z}}(r)=\left\\{\begin{array}[]{ll}\left[1-\frac{1}{2}\left(\frac{\gamma^{(i)}(r)}{\rho}\right)^{2}\right]^{-1/2}&;\
\hbox{$S_{z}=\pm 1$}\\\ 1&;\ \hbox{$S_{z}=0$}\end{array}\right.$ (37)
where
$\displaystyle\gamma^{(i)}(r)=\frac{1}{\pi^{2}}\int n^{(i)}(k,{\cal
T},\rho^{(i)})J_{0}(kr)k^{2}dk.$ (38)
From the minimization of the two-body cluster energy we get a set of coupled
and uncoupled Euler-Lagrange differential equations. The Euler-Lagrange
equations for uncoupled states are
$\displaystyle
g_{\alpha}^{(1)^{\prime\prime}}-\\{\frac{a_{\alpha}^{(1)^{\prime\prime}}}{a_{\alpha}^{(1)}}+\frac{m}{\hbar^{2}}[V_{c}-3V_{\sigma}+(V_{\tau}-3V_{\sigma\tau})(4T-3)$
$\displaystyle+(V_{T}-3V_{\sigma T})[T(6T_{z}^{2}-4)]+2V_{\tau
z}T_{z}+\lambda]$
$\displaystyle+\frac{m}{\hbar^{2}}(V_{l2}-3V_{{l2}\sigma}+(V_{{l2}\tau}-3V_{{l2}\sigma\tau})(4T-3))\frac{c_{\alpha}^{(1)^{2}}}{a_{\alpha}^{(1)^{2}}}\\}g_{\alpha}^{(1)}=0,$
(39)
while for the coupled states, these equations are written as follows,
$\displaystyle
g_{\alpha}^{(2)^{\prime\prime}}-\\{\frac{a_{\alpha}^{(2)^{\prime\prime}}}{a_{\alpha}^{(2)}}+\frac{m}{\hbar^{2}}[V_{c}+V_{\sigma}+2V_{t}-V_{{ls}}+(V_{\tau}+V_{\sigma\tau}+2V_{t\tau}-V_{{ls}\tau})(4T-3)$
$\displaystyle+(V_{T}+V_{\sigma T}+2V_{tT})[T(6T_{z}^{2}-4)]+2V_{\tau
z}T_{z}+\lambda]$
$\displaystyle+\frac{m}{\hbar^{2}}[V_{l2}+V_{{l2}\sigma}+(V_{{l2}\tau}+V_{{l2}\sigma\tau})(4T-3)]\frac{c_{\alpha}^{(2)^{2}}}{a_{\alpha}^{(2)^{2}}}$
$\displaystyle+\frac{m}{\hbar^{2}}[V_{{ls}2}+V_{{ls}2\tau}(4T-3)]\frac{d_{\alpha}^{(2)^{2}}}{a_{\alpha}^{(2)^{2}}}+\frac{b_{\alpha}^{2}}{r^{2}a_{\alpha}^{(2)^{2}}}\\}g_{\alpha}^{(2)}$
$\displaystyle+\\{\frac{1}{r^{2}}-\frac{m}{2\hbar^{2}}[V_{ls}-2V_{l2}-2V_{{l2}\sigma}-3V_{{ls}2}$
$\displaystyle+(V_{{ls}\tau}-2V_{{l2}\tau}-2V_{{l2}\sigma\tau}-3V_{{ls}2\tau})(4T-3)]\\}\frac{b_{\alpha}^{2}}{a_{\alpha}^{(2)}a_{\alpha}^{(3)}}g_{\alpha}^{(3)}=0,$
(40) $\displaystyle
g_{\alpha}^{(3)^{\prime\prime}}-\\{\frac{a_{\alpha}^{(3)^{\prime\prime}}}{a_{\alpha}^{(3)}}+\frac{m}{\hbar^{2}}[V_{c}+V_{\sigma}-4V_{t}-2V_{ls}+(V_{\tau}+V_{\sigma\tau}-4V_{t\tau}-2V_{{ls}\tau})(4T-3)$
$\displaystyle+(V_{T}+V_{\sigma T}-4V_{tT})[T(6T_{z}^{2}-4)]+2V_{\tau
z}T_{z}+\lambda]$
$\displaystyle+\frac{m}{\hbar^{2}}[V_{l2}+V_{{l2}\sigma}+(V_{{l2}\tau}+V_{{l2}\sigma\tau})(4T-3)]\frac{c_{\alpha}^{(3)^{2}}}{a_{\alpha}^{(3)^{2}}}$
$\displaystyle+\frac{m}{\hbar^{2}}[V_{{ls}2}+V_{{ls}2\tau}(4T-3)]\frac{d_{\alpha}^{(3)^{2}}}{a_{\alpha}^{(3)^{2}}}+\frac{b_{\alpha}^{2}}{r^{2}a_{\alpha}^{(2)^{2}}}\\}g_{\alpha}^{(3)}$
$\displaystyle+\\{\frac{1}{r^{2}}-\frac{m}{2\hbar^{2}}[V_{ls}-2V_{l2}-2V_{{l2}\sigma}-3V_{{ls}2}$
$\displaystyle+(V_{{ls}\tau}-2V_{{l2}\tau}-2V_{{l2}\sigma\tau}-3V_{{ls}2\tau})(4T-3)]\\}\frac{b_{\alpha}^{2}}{a_{\alpha}^{(2)}a_{\alpha}^{(3)}}g_{\alpha}^{(2)}=0,$
(41)
where
$\displaystyle g_{\alpha}^{(i)}(r)=f_{\alpha}^{(i)}(r)a_{\alpha}^{(i)}(r).$
(42)
The primes in the above equations mean differentiation with respect to $r$.
The Lagrange multiplier $\lambda$ is introduced by the normalization
constraint, Eq. (34). Now, we can calculate the correlation functions by
numerically solving these differential equations and then using these
correlation functions, the two-body energy is obtained. Finally, we can
compute the energy and then the free energy of the system.
## III Results and discussion
We have presented the effective masses of the spin-up and spin-down nucleons
as functions of the spin polarization ($\delta$) at $\rho=0.5fm^{-3}$ and
${\cal T}=20$ MeV in Fig. 1. It is seen that the difference between the
effective masses of spin-up and spin-down nucleons increases by increasing the
polarization. We also see that the effective mass of spin-up nucleons
increases by increasing the polarization whereas the effective mass of spin-
down nucleons decreases by increasing the polarization. These behaviors have
been also seen for the effective mass of the neutron in the case of spin
polarized hot neutron matter apv ; dapv ; bprrv ; bordbig4 . A similar
qualitative behavior of the nucleon effective mass as a function of the
isospin asymmetric parameter
$\beta=\frac{\rho_{n}-\rho_{p}}{\rho_{n}+\rho_{p}}$ has been already found in
non-polarized isospin asymmetric nuclear matter, as it has been already
discussed in Refs. bomb1 ; li .
The free energy per nucleon of the polarized hot nuclear matter versus the
total number density ($\rho$) for different values of the spin polarization
($\delta$) at ${\cal T}=10$ and $20$ MeV is shown in Fig. 2. It can be seen
that for each value of the temperature, the free energy increases by
increasing both density and polarization. From Fig. 2, it is seen that the
free energy of the polarized hot nuclear matter decreases by increasing the
temperature. We have seen that above a certain values of the temperature and
spin polarization, the free energy does not show any bound states for the
polarized hot nuclear matter. From Fig. 2 we also see that for all given
temperatures there is no crossing of the free energy curves for different
polarizations and the difference between the free energy of the nuclear matter
at different polarizations increases by increasing the density. This indicates
that the spontaneous transition to the ferromagnetic phase does not occur in
the hot nuclear matter. We have also compared the free energy per nucleon of
the unpolarized case of the nuclear matter at different temperatures in Fig.
3. It is seen that the free energy of unpolarized nuclear matter decreases by
increasing the temperature. We can see that above a certain temperature, the
free energy does not show the bound state (the minimum point of the free
energy) for the unpolarized nuclear matter.
For the polarized hot nuclear matter, the magnetic susceptibility, $\chi$,
which characterizes the response of the system to the magnetic field can be
calculated by using the following relation,
$\displaystyle\chi=\frac{\mu^{2}\rho}{\left(\frac{\partial^{2}F}{\partial\delta^{2}}\right)_{\delta=0}},$
(43)
where $\mu$ is the magnetic moment of the nucleons. Fig. 4 shows the ratio
${{\chi_{F}}/{\chi}}$ as a function of the temperature at $\rho=0.16fm^{-3}$,
where $\chi_{F}$ is the magnetic susceptibility for a noninteracting Fermi
gas. As it can be seen from this figure, this ratio is inversely proportional
to absolute temperature without any anomalous change in its behavior. This
indicates that hot nuclear matter is paramagnetic. The ratio
${{\chi_{F}}/{\chi}}$ has been also shown versus the total number density at
temperature ${\cal T}=20$ MeV in Fig. 5. A magnetic instability would require
${{\chi_{F}}/{\chi}<0}$. It is seen that the value of ${{\chi_{F}}/{\chi}}$ is
always positive and monotonically increasing up to highest density and does
not show any spontaneous phase transition to the ferromagnetic phase for the
hot nuclear matter.
The difference between the entropy per nucleon of the fully polarized and
unpolarized cases of the nuclear matter is plotted as a function of the total
number density at ${\cal T}=20$ MeV in Fig. 6. It is seen that for all given
values of the density, this difference is negative. This shows that the fully
polarized case of hot nuclear matter is more ordered than the unpolarized
case. We also see that the magnitude of this deference decreases by increasing
the density. The entropy per nucleon of the polarized hot nuclear matter
versus the spin polarization for fixed density $\rho=0.5fm^{-3}$ and
temperature ${\cal T}=20$ has been presented in Fig. 7. It is shown that the
entropy decreases by increasing the polarization. It is also shown that the
highest value of the entropy occurs for the unpolarized case of the hot
nuclear matter. For the polarized hot nuclear matter, the following condition
for the effective mass prevents the anomalous behavior of the entropy versus
the spin polarization apv ,
$\displaystyle\frac{m^{*}(\rho,\delta=1.0)}{m^{*}(\rho,\delta=0.0)}<2^{2/3},$
(44)
where $m^{*}(\rho,\delta=1.0)$ and $m^{*}(\rho,\delta=0.0)$ are the effective
masses of the fully polarized and unpolarized nuclear matter, respectively.
This condition was first derived in Ref. apv for the particular case of the
Skyrme interaction where the effective mass is independent of the momentum and
temperature and therefore the single particle potential is purely parabolic.
In our approach, the effective mass depends on both density and temperature
but is independent of the momentum. In other words, a similar rigorous
condition can not be obtained straightforwardly. However, within this
approximation, one can use the condition of Eq. (44). From our result for the
effective mass at ${\cal T}=20MeV$ for $\rho=0.5fm^{-3}$ (Fig. 1), we have
found that this ratio is $1.24$. We see that this value is smaller than the
above limiting value which indicates that the entropy of polarized case of the
hot nuclear matter is always smaller than the entropy of unpolarized case.
This so-considered ”natural” behavior was also found in the case of Gogny dapv
and in the BHF analysis of Ref. bprrv . In contrast, for Skyrme forces the
entropy per particle of the polarized phase is seen to be higher than the non-
polarized one above a certain density dapv .
Finally, we have plotted the pressure of the polarized hot nuclear matter as a
function of the total number density ($\rho$) for different polarizations at
${\cal T}=10$ and $20$ MeV in Fig. 8. For all values of temperature and
polarization, it is seen that the pressure increases by increasing the
density. For this system, we see that at each temperature the equation of
state becomes stiffer as the polarization increases. For each polarization, it
is found that the pressure of the polarized hot nuclear matter increases by
increasing the temperature.
## IV Summary and Conclusions
The lowest order constrained variational (LOCV) method has been used for
calculating the susceptibility of the polarized hot nuclear matter and some of
the thermodynamic properties of this system such as the effective mass, free
energy, entropy and the equation of state. In our calculations, we have
employed the $AV_{18}$ potential. Our results show that the spontaneous
transition to the ferromagnetic phase does not occur for the hot nuclear
matter. We have seen that the spin polarization substantially affects the
thermodynamic properties of the hot nuclear matter.
###### Acknowledgements.
This work has been supported by Research Institute for Astronomy and
Astrophysics of Maragha. We wish to thank Shiraz University and Zanjan
University Research Councils.
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Figure 1: The effective mass of spin-up (full curve) and spin-down (dashed
curve) nucleons versus the spin polarization ($\delta$) for density
$\rho=0.5fm^{-3}$ at ${\cal T}=20$ MeV.
Figure 2: The free energy per nucleon of the polarized hot nuclear matter as a
function of the total number density ($\rho$) for different values of the spin
polarization ($\delta$) at ${\cal T}=10$ (a) and ${\cal T}=20$ MeV (b).
Figure 3: The free energy per nucleon of the nuclear matter versus the total
number density ($\rho$) for unpolarized case at ${\cal T}=0$, $10$ and $20$
MeV .
Figure 4: The magnetic susceptibility of the hot nuclear matter versus the
temperature at $\rho=0.16fm^{-3}$ .
Figure 5: The magnetic susceptibility of the hot nuclear matter versus the
total number density ($\rho$) at ${\cal T}=20$ MeV.
Figure 6: As Fig. 4 but for the entropy difference of the fully polarized and
the unpolarized cases.
Figure 7: The entropy per nucleon as a function of the spin polarization
($\delta$) for density $\rho=0.5fm^{-3}$ at ${\cal T}=20$ MeV .
Figure 8: The equation of state of the polarized hot nuclear matter for
different values of the spin polarization ($\delta$) at ${\cal T}=10$ (a) and
${\cal T}=20$ MeV (b).
|
arxiv-papers
| 2009-09-08T08:53:45 |
2024-09-04T02:49:05.117255
|
{
"license": "Public Domain",
"authors": "M. Bigdeli, G.H. Bordbar and Z. Rezaei",
"submitter": "Gholam Hossein Bordbar",
"url": "https://arxiv.org/abs/0909.1427"
}
|
0909.1444
|
Scaling functional patterns of skeletal and cardiac muscles: New non-linear
elasticity approach
.
Valery B. Kokshenev
.
Departamento de Fisica, Instituto de Ciencias Exatas, Universidade Federal de
Minas Gerais, Caixa Postal 702, 30123-970, Belo Horizonte, Minas Gerais,
Brazil. Email: valery@fisica.ufmg.br
.
Submitted to the Physica D 15 August 2009
.
Abstract: Responding mechanically to environmental requests, muscles show a
surprisingly large variety of functions. The studies of in vivo cycling
muscles qualified skeletal muscles into four principal locomotor patterns:
motor, brake, strut, and spring. While much effort of has been done in
searching for muscle design patterns, no fundamental concepts underlying
empirically established patterns were revealed. In this interdisciplinary
study, continuum mechanics is applied to the problem of muscle structure in
relation to function. The ability of a powering muscle, treated as a
homogenous solid organ, tuned to efficient locomotion via the natural
frequency is illuminated through the non-linear elastic muscle moduli
controlled by contraction velocity. The exploration of the elastic force
patterns known in solid state physics incorporated in activated skeletal and
cardiac muscles via the mechanical similarity principle yields analytical
rationalization for locomotor muscle patterns. Besides the explanation of the
origin of muscle allometric exponents observed for muscles in legs of running
animals and wings of flying birds, the striated muscles are patterned through
primary and secondary activities expected to be useful in designing of
artificial muscles and modeling living and extinct animals.
.
PACS: 89.75.Kd, 89.75.Da, 87.10.Pq, 87.19.Ff
Key words: Dynamic patterns, Scaling laws, Non-linear elasticity, Muscles.
## I Introduction
The mechanical role of muscles varies widely with their architecture and
activation conditions. Striated (skeletal and cardiac) muscles are diverse in
their contractive interspecific and intraspecific functional properties
observed among and within animal species, nevertheless, ”the smaller muscles
and muscles of smaller animals are quicker”. This generic feature of skeletal
muscles was established by Hill [1]. More recently, the physiological
adaptation of skeletal muscles resulting in beneficial changes in muscle
function has been recognized by a number of investigators. It was learned that
long-fibre muscles commonly contract at over larger length ranges and
relatively higher velocities producing the greatest muscle forces the lowest
relative energetic costs [2]. Muscles having shorter fibres expose smaller
length change, but their cost of force generation is relatively less, e.g.
[3]. Searching for determinants of evolution of shape, size, and force output
of cardiac and skeletal muscle, a little is known about the regulation of
directional processes of mass distribution [4,5]. Although skeletal muscles
grow in length as the bones grow, most studies only involve force increasing
with cross-sectional area. Following the idea that the muscle force production
function is a critical evolutionary determinant [5], I develop a physical
study of muscle form adaptation to a certain primary activity with growth of
size preserving muscle shape.
When designing architecture of the striated muscle built from repeating units
(fibres and sarcomeres) at least three distinct muscle activities should be
distinguished [5]: (i) the _concentric_ contraction defined as the production
of active tension while the muscle is shortening and performing positive work,
(ii) the _eccentric_ contraction defined as the contraction during lengthening
performing negative work in a controlled fashion, and (iii) the _isometric_
contraction when the muscle force output is produced without changing of
length and performing net work. The corresponding _mechanical work patterns_
called by Russel et al. [5] as ”concentric work” and ”eccentric work” (that
might be extended by ”isometric work”) were carefully studied via in vivo
measurements of length-force cycling of individual skeletal muscles in active
animals, such as (i) the pectoralis in flying birds, (ii) leg extensors in
running cockroaches, and (iii) gastrocnemius in the level running turkey. The
corresponding _muscle locomotor patterns_ were called as (i) _motor_ , (ii)
_brake_ , and (iii) _strut_ functions [6].
The seminal research by Hill [1] on dynamics of electrically stimulated
_isolated_ muscles was restricted to a single isotonic shortening. The studies
of the corresponding motor function resulted in famous force-inverse-velocity
master curve remaining the major dynamic constraint of all real (slow-fibre,
fast-fibre, and superfast) muscles [7] and computationally modeled muscles,
e.g. [8]. Besides, other two fundamental rules of muscle dynamics were noted
by Hill [1]. Examining hovering humming and sparrow birds, he recognized that
the ”frequencies of wings are roughly in inverse proportion to the cube roots
of the weights, i.e. to linear size”. Moreover, because the linear
proportionality between the stroke period and body length was found equally in
electrically stimulated isolated muscles, the intrinsic _frequency-length_
feature constrained by scaling rule $T_{m}^{-1}\varpropto L_{m}^{-1}$ beyond
the nervous control is likely more universal than previously appreciated.
Second _velocity-length_ Hill’s constraint states that ”the intrinsic speed of
muscle has to vary inversely to length”, i.e. $V_{m}\varpropto L_{m}^{-1}$.
Both Hill’s scaling rules remain a challenge to viscoelastic models of
transient-state mechanics and other existing theories of muscle contraction
[9].
The earliest theories of muscle motor function supposed muscle to be an
elastic body which, when stimulated, was converted in an active state
containing elastic energy causing the muscle to shorten. Such _elastic-energy
theories_ failed to explain mechanisms of the force production in terms of
viscoelastic characteristics. To a certain extent, poor experimental
approaches providing often conflicting clues to muscle structure in relation
to function may explain a little progress in understanding of contractile
properties of a muscle [9,4]. Moreover, physiological muscle properties
accounted for theories of muscle contractions developed at both molecular and
macroscopic scales are primarily focused on the reproduction of force-velocity
curve [9]. Besides, the existing phenomenological frameworks such as Hill-type
muscle models only mimic the proper mechanical characteristics of muscles by
means of passive viscoelastic springs attached to muscle contractive element
in series [10,11,3] or in parallel [12] and recruited when muscle is
activated. By ignoring the proper muscle function of force production and
force transmission throughout the muscle organ, these models are able to
explain no one of Hill’s principal constraints in muscle dynamics. On the
other hand, there exist experimental evidences of the adaptive ability of
skeletal muscle to exchange _elastic_ strain energy during force production
[2]. In line with this concept, it has recently communicated on a possibility
of the modeling of the adaptive muscle elasticity by elastic force patterns
[13].
In the present paper, I develop an integrative theoretical framework to the
problem of forces, structure, and contractive non-linear dynamics in striated
muscles. Instead of Hill-type modeling of in vitro motor function, e.g. [3],
brake function, e.g. [12, 2], and strut function, e.g. [14], or study of
muscle design by means of simulation of phenomenological force-length and/or
force-velocity constraints [8], the powerful method of continuum mechanics
generally providing macroscopic characterization and modeling of soft tissues,
e.g. [15, 16], is employed. By further exploration of the elastic force
patterns, I propose a self-consistent depiction of the three velocity-distinct
characteristic points well distinguished in all in vivo force-length loops of
the naturally activated muscles. Unlike the earliest elastic theories based on
minimization of energy, I develop the physical concept of similarity between
the force output and reaction active elastic forces that permits to avoid the
details of muscle activation process. The theory is validated by a comparison
to phenomenological scaling rules including both mentioned Hill’s dynamic
constraints and therefore may be hopefully helpful in designing artificial
muscles [15] and modeling living and extinct organisms [17].
## II Theory
### II.1 Theoretical Background
#### II.1.1 McMahon’s scaling models
The engineering models by McMahon [18, 19] develop previous Hill’s approach to
the problem of scaling of parameters of animal performance to _body weight_
$W=Mg$. Using Hill’s geometric similarity models [1,19] equally applied to
animal body, long bone, or individual muscle, each one was approximated by a
cylinder of longitudinal _length_ $L$ and _cross-sectional area_ $A$ (or
diameter $D\backsim\sqrt{A}$). Then, the assumption on the _weight-invariance_
of for the _tissue density_ , namely
$\rho_{tiss}=\frac{M}{AL}\varpropto W^{0}\text{,}$ (1)
was adopted. In mammalian long-bone allometry, this invariant was verified and
observed with a high precision [20]. Mechanical models of bending bones and
shortening muscles were introduced by McMahon via the weight-invariant
_elastic modulus_ $E_{tiss}$, _stress_ $\sigma_{tiss}$, and _strain_
$\varepsilon_{tiss}$, namely
$E_{tiss}=\frac{\sigma_{tiss}}{\varepsilon_{tiss}}\varpropto W^{0}\text{, with
}\sigma_{tiss}=\frac{\Delta F}{A}\text{ and }\varepsilon_{tiss}=\frac{\Delta
L}{L}\text{.}$ (2)
Here $\Delta L$ ($=L-L_{0}$) is the _length change_ accompanied by the _force
change_ $\Delta F$ ($=F-F_{0}$) counted off from the _resting length_ $L_{0}$.
While searching for functional mechanical patterns of biological systems
determined by _maximal_ forces using Eqs. (1) and (2), the maximal
stress/strain scaling relations
$\sigma_{geom}^{(\max)}\varpropto W^{1/3}\text{,
}\sigma_{elast}^{(\max)}\varpropto W^{1/4}\text{, and
}\sigma_{stat}^{(\max)}\varpropto W^{1/5}\text{,}$ (3)
could be readily derived from McMahon’s _geometric_(isometric), _elastic_
(buckling stress) and _static_ (bending elastic stress) _similarity models_
distinguished through McMahon’s scaling relations
$L_{geom}\backsim D\text{, }L_{elast}\varpropto D^{2/3}\text{, and
}L_{stat}\varpropto D^{1/2}\text{.}$ (4)
Instead, the _maximum_ stress and strain
$\sigma_{tiss}^{(\max)}\text{ }\varpropto\varepsilon_{tiss}^{(\max)}\varpropto
W^{0}\text{,}$ (5)
were postulated (see Table 4 in [19]) extending groundlessness his exact
result for the _mean_ stress $\sigma_{elast}^{(mean)}\varpropto W^{0}$,
obtained within the static stress similarity model (see Fig. 1 in [19]). The
improved self-consistent maximal stresses shown in Eqs. (3) follow
straightforwardly from McMahon’s cross-sectional areas
$A_{geom}^{(isom)}\varpropto W^{2/3}\text{, }A_{elast}^{(buck)}\varpropto
W^{3/4}\text{, and }A_{static}^{(bend)}\varpropto W^{4/5}$ (6)
applied to Eq. (2), along with McMahon’s idea on the dominating gravitational
forces in bones, muscles, and bodies, i.e. $\Delta F\backsim gM_{b}\backsim
gM_{m}\backsim W$. As shown in [20], the structure of long bones is driven by
peak muscle forces, but not by gravity.
#### II.1.2 Muscle shape and structure
After Alexander [21], the _physiologic_ cross-sectional area $A_{0m}$ (PCSA)
of the isolated skeletal _muscle_ $m$__ of _mass_ $M_{m}$ composed of $N$
bundles of masses $m_{i}$ was commonly estimated, e.g. [22], with the help of
the cylinder-geometry relation $A_{i}=m_{i}/\rho_{musc}L_{i}$, where $L_{i}$
is directly measured muscle fibre length. The spindle-like shape of the muscle
as whole organ was therefore determined by the muscle PCSA, namely __
$\text{
}A_{0m}={\displaystyle\sum\limits_{i=1}^{N}}A_{i}=\frac{M_{m}}{\rho_{musc}L_{0m}}\text{,
and
}\frac{1}{L_{0m}}=\frac{1}{M_{m}}{\displaystyle\sum\limits_{i=1}^{N}}\frac{m_{i}}{L_{i}}\text{,
with }M_{m}={\displaystyle\sum\limits_{i=1}^{N}}m_{i}\text{,}$ (7)
resulted in the sum of areas of muscle and the muscle length $L_{0m}$ of the
parallel-linked contractible subunits described statistically by the length-
unversed sum weighed by masses. Such a coarse-grained characterization of the
_muscle structure_ generally ignores the arrangement of muscle fibres relative
to generated force axis, distinguished by _pinnate angles_.
In scaling models, the evolution of the muscle structures across different-
sized animals of _body mass_ $M$ is observed statistically via _allometric
exponents_ $a_{m}$, $l_{m}$, and $\alpha_{m}$ determined by common rules
[21,23,25]:
$A_{0m}\varpropto M_{m}^{a_{m}}\text{, }L_{0m}\varpropto M_{m}^{l_{m}}\text{,
and }M_{m}\varpropto M^{1+\alpha_{m}},$ (8)
where the _muscle mass index_ $\alpha_{m}$ plays the same role as Prangel’s
index $\beta$ in bones, as noted in [26].
When the muscle-density invariance employed implicitly in Eq. (7) and
specified in Eq. (1) is applied to different skeletal muscles, the muscle
shape approximated by cylinder geometry is also preserved. Consequently, the
_muscle functional volume_
$A_{m}L_{m}=A_{0m}L_{0m}=\frac{M_{m}}{\rho_{0m}}\text{, with
}\rho_{0m}=\rho_{musc}\varpropto M_{m}^{0}\varpropto M^{0}\text{,}$ (9)
holding in all muscle work loops plays the role of the mechanical muscle
invariant. This condition is ensured by functional change
$\Delta\rho_{musc}/\rho_{musc}$ not exceeding $5\%$ [24]. Hence, the function-
independent _muscle-shape constraint_ [13]
$a_{m}+l_{m}=1+\alpha_{m}$ (10)
straightforwardly follows from Eqs. (8) and (9). Likewise the case of hindlimb
mammalian bones of the mean structure $a_{b}^{(\exp)}=2d_{b}^{(\exp)}=0.752$,
$l_{b}^{(\exp)}=0.298$, and $\beta^{(\exp)}=0.04$ [20,26], Eq. (10) is also
empirically observable in muscle allometry (see analysis in Table 5 below).
### II.2 General Muscle Characterization
#### II.2.1 Maximal force and stress
In in vivo work loops, the muscle locomotor patterns can be generally
specified regardless of details of activation-deactivation conditions. In Fig.
1, the linear-slope characteristics $L_{1m}$ can be introduced in the force-
length cycling by the domains: $L_{2m}<L_{1m}<L_{3m}\approx L_{0m}$, for the
motor function, $L_{2m}>L_{1m}>L_{3m}$, for the brake function, and by
$L_{2m}\gtrsim L_{1m}\gtrsim L_{3m}\approx L_{0m}$, for the strut function
showing nearly isometric muscle contractions.
.
Place Fig. 1
.
Moreover, such a qualitative general characterization of the activated
individual muscle $m$ of _resting length_ $L_{0m}$ can be rationalized on the
basis of common _two-point_ force-length characterization, namely
$F_{musc}^{(\exp)}(L_{2m})=F_{musc}^{(\max)}=F_{2m}\text{ and
}F_{musc}^{(\exp)}(L_{1m})=F_{1m}\text{,}$ (11)
__ introduced by the maximum force __ $F_{2m}^{(\max)}$ and the _optimum_
muscle length [28, 29] $L_{1m}$ . The _instant dynamic length_
$L_{m}=L_{1m}\pm\Delta L_{1m}$ is counted off from the characteristic point
$L_{1m}$ via the optimum length change $\Delta L_{1m}$ (11) shown in Fig. 1
for all functions.
First, the linearization of the in vivo muscle force-length curve allows one
to determine _trial_ peak stress and corresponding strain by
$\sigma_{musc}^{(\max)}=\frac{F_{musc}^{(\max)}}{A_{2m}}\text{ and
}\varepsilon_{musc}^{(\max)}=\frac{\Delta L_{1m}^{(\max)}}{L_{2m}}\text{, with
}\Delta L_{1m}^{(\max)}=|L_{2m}-L_{1m}|\text{. }$ (12)
The corresponding force change $\Delta F_{musc}^{(\max)}$ observed near the
_optimum force_ $F_{1m}^{(\max)}$ (11) provides
$F_{musc}^{(\max)}=F_{musc}^{(\exp)}(L_{1m})+\Delta
F_{musc}^{(\max)}=F_{1m}+K_{musc}^{(\max)}\Delta L_{1m}^{(\max)}\text{ }$ (13)
that determinates _effective_ _muscle stiffness_ and _effective modulus_ ,
respectively
$\displaystyle K_{musc}^{(\max)}$ $\displaystyle\equiv
K_{2m}=\left|\frac{dF_{musc}}{dL_{m}}\right|_{F_{1m}^{(\max)}}\thickapprox\frac{\Delta
F_{musc}^{(\max)}}{\Delta L_{1m}^{(\max)}}=\frac{\Delta
F_{musc}^{(\max)}}{F_{musc}^{(\max)}}E_{musc}^{(\max)}\frac{A_{2m}}{L_{2m}}\text{,
}$ $\displaystyle\text{and }E_{musc}^{(\max)}$ $\displaystyle\equiv
E_{2m}=\frac{\sigma_{musc}^{(\max)}}{\varepsilon_{musc}^{(\max)}}\text{,}$
(14)
following from Eqs. (12) and (13).
#### II.2.2 Active stiffness and resonant muscle mechanics
Secondly, treating the maximum-force crossover state (11) as the generic
transient-neutral __ state [26] the _resonant_ _frequency_
$1/T_{musc}^{(\max)}=T_{2m}^{-1}$ related to point $2$ in Fig. 1 associated
with maximum efficiency of muscle cycling, e.g. [29], can also be introduced
as _natural frequency_ [19], namely
$T_{2m}^{-1}\thicksim
2\pi\sqrt{\frac{K_{2m}}{M_{m}}}\thicksim\sqrt{\frac{E_{musc}^{(\max)}}{\rho_{0m}}}\left(\frac{\Delta
F_{musc}^{(\max)}}{F_{musc}^{(\max)}}\right)^{1/2}\frac{1}{L_{2m}}\text{,}$
(15)
and analyzed by Eqs. (9) and (14).
One can see that Eq. (15) yields first Hill’s general constraint discussed in
Introduction. However, the following three conditions are required: (i) the
preservation of dynamic functional volume __(9), (ii) the weight-invariance of
the elastic modulus $E_{musc}^{(\max)}$ (2), and (iii) the existence of force
similarity between the exerted force $F_{musc}^{(\max)}$ and its change
$\Delta F_{musc}^{(\max)}$ (13). Therefore, the muscle _force-similarity
principle_ , namely
$F_{musc}\cong\Delta F_{musc}\cong F_{prod}\cong F_{elast}\cong\Delta
F_{elast}\text{,}$ (16)
implying a coexistence of all forces in biomechanically equivalent states [26]
must be adopted. Here the _active elastic force_ $\Delta F_{elast}$ (shown
schematically as $F_{act}$ in Fig. 1D) is also included. The total state-
transient elastic force $F_{elast}$ is the superposition of common _passive
elastic force_ $F_{pass}$ provoked by external loads and _active elastic
force_ $\Delta F_{elast}$ caused by the _production_ force $F_{prod}$. The
correspondence sign $\cong$ indicates that though the involved physical
characteristics belong to the same mechanical state, they may differ in both
physical and numerical parameters stipulating this state.
Given that the peak _active muscle stress_ $\sigma_{m}$ always exceeds the
corresponding passive stress, e.g. [14], in further I focus on the fully
activated _transient states_ described by
$\sigma_{m}=\frac{\Delta F_{elast}}{A_{m}}=E_{m}\frac{\Delta
L_{m}}{L_{m}}\text{. }$ (17)
Unlike Eqs. (2) and (12), $\sigma_{m}$ is the true intrinsic _elastic stress_
in a certain (not specified) dynamic state. This reveals the maximum-amplitude
_elastic force_ of the fully activated muscle
$\Delta F_{elast}\equiv\Delta F_{m}=K_{m}\Delta L_{m}=E_{m}A_{m}\frac{\Delta
L_{m}}{L_{m}}$ (18)
and in turn provides the corresponding _active_ _muscle stiffness_
$K_{m}=E_{m}\frac{A_{m}}{L_{m}}\text{.}$ (19)
The underlying mechanical _sarcomere elastic stiffness_ $K_{s}$ is related via
the muscle-volume average, namely
$K_{m}=\frac{1}{A_{m}L_{m}}\int K_{s}(r_{m})\text{ }d^{3}r_{m}\text{,}$ (20)
originated from end-to-end intercellular overlapping [31, 12].
The _muscle energy change_
$\Delta U_{m}\backsim K_{m}\Delta L_{m}^{2}\cong E_{m}A_{m}\frac{\Delta
L_{m}^{2}}{L_{m}}$ (21)
stored or released during active-period contraction provides the mechanical
_cost of energy_
$CU_{m}=\frac{\Delta U_{m}}{\Delta L_{m}}\cong E_{m}A_{m}\frac{\Delta
L_{m}}{L_{m}}\text{.}$ (22)
These relations demonstrate how the observable mechanical characteristics can
be linked to the underlying muscle elastic forces using the force-similarity
principle (16). In turn, the _contraction velocity_
$V_{m}=\overline{V_{m}(t)}\equiv\frac{1}{\Delta t_{m}}\int_{0}^{\Delta
t_{m}}\left[\frac{dL_{m}(t)}{dt}\right]dt\backsim\left[\frac{dL_{m}(t)}{dt}\right]_{t=\Delta
t_{m}}\cong\frac{L_{m}}{T_{m}}$ (23)
is defined by the instant velocity $V_{m}(t)$ averaged over _activation time_
$\Delta t_{m}$.
#### II.2.3 Fast and slow activated muscles
According to the most general classification of diverse muscles, three types
are conventionally distinguished: red (slow fibre) muscles, white (fast fibre)
muscles, and intermediate type, mixed fibre muscles. Although collective
mechanisms of muscle contractions are poor understood, e.g. [32], physically,
the two limiting situations of dynamic accommodation of local forces generated
by cross bridge attachments can be generally rationalized. As schematically
drawn in Fig. 1D, in an activated muscle, the dynamic process of equilibration
between the production intrinsic forces and external loads (not shown) is
followed by the spatiotemporal relaxation of elastic forces. For the simplest
case of _slow muscles_ , the dynamic equilibration occurs via the slow channel
of relaxation, assumably common for both active, $F_{prod}^{(slow)}$, and
passive elastic forces. Since passive forces in solids are short of range
[33], both the forces are proportional to muscle _surface_. In contrast, it is
plausible to adopt that in _fast muscles_ the fast-twitch fibres transmit the
locally generated forces in all directions, i.e. along and across fibres,
resulting in the overall maximum force output $F_{prod}^{(fast)}$ to be linear
with dynamic muscle _volume_. Basing on such a general physical picture, a
function-independent and regime- independent characterization of the force
production function, namely
$F_{prod}^{(fast)}\varpropto A_{rm}L_{rm}\text{ and
}F_{prod}^{(slow)}\varpropto A_{rm}\text{, with }r=1,2\text{, and }3\text{,}$
(24)
is proposed via the force-size scaling rules for all three distinct states
shown in Fig. 1 and hereafter distinguished by symbol $r$.
The widely adopted by biologists linear-displacement regime is discussed in
Eq. (2) via $\Delta L\varpropto L$ resulted in the weight-independent strain
(5). The corresponding optimum-velocity regime $r=1$, attributed to the
instant length-independent elastic strains,
$\varepsilon_{m}^{(opt)}=|L_{m}-L_{3m}|/L_{m}\varpropto L_{m}^{0}$ (2) with
$L_{m}$ lying between $L_{1m}$ and $L_{3m}\approx L_{0m}$, is now clarified by
the scaling equations
$E_{1m}^{(fast)}=E_{fast}^{(opt)}\varpropto L_{m}^{1}\text{ and
}E_{1m}^{(slow)}=E_{slow}^{(opt)}\varpropto L_{m}^{0}$ (25)
characteristic of fast and slow muscles. Such a muscle description follows
from the similarity (16) between the active elastic force $\Delta
F_{1m}=\Delta F_{elast}^{(opt)}=E_{1m}A_{1m}\varepsilon_{1m}$ (18) and
corresponding production force (24). The _optimum_ force-velocity muscle
mechanics is rationalized below in Table 1 and then tested by empirical data.
Similarly, the bilinear-displacement regime $r=2$ introduced by the dynamic
length change $\Delta L_{2m}=|L_{m}-L_{1m}|\varpropto L_{m}^{2}$, with $L_{m}$
lying between $L_{2m}$ and $L_{1m}$, and the _maximum_ active elastic force
$\Delta F_{2m}=\Delta
F_{elast}^{(\max)}=E_{2m}A_{2m}\varepsilon_{2m}^{(\max)}$ (18) results in the
maximal elastic moduli
$E_{fast}^{(\max)}=E_{2m}^{(fast)}\varpropto L_{m}^{0}\text{ and
}E_{slow}^{(\max)}=E_{2m}^{(slow)}\varpropto L_{m}^{-1}\text{,}$ (26)
adjusted with the muscle production function (24) via the force similarity
principle (16). Finally, the high-velocity trilinear regime $r=3$ is suggested
by the moderate-force and moderate-elastic muscle determined by
$E_{fast}^{(\operatorname{mod})}=E_{3m}^{(fast)}\varpropto L_{m}^{-1}\text{
and }E_{slow}^{(\operatorname{mod})}=E_{3m}^{(slow)}\varpropto
L_{m}^{-2}\text{. }$ (27)
This condition specifies point $3$ in Fig. 1, along with the underlying cubic-
power muscle displacements $\Delta L_{3m}\varpropto L_{m}^{3}$ scaled by
dynamic $L_{m}$ lying above or below the characteristic length $L_{3m}$ in any
muscle acting as motor, brake or strut (see Fig. 1).
### II.3 Muscle Functions
Likewise the naturally curved mammalian long bones biomechanically adapted to
the maximum longitudinally bending [20, 26], the muscle _motor function_ is
assigned to locomotor muscles showing concentric positive work exerted by
elastic bending forces. Given that the _elastic force patterns_ coincide for
bending and torsion [26], both kinds of unpinnate and uni-pinnate skeletal
muscles, having respectively close to zero and non-zero fixed pinnate angles,
may be expected to be structured by the same motor function. The specific-
function mechanical characterization is described in Appendix B and results
are summarized in Table 2.
## III Results
### III.1 Assumptions and predictions
The following assumptions are made regarding elastic striated muscles:
1\. The muscles are considered at macroscopic scale as individual homogeneous
organs. Within the continuum mechanics, the coarse-grained approach ignores
the details of heterogeneous microstructure and pinnate angles.
2\. When activated under different boundary loaded conditions, the muscles do
not undergo changes in shape and whole volume. The emerging elastic fields
follow patterns established for long solid cylinders.
3\. The mechanical similarity adopted between the extrinsic forces exerted by
the muscle and intrinsic elastic reaction forces, as well as the dynamic
similarity adopted for contraction velocities and frequencies are observable
in all biomechanically equivalent states.
4\. The natural ability of the non-linear elastic tuning of fast and slow
muscles to distinct locomotor states can be characterized by the elastic
moduli sensitive to evolving dynamic variable associated here with the regime-
characteristic muscle length.
.
The function-independent mechanical characterization of muscles is provided in
Table 1.
.
Place Table 1.
.
The specific case of muscle structure accommodation in the bilinear regime is
described in Table 2.
.
Place Table 2
.
The rules of mass distribution across and along the muscle axis provided in
Table 2 in terms of the muscle-structure scaling exponents [$a_{2m}$,
$l_{2m}$] are characteristic for slow, fast, and mixed muscles producing
maximum force. In Table 3, these scaling rules are compared with the finding
for the optimal-force state [$a_{1m}$, $l_{1m}$] and moderate-force state
[$a_{3m}$, $l_{3m}$].
.
Place Table 3
.
The dynamic characteristics of distinct-velocity contractions are predicted in
Table 4.
.
Place Table 4
.
The consequences of the theoretical scaling framework are:
1\. The peak forces generated in all regimes scales as muscle volume or
surface in fast or slow muscles, respectively.
2\. A general, function-independent mechanical description of the striated
muscle activated in the liner-displacement regime is predicted for each type
of muscles (Table 1).
3\. The muscle-type independent locomotor functions and related mechanical and
dynamic characteristics of the striated muscle activated in the bilinear
regime are predicted (Table 2).
4\. The muscle-type independent varied dynamic structures are predicted for
all muscle regimes and functions (Table 3).
5\. The function-independent dynamic scaling characteristics are obtained in
Table 4 for all type of muscles.
In what follows, all theoretical findings are tested by the available from the
literature data.
## IV Discussion
”What determines the shape, size, and force output of cardiac and skeletal
muscle?” (Louis Sullivan quoted in [5]). The provided coarse-grained study of
conservative striated muscles suggests that the size-dependent peak elastic
forces determine fiber-type-independent patterns of the functionally adapted
structures preserving muscle shape. The size-dependent peak force output is
determined by the muscle volume and area for white and red muscles, regardless
of muscle structure and function.
### IV.1 Function against structure
#### IV.1.1 General muscle characterization
Being composed of bundles of muscle fibres including all other contractible
components (neural, vascular, and collagenous reticulum), the striated muscle
is thought of as a heterogeneous _continuum medium_ transmitting the produced
tension internally and externally, e.g. [34]. Primarily, I address the problem
of mechanical design of striated muscle to a general, function-independent
characterization of the individual muscle organ loaded by tension, reaction,
and gravity through tendons, ligaments, and bones. My non-energetic approach
is physically grounded by the existence of linear force-displacement regions
(shown by the solid arrows in Figs. 1A, 1B, and 1C) in all in vivo work loops
regardless of dynamic details of approaching to the maximum exerted force
$F_{musc}^{(\max)}$. Hence, the mechanical characterization of the maximum-
force activated muscle arises from the muscle __ stiffness $K_{m}^{(\max)}$
(14) underlaid by sarcomere stiffness $K_{s}^{(\max)}$ (20). Consequently, all
forces involved in muscle contraction following by active and passive elastic
strains allow common mechanical description (shown in Fig. 1D) not depending
on their biochemical, inertial, or reaction origin.
The analytical justification of Hill’s first frequency-length constraint
arises from the analysis of Eq. (15) that requires eventually the usage of the
similarity between all intrinsic muscle forces, Eq. (16). The constraint
$T_{m}^{-1}\varpropto L_{m}^{-1}$ and other mechanical characteristics for
slow muscles accumulated in Table can be applied to _steady-speed_ regimes of
locomotion modes where all forces are generally equilibrated and controlled by
slow-fibre muscles [35]. In the case of non-steady transient locomotion when
fast-twitch fibres and nervous control are additionally requested [35], Hill’s
first constraint transforms [by Eqs. (15) and (25)] into a new one,
$T_{m}^{-1}\varpropto L_{m}^{-1/2}\varpropto 1/V_{fast}^{(opt)}$ (Tables 1 and
4), well known for animals running with maximal _optimum_ speed [36,37]
$V_{run}^{(\max)}\backsim
V_{fast}^{(opt)}\varpropto\sqrt{L}\varpropto\sqrt{L_{m}}$. We have therefore
demonstrated how the suggested _dynamic similarity_ establishes a link between
the body-propulsion speed and locomotor-muscle contraction velocity, also
described by Rome et al. [38]. Being united with the muscle-force similarity,
both constraints yield _mechanical similarity_ , the key principle explored in
this research.
#### IV.1.2 Maximum force output against structure and velocity
In muscle physiology, the functional effect of muscle conceptual architecture
simply states that muscle force output is proportional to PCSA. The proposed
study of adaptation of the muscle structure via the force production function
seems to be in qualitative agreement with this statement, because in all cases
exposed in Eq. (24) the muscle force output is _proportional_ to $A_{m}$. Such
a simplified treatment of the fast-muscle mechanics (formally substituted by
that for slow muscles) arrived at the widely adopted opinion that the peak
muscle stress $F_{prod}^{(slow)}/A_{m}$ , specifying the case of slow muscles
in linear dynamic regimes with $\sigma_{m}^{(slow)}\varpropto L_{m}^{0}$
(Table 1), is generic for any muscle, as already discussed in Eq. (5).
Although the proposal on scaling of the maximum production force (and active
stress) with muscle size (24) is a challenge for further research, the
provided fairly general physical grounds are supported by empirical
observations by Marden and Allen [39]. They established statistically that the
maximum force output in all biological (and human-made) motors falls into two
fundamental scaling laws: (i) in fast-cycling motors, presented by flying
insects, bats and birds, swimming fishes, and running animals it scales as
(_motor mass)_ 1 and (ii) in slow-cycling motors, such as myosin molecules,
muscle cells, and some (unspecified) ”whole muscles” the force at output
scales as (_motor mass_)2/3. The ”motor mass” was associated with muscle (and
fuel) mass. That fact that the authors observed muscle motors from sarcomere
to whole muscle organ passing through the single-fibre level of muscle
organization, makes a basis for the discussed below _micro-macro scale
correspondence_.
The proposed treatment of the in vivo force-length curves is provided for
three distinct force-velocity characteristic points (shown in Fig. 1)
correlated by the inequalities
$F_{2m}>F_{1m}>F_{3m}\text{ and }V_{2m}<V_{1m}<V_{3m}\text{.}$ (28)
These three generic function-independent states are associated with the linear
($r=1$), bilinear ($r=2$), and trilinear ($r=3$) muscle dynamics determined
via the muscle elastic moduli $E_{rm}$ in Eqs. (25), (26), and (27),
respectively. The mechanical characterization of slow and fast striated
muscles is therefore provided in terms of the maximum ($\Delta F_{2m}$),
optimum ($\Delta F_{1m}$) and moderate ($\Delta F_{3m}$) active elastic forces
developed at the measurable maximum ($V_{3m}$), optimum ($V_{1m}$), and
moderate ($V_{2m}$) contraction velocities (Table 4). The stabilization of the
dynamic regimes is expected at the natural frequencies, which also are scaled
in Table 4 to the dynamic length $L_{rm}$.
#### IV.1.3 Muscle functions against size and shape
Searching for answer on ”what features make a muscular system well-adapted to
a specific function?” [28], it has been shown preliminary [13] that such
features are related to natural conditions of the stabilization or tuning to
the moderate-velocity regime $r=2$ via the mean dynamic length of the fast-
twitch fibers adapted by the best way to one of the patterns of muscle
locomotor functions. In this study such features specify the role of slow-
twitch fibers.
The elastic-force patterns underlying concentric, eccentric, isometric, and
cardiac contractions are suggested in Eqs. (35), (39), (42), and (44),
respectively. The solutions to the muscle-force and muscle-shape constraints
are accumulated in Table 2 as patterned functions well distinguished by the
muscle _structure parameter_ ($\eta_{m}=d\ln A_{m}/d\ln L_{m}$) established
for the motor ($\eta_{1}=4$), brake ($\eta_{2}=3$), strut ($\eta_{3}=\infty$)
skeletal muscles, an extended by the pump ($\eta_{5}=1$) cardiac muscle and
one spring ($\eta_{4}=2$) striated muscle. These structurally adapted muscles
are thought of as to be suited to efficient work during powering when,
respectively, shortening ($m=1$), lengthening ($m=2$), or remaining in the
nearly isometric dynamic state ($m=3$), high-pressure-resistant state ($m=4$),
and likely energy-saving state ($m=5$). The found new pump function is in
accord with the observation by Russel et al. [5] that ”the heart chamber,
unlike skeletal muscles, can extend in both longitudinal and transverse
directions, and cardiac cells can grow in length and width”, that implies
$\eta_{5}<$ $\eta_{1},\eta_{2}$, or $\eta_{3}$. Given that only a few patterns
exist in elastic theory of solids [26], it is not striking that the spring,
brake, and motor functions resembles McMahon’s ”geometric”, ”elastic”, and
”static” stress similarities discussed in Eqs. (3) and (4).
In Table 3, conceivable stable dynamic structures corresponding to muscle
activity in different dynamic regimes are analyzed. As in the case of Table 2,
the solutions of dynamic constraints follow from the similarity between force
output (16) and elastic-force patterns. The resulting _dynamic_ states are
discussed in terms of the scaling exponents for the muscle _dynamic structure_
[$A_{rm}$, $L_{rm}$] preserving muscle shape and volume (9). Other related
observable mechanical characteristics are exemplified in Tables 1 and 2. The
major outcome of the analysis in Tables 2 and 3 is that both slow-twitch and
fast-twitch fibres belonging to the same muscle $m$ should manifest concerted
behavior coordinated by the dynamic active elastic forces.
Another significant feature of the analysis in Table 3 is a striking
prediction of the mechanical functions which are expected to be shown by a
given striated muscle $m$ of certain specialization (_primary functions_
indicated by regime $r=2$, see proof below) when its cycling dynamics is
switched to regimes $r=1$ and $3$ by tuning to the corresponding natural
frequencies $T_{rm}^{-1}$. In case of regime $r=1$, both types of arbitrary
slow muscle tuned to $T_{1slow}^{-1}$ and fast muscle tuned to
$T_{1fast}^{-1}$ (Table 4) are expected to show maximum workloop efficiency
when acting as controlled spring. In the efficient nonlinear regime $r=3$ the
slow and fast struts ($m=3$ in Table 2) will not show another function, but
any type of brakes ($m=2$ in Table 2) will work as motors, whereas motor are
expected to expose a new function, say, $m=6$ [determined by
$\eta_{6}=(6/7)/(1/7)=6$] that is closer to the brake activity ($\eta_{2}=3$)
than the strut ($\eta_{3}=\infty$). The cardiac muscles seem to display a
crucial dynamic state, say $m=0$ with $\eta_{0}=0$, which flatters the heart.
Such predicted _secondary functions_ and unusual ($m=0$ and $6$) muscles
adapted to new functions is a challenge deserving further study by
experimentalists.
### IV.2 Direct observation of muscle specialization
”If a muscle is specialized for a particular mechanical role how this is
reflected in it architecture?” [40]. The stated problem is approached here by
the comparative analysis between the muscle allometric exponents and those
predicted for particular efficient activities describing the trends of biomass
accommodation via PCSA and along a muscle.
#### IV.2.1 Isolated muscles in hindlimb of mammals and birds
In Table 5, the _morphometric data_ on the allometric exponents for the mean
cross-sectional area $A_{0m}^{(\exp)}$ and length $L_{0m}^{(\exp)}$ of four
skeletal muscles in the mammalian hindlimb for $35$ quadrupedal species of
body-mass domain exceeding four orders in magnitude are studied.
.
Place Table 5
.
First, let us verify the cylinder-shape similarity of skeletal muscles
described by Eq. (9). The muscle mass index $\alpha_{0m}$ estimated in Eq.
(10) via experimental data $a_{0m}^{(\exp)}$ and $l_{0m}^{(\exp)}$ is compared
in Table 5 with the measured indexes $\alpha_{0m}^{(\exp)}$.
.
Place Fig. 2
.
In Figs. 2 and 3, the method of determination of the primary mechanical
function is illustrated: the adapted muscle structure is indicated by the
appropriate theoretical point located most closely to the datapoint.
.
Place Fig. 3
.
The found reliable estimates $\alpha_{0m}^{(est)}$ were used then in the
muscle-function analysis in Figs. 2 and 3. The established small indices
$\alpha_{0m}$ generally validate the muscle biomechanics by proving a high-
precision observation of locomotory muscle patterns via muscle morphometry and
_functional physiology_. This implies that the effect of biomechanical
adaptation of muscle design to active elastic forces predominates over effects
of biological adaptation assigned to small $\alpha_{0m}^{(\exp)}$.
Secondly, the analysis in Figs. 2 and 3 indicates strong correlations between
the morphometrically characterized structure of skeletal muscle and one of the
primary locomotor functions described in Table 2. The primary functions
indicated in Table 5 are found with a high degree of certainty. Indeed, as
illustrated in Fig. 2, the deviations of distances measured along the dashed
line, corresponding to a given muscle, between the datapoint and distant
challengers for the primary function, from the smallest distance indicating
the primary candidate, always exceed the experimental uncertainty.
Thirdly, the found muscle mechanical specifications do not conflict with the
_physiological categorization_ established for joint extensors and flexors,
which muscle structures are shown to be adapted to the brake and motor
functions via activation of eccentric and concentric elastic forces. The found
structure parameter $\eta_{plant}\thickapprox 18$ indicates the foot support
activity for plantaris as the primary function (Table 5) that is in accord
with in vivo workloop presented in Fig. 1C. As shown in Table 3, the struts
are most conservative muscles no changing their support function in non-linear
regimes. In contrast, the gastrocnemius in mammals manifests their motor,
strut, and brake functions in, respectively, uphill, level, and incline
running of animals. Through the motor adapted structure with
$\eta_{gast}\thickapprox\eta_{1}=4$, the analysis in Fig. 3 establishes the
motor activity for gastrocnemius as the primary function naturally selected
for the significant mechanical task of uphill running exploring the bilinear
muscle dynamics. The effective trilinear gastrocnemius-displacement dynamics
is most close to the brake-like activity $\eta_{6}=6$, attributed to the
secondary function of the motor experimentally observed in gastrocnemius of
incline running turkey [27] and hopping tammar wallabies [24].
In Fig. 4, the overall muscle peak stress data measured in limb muscles of
animals in strenuous activity, reviewed by Biewener [25], are re-examined and
re-analyzed accounting for the primary functions of hindlimb muscles
established in Table 5.
.
Place Fig. 4
.
The uphill-motor specialization of gastrocnemius is independently supported by
the compressive-stress analysis made in Fig. 4 for fast running, jumping, and
hopping mammals. The stress scaling exponents ($s_{m}$) predicted for the
motor ($s_{1}=1/5$), strut ($s_{3}=0$), and control ($s_{4}=0$) functions are
shown to be distinguishable in work-specific mammalian muscles described in
Table 2. Hence, although the overall-function data by Biewener [25] indeed
expose almost weight-independent muscle stress, earlier postulated by McMahon
in Eq. (5) and only in part justified here by the slow-fibre muscles (Table 1)
and strut muscles (Table 2), the analyses in Fig. 4 demonstrates how the
function-specific muscle stress may serve as a new tool for the direct
observation of muscle specialization ignored in all previous overall-function
analyses.
I have also investigated an interesting question: whether the primary function
established for a certain leg muscle in mammals specialized to fast running
coincides with that for the same muscle in birds? The pioneering data on
individual leg muscles in $8$ running birds, ranging in size from $0.1$ _kg_
quail to $40$ _kg_ ostrich, are analyzed in Table 6 and Fig. 5.
.
Place Table 6
.
Place Fig. 5
.
In running and non-running birds (Fig. 5), the _gastrocnemius_ is employed as
the brake and spring, in contrast to the motor function in mammals (Table 5).
This is in accord with Bennett [23], who noted that ”the full force-generated
capacity of gastrocnemius is only used occasionally, such as during take-off,
when a bird attempts to throw itself into the air”. This explains our indirect
observation: the primary function of the gastrocnemius in running specialists
is attributed to the foot flexor in mammals and ankle extensor in birds (Table
6). In _non-running_ birds, the legs are designed to control the ground
locomotion (Fig. 5), whereas the wings may share motor and brake functions
(Table 3), in accord with the review by Dickinson et al. [6].
#### IV.2.2 Micro-macro scale correspondence
There are many striking examples when skeletal muscles expose adaptation to a
specific function, e.g. [43, 3]. The striated muscles anatomically suited to
concentric or eccentric work [2] are structurally distinct having,
respectively, long thin cells or short wide cells [5]. This observation
suggests the _microscopic level_ of muscle-cell adaptation introduced here by
$A_{cell}^{(conc)}>A_{cell}^{(ecent)}\text{ and
}L_{cell}^{(ecent)}>L_{cell}^{(conc)}$ (29)
for the _cellular_ cross-sectional area $A_{cell}$ ($\equiv A_{s}$) and length
$L_{cell}$ ($\equiv L_{s}$). Adopting these function specific trends, one may
expect to observe the cell-structure parameters $\eta_{s}=4$ and $3$ for
sarcomeres accommodated to efficient shortening or stretching of muscle as a
whole.
A general question arises whether allometric coefficients of proportionality
omitted above in all structure-function power-law (scaling) relations are also
attributed to active elastic strains accompanying maximum force production?
Or, alternatively, other microscopically justified mechanisms, c.f. [44], or
additional parameters (such as pinnate angle) may result in different general
macroscopic consequences? Given the highly conservative nature of contractive
units of _skeletal_ muscles and their well pronounced organization [25], the
_specific-function trends_ of the muscle cross-sectional area
$A_{strut}^{(\text{{isom}})}>A_{motor}^{(conc)}>A_{brake}^{(eccen)}>A_{contr}^{(sprin)}\text{
}$ (30)
and muscle-fibre length
$L_{contr}^{(sprin)}>L_{brake}^{(eccen)}>L_{motor}^{(conc)}>L_{strut}^{(\text{{isom}})}$
(31)
are generally expected from Table 2. The suggested trends become observable
via the primary functions established in Table 5 for gastrocnemius ($m=1$),
DDF ($m=1$), CDE ($m=2$), and plantaris ($m=3$), when the regression data [22]
on passive-muscle structure [$A_{0m}^{(\exp)}(M)$, $L_{0m}^{(\exp)}(M)$] are
taken additionally into consideration:
$A_{plant}^{(\exp)}>A_{gast}^{(\exp)}\gtrsim
A_{DDF}^{(\exp)}>A_{CDE}^{(\exp)}$ and
$L_{CDE}^{(\exp)}>L_{gast}^{(\exp)}\gtrsim
L_{DDF}^{(\exp)}>L_{plant}^{(\exp)}$, starting with $M>1$ $kg$.
Similarly, the trend for active stiffness
$K_{strut}^{(\max)}>K_{motor}^{(\max)}>K_{brake}^{(\max)}\text{ and,
generally, }K_{fast}^{(\max)}>K_{slow}^{(\max)}\text{ }$ (32)
straightforwardly follows from Table 2. Given that the _optimum velocity_ for
fast fibres $V_{1m}\varpropto L_{m}^{1/2}$ (Table 1), Eq. (31) provides
$V_{brake}^{(opt)}>V_{motor}^{(opt)}>V_{strut}^{(\text{{opt}})}$ (33)
Moreover, a crude estimate for the _cost energy_
$CU_{motor}^{(\max)}>CU_{strut}^{(\max)}>CU_{brake}^{(\max)}$ (34)
follows from $CU_{fast}^{(\max)}\varpropto M_{m}$ (22) and the experimental
data by Pollock and Shadwick [22],
$M_{1}^{(\exp)}>M_{3}^{(\exp)}>M_{2}^{(\exp)}$, considered at the same body
mass $M$. The finding (34) is in accord with the experimental observation
[44]: muscles contracting nearly isometrically (strut function) generate force
more economically than muscles involved in concentric work (via motor
function).
#### IV.2.3 Muscle dynamics of mammalian legs and dragonfly wings
Given that _mammalian leg extensors_ are active mostly during lengthening [2],
the brake primary function ($m=2$ in Table 2) could be assigned to leg muscles
specified by effective length $L_{leg}\varpropto M^{1/4}$ ($\alpha_{leg}=0$ is
adopted). In accord with Hill’s second constraint, underlaid by the proper
frequency $T_{3m}^{-1}\varpropto L_{m}^{-2}$ (Table 4), the theory predicts
$V_{leg}^{(\max)}\varpropto L_{leg}^{-1}\varpropto M^{-1/4}$ that results in
$1/T_{leg}^{(\max)}\varpropto L_{leg}^{-2}\varpropto M^{-1/16}$. Similarly,
for the wing-motor muscles in _flying birds_ ($m=1$ in Table 2) one should
expect $V_{wing}^{(\max)}\varpropto L_{wing}^{-1}\varpropto M^{-1/5}$, for
contraction velocity, and $1/T_{wing}^{(\max)}\varpropto M^{-1/25}$, for the
frequency or, alternatively, $1/T_{wing}^{(opt)}\varpropto M^{-1/5}$, in the
optimum-velocity regime (see Table 4). Hence, analytically revealed Hill’s
constraint becomes observable via the empirical regression data by Medler
[43]: on the maximum-amplitude contraction velocities for the locomotor
muscles in leg of terrestrial animals, $V_{leg}^{(\exp)}\varpropto M^{-0.25}$,
and that for wings in flying birds, bats, and insects,
$V_{wing}^{(\exp)}\varpropto M^{-0.20}$. Moreover, the experimental data by
Schilder and Marden [45] of the wingbeat frequency
$1/T_{wing}^{(\exp)}\varpropto M_{m}^{-0.20}$ scaled by mass $M_{m}$ (and
length $L_{0m}$) of the basalar muscle in dragonflies indicate that the motor-
type muscles ($L_{0m}\varpropto M_{m}^{1/5}$, see analysis in Fig. 6) were
studied self-consistently in the optimum, steady-velocity motion regime.
.
Place Fig. 6
.
In the same optimum-velocity regime (Table 1), the maximum-amplitude _static
force_ $F_{stat}^{(\exp)}\cong\Delta F_{1m}^{(slow)}\varpropto M_{m}^{2/3}$
and net _lever-system force_ $F_{ind}^{(\exp)}\cong\Delta
F_{1m}^{(fast)}\varpropto M_{m}$ reported by Schilder and Marden [45] may be
associated with the slow and fast activated fibres in the basalar muscles
tuned elastically to the linear regime through the dynamic PCSA
$A_{1m}^{(dyn)}\varpropto M_{m}^{2/3}$ and length $L_{1m}^{(dyn)}\varpropto
M_{m}^{1/3}$. The observed dynamic force output $F_{dyn}^{(\exp)}\varpropto
M_{m}^{0.83}$ can be therefore suggested as the mixed-fibre force
$F_{dyn}^{(pred)}\cong\Delta F_{1m}^{(mix)}\varpropto M_{m}^{5/6}$ (Table 1),
i.e. as a compromise of the forces $F_{stat}^{(\exp)}$ and $F_{ind}^{(\exp)}$.
These estimates challenge further analysis of the reported dynamic forces.
## V Conclusion
A theoretical framework for mechanical characterization of the three transient
activated states of the striated muscles passing in force-length cycles
through the three distinct dynamic regimes is proposed. The explicit
analytical description of muscle locomotor functions and related mechanical
characteristics is provided on the basis of two concepts: (i) the preservation
of spindle-type shape in skeletal muscles and egg-type shape in cardiac
muscles related to the preservation of dynamic muscle volume and (ii) the
mechanical similarity between action and reaction forces emerging in
biomechanically equivalent states. Exploring known patterns of elastic forces
in continuum mechanics, the macroscopic study of the force production and its
functional and structural accommodation in the loaded muscle organ as a whole
provides the following major points.
1\. It is demonstrated how the dynamic (frequency-velocity) constraints for
muscle contractions, first observed by Hill in hovering birds and then
revealed in locomotor muscles of running animals and flying birds, bats, and
insects, can be derived from the generic principle of mechanical (force and
velocity) similarity.
2\. It is shown how relations in classical mechanics of solids can be explored
in soft tissues. The study is grounded by the active-force muscle stiffness
reliably derived in all muscle work loops nearby and below the maximum-
amplitude exerted forces. The muscle stiffness, underlaid by sarcomere
stiffness, is shown to be dependent on muscle geometry and dynamic functional
variable, underlaid by elastic moduli, which encompass all contractive
elements acting as an elastic continuum medium.
3\. The theoretical prediction that the fast and slow muscles should generate
maximum forces linear, respectively, with the muscle volume and cross-
sectional area, regardless of muscle function and structure, is in part
validated by the direct empirical observation of maximal forces exerted by
animals and by the provided indirect observation of the adapted (primary)
muscle functions in legs of mammals and birds.
4\. The macroscopic structures of locomotor skeletal muscles observable
directly by muscle allometry are found to be adapted to the maximum-force
state, following moderate-velocity dynamic regime, instead of the expected
optimum velocity regime. Such a bilinear-displacement muscle dynamics
involving both fast-twitch and slow-twitch powering muscle fibres sheds light
on the origin of allometric power laws and muscle specialization. The adapted
structures are examined via available empirical data: the legs are brakes in
mammals and springs in non-running birds, whereas the wings are motor-brake
engines in flying species. Suggested pump function for the cardiac muscles
needs further experimental tests.
5\. The provided study of the muscle specialization in mammalian hindlimb
indicates that the properly tuned force production function is a dominated
factor in the accommodation of muscle structure. This finding also indicates
the predomination role of mechanical effects over biological adaptive
mechanisms assigned to the relatively small muscle-mass index. As the result,
a new investigation tool for indirect statistical observation of the
biomechanical adaptation of individual locomotor muscles is proposed through
the regression analysis of in vivo muscle stresses in synergists scaled across
different-sized animals.
6\. The assumption on that the muscle tuning muscle ability of animals can be
modeled by active elastic forces via non-linear muscle elastic moduli is
validated by the observation of the theoretical predictions for muscle
dynamics of legs and wings in running and flying specialists. Predictions are
made for the experimental modelling the primary and secondary function by
tuning the cycling muscle to the corresponding natural frequency and
controlling its efficiency.
7\. The conservative character of architecture and related mechanical
characteristics of striated muscles suggests general trends following from
mechanical and shape constraints. The trends dictated by primary functions
explain, in particular, why the muscles having larger fibre and sarcomere
lengths and suited to efficient eccentric work, tend toward higher optimum
contraction velocities, but show lower maximum stiffness and mechanical energy
cost.
8\. As an intriguing outcome of the analysis of maximal contraction muscle
velocities and frequencies, the maximum-speed steady locomotion is revealed to
be _controlled_ by non-linear elasticity of slow-fibre muscles generating
moderated force. This finding deserves further evaluation in finite muscle
element analysis studying top speeds of living and extant animals.
.
Acknowledgments
I thank Andrew A. Biewener and James H. Marden for careful reading of the
draft of this paper and helpful critical comments. Rudolf J. Schilder and
James H. Marden are appreciated for giving datapoints in Fig. 6. The financial
support by CNPq is also acknowledged.
.
Appendix A. List of abbreviations
.
PCSA - physiologic cross-sectional area
.
Mathematical signs and symbols
$=$ \- common equality sign
$\equiv$ \- identity sign implying ”by definition”
$\approx$ \- approximate equality sign
$\sim$ \- proportionality relation symbol omitting only numerical coefficients
$\cong$ \- here used as similarity sign supporting only physical dimension
units
$\propto$ \- here used as scaling rule symbol not supporting dimension units
.
Physical and geometrical notations
$\alpha_{m}$\- muscle-mass allometric index
$\varepsilon_{m}^{(opt)}$\- muscle strain in the optimum dynamic regime
$\eta_{m}$\- muscle geometry parameter
$\rho_{tiss}$\- tissue density
$\sigma_{tiss}^{(\max)}$\- peak tissue stress
$\Delta L$ \- length change
$\Delta F$ \- force change
$\Delta t_{m}$\- activation timing of muscle $m$
$A_{rm}$\- cross-sectional area of muscle $m$ in passive ($r=0$) and active
($r\neq 0$) states
$a$\- scaling exponent for cross-sectional area
$D$ \- diameter of ideal cylinder
$E_{rm}$ \- active-muscle elastic modulus establishing the dynamic regime
$r\neq 0$
$e$\- strain scaling exponent
$\Delta F_{elast}^{(\max)}=\Delta F_{m}^{(\max)}$\- maximum active elastic
force
$F_{prod}^{(fast)}$\- production force by fast muscle
$F_{motor}^{(conc)}$\- elastic force adapted to concentric work in motor
muscle
$F_{musc}^{(\max)}$\- maximum force exerted by muscle
$K_{m}$\- active muscle stiffness
$K_{s}$\- sarcomere/cellular stiffness
$L$\- length of an ideal cylinder
$L_{m}$ \- variable muscle length in non-specified dynamics
$L_{rm}$ \- dynamic muscle length in the regime $r$
$l$ \- length exponent
$m$ \- muscle in unspecified function
$M$ \- body mass of animals
$M_{m}$ \- muscle mass
$r$ \- numerical parameter indicating transient dynamic states via optimum-
velocity ($r=1$), moderate-velocity ($r=2$), and high-velocity ($r=3$) dynamic
regimes, distinct of passive muscle state ($r=0$).
$T_{rm}$\- period of cycling in the adapted regime $r$
$V_{rm}$\- muscle contraction velocity in the dynamic regime $r$
$W$ \- body weight
.
Appendix B. Scaling Muscle Functions
.
The _motor function_ is associated with the active force $F_{prod}^{(\max)}$
generated during muscle shortening at moderate contraction velocity at the
turning points $2$ in Figs. 1A, 1B, and 1C. In _fast-fibre_ muscles, the
corresponding _concentric force_
$F_{motor}^{(conc)}=F_{elast}^{(\max)}=\Delta F_{2m}^{(conc)}\sim
E_{2m}^{(fast)}A_{2m}^{3/2}L_{2m}^{-1}\text{ }\cong F_{prod}^{(fast)}$ (35)
is described by the known universal pattern of the maximal elastic forces [33]
equally applied to pure bending, pure torsion, or complex bending-torsion
loads subjected to long cylinder of length $L_{2m}$ and cross-sectional area
[26] $A_{2m}$. The exploration of Eq. (35) though Eqs. (8), (16), (24), and
(26) results in the fast-muscle-force constraint
$3a_{m}/2-l_{m}=1+\alpha_{m}$. It is remarkable that the case of slow-fibre
muscle, namely
$F_{motor}^{(conc)}=F_{elast}^{(\max)}=\Delta F_{2m}^{(conc)}\sim
E_{2m}^{(slow)}A_{2m}^{3/2}L_{2m}^{-1}\text{ }\cong F_{prod}^{(slow)}$ (36)
results in the slow-muscle-force constraint $3a_{m}/2-2l_{m}=a_{m}$, which is
exactly the same as fast muscle, in view of function-independent Eq. (10).
Therefore, any muscle tuned to the motor locomotor function should expose its
_dynamic structure_ scaled by
$a_{motor}^{(conc)}=\frac{4}{5}(1+\alpha_{motor})\text{ ,
}l_{motor}^{(conc)}=\frac{1}{5}(1+\alpha_{motor})\text{, }$ (37)
regardless of the fibre type content. This finding follows from both the
muscle force constraints solved with the help of the function-independent
muscle-shape constraint (10). Moreover, as shown in [26], the principal
component of the compressive stress $\sigma_{m}^{(conc)}$ specifying Eq. (17)
may be caused by the peak transverse-tensile _strains_
$\varepsilon_{motor}^{(conc)}=\frac{\Delta D_{m}^{(\max)}}{L_{m}}\varpropto
M_{m}^{e_{m}}\text{, with
}e_{m}=e_{motor}^{(conc)}=\frac{a_{m}}{2}-l_{m}\text{,}$ (38)
where $\Delta D_{m}^{(\max)}\thicksim D_{m}\backsim A_{m}^{1/2}$ is transverse
muscle deformation.
Likewise, the maximum elastic _eccentric force_
$F_{brake}^{(eccen)}=\Delta F_{2m}^{(eccen)}\sim
E_{2m}^{(eccen)}A_{2m}^{2}L_{2m}^{-2}\cong F_{prod}^{(fast)}\text{,}$ (39)
associated with the _brake muscle function_ (Fig. 1B) provides the maximum
elastic stress
$\sigma_{brake}^{(\max)}=\frac{F_{brake}^{(eccen)}}{A_{m}}\varpropto
M_{m}^{s_{m}}\text{, with }s_{m}=s_{brake}^{(eccen)}=a_{m}-2l_{m}\text{,}$
(40)
following from Eqs. (17) and (39). The unique solution to both fast-muscle-
force constraint, $2a_{m}-2l_{m}=1+\alpha_{m}$, and slow-muscle-force
constraint, $2a_{m}-3l_{m}=a_{m}$, is
$a_{brake}^{(eccen)}=\frac{3}{4}(1+\alpha_{brake})\text{ ,
}l_{brake}^{(eccen)}=\text{
}s_{brake}^{(eccen)}=\frac{1}{4}(1+\alpha_{brake})\text{.}$ (41)
The _strut muscle function_ treated as antagonistic to both motor and brake
functions drives nearly isometric contractions characteristic of small, but
non-zero length change ($\Delta L_{m}\ll L_{m}$) achieved near peak forces
(see Fig. 1C). This suggests the nearly _isometric force_
$F_{strut}^{(isom)}=E_{2m}^{(isom)}\varepsilon_{2m}^{(isom)}A_{2m}\cong
F_{prod}^{(fast)}\text{, with }\varepsilon_{2m}^{(isom)}=\Delta
L_{2m}^{(isom)}/L_{2m}\text{,}$ (42)
in fast muscles. Again, one solves the _muscle strut_ constraints
$2a_{m}+l_{m}=1+\alpha_{m}$ and $2a_{m}+2l_{m}=a_{m}$ resulting in
$a_{strut}^{(isom)}=1+\alpha_{strut}\text{ and
}l_{strut}^{(isom)}=s_{strut}^{(isom)}=0\text{, with }\Delta
L_{2m}^{(isom)}\varpropto L_{2m}^{2}\text{,}$ (43)
for any type of muscles.
A new antagonist (to strut muscle) tuned to the _cardiac_ type contractions
via active elastic force
$F_{pump}^{(card)}=\Delta F_{2m}^{(card)}\thicksim
E_{2m}^{(card)}L_{2m}^{2}\cong F_{prod}^{(fast)}$ (44)
is associated with, say, _pump function_ providing the fast-muscle-force
constraint $2l_{m}=1+\alpha_{m}$. This yields
$a_{pump}^{(card)}=l_{pump}^{(card)}=\text{
}s_{pump}^{(card)}=e_{pump}^{(card)}=\frac{1}{2}(1+\alpha_{pump})\text{,}$
(45)
equally applied to slow-fibre muscles resulting in the slow-force constraint
$l_{m}=a_{m}$.
To complete the intrinsic-force description, the spring-type _control
function_ associated with the optimum-regime elastic force
$F_{contr}^{(sprin)}=F_{elast}^{(opt)}\varpropto
E_{1m}^{(slow)}M_{m}^{2/3}\varpropto
E_{1m}^{(slow)}A_{m}^{2/3}L_{m}^{2/3}\cong F_{prod}^{(slow)}$ (46)
in slow-fiber muscles results in
$a_{cont}^{(sprin)}=\frac{2}{3}(1+\alpha_{cont})\text{,
}l_{cont}^{(sprin)}=\frac{1}{3}(1+\alpha_{cont})\text{, with
}s_{cont}^{(sprin)}=e_{cont}^{(sprin)}=0\text{,}$ (47)
that follows from the slow-force and fast-force constraints
$2(a_{m}+l_{m})/3=a_{m}$ and $(2a_{m}+5l_{m})/3=1+\alpha_{m}$ and therefore is
valid for any type of muscle tuned to velocity-optimum regime. All obtained
specific-function mechanical characteristics are summarized in Table 2.
.
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Figure Captions
.
Fig. 1. The qualitative analysis of the in vivo muscle force-length data. The
muscle _motor function_ is presented by gastrocnemius powering during
shortening in uphill running turkey (_inset A_ , adapted from [27]). The
lateral gastrocnemius and plantaris act as brake (_inset B_) and strut (_inset
C_) in hopping tammar wallabies [24]. The solid (and dashed) _arrows_ indicate
rasing (and decreasing) of the exerted force near its maximum magnitude
$F_{\max}$. The regions of the linear force-length domain are displayed by the
force change $\Delta F_{1m}$ and length change $\Delta L_{1m}$, both estimated
from $L_{1m}$, and the starting datapoint $F_{1m}$ of the force enhancement
achieved at the optimum contraction velocity $V_{opt}$. Similar to physical
pendulum, the resting length $L_{0m}$ is expected to be passed at near maximum
velocities $V_{\max}$ and lower forces $F_{3m}$. The origin of intrinsic
muscle forces (_inset D_): in both cases of the powering shortening (_motor_)
and lengthening (_brake_) muscles the resulted force $F_{\max}$ is a
superposition of the production force output $F_{prod}$ and reaction passive
$F_{pass}$ and active $F_{act}$ _elastic_ forces [see also text below Eq.
(16)].
.
Fig. 2. The indirect observation of the primary activity of mammalian
plantaris. The _solid symbol_ is the datapoint [22] presented in Table 5 and
the bars indicate experimental error. The _open symbols_ are theoretical
estimates for stable dynamic structures established for the motor, brake,
strut, or control functions described in Table 2, with
$\alpha_{m}=\alpha_{0m}^{(est)}$ taken from Table 5.
.
Fig. 3. The observation of the primary mechanical function in some isolated
individual muscles in mammals. The analysis and notations correspond to those
in Fig. 2. The experimental (and theoretical) data for gastrocnemius, DDF
(deep digital flexor), and CDE (common digital extensor) are shown,
respectively, by the closed (and open) inverted triangles, regular triangles,
and circles. All the data are taken from Table 5.
.
Fig. 4. The qualitative study of the in vivo data on the peak stress in
individual leg muscles of animals in strenuous activity. The symbols employed
above in Figs. 2 and 3 are extended by the open circles (triceps) for the data
on peak muscle stress taken from Table 1 in [25], with the exclusion of the
slow-mode data on cantering goat and trotting cat. The data [41] on the
activated isometric stress in isolated white rabbit tibialis are added. The
_dashed line_ shows the brake-functional stress indicated by the stress
scaling exponent $s=1/4$. The _solid lines_ are drawn by $115\cdot M^{1/5}$,
for the motor function, and by $215$ _kPa_ , for the strut and spring
functions. All coefficients are adjusted by eye.
.
Fig. 5. The analysis of the primary mechanical functions for leg muscles in
running and non-running birds. The measured (and estimated) data taken from
Table 6 (and Table 2) for gastrocnemius, femorotibialis, and digital flexors
are shown by the closed (and open) inverted triangles, circles, and regular
triangles, respectively. The semi-open triangles are the data by Bennett [23]
for non-running birds.
.
Fig. 6. The qualitative scaling of the basalar structure to muscle mass in
male dragonflies (Odonata and Anisoptera, listed in Fig. 5 in [45]. The
datapoints for muscle length $L_{0m}^{(\exp)}$ is a courtesy by the authors.
The estimated muscle cross-sectional area $A_{0m}^{(est)}$ is obtained on the
basis of Eq. (1) taken with $\rho_{musc}^{(\exp)}=$ $1060$ $\emph{kg/m}^{3}$
[46]. The _solid lines_ are $L_{motor}=0.052\cdot M_{m}^{1/5}$ and
$A_{motor}=0.018\cdot M_{m}^{4/5}$. The _dashed lines_ indicated by the
scaling exponents are drawn according to muscle specialization shown in Table
2. All pre-exponential coefficients are adjusted by eye.
Tables
.
Optimum muscle characteristics, (equations) | Fast fibres | Slow fibres | Mixed fibres
---|---|---|---
Optimum length change, $\Delta L_{1m}$, (25) | $L_{m}$ | $L_{m}$ | $L_{m}$
Production/active-elastic force, $\Delta F_{1m}$, (16), (18), (25) | $A_{m}L_{m}$ | $A_{m}$ | $A_{m}L_{m}^{1/2}$
Optimum stiffness, $K_{1m}=E_{1m}A_{1m}/L_{1m}$, (19) | $A_{m}$ | $A_{m}L_{m}^{-1}$ | $A_{m}L_{m}^{-1/2}$
Optimum elastic stress, $\sigma_{1m}=\Delta F_{1m}/A_{1m}$, (17) | $L_{m}$ | $L_{m}^{0}$ | $L_{m}^{1/2}$
Contraction frequency, $T_{1m}^{-1}\thicksim\sqrt{E_{1m}/\rho_{0m}}/L_{1m}$, (15) | $L_{m}^{-1/2}$ | $L_{m}^{-1}$ | $L_{m}^{-3/4}$
Optimum velocity, $V_{1m}=V_{musc}^{(opt)}$, (23) | $L_{m}^{1/2}$ | $L_{m}^{0}$ | $L_{m}^{1/4}$
Optimum power, $P_{1m}=F_{1m}V_{1m}$ | $A_{m}L_{m}^{3/2}$ | $A_{m}$ | $A_{m}L_{m}^{3/4}$
Table 1. General mechanical characteristics of the striated muscles tuned to
linear-displacement dynamic regime scaled to dynamic fiber length
$L_{m}=L_{1m}$. The _mixed-fibre_ scaling dynamic exponents (shown in the last
column) are modeled by the common means for the fast-muscle and slow-muscle
exponents (established in the second and third columns), i.e.
$F_{mix}\backsim\sqrt{F_{fast}F_{slow}}$; $A_{m}$ and $L_{m}$ are attributed
to the stabilized _dynamic_ muscle geometry constrained by muscle volume (9).
.
| Locomotor pattern, regime
---
(equation)
| Motor, r=2
---
(37)
| Brake, r=2
---
(41)
| Strut, r=2
---
(43)
| Control, r=1
---
(47)
| Pump, r=2
---
(45)
| Force pattern, muscle
---
(equation)
| $F_{{\small motor}}^{{\small(conc)}}$, m=1
---
(35)
| $F_{{\small brake}}^{{\small(eccen)}}$, m=2
---
(39)
| $F_{{\small strut}}^{{\small(isom)}}$, m=3
---
(42)
| $F_{{\small contr}}^{{\small(sprin)}}$, m=4
---
(46)
| $F_{{\small pump}}^{{\small(card)}}$, m=5
---
(44)
Maximum force output, (24) | $1+\alpha_{1}$ | $1+\alpha_{2}$ | $1+\alpha_{3}$ | $\frac{2}{3}(1+\alpha_{4})$ | $1+\alpha_{5}$
Muscle fibre length, (8) | $\frac{1}{5}(1+\alpha_{1})$ | $\frac{1}{4}(1+\alpha_{2})$ | $0$ | $\frac{1}{3}(1+\alpha_{4})$ | $\frac{1}{2}(1+\alpha_{5})$
Cross-sectional area, (8) | $\frac{4}{5}(1+\alpha_{1})$ | $\frac{3}{4}(1+\alpha_{2})$ | $1+\alpha_{3}$ | $\frac{2}{3}(1+\alpha_{4})$ | $\frac{1}{2}(1+\alpha_{5})$
Structure parameter, $\eta_{m}$=$a_{m}l_{m}^{-1}$ | $4$ | $3$ | $\ \infty$ | $2$ | $1$
Length change*, (13) | $\frac{2}{5}(1+\alpha_{1})$ | $\frac{1}{2}(1+\alpha_{2})$ | $0$ | $\frac{1}{3}(1+\alpha_{4})$ | $1+\alpha_{5}$
Maximum stress/strain*, (12) | $\frac{1}{5}(1+\alpha_{1})$ | $\frac{1}{4}(1+\alpha_{2})$ | $0$ | $0$ | $\frac{1}{2}(1+\alpha_{5})$
Maximum stiffness*, (19) | $\frac{3}{5}(1+\alpha_{1})$ | $\frac{1}{2}(1+\alpha_{2})$ | $1+\alpha_{3}$ | $\frac{1}{3}(1+\alpha_{4})$ | $0$
Natural frequency*, (15) | $-\frac{1}{5}(1+\alpha_{1})$ | $-\frac{1}{4}(1+\alpha_{2})$ | $0$ | $-\frac{1}{3}(1+\alpha_{4})$ | $-\frac{1}{2}(1+\alpha_{5})$
Energy change*, (21) | $\frac{7}{5}(1+\alpha_{1})$ | $\frac{3}{2}(1+\alpha_{2})$ | $1+\alpha_{3}$ | $1+\alpha_{4}$ | $2(1+\alpha_{5})$
Moderate velocity* (23) | $0$ | $0$ | $0$ | $0$ | $0$
Table 2. The locomotor functions and their mechanical characteristics scaled
to dynamic structures. The all-type powering individual muscles $m=1,2,3,$ and
$5$ are tuned to the maximum-force bilinear dynamic regime $r=2$ [described in
Eq. (26)] and muscles $m=4$ act in the linear regime $r=1$ [Eq. (25)]. *The
data shown for fast muscles. The allometric exponents are related to animal’s
body mass via Eq. (8).
.
Dyn. regimes | Force | Funct. | Structure | Funct. | Structure | Funct. | Structure | Funct. | Structure
---|---|---|---|---|---|---|---|---|---
$r=1,2,3$ | ${\small F}_{{\small prod}}^{{\small\max}}$ ${\small\varpropto}$ | ${\small F}_{{\small motor}}^{{\small conc}}{\small\varpropto}$ | | ${\small a}_{m}$ | ${\small l}_{m}$
---|---
${\small F}_{{\small brake}}^{{\small eccen}}{\small\varpropto}$ | | ${\small a}_{m}$ | ${\small l}_{m}$
---|---
${\small F}_{{\small strut}}^{{\small isom}}{\small\varpropto}$ | | ${\small a}_{m}$ | ${\small l}_{m}$
---|---
${\small F}_{{\small plun}}^{{\small card}}{\small\varpropto}$ | | ${\small a}_{m}$ | ${\small l}_{m}$
---|---
${\small E}_{1m}^{(slow)}{\small\varpropto L}_{m}^{{\small 0}}$ | ${\small A}_{m}$ | ${\small A}_{m}^{\frac{3}{2}}{\small L}_{{\small m}}^{{\small-}1}$ | | $\frac{{\small 2}}{3}$ | $\frac{{\small 1}}{3}$
---|---
${\small A}_{m}^{2}{\small L}_{m}^{-2}$ | | $\frac{{\small 2}}{3}$ | $\frac{{\small 1}}{3}$
---|---
${\small A}_{m}$ | | ${\small nc}$ | ${\small nc}$
---|---
${\small L}_{m}^{2}$ | | $\frac{{\small 2}}{3}$ | $\frac{{\small 1}}{3}$
---|---
${\small E}_{1m}^{(fast)}{\small\varpropto L}_{m}$ | ${\small A}_{{\small m}}{\small L}_{{\small m}}$ | ${\small A}_{m}^{\frac{3}{2}}$ | | $\frac{{\small 2}}{3}$ | $\frac{{\small 1}}{3}$
---|---
${\small A}_{m}^{2}{\small L}_{m}^{-1}$ | | $\frac{{\small 2}}{3}$ | $\frac{{\small 1}}{3}$
---|---
${\small A}_{m}{\small L}_{m}$ | | ${\small nc}$ | ${\small nc}$
---|---
${\small L}_{m}^{3}$ | | $\frac{{\small 2}}{3}$ | $\frac{{\small 1}}{3}$
---|---
${\small E}_{2m}^{(slow)}{\small\varpropto L}_{{\small m}}^{{\small-1}}$ | ${\small A}_{m}$ | ${\small A}_{m}^{\frac{3}{2}}{\small L}_{{\small m}}^{{\small-2}}$ | | $\frac{\mathbf{4}}{\mathbf{5}}$ | $\frac{\mathbf{1}}{\mathbf{5}}$
---|---
${\small A}_{m}^{2}{\small L}_{m}^{-3}$ | | $\frac{\mathbf{3}}{\mathbf{4}}$ | $\frac{\mathbf{1}}{\mathbf{4}}$
---|---
${\small L}_{m}^{-1}{\small A}_{m}$ | | $\mathbf{1}$ | $\mathbf{0}$
---|---
${\small L}_{m}^{1}$ | | $\frac{\mathbf{1}}{\mathbf{2}}$ | $\frac{\mathbf{1}}{\mathbf{2}}$
---|---
${\small E}_{2m}^{(fast)}{\small\varpropto}L_{m}^{0}$ | ${\small A}_{{\small m}}{\small L}_{{\small m}}$ | ${\small A}_{{\small 0}}^{\frac{3}{2}}{\small L}_{{\small 0}}^{{\small-1}}$ | | $\frac{\mathbf{4}}{\mathbf{5}}$ | $\frac{\mathbf{1}}{\mathbf{5}}$
---|---
${\small A}_{0}^{2}{\small L}_{0}^{-2}$ | | $\frac{\mathbf{3}}{\mathbf{4}}$ | $\frac{\mathbf{1}}{\mathbf{4}}$
---|---
${\small A}_{0}$ | | $\mathbf{1}$ | $\mathbf{0}$
---|---
${\small L}_{0}^{2}$ | | $\frac{\mathbf{1}}{\mathbf{2}}$ | $\frac{\mathbf{1}}{\mathbf{2}}$
---|---
${\small E}_{3m}^{(slow)}{\small\varpropto L}_{m}^{{\small-2}}$ | ${\small A}_{m}$ | ${\small A}_{m}^{\frac{3}{2}}{\small L}_{{\small m}}^{{\small-3}}$ | | $\frac{{\small 6}}{7}$ | $\frac{{\small 1}}{7}$
---|---
${\small A}_{m}^{2}{\small L}_{m}^{-4}$ | | $\frac{{\small 4}}{5}$ | $\frac{{\small 1}}{5}$
---|---
${\small L}_{m}^{-2}{\small A}_{m}$ | | ${\small 1}$ | ${\small 0}$
---|---
${\small L}_{m}^{0}$ | | ${\small 0}$ | ${\small 1}$
---|---
${\small E}_{3m}^{(fast)}{\small\varpropto L}_{m}^{{\small-1}}$ | ${\small A}_{{\small m}}{\small L}_{{\small m}}$ | ${\small A}_{m}^{\frac{3}{2}}{\small L}_{{\small m}}^{{\small-2}}$ | | $\frac{{\small 6}}{7}$ | $\frac{{\small 1}}{7}$
---|---
${\small A}_{m}^{2}{\small L}_{m}^{-3}$ | | $\frac{{\small 4}}{5}$ | $\frac{{\small 1}}{5}$
---|---
${\small A}_{m}{\small L}_{m}^{-1}$ | | ${\small 1}$ | ${\small 0}$
---|---
${\small L}_{m}^{1}$ | | ${\small 0}$ | ${\small 1}$
---|---
Table 3. Locomotor functions predicted by dynamic structured for slow and fast
striated muscles tuned to distinct dynamic regimes. The primary functions
($r=2$) are shown by bold type. The analysis of functional muscle structures
made in terms of elastic-force patterns: the active-muscle optimum-velocity
($r=1$), moderate-velocity ($r=2$), and high-velocity ($r=3$) dynamic regimes
are described in the first column via the muscle elastic moduli $E_{rm}$ [Eqs.
(25), (26), and (27)] and specified by slow and fast force output [Eq. (24)],
shown in the second column. The third and next odd columns show the elastic
force functional scaling in concentric, eccentric, isometric, and pump
contractions. The corresponding solutions to scaling equations underlaid by
the force similarity principle (16) are shown for simplicity with
$\alpha_{rm}=0$, in the forth and next even columns. _Notation:_ $nc$
indicates non-conclusive solution.
.
Dynamic regimes | Optimum, | $r=1$ | Moderate, | $r=2$ | Maximum, | $r=3$
---|---|---|---|---|---|---
Muscle type | slow | fast | slow | fast | slow | fast
Natural frequency, Eq. (15) | $L_{m}^{-1}$ | $L_{m}^{-1/2}$ | $L_{m}^{-3/2}$ | $L_{m}^{-1}$ | $L_{m}^{-2}$ | $L_{m}^{-3/2}$
Contraction velocity, Eq. (23) | $L_{m}^{0}$ | $L_{m}^{1/2}$ | $L_{m}^{-1/2}$ | $L_{m}^{0}$ | $L_{m}^{-1}$ | $L_{m}^{-1/2}$
Table 4. Dynamic characterization of the red (slow) and white (fast) striated
muscles in the optimum-, moderate-, and maximum-velocity dynamic regimes
$r=1,2$, and $3$ described in Table 3.
.
Individual mammalian muscles | $a_{0m}^{(\exp)}$ | $l_{0m}^{(\exp)}$ | ${\small\alpha}_{0m}^{(\exp)}$ | ${\small\eta}_{{\small 0}m}$ | ${\small\alpha}_{0m}^{(est)}$ | $\mathit{a}_{m}$ | $\mathit{l}_{m}$ | Prim. functions
---|---|---|---|---|---|---|---|---
Gastrocnemius (and soleus) | ${\small 0.77\pm}.{\small 02}$ | ${\small 0.21\pm.02}$ | -${\small 0.03}$ | ${\small 3.7}$ | -${\small 0.02}$ | ${\small 0.78}$ | ${\small 0.20}$ | motor, ${\small m=1}$
Deep digital flexor (DDF)∗) | ${\small 0.85\pm.03}$ | ${\small 0.18\pm.02}$ | ${\small\ 0.03}$ | ${\small 4.7}$ | ${\small\ 0.03}$ | ${\small 0.82}$ | ${\small 0.21}$ | motor, ${\small m=1}$
Comm. digit. extensor (CDE) | ${\small 0.69\pm.04}$ | ${\small 0.24\pm.02}$ | -${\small 0.07}$ | ${\small 2.9}$ | -${\small 0.07}$ | ${\small 0.70}$ | ${\small 0.23}$ | brake, ${\small m=2}$
Plantaris (SDF) | ${\small 0.91\pm.04}$ | ${\small 0.05\pm.04}$ | -${\small 0.03}$ | ${\small 18}$ | -${\small 0.04}$ | ${\small 0.96}$ | ${\small 0.00}$ | strut, ${\small m=3}$
Ankle-joint muscle group | ${\small 0.81\pm.03}$ | ${\small 0.17\pm.03}$ | -${\small 0.03}$ | ${\small 4.8}$ | -${\small 0.03}$ | ${\small 0.78}$ | ${\small 0.19}$ | motor, ${\small g=1}$
Table 5. The analysis of the allometric data by Pollock and Shadwick [22]
provided on the basis of Eq. (10) and Table 2. The shown statistical error is
approximated by the symmetrized $95\%$ confidence interval. The methodology of
the analysis is illustrated in Fig. 2. The primary functions found in Figs. 2
and 3 are described following Table 2, with $\alpha_{m}=\alpha_{0m}^{(est)}$.
The overall muscle group ($g=1$) is determined as the standard mean over all
muscles. ∗)DDF includes individual flexor hallucis and flexor digitorum
longus; SDF means superficial digital flexor.
.
Running birds | $\ a_{0m}^{(\exp)}$ | $\alpha_{0m}^{(\exp)}$ | $l_{0m}^{(est)}$ | $a_{0m}^{(\exp)}/l_{0m}^{(est)}$ | $a_{2m}$ | $l_{2m}$ | Primary function/force
---|---|---|---|---|---|---|---
Gastrocnemius | $0.81\pm 0.14$ | $\ 0.14$ | $0.33$ | $2.5$ | $0.85$ | $0.29$ | brake/eccentric
Digital flexors (DF) | $0.76\pm 0.22$ | $-0.03$ | $0.21$ | $3.6$ | $0.78$ | $0.19$ | motor/concentric
Femorotibialis | $0.80\pm 0.12$ | $-0.02$ | $0.18$ | $4.4$ | $0.78$ | $0.20$ | motor/concentric
Overall group | $0.79\pm 0.16$ | $\ 0.03$ | $0.24$ | $3.3$ | $0.77$ | $0.26$ | brake/eccentric
Table 6. The analysis of the allometric data by Maloiy et al. [42]. The shown
large error is due to relatively wide confidence limits. The mean exponents
$l_{0m}^{(est)}$ are estimated via Eq. (10). The overall muscle group is
determined as the standard mean over all muscles. The indicated primary
functions and active elastic forces are described by the evaluated dynamic-
structure exponents $a_{2m}$ and $l_{2m}$ found as most close to the
experimental resting-volume data on $a_{0m}^{(\exp)}$ and $l_{0m}^{(\exp)}$
and therefore assigned to regime $r=2$ (Table 2).
|
arxiv-papers
| 2009-09-08T09:53:50 |
2024-09-04T02:49:05.124122
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Valery B. Kokshenev",
"submitter": "Valery B. Kokshenev",
"url": "https://arxiv.org/abs/0909.1444"
}
|
0909.1503
|
Axially Symmetric Cosmological Mesonic Stiff Fluid Models in Lyra’s Geometry
Ragab M. Gad111Email Address: ragab2gad@hotmail.com
Mathematics Department, Faculty of Science,
Minia University, 61915 El-Minia, EGYPT.
###### Abstract
In this paper, we obtained a new class of axially symmetric cosmological
mesonic stiff fluid models in the context of Lyra’s geometry. Expressions for
the energy, pressure and the massless scalar field are derived by considering
the time dependent displacement field. We found that the mesonic scalar field
depends on only $t$ coordinate. Some physical properties of the obtained
models are discussed.
Keywords: Lyra’ geometry; axially symmetric space-time; mesonic scalar field;
stiff fluid.
## 1 Introduction
After Einstein proposed his theory of general relativity he succeeded in
geometrizing the phenomena of gravitation by expressing gravitational in terms
of metric tensor $g_{ij}$. This idea of geometrizing gravitation inspired many
physicists to generalize the theory in order to incorporate electrodynamics as
a purely geometrical construct. One of the first attempts in this direction
was made in 1918 by Weyl [1] who suggested a theory based on a generalization
of Riemannian geometry, by formulating a new kind of gauge theory involving
metric tensor, to geometrize gravitation and electromagnetism. This theory was
criticized due to non-integrability of length of vector under parallel
displacement 222In the theory of general relativity if a vector undergoes a
parallel displacement, its direction may change, but not its length. While in
Weyl’s geometry not only the direction but the length may change and depends
on the path between two points, which show that the length is not integrable..
Also as pointed out by Einstein that this theory implies that frequency of
spectral lines emitted by atoms would not remain constant but would depend on
their past histories, which is in contradiction to observed uniformity of
their properties [2]. Nevertheless, completely apart from these criticisms,
Weyl’s geometry provides an interesting example of non-Riemannian connections.
Folland [3] gave a global formulation of Weyl manifolds clarifying
considerably many of Weyl’s basic ideas thereby.
In 1951 Lyra [4] proposed a modification of Riemannian geometry by introducing
a gauge function into the structure less manifold which is in close
resemblance to Weyl’s geometry [5]. This modification was to overcome the
problems appeared in Weyl’s geometry and is more in keeping with the spirit of
Einstein’s principle of geometrization, since both the scalar and tensor
fields have more or less intrinsic geometrical significance. In this way
Riemannian geometry was given a new modification and the modified geometry was
named as Lyra’s geometry. For physical motivation of Lyra’s geometry we refer
to the literature [6]-[8]. However, in contrast to Weyl’s geometry, in Lyra’s
geometry the connection is metric preserving and length transfers are
integrable as in Riemannian geometry. Subsequently, Sen [9] and Sen and Dunn
[2] proposed a new scalar tensor theory of gravitation. They constructed an
analog of Einstein field equation based in Lyra’s geometry, see equation
(2.6). Sen [9] found that static model with finite density in Lyra’s geometry
is similar to the static Einstein model, but a signification differences was
that the model exhibited red shift. Halford [10] pointed out that the constant
displacement vector field $\phi_{i}$ in Lyra’s geometry plays the role of a
cosmological constant in the normal general relativistic treatment. Halford
[11] showed that the scalar tensor treatment based in Lyra’s geometry predicts
the same effects, within observational limits, as in Einstein theory. Several
attempts have been made to cast the scalar tensor theory of gravitation in
wider geometrical context [12].
Many Authors [13] have studied cosmological models based on Lyra’s geometry
with a constant displacement field vector in the time-direction. Singh and his
collaborators [14] have studied Bianchi type I, III, Kantowski-Sachs and new
class of models with a time dependent displacement field. They have made a
comparative study of Robertson-Walker models with a constant deceleration
parameter in Einstein’s theory with a cosmological terms and in the
cosmological theory based on Lyra’s geometry.
Recently, several authors [15]\- [21] studied cosmological models based on
Lyra’s geometry in various contexts. With these motivations, in this paper, we
obtained exact solutions of the field equations for mesonic stiff fluid models
in axially symmetric space-times within the frame work of Lyra’s geometry for
time varying displacement field vector.
Axially symmetric cosmological models have been studied in both Riemannian and
Lyra geometries. In context of general relativity theory, by adopting the
comoving coordinate system, these models with string dust cloud source are
studied by Bhattacharaya and Karade [22]. They shown that some of these models
are singular free even at an initial epoch. In the context of Lyra’s geometry
these models are studied in the presence of cosmic string source and thick
domain walls [23] and in the presence of perfect fluid distribution [24].
This paper is organized as follows: The metric and field equations are
presented in section 2. Section 3 deals with solving the field equations.
Finally, in section 4, concluding remarks are given.
## 2 Fundamental Concepts and Field Equations
Consider the axially symmetric metric [22] in the form
$ds^{2}=dt^{2}-A^{2}(t)(d\chi^{2}+f^{2}(\chi)d\phi^{2})-B^{2}(t)dz^{2},$ (2.1)
with the convention $x^{0}=t$, $x^{1}=\chi$, $x^{2}=\phi$, $x^{3}=z$ and $A$
and $B$ are functions of $t$ only while $f$ is a function of the coordinate
$\chi$ only.
The volume element of the model (2.1) is given by
$\mathcal{V}=\sqrt{-g}=A^{2}fB$ (2.2)
The four-acceleration vector, the rotation, the expansion scalar and the shear
scalar characterizing the four velocity vector field, $u^{a}$, respectively,
have the usual definitions as given by Raychaudhuri [25]
$\begin{array}[]{ccc}\dot{u}_{i}&=&u_{i;j}u^{j},\\\
\omega_{ij}&=&u_{[i;j]}+\dot{u}_{[i}u_{j]},\\\ \Theta&=&u^{i}_{;i},\\\
\sigma^{2}&=&\frac{1}{2}\sigma_{ij}\sigma^{ij},\end{array}$ (2.3)
where
$\sigma_{ij}=u_{(i;j)}+\dot{u}_{(i}u_{j)}-\frac{1}{3}\Theta(g_{ij}+u_{i}u_{j}).$
In view of the metric (2.1), the four-acceleration vector, the rotation, the
expansion scalar and the shear scalar given by (2.3)can be written in a
comoving coordinates system as
$\begin{array}[]{ccc}\dot{u}_{i}&=&0,\\\ \omega_{ij}&=&0,\\\
\Theta&=&\frac{2\dot{A}}{A}+\frac{\dot{B}}{B},\\\
\sigma^{2}&=&\frac{1}{9}\big{(}11\big{(}\frac{\dot{A}}{A}\big{)}^{2}+5\big{(}\frac{\dot{B}}{B}\big{)}^{2}+\frac{2\dot{A}\dot{B}}{AB}\big{)}.\end{array}$
(2.4)
The non vanishing components of the shear tensor$\sigma_{ij}$ are
$\begin{array}[]{ccc}\sigma_{11}&=&A(\frac{1}{3}\Theta A-\dot{A}),\\\
\sigma_{22}&=&Af^{2}(\frac{1}{3}\Theta A-\dot{A}),\\\
\sigma_{33}&=&B(\frac{1}{3}\Theta B-\dot{B}),\\\
\sigma_{44}&=&-\frac{2}{3}\Theta.\par\end{array}$ (2.5)
The field equations in normal gauge for Lyra’s geometry as obtained by Sen [9]
(in gravitational units $c=8\pi G=1)$ read as
$R_{ij}-\frac{1}{2}Rg_{ij}=-T_{ij}-\frac{3}{2}\phi_{i}\phi_{j}+\frac{3}{4}g_{ij}\phi_{\alpha}\phi^{\alpha},$
(2.6)
the left hand side is the usual Einstein tensor, whereas $\phi_{i}$ is a time-
like displacement field vector defined by
$\phi_{i}=(0,0,0,\lambda(t)),$
and $T_{ij}$ is the energy momentum tensor corresponding to perfect fluid and
massless mesonic scalar field and is given by
$T_{ij}=(\rho+p)u_{i}u_{j}-pg_{ij}+V_{,i}V_{,j}-\frac{1}{2}g_{ij}V_{,k}V^{,k}.$
(2.7)
Here $p$ is the pressure, $\rho$ the energy density and $u_{i}$ the four
velocity vector satisfying the relation in co-moving coordinate system
$g_{ij}u^{i}u^{j}=1,\qquad u^{i}=u_{i}=(1,0,0,0).$
However, $V$ is the massless scalar field and we assume it to be a function of
$t$ and $\chi$ coordinates. The Scalar field $V$ is governed by the Klein-
Gordan wave equation
$g^{ij}V_{;ij}=0.$
For the line element (2.1), the field equations (2.6) with equation (2.7) lead
to the following system of equations
$\frac{\ddot{A}}{A}+\frac{\ddot{B}}{B}+\frac{\dot{A}\dot{B}}{AB}+\frac{3}{4}\lambda^{2}=-(p+\frac{1}{2}\dot{V}^{2})-\frac{V^{\prime
2}}{2A^{2}},$ (2.8)
$\frac{\ddot{A}}{A}+\frac{\ddot{B}}{B}+\frac{\dot{A}\dot{B}}{AB}+\frac{3}{4}\lambda^{2}=-(p+\frac{1}{2}\dot{V}^{2})+\frac{V^{\prime
2}}{2A^{2}},$ (2.9)
$\frac{2\ddot{A}}{A}+(\frac{\dot{A}}{A})^{2}-\frac{f^{\prime\prime}}{fA^{2}}+\frac{3}{4}\lambda^{2}=-(p+\frac{1}{2}\dot{V}^{2})+\frac{V^{\prime
2}}{2A^{2}},$ (2.10)
$(\frac{\dot{A}}{A})^{2}+\frac{2\dot{A}\dot{B}}{AB}-\frac{f^{\prime\prime}}{fA^{2}}-\frac{3}{4}\lambda^{2}=(\rho+\frac{1}{2}\dot{V}^{2})+\frac{V^{\prime
2}}{2A^{2}},$ (2.11)
$\dot{\rho}+(\rho+p)(\frac{2\dot{A}}{A}+\frac{\dot{B}}{B})=0,$ (2.12)
$\ddot{V}-\frac{1}{A^{2}}V^{\prime\prime}+(\frac{2\dot{A}}{A}+\frac{\dot{B}}{B})\dot{V}-\frac{ff^{\prime}}{f^{2}A^{2}}V^{\prime}=0.$
(2.13)
Here the over heat dot denotes differentiation with respect to $t$ and over
head prime denotes differentiation with respect to $\chi$.
From equations (2.8) and (2.9), we get
$V^{\prime}=0.$ (2.14)
Consequently, the mesonic scalar field does not exit in the direction of
$\chi$.
The functional dependence of the metric together with equations (2.9) and
(2.10), using equation (2.14), imply that
$\frac{f^{\prime\prime}}{f}=k^{2},\qquad k^{2}=\text{constant}.$ (2.15)
If $k=0$, then the solution of this differential equation is
$f(\chi)=k_{1}\chi+k_{2}$, $k_{1}$ and $k_{2}$ are constants of integration.
Without loss of generality, we choose $k_{1}=1$ and $k_{2}=0$. Thus we shall
have
$f(\chi)=\chi.$ (2.16)
In the case $f(\chi)=\chi$ the field equations (2.8)-(2.13), using equation
(2.14), reduce to
$\frac{\ddot{A}}{A}+\frac{\ddot{B}}{B}+\frac{\dot{A}\dot{B}}{AB}+\frac{3}{4}\lambda^{2}=-(p+\frac{1}{2}\dot{V}^{2}),$
(2.17)
$\frac{2\ddot{A}}{A}+(\frac{\dot{A}}{A})^{2}+\frac{3}{4}\lambda^{2}=-(p+\frac{1}{2}\dot{V}^{2}),$
(2.18)
$(\frac{\dot{A}}{A})^{2}+\frac{2\dot{A}\dot{B}}{AB}-\frac{3}{4}\lambda^{2}=(\rho+\frac{1}{2}\dot{V}^{2}),$
(2.19) $\dot{\rho}+(\rho+p)(\frac{2\dot{A}}{A}+\frac{\dot{B}}{B})=0,$ (2.20)
$\ddot{V}+(\frac{2\dot{A}}{A}+\frac{\dot{B}}{B})\dot{V}=0.$ (2.21)
We can easily find from (2.21) that
$\dot{V}=\frac{n}{A^{2}B},$ (2.22)
where $n(\neq 0)$ is a constant of integration.
From equations (2.17) and (2.18), we get
$\frac{\ddot{B}}{B}+\frac{\dot{A}\dot{B}}{AB}=\frac{\ddot{A}}{A}+(\frac{\dot{A}}{A})^{2}.$
(2.23)
We assume $A$ to be some arbitrary function of $B$, say
$A=\psi(B).$ (2.24)
So equation (2.23) becomes
$\big{(}\frac{\psi_{B}}{\psi}-\frac{1}{B}\big{)}\ddot{B}+\big{[}\frac{\psi_{BB}}{\psi}+(\frac{\psi_{B}}{\psi})^{2}-\frac{\psi_{B}}{B\psi}\big{]}\dot{B}^{2}=0,$
(2.25)
where $\psi_{A}=\frac{d\psi}{dA}$. Equation (2.25) results in the following
possibilities.
(i-1)
$\frac{\psi_{B}}{\psi}-\frac{1}{B}=0,\quad\text{and}\qquad\frac{\psi_{BB}}{\psi}+(\frac{\psi_{B}}{\psi})^{2}-\frac{\psi_{B}}{B\psi}=0,$
(2.26)
(i-2)
$\ddot{B}=0,\quad\text{and}\qquad\frac{\psi_{BB}}{\psi}+(\frac{\psi_{B}}{\psi})^{2}-\frac{\psi_{B}}{B\psi}=0,$
(2.27)
(i-3)
$\dot{B}=0.$ (2.28)
## 3 Solutions of Field Equations
We have only five highly non-linear field equations (2.17)-(2.21) in sex
unknowns , $A,B,p,\rho,V$ and $\lambda$. In order to obtain its exact
solution, we assume one more physically reasonable condition amongst these
variables. We consider here the effective "stiff fluid" distribution, that is,
a perfect fluid with the equation of state:
$p=\rho.$ (3.1)
The equation of state (3.1) was apparently first proposed by Zeldovich [26].
It should have applied in the early Universe, the justification being the
observation that with (3.1) the velocity of sound equals the velocity of
light, so no material in this Universe could be more stiff.
Using the condition (3.1) in equation (2.20) and by integrating, we get
$p=\rho=\frac{m}{A^{4}B^{2}},$ (3.2)
where $m(\neq 0)$ is a constant of integration.
Case (i-1):
From equation (2.26), using the first equation in the second equation, then by
integrating, we get
$\psi=c_{1}B+c_{2},$ (3.3)
where $c_{1}(\neq 0)$ and $c_{2}$ are integration constants. Using this result
in the first equation in (2.26), we get $c_{2}=0$. So that equation (3.3)
becomes
$\psi=c_{1}B.$ (3.4)
Using this result in equation (2.25), we have
$A=c_{1}B.$ (3.5)
Now using this equation and the condition (3.1), in equations (2.17)-(2.19),
we get
$\frac{2\ddot{B}}{B}+(\frac{\dot{B}}{B})^{2}+\frac{3}{4}\lambda^{2}=-\frac{3}{2}\rho,$
(3.6) $3(\frac{\dot{B}}{B})^{2}-\frac{3}{4}\lambda^{2}=\frac{3}{2}\rho.$ (3.7)
These equations yield
$B=(at+b)^{\frac{1}{3}},$ (3.8)
where $a(\neq 0)$ and $b$ are constants of integration.
According to equations (3.5) and (3.8) the line element (2.1) can be written
in the following form
$ds^{2}=dt^{2}-(at+b)^{\frac{2}{3}}(c_{1}^{2}d\chi^{2}+c_{1}^{2}\chi^{2}d\phi^{2}+dz^{2}),$
(3.9)
Physical properties of the model
Using equations (3.5) and (3.8) in equations (2.19)-(2.21), take into account
(3.1), the expressions for density $\rho$, pressure $p$, massless scalar field
$V$ and displacement field $\lambda$ are given by
$\rho=p=\frac{c}{(at+b)^{2}},\qquad c=\frac{m}{c_{1}^{4}},$
which shows that $\rho$ and $p$ are not singular,
$V=\frac{n}{ac_{1}^{2}}\log(at+b)+c_{3},$
$\lambda^{2}=\frac{c_{4}}{(at+b)^{2}},\qquad
c_{4}=\frac{4a^{2}}{9}-\frac{2(2m+n^{2})}{3c_{1}^{4}}.$
It is observed, from equations (3.5) and (3.8) that $A(t)$ and $B(t)$ can be
singular only for $t\rightarrow\infty$. Thus the line element (3.9) is
singular free even at $t=0$.
For the line element (3.9), using equations (2.2), (2.4) and (2.5), we have
the following physical properties:
The volume element is
$\mathcal{V}=c_{1}^{2}\chi(at+b).$
This equation shows that the volume increases as the time increases, that is,
the model (3.9) is expanding with time.
The expansion scalar, which determines the volume behavior of the fluid, is
given by
$\Theta=\frac{a}{at+b}.$
The only non-vanishing component of the shear tensor, $\sigma_{ij}$, is
$\sigma_{44}=-\frac{2a}{3(at+b)}.$
Hence, the shear scalar $\sigma$ is given by
$\sigma^{2}=2\big{(}\frac{a}{3(at+b)}\big{)}^{2}.$
Since $\lim_{t\rightarrow\infty}(\frac{\sigma}{\Theta})\neq 0$, then the model
(3.9) does not approach isotropy for large value of $t$. Also the model does
not admit acceleration and rotation, since $\dot{u}_{i}=0$ and $\omega_{ij}=0$
.
Case (i-2):
The first equation in (2.27) gives
$B=b_{1}t+b_{2},$ (3.10)
where $b_{1}(\neq 0)$ and $b_{2}$ are constants of integration. Using this
result in the second equation in (2.27) and take into account (2.24), we get
$A^{2}=b_{3}(b_{1}t^{2}+2b_{2}t)+b_{4},$
where $b_{3}(\neq 0)$ and $b_{4}$ are constants of integration. This equation
can be written in the form
$A=(b_{3}B^{2}+b_{5})^{\frac{1}{2}},$ (3.11)
where the constant $b_{5}$ depends on the constants $b_{1}$ and $b_{2}$.
Using equations (3.10) and (3.11) in equations (2.17)-(2.21), take into
account the condition $\rho=p$, we get
$\rho=p=\frac{m}{(b_{1}t+b_{2})^{2}}[b_{3}(b_{1}t+b_{2})^{2}+b_{5}]^{2},$
$V=\frac{n}{b_{3}b_{5}}\log\frac{b_{1}t+b_{2}}{\sqrt{b_{3}(b_{1}t+b_{2})^{2}+b_{5}}}+b_{6},$
where $b_{6}$ is the constant of integration,
$\lambda^{2}=\frac{2(2b_{1}^{2}b_{3}^{2}(b_{1}t+b_{2})^{3}-2m-n^{2})}{3(b_{3}(b_{1}t+b_{2})^{2}+b_{5})^{2}(b_{1}t+b_{2})^{2}}-\frac{8b_{1}^{2}b_{3}}{3(b_{3}(b_{1}t+b_{2})^{2}+b_{5})}.$
In this case the line element (2.1) takes the following form
$ds^{2}=dt^{2}-(b_{3}(b_{1}t+b_{2})^{2}+b_{5})(d\chi^{2}+\chi^{2}d\phi^{2})-(b_{1}t+b_{2})^{2}dz^{2}.$
(3.12)
We shall now give the expression for kinematic quantities. A straightforward
calculation leads to the expressions for element volume $\mathcal{V}$,
expansion scalar $\Theta$ and shear tensor $\sigma_{ij}$ of model (3.12) are
given, respectively, by
$\mathcal{V}=\chi(b_{1}t+b_{2})(b_{3}(b_{1}t+b_{2})^{2}+b_{5}),$
which shows that the model is expanding with time,
$\Theta=\frac{2b_{1}b_{3}(b_{1}t+b_{2})}{(b_{3}(b_{1}t+b_{2})^{2}+b_{5})}+\frac{b_{1}}{(b_{1}t+b_{2})},$
$\sigma_{11}=\frac{b_{1}(b_{3}(b_{1}t+b_{2})^{2}+b_{5})}{3(b_{1}t+b_{2})}-\frac{b_{1}b_{3}(b_{1}t+b_{2})}{3},$
$\sigma_{22}=\chi^{2}\sigma_{11},$
$\sigma_{33}=\frac{2b_{1}b_{3}(b_{1}t+b_{2})^{3}}{b_{3}(b_{1}t+b_{2})^{2}+b_{5}}-\frac{2b_{1}(b_{1}t+b_{2})}{3},$
$\sigma_{44}=-\frac{4b_{1}b_{3}(b_{1}t+b_{2})}{3(b_{3}(b_{1}t+b_{2})^{2}+b_{5})}-\frac{2b_{1}}{3(b_{1}t+b_{2})},$
and other components of the shear tensor, $\sigma_{ij}$, being zero. Hence
$\sigma^{2}=\frac{1}{9}\Big{(}11\frac{b_{1}^{2}b_{3}^{2}(b_{1}t+b_{2})^{2}}{(b_{1}t+b_{2})^{2}+b_{5})^{2}}+5\big{(}\frac{b_{1}}{b_{1}t+b_{2}}\big{)}^{2}+\frac{2b^{2}_{1}b_{3}}{b_{3}(b_{1}t+b_{2})^{2}+b_{5}}\Big{)}.$
Moreover, this model represents non-rotating and has vanishing acceleration.
Since $\lim_{t\rightarrow\infty}(\frac{\sigma}{\Theta})\neq 0$, then the model
(3.12) do not approach isotropy for large value of $t$.
Case (i-3):
Equation (2.28) can be easily integrated to give
$B=\ell,$ (3.13)
where $\ell(\neq 0)$ is the constant of integration. Using this equation in
equations (2.17)-(2.19) and take into account the condition (3.2),we have
$A=(\ell_{1}t+\ell_{2})^{\frac{1}{2}},$ (3.14)
where $\ell_{1}(\neq 0)$ and $\ell_{2}$ are constants of integration. Using
equations (3.13) and (3.14) in equations (2.17)-(2.21), take into account the
condition $\rho=p$, we get
$\rho=p=\frac{m}{\ell^{2}(\ell_{1}t+\ell_{2})^{2}},$
$V=\frac{n}{\ell\ell_{1}}\log(\ell_{1}t+\ell_{2})+\ell_{3},$
where $\ell_{3}$ is the constant of integration,
$\lambda^{2}=\frac{\ell_{4}}{(\ell_{1}+\ell_{2})^{2}},\qquad\ell_{4}=\big{(}3\ell_{1}^{2}-\frac{2(2m+n^{2})}{3\ell^{2}}.$
In this case the line element (2.1) can be written in the form
$ds^{2}=dt^{2}-(\ell_{1}t+\ell_{2})(d\chi^{2}+\chi^{2}d\phi^{2})-\ell^{2}dz^{2}.$
(3.15)
From equation (3.14), one can observed that $A(t)$ is singular only when
$t\rightarrow\infty$. Consequently, the line element (3.15) is singular free
even $t=0$.
For the line element (3.15), the volume element $\mathcal{V}$ and the
kinematics properties (acceleration $\dot{u}_{i}$, rotation $\omega_{ij}$,
expansion scalar $\Theta$, shear tensor $\sigma_{ij}$ and shear scalar
$\sigma$) respectively found to have the following expressions:
$\mathcal{V}=\ell\chi(\ell_{1}t+\ell_{2}),$
which shows that the model is expanding with time,
$\dot{u}=0,$ $\omega_{ij}=0,$ $\Theta=\frac{\ell_{1}}{\ell_{1}t+\ell_{2}},$
$\sigma_{11}=-\frac{\ell_{1}}{6},$ $\sigma_{22}=\chi^{2}\sigma_{11},$
$\sigma_{33}=\frac{\ell\ell_{1}}{3(\ell_{1}t+\ell_{2})},$
$\sigma_{44}=-\frac{2\ell_{1}}{3(\ell_{1}t+\ell_{2})}.$
$\sigma^{2}=\frac{11\ell_{1}^{2}}{36(\ell_{1}t+\ell_{2})}.$
As in the above two cases,
$\lim_{t\rightarrow\infty}(\frac{\sigma}{\Theta})\neq 0$, the model (3.15) do
not approach isotropy for large value of $t$.
## 4 Conclusions
This paper deals with axially symmetric space-time in the presence of mesonic
stiff fluid distribution within the framework of Lyra’s geometry for time
dependent displacement field. We have presented a new class of exact solutions
of Einstein’s field equations for this space-time. The obtained models
represent shearing, non-rotating and expanding with time $t$. Moreover, these
models are singular free even at the initial epoch $t=0$ and have vanishing
accelerations. For all models, we found also that
$\lim_{t\rightarrow\infty}(\frac{\sigma}{\Theta})\neq 0$, this means that they
are not approach isotropy for large time $t$. We found also that the mesonic
scalar field in axially symmetric space-time exists only in the t-direction.
## References
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|
arxiv-papers
| 2009-09-08T14:44:47 |
2024-09-04T02:49:05.134365
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ragab M. Gad",
"submitter": "Ragab Gad",
"url": "https://arxiv.org/abs/0909.1503"
}
|
0909.1516
|
Salient features in locomotor evolutionary adaptations of proboscideans
revealed via the differential scaling of limb long bones
.
By Valery B. Kokshenev and Per Christiansen
.
Submitted to the Journal of the royal Society Interface 03 June 2009
.
Abstract. The standard differential scaling of proportions in limb long bones
(length against circumference) is applied to a phylogenetically wide sample of
the Proboscidea, Elephantidae and the Asian (Elephas maximus) and African
elephant (Loxodonta africana). In order to investigate allometric patterns in
proboscideans and terrestrial mammals with parasagittal limb kinematics, the
computed slopes (slenderness exponents) are compared with published values for
mammals and studied within a framework of theoretical models of long bone
scaling under gravity and muscle forces. Limb bone allometry in E. maximus and
the Elephantidae are congruent with adaptation to bending and/or torsion
induced by muscular forces during fast locomotion, as in other mammals,
whereas limb bones in L. africana appear adapted for coping with the
compressive forces of gravity. Consequently, hindlimb bones are expected to be
more compliant than forelimb bones in accordance with in vivo studies on
elephant locomotory kinetics and kinematics, and the resultant negative limb
compliance gradient in extinct and extant elephants, which contrasts to other
mammals, suggests an important locomotory constraint preventing achievement of
a full-body aerial phase during locomotion. Differences in ecology may be
responsible for the subtle differences observed between African and Asian
elephant locomotion, and the more pronounced differences in allometric and
mechanical patterns established in this study.
Key words: long bone scaling models; standard differential scaling; limb
gradient functions; proboscideans; extinct and extant elephants.
## I Introduction
Differential scaling of the proportions of the limb long bones in terrestrial
mammals has been studied by many researchers (most often bone length $L$ and
circumference $C$ or diameter $D$) with the aim to establish correlations
between design and posture of mammalian limbs coping with support of mass and
locomotion in the gravitational field. Many attempts to formulate generally
applicable allometric power laws ($L\varpropto M^{l}$ and $C=\pi D\varpropto
M^{d}$, where $M$ is body mass) have been the subject of a long standing
debate and controversy (e.g., McMahon 1973, 1975a, b; Alexander 1977;
Alexander et al. 1979a; Biewener 1983, 2005; Economos 1983; Bertram & Biewener
1990; Christiansen 1999a, b, 2002, 2007; Kokshenev, 2003, 2007; Kokshenev et
al. 2003). Among the proposed theoretical frameworks, the three most common of
which are: the _geometric_ (or isometric) _similarity model_ (GSM, with
$l_{0}=d_{0}=1/3$); the _elastic similarity model_ (ESM, with
$l_{0}=1/4,d_{0}=3/8$); and the static _stress similarity model_ (SSM, with
$l_{0}=1/5,d_{0}=2/5$). Subsequently, these similarity models have typically
been explored in analysis of allometric power laws using measured bone lengths
and diameters resulted in allometric exponents ($l$ and $d$), when body masses
are available. In cases where body masses are unknown, a _slenderness
exponent_ $\lambda$ can be computed (via $L\varpropto C^{\lambda}$) and
compared with that predicted as $\lambda_{0}=l_{0}/d_{0}=1$, $2/3$, or $1/2$
by the GSM, ESM, and SSM, respectively (McMahon 1975a).
Originally proposed to explain animal design to experience similar elastic
forces and stresses under gravity, the corresponding ESM (or ”buckling” model)
and SSM (McMahon 1973, 1975a, b) were found not to apply to terrestrial
mammals as a group, neither overall body proportions (Economos 1983; Silva
1998), nor allometric scaling of long bones (Alexander et al. 1979a; Biewener
1983, 2005; Economos 1983; Christiansen 1999a, b; Kokshenev et al. 2003).
Instead, body proportions and long bones were found closer to isometry, i.e.,
to GSM with $\lambda_{0}=1$, and in large mammals and in large mammals they
both become progressively more robust with increments in body size. Testing
McMahon’s models, many studies indicated that muscle forces, showing size-
dependent fluctuations among terrestrial mammals, are highly important (e.g.,
Alexander 1985, Biewener 1983, 1989, 1990, 2005). Direct evidences of muscle
forces affecting proportions of the allometric scaling of the major long bones
in Artiodactyla were provided by Selker and Carter (1989). It was analytically
shown that the failure of any one predicted power law was shown not caused by
the failure of underlying elastic force patterns but due to McMahon’s
simplifications for evolutionary adaptive properties for maintaining similar
skeletal functional stresses apart of muscles and under the dominating
influence of gravity (Kokshenev et al. 2003). Then, using the overall-bone
slenderness exponent $\lambda_{\exp}^{(mam)}=0.80\pm 0.02$, resulting from re-
analysis of bone proportions in a wide-ranging sample of terrestrial mammals
from Christiansen (1999a), also not matching any of McMahon’s original
predictions, the dominating role of muscle forces in long-bone scaling was
demonstrated from first physical principles (Kokshenev 2003).
Naturally, bone shape is a hereditary property, but bone is a phenotypically
plastic tissue, capable of reacting powerfully to its mechanical environment.
In the growing fetus, bone shape, size and position is initially determined by
the early cartilaginous anlagen during embryonic skeletogenesis, which are
subsequently gradually replaced by endochondral ossification during ontogeny
(Favier & Dollé 1997; Currey 2003; Provet & Schipani 2005). However, bone
shape is heavily influenced by a mechanical response to the environment during
ontogeny and throughout an animal’s life. It has been demonstrated that strain
rate and magnitude, surrounding tissue formation, and fetal muscle
contractions are prerequisites for normal bone formation during ontogeny
(Rodriguez et al. 1992; Mosley et al. 1997; Mosley & Lanyon 1998; Lamb et al.
2003). In post-natal and adult mammals, bone is capable of reacting to changes
in mechanical stresses enforced by physical activity and muscle mechanics with
rapid alterations of size and shape (Biewener 1983, 1989, 1990; Carrano &
Biewener 1999; Curry 2003; Firth et al. 2005; Warden et al. 2005; Franklyn et
al. 2008), and hereditary properties determined by the genome appear primarily
responsible for bone patterning during fetal ontogeny and is less relevant for
bone size and shape in the adult animal (Mariani & Martin 2003). These factors
are seemingly beyond macroscopic elastic theories for formulating allometric
power laws for bone scaling. Moreover, the concept of uniform elastic
similarity also seems to be inconsistent with a diversity of functional local
elastic forces and stresses, which are not constant in bones during locomotion
and therefore are not likely to be reflected in scaling analysis of external
bone dimensions (Doube et al. 2009). Nevertheless, the authors hopefully
believe that scaling predictions arising from the macroscopic spatial
continuous mechanics applied to bone tissue under broad spectrum of loading
conditions (Kokshenev et al. 2003), and therefore reflecting most general
trends in proportional limb bone adaptations to environmental conditions, can
be reliably verified at least by the overall limb and bone allometric data.
The above bone scaling studies and other statistical and experimental studies
(e.g., Biewener et al. 1983a, b, 1989; Rubin & Lanyon 1984; Selker & Carter,
1989; Streicher & Muller 1992; Carrano & Biewener 1999) have stimulated
formulation of novel theoretical concepts in light of dynamic bone strain
similarity (Rubin & Lanyon, 1984) or mechanical (strain and stress) similarity
(Kokshenev 2007). Basic conceptions of McMahon’s (1973, 1975a) elastic
similarity hypothesis have also been reconsidered (Kokshenev et. al., 2003,
Kokshenev 2003). The framework of bone scaling is, by default, limited to long
bones approximated by cylinders with $L\gg D$ justified by the ratio
$L/D\backsim 10$, at least for mammalian humerus, radius, ulna, femur, and
tibia. Moreover, the justification for application of the elastic theory
patterns established for arbitrary loaded long solid cylinders (Kokshenev
2007) is generally based on the assumption that the long bones play the
primary role in body support. Consequently, the positive allometry of long
bone structure in relation to body mass observed in regression analysis may be
expected to be better understood by biomechanical adaptation of bones to
maximal external loads emerging during fast locomotion and studied via bone-
reaction elastic forces and stresses, whereas non-mechanical ontogeny effects
of limb bone adaptation associated with a small Prange’s index are relatively
small (Kokshenev 2007).
In the present study we analyze the surprisingly varied differential scaling
of the limb long bones in a taxonomically narrow clade of mammals, the extant
proboscideans (Proboscidea, Elephantidae); the Asian elephant (Elephas
maximus), and African savannah elephant (Loxodonta africana). These gigantic
land mammals have a more upright limb posture, notably much more upright
propodials and different locomotor mechanics from other terrestrial mammals in
that fast locomotion is ambling with no suspended phase in the stride, but
with duty factors $\beta>0.5$ (Gambaryan 1974; Alexander et al. 1979b;
Hutchinson et al. 2003, 2006). We compare theoretical predictions with the
data from a phylogenetically wide sample of extinct proboscideans from
Christiansen (2007) completed here by the Elephantidae family, as well as
allometry results from scaling studies of running mammals with parasagittal
limb kinematics, in the hope of establishing generic allometric patterns
distinguishing limb postures characteristic of high-power locomotion in
proboscideans and mammals.
Modern elephants are characteristic in having very long limb bones for their
body size (Christiansen 2002), and a very upright, though not strictly
columnar (Ren et al. 2007), limb posture, in which the two propodial bones
(femur and in particular humerus) are kept at a distinctly greater angle
compared to the ground than is the case in other large, quadrupedal mammals.
The steeply inclined propodials imply that during standing and at low speeds,
the primary forces affecting the limb bones will be axially compressive.
However, with increments in speeds, joint flexion increases during the support
phase, and during the recovery phase, joint flexion can be high (Ren et al.
2007). Although there is no difference in joint flexion between juvenile and
adults or between Asian and African elephants, the propodial bones are still
markedly more inclined compared to horizontal even during the fast locomotion
than is in the case for other quadrupedal running mammals. During the support
phase in locomotion, the ankle, unlike the more mobile wrist, displays spring-
like mechanical properties, reminiscent of, albeit less than in quadrupedal
running mammals, matching the more compliant hind limbs during locomotion.
This also is consistent with the tendons of the hind-foot scaling with
positive allometry during ontogeny, whereas those of the forelimb scale with
negative allometry, and thus become progressively more gracile (Miller et al.
2008), thereby supporting the observation that the hindlimbs are more
compliant with bouncing kinematics than the stiffer, vaulting forelimbs,
during fast locomotion. In view of the progress in application of bone scaling
models (Kokshenev 2003, 2007), a general problem arises whether kinematic
evidences on the mechanical influence on bone ontogeny can be independently
revealed by the differential limb bone scaling?
## II Materials and Methods
### II.1 Theoretical background
A theoretical analysis of non-critical elastic forces emerging in long bones
of adult mammals resulted in mode-independent relationships for bone scaling
exponents
$d=\frac{1}{3}+b\text{ and }l=\frac{1}{3}-b\text{, with
}\lambda(b)=\frac{l}{d}=\frac{1-3b}{1+3b}$ (1)
discussed in Eq. (8) in Kokshenev 2007. Here $b$ is Prange’s index scaling
bone mass to body mass. Since scaling index $b$ is consistently small for
mammals (see e.g. table 1 in Kokshenev 2003), equation (1) matches previous
observations mentioned in Introduction of the closeness of mammalian bone
allometry to isometry. Accordingly, small deviations from the force-isotopic
GSM are described via the directly observable model-independent index $b$. The
differential scaling data for mammals (Christiansen 1999a, 1999b) supports the
physically justified inequalities $d>1/3>l$ (Kokshenev 2007) providing the
constraint $\lambda(b)<1$, resulting in $b<1/6$ following from equation (1).
The empirically established constraint $b_{\exp}^{(mam)}=0.04\pm 0.01$, for
overall-averaged long bones in mammalian limbs (see table 1 in Kokshenev
2007), results in the _model-independent_ pattern, namely
$d_{pre}^{(mam)}=0.37\pm 0.01\text{ and }l_{pre}^{(mam)}=0.29\pm 0.01\text{,
or }\lambda_{pre}^{(mam)}=0.785\pm 0.025$ (2)
predicted for mammals capable of true running with a fully suspended aerial
phase in the stride $[1]$.11footnotetext: It seems to be interesting to
compare the proposed empirical exponent
$\lambda_{\operatorname{mod}}^{(mam)}=0.785$, substituting McMahon’s
$\lambda_{0}=1/2$, with the theoretical estimate
$\lambda_{\operatorname{mod}}=7/9$ ($\thickapprox 0.778$) obtained on the
basis of a pattern of non-axial elastic forces in bone resisting functionally
relevant limb muscles (Kokshenev 2008). The bone-muscle scaling theory will be
discussed elsewhere.
As for McMahon’s models discussed in Introduction and revised by Kokshenev
(2003, 2007) and Kokshenev et al. (2003), they can be broadly interpreted as
follows. If both gravitational and muscular competitive forces driven by
structural bone adaptation to complex (axial and non-axial) compression were
equally important in bone interspecific allometry, the observed statistically
overall-bone slenderness exponent $\lambda_{\exp}$ is expected to be nearly
isometric, i.e., close to maximum $\lambda_{0}=1$, following from the GSM. If,
however, gravitational forces were dominating, bone proportions could be
expected to become optimized for exploitation of long bone stiffness during
modes of fast locomotion in mammals with near parasagittal limb kinematics,
and would result in $\lambda_{0}=2/3$, as predicted by the ESM, as predicted
by the ESM, also known as the buckling model. Although the ESM was originally
introduced for the states of elastic instability (McMahon 1973) or
thermodynamic instability (Kokshenev et al. 2003), the domain of resulting
scaling relations extends far below critical amplitudes of forces (and
critical stresses and strains), which should still exclude non-axial elastic
forces (Kokshenev 2007). In contrast, the adaptation to faster forms of
locomotion via more compliant long bones subjected to the non-axial
compression could be observed via the exponent $\lambda_{0}=1/2$, predicted by
the SSM, treated as a ”bending-torsion” model (Kokshenev 2007). As an example
of theoretical rationalization underlying these two distinct patterns of
bipedal locomotion in the sagittal plane, a transition from stiff-limbged
slow-walking to the compliant-limbged fast walking following by running was
illuminated in terms of a dynamic instability of the trajectory of center of
gravity in humans (Kokshenev 2004).
When muscle forces play a dominating role in the formation of bone
proportions, the appropriate modified model for bone evolution (hereafter
termed SSMM) predicts $\lambda_{pre}^{(bend)}=0.80\pm 0.03$, as derived from
both allometric mammalian limb bone and muscle data (see equation (18) in
Kokshenev 2003). Since this prediction is congruent with the one made in
equation (2), we infer that mammalian long bones are designed to resist peak
bending and/or torsional bone compressions produced by muscles during fast
running modes, that was initially established via bending functional bone
stress (Rubin & Lanyon 1982) and then explained analytically (see figure 1 in
Kokshenev 2007). For mammals as group, we therefore use the SSMM estimate
$\lambda_{\operatorname{mod}}^{(mam)}=0.785$ predicted semi-empirically in
equation (2) and theoretically in $[1]$.
### II.2 Materials and analysis
We used previously published data on external limb long bone dimensions from
$19$ species and $217$ specimens of proboscideans (Christiansen 2007), which
was supplemented by new data collected for the purpose of this study. We
compared this to published data from $79$ and $98$ species of running mammals
from Christiansen 1999a and 1999b, respectively. We conducted regression
analysis on $Log_{10}$ transformed external limb long bone articular lengths
and diaphysial diameters using the standard _least squares_ (LS) and _reduced
major axis_ (RMA) methods. All species with multiple specimens were averaged
prior to analysis. The significances of the regression parameters were
evaluated by computing the correlation coefficient, the standard error of the
estimate, and the F-statistic of the regressions, and the $95\%$ confidence
intervals for the regression intercepts and slopes (see online electronic
supplementary material). The regression analysis included the entire
Proboscidea; the Elephantidae (Elephas sp., Loxodonta africana, and Mammuthus
sp.). We computed separate regression analyses for the extant Asian elephant
(Elephas maximus) and African savannah elephant (Loxodonta africana), since
these are, by default, the only taxa for which locomotory information exists;
we included no data from the forest elephant (Loxodonta cyclotis) within the
African elephant, because this taxon most likely constitutes a separate
species (Barriel et al. 1999).
## III Results
Searching for generic morphometric patterns in long bones via external
dimensions, to which the similarity models (McMahon 1973, 1975a; Kokshenev
2003, 2007; Kokshenev et al. 2003) are broadly addressed, we study the
slenderness bone exponent $\lambda$ as directly observed by the slopes in
plots $Log_{10}L$ vs $Log_{10}C$. In table 1, the results of bone-size
regression analysis in species-averaged specimens of proboscideans are
compared with those for mammals.
.
Species | Ele | phant | idae | Pro | bosci | deans | Ma | mmals |
---|---|---|---|---|---|---|---|---|---
Limb bones | $N$ | $\lambda$ | $r$ | $N$ | $\lambda$ | $r$ | $N$ | $\lambda$ | $r$
Humerus | 7 | 0.912 | 0.990 | 16 | 1.134 | 0.831 | 189 | 0.7631 | 0.9738
Radius | 6 | 0.813 | 0.853 | 10 | 1.078 | 0.878 | 189 | 0.7530 | 0.9957
Ulna | 6 | 0.727 | 0.888 | 14 | 0.929 | 0.866 | 189 | 0.849∗ | 0.9600
Femur | 7 | 0.747 | 0.966 | 14 | 0.802 | 0.816 | 189 | 0.8431 | 0.9763
Tibia | 6 | 0.751 | 0.925 | 11 | 0.772 | 0.857 | 188 | 0.7641 | 0.9499
Limb bone, LS | 6 | 0.790 | 0.924 | 13 | 0.943 | 0.850 | 189 | 0.795 | 0.971
Limb bone, RMA | 6 | 0.856 | 0.924 | 13 | 1.165 | 0.850 | 189 | 0.778 | 0.971
Table 1. The statistical data on the slenderness of individual and effective
limb bones in animals. The data for Elephantidae and Proboscidea are shown on
the basis of regression data provided in online electronic supplementary
material, extending table 2 in Christiansen 2007, and these for mammals are
taken from table 2 in Christiansen 1999b. The mean _slenderness exponents_
$\lambda=dLog_{10}L/dLog_{10}C$ presented by the slopes ($\lambda$)
exemplified in figure 1 are observed in $N$ _species_ through the LS
regression of with the _correlation coefficient_ $r$. The limb bone LS
characterization corresponding to the overall-bone mean data is introduced by
the standard mean over all 5 bones. The RMA data in the last row are shown
only for the resulting limb bone data. The bold numbers are the data used
below in figures. The italic numbers indicate the slope data contrasting to
mammalian data with $\lambda<1$ (see also discussion in section 2.1 in the
Methods). ∗) The data estimated with the help of ratio $l/d$ for the ulna
allometric exponents taken from table 2 in Christiansen 1999a.
.
In figure 1 and table 2, we analyze modern elephants.
.
Place figure 1
.
Species | E. | maxi | mus | Loxo | donta | | afri | cana
---|---|---|---|---|---|---|---|---
Limb bones | $n$ | $\lambda$ | $r$ | $n$ | $\lambda$ | $r$ | $p_{\min}$ | $p_{\max}$
Humerus | 22 | 0.754 | 0.985 | 14 | 0.616 | 0.978 | 0.01 | 0.02
Radius | 19 | 1.014 | 0.987 | 8 | 0.675 | 0.994 | — | 0.001
Ulna | 20 | 0.818 | 0.981 | 11 | 0.644 | 0.979 | 0.01 | 0.02
Femur | 25 | 0.758 | 0.972 | 13 | 0.618 | 0.986 | 0.01 | 0.02
Tibia | 20 | 0.913 | 0.970 | 10 | 0.688 | 0.939 | 0.02 | 0.05
Forelimb bone | 20 | 0.862 | 0.984 | 11 | 0.645 | 0.984 | 0.01 | 0.02
Hindlimb bone | 23 | 0.836 | 0.971 | 12 | 0.653 | 0.963 | 0.015 | 0.035
Limb bone, LS | 21 | 0.851 | 0.979 | 11 | 0.648 | 0.975 | 0.013 | 0.028
Limb bone, RMA | 21 | 0.869 | 0.979 | 11 | 0.665 | 0.975 | 0.013 | 0.028
Table 2. The statistical data on $Log_{10}L$ vs $Log_{10}C$ regression for
individual limb bones in Elephas maximus and Loxodonta africana. Notations of
table 1 are extended by the t-test comparisons of slopes in $n$ _specimens_
shown by $p_{min}<p<p_{max}$ (for details see the electronic supplementary
material). Characteristic of a given group of species _forelimb_ and _hindlimb
bones_ are introduced through the fore-bone (humerus, radius, and ulna) and
hind-bone (femur and tibia) standard means, respectively. The effective _limb
bone_ is determined by the overall (5-bone) standard mean. Other notations are
the same as in table 1.
.
The main results obtained by the LS regression in tables 1 and 2 are displayed
and analyzed in figure 2.
.
Place figure 2
.
In figure 2, the model predictions for bone slenderness exponent are compared
with those of the entire group of proboscideans, Elephantidae, mammals, and
modern elephants. As seen from the data presented by the bone-averaged
exponents (in table 2) resulting from the LS and RMA species-average
statistics (shown by bars), the limb bones of the family Elephantidae are
structurally designed likewise those in mammals. The data for extant Elephas
maximus are also quite similar to mammals, whereas the bone exponents in
Loxodonta africana are distinctly lower (see also figure 1). This implies that
the data for Elephas maximus as well as the family Elephantidae, are better
explained by adaptations to peak muscular forces during locomotion, whereas
the limb bones in Loxodonta africana are indicative of adaptation to cope with
the forces of gravity $[2]$.22footnotetext: Our sample of Elephas maximus has
more large juveniles included than in Loxodonta africana, and all resulting
slopes are generally higher than in Loxodonta africana (table 2). Comparing
only adult specimens, therefore ignoring ontogenetic adaptations, the radius
and femur slopes remain significantly lower in Loxodonta africana than Elephas
maximus, whereas the exponents for humerus, ulna and tibia become non-
significantly different (see online supplementary material). In Loxodonta
africana, all exponents for adult specimens only are similar to the full
sample including large juveniles, whereas the slopes in Elephas maximus become
significantly higher (femur) and lower (tibia). Overall, the LS-average
slenderness exponent in adult Elephas maximus ($\lambda_{\exp}=0.839$) remains
significantly higher ($p<0.05$) than in adult Loxodonta africana
($\lambda_{\exp}=0.662$). The Elephantidae and individual species within this
family also have thinner long bones than more primitive proboscideans (Haynes
1991; Christiansen 2007).
.
Place figure 3
.
In figure 3, we examine the allometry exponents of individual bones, with the
aim to interpret their adaptation to the patterns of peak elastic forces and
stresses. Within the Elephantidae, the similarity in variation of bone
slenderness in those three groups is very evident from the parallel lines, as
shown in figure 3. When the mechanical origin of the model predictions
provided in Methods is taken in consideration, the limb bones in Elephas
maximus indicate adaptation to complex muscular and, in part, gravity
stresses.
It is well established in comparative zoology that the humerus of running
parasagittals is loaded differently owing to large muscle attachments and an
inclined angle compared to the epipodials, but in elephants the humerus and
femur are not very steeply inclined. This is broadly congruent with the
observation in figure 3 that the femur, which being almost vertical when the
animal stands motionless and much more inclined in running mammals than in
fast moving elephants, should involve lower bending and torsional moments in
Elephas maximus than in other mammals and much less muscular moments in
Loxodonta africana. Indeed, as predicted by the ESM beyond experimental error,
the limb bones in Loxodonta africana appear to be adapted for axially
compressive stress generated by gravity. Such a distinction in allometry
slopes induced by mechanical adaptation, clearly distinguishes Loxodonta
africana from Elephas maximus, which, in turn appears to bear a mechanical, if
not morphological similarity to the limb bones of mammals capable of true
running. Accordingly, the limbs of Elephas maximus would appear to be more
adapted for resisting the forces from limb muscles, which broadly is more
consistent with faster forms of locomotion.
As seen in figure 3, the radius and femur in modern elephants expose different
loading trends of those in mammals. A consequent distinct mechanical
characterization of the hindlimbs and forelimbs is displayed in figure 4.
Place figure 4
.
In figure 4, the differently designed forelimb and hindlimb bones are
schematically shown through the hind-fore-limb bone vector, indicating the
existence of a gradient in the limb bone functions, likewise gradients in
muscle functions in mammals (Biewener et al. 2006) increasing the stability of
running (Daley et al. 2007). One can see that the Elephantidae contrasts to
mammals in limb bone functions.
## IV Discussion
### IV.1 Surveying kinematic empirical data
Recent studies have provided new insights and have significantly enhanced our
understanding of elephant locomotion. Typically called graviportal, elephants
are unable to run with a suspended phase in the stride, and even during fast
locomotion, one limb is placed firmly on the ground (Gambaryan 1974). New
studies have, however, indicated that fast moving elephants are not merely
walking, but that kinetics and kinematics during fast locomotion differ from
walking, and numerous locomotor parameters are similar to those of running
mammals, for instance offsetting the of phase of forelimb and hindlimb
footfalls; achieving the transient walk-run duty factors $0.5$; and
maintenance of pendular locomotor kinematics, typical of walking gaits, for
the forelimbs, whereas the hind limbs move with a more spring-like (bouncing)
action (Hutchinson et al. 2006; Ren & Hutchinson 2007). Accordingly, at high
speeds elephant locomotor kinematics are indicative of walking gaits whereas
kinetic analyses indicate running, as in other quadrupedal mammals. The
primary differences are that elephants never achieve a full-body aerial phase
during any form of locomotion, although at speeds of just over $\char
126\relax 2$ $ms^{-1}$, the hind limbs begin exhibiting an aerial phase with
bouncing kinetics, implying that the limbs become compliant, whereas the
forelimbs maintain a more straight morphology, consistent with more vaulting
mechanical properties (Ren & Hutchinson 2007). Another significant difference
from running, quadrupedal mammals, there is no marked change of gait at high
speeds, even at Froude numbers $Fr>3$, a value when other quadrupedal animals
have changed gaits to a bouncing, running gats with a full-body aerial phase
(Alexander 1983, 1989; Alexander & Jayes 1983).
During progressively faster locomotion, elephants initially increase speeds
primarily by increments in stride frequency, but at high speeds, further speed
increase is facilitated primarily by increments in stride length (Hutchinson
et al. 2006). This slow-to-fast gait transition is similar to walking-trotting
transition in quadrupedal running mammals, where increments in speed is a
function of those in both stride length and frequency, whereas in the case of
running gaits with bouncing limb kinetics and a full-body aerial phase is
characterized primarily by increments in stride length (e.g., Heglund et al.
1974; Pennycuick 1975, Biewener 1983; Alexander 1983, 1989; Alexander & Jayes
1983). Locomotor kinematic parameters in Asian and African elephants are
broadly similar, but statistically significant differences exist, pertaining
to relative stride lengths, stride frequencies, stance phase, and duty factor
with speed. African elephants have higher duty factors, shorter stride lengths
and higher stride frequencies than Asian elephants (Hutchinson et al. 2006).
Interestingly, large elephants, such as full grown bulls, appear incapable of
reaching the same locomotor intensity as small elephants, and have duty
factors $\beta<0.5$, implying that no limb pair ever exhibits an aerial phase;
they are, in effect, only walking during fast locomotion. Consequently, this
observation may imply that large individuals may exploit the compliance of
bone tissue to a lesser extent.
### IV.2 What does the overall-bone statistics tell us?
The theoretical concepts provided in the Methods section permits an
interpretation of the limb bone allometric data in terms of the adapted
loading patters. Beyond any modeling, the regression analysis illustrated in
figure 1 indicates that the individual limb bones in distinct extant elephants
are likely similar in bone ontogeny but they are evidently different in their
mechanical adaptation as revealed by distinct slopes in size proportions. When
known theoretical models are employed in terms of the overall-bone slenderness
exponent (in figure 2), one can see that the limbs in Elephas maximus, as well
as in Elephantidae, show a similarity to the limbs of other mammals, broadly
exploiting muscular forces during efficient locomotion. This observation
agrees with in vivo data on the elephant locomotor kinematics having generally
many patterns in common with typical tetrapods (e.g., Hutchinson at al. 2006).
In contrast, our analysis show that Loxodonta africana successfully employs
body gravitation and reaction gravitation forces for efficient walking. More
specifically, the analysis in figure 2 indicates that the overall-bone
averaged slenderness allometry exponent associated with the structure of an
effective limb bone (defined in table 2) can be understood in mammals and the
Elephantidae (but not intraspecifically in Loxodonta africana), by its
adaptation to peak muscular forces generated during fast locomotion, whereas
the limb bones in Loxodonta africana are indicative of adaptation to cope with
the forces of gravity more successfully exploiting in walking. This finding
also implies that both propodial and epipodial limb bones in mammals are
established to be adapted for peak functional bending and torsional stresses
(figure 3) by the exploiting of bone compliance, contrasting to more stiffer
limbs in Asiatic elephants, as predicted by the ESM (figure 4).
When Elephas maximus and Loxodonta africana are compared, no contrasting
adaptation of any individual limb bone is revealed in figure 3, because all
bone lines are parallel. This is not the case of lines lying between mammals
as group and elephants. This observation suggests a similarity between bone
joint angles generally established for modern elephants during fast locomotion
by direct observations, e.g., by Ren et al. (2007). Consequently, bone angles,
which are evidently similar in elephants, and distinct from other mammals,
indicate differing loading conditions related to limb postures in elephants,
extant and extinct, from those of other mammals. More specifically, the
epipodial femur, shows in figure 3 its adaptation to complex compression
(bending and/or torsion) during bouncing kinematics of the hindlimb involved
in fast locomotion, thereby exploiting rather the bone compliance, than the
bone stiffness associated with more isometric bone proportion scaling exposing
by the forelimb radius, in Elephas maximus and forelimb humerus in
Elephantidae, thereby contrasting to parasagittal femur and radius. Such
mechanical trends are consistent with the kinematic data (Hutchinson et al.
2006) that the hind limbs of modern elephants during fast locomotion are
broadly more compliant than the fore limbs. This empirical finding in modern
elephants is now generalized over extinct elephants.
The contrasting postures between mammals as group and elephants can be
understood by the different design of fore limbs and hind limbs revealed in
figure 4 by the opposed directions of the gradient of the limb functions in
locomotor kinematics. As for the small negative gradient in Loxodonta africana
with respect to Elephas maximus, it can be ignored, because the typical
statistical error (shown by the bars) exceeds the length of the limb bone
vector. In other words, the crossover of the bone lines between the modern
elephants can be referred to a small statistical uncertainty that can be
ignored, providing a qualitative agreement with the overall similarity in limb
postures revealed in figure 3. On the contrary, the observation in figure 4 of
the gradient of limb functions for the Elephantidae exceeding the statistical
error may suggest a trend for forelimb bones to be more isometric, and also
that the forelimb bones in Elephas maximus are therefore stiffer than bones in
the hind limbs. This also is congruent with the differences in limb locomotor
kinematics (Hutchinson et al. 2006), outlined above.
### IV.3 Positive gradient of limb stiffness as major locomotor constraint in
elephants
Our study of the allometry of individual limb bones reveals different patterns
in limb mechanical adaptation in proboscideans and begs the question how they
correlate with kinematic patterns characteristic of modern elephants? Such a
correlation is expected, since the mean data on duty factor $\beta$ in limbs
of modern elephants (Hutchinson et al. 2006) are underlaid by the overall-bone
(and overall-muscle) limb characterization described here though the limb bone
slenderness exponent $\lambda.$
Near the walk-run transition in mammals, with the Froude number
$Fr\thickapprox 1$, the scaling predictions for the forelimb duty factor
$\beta_{FL}=0.52$ and the hindlimb duty factor $\beta_{HL}=0.53$ (estimated on
the basis of empirical scaling relations by Alexander & Jayes 1983), provide
negative _limb duty factor gradient_ $\Delta\beta$
($\equiv\beta_{FL}-\beta_{HL}$) for mammals, contrasting with the positive
gradient $\Delta\beta_{ele}$ for elephants (see table 6 in Hutchinson et al.
2006). These data can be related to our LS bone exponent $\lambda_{FB}=0.788$
in mammalian forelimb and $\lambda_{HB}=0.804$ in mammalian hindlimb,
corresponding to the mammalian overall-bone exponent
$\lambda_{\exp}^{(mam)}=0.795$ (table 1) and the negative _limb bone gradient_
$\Delta\lambda_{\exp}^{(mam)}=-0.016$ (with,
$\Delta\lambda\equiv\lambda_{FL}-\lambda_{HL}$). For elephants, the duty
factor gradient $\Delta\beta>0$ also correlates to the bone gradient
$\Delta\lambda>0$ (figure 4). Indeed, as follows from tables 1 and 2,
$\Delta\lambda_{\exp}^{(ele)}=0.026$, for Elephas maximus, and
$\Delta\lambda_{ele}=0.068$, for Elephantidae. As for the discrepancy in signs
between $\Delta\beta_{\exp}^{(ele)}=0.026$ (table 4 by Hutchinson et al. 2006)
and $\Delta\lambda_{\exp}^{(ele)}=-0.008$ for the Loxodonta africana, it was
referred above to the statistical error.
Employing the similarity in limb functions and their gradients observed
directly (kinematically) in modern elephants and indirectly (allometrically)
via limb bones of extinct and extant elephants, we develop a simple linear
model predicting limb duty factors for Elephantidae, which includes more
primitive groups of proboscideans (see the electronic supplementary material).
The linearization procedure of the mean limb data and their gradients known
for living elephants (shown, respectively, by the dashed lines in figure 5 and
its inset) results in the duty factor predictions for Elephantidae, as
explained in figure 5.
.
Place figure 5
.
The geometrical visualization of locomotory constraints imposed on animal
limbs in a certain locomotion mode can be displayed on a $\lambda$-$\beta$
diagram presented by figure 5. Such a characterization makes a link between
the limb bone and limb bone gradient bone proportions (figure 4) and the limb
and limb gradient kinematics (figure 4 by Hutchinson et al. 2006).
The major limb functional difference in mammals as a group and elephants
indicated by different orientation of the characteristic vectors shown in
figure 5 is due to the difference in signs of the limb stiffness-compliant
gradient transferred between fore and hind limbs during the animal’s forward
propulsion of the body. Running mammals, having hindlimbs which are stiffer
than the forelimbs and therefore transfer the positive compliance limb
gradient (or negative stiff limb gradient), but they are able to achieve a
full-body aerial phase during fast locomotion. In contrast, elephants,
transferring negative limb compliance gradient (hindlimb bones are more
compliant than forelimb ones) do not achieve a full-body aerial phase during
any form of locomotion, though are able to sufficiently reduce the positive
stiff limb bone gradient by limb muscles when showing a negative limb duty
factor gradient $\Delta\beta$ during both slow and fast walking gaits (see
figure 4B by Hutchinson et al. 2006). However, it is not enough for changing
of the positive direction of the limb gradient vector in the $\lambda$-$\beta$
diagram, contrasting to mammals (figure 5), since elephants, being naturally
constrained in limb bone proportions, are not able to change sign of the
positive gradient in limb bone slenderness $\Delta\lambda$. Hence, the
preserved excessive positive hindlimb-forelimb stiffness gradient ensured by
the corresponding mass-independent and speed-independent positive limb bone
slenderness, explains the inability of elephants to perform true running with
a full-body aerial phase discussed by Hutchinson et al. 2003. Consequently, in
order to move fast they increase stride frequency, mostly exploiting forelimb
stiffness, instead of compliance, and in order to increase the stride length,
they are forced to use hindlimb bone compliance, instead of stiffness.
### IV.4 Asian compared with African elephants
Being similar in fast locomotion gaits with respect to the lateral sequence
footfall pattern, the Asian, African and most likely extinct elephants (as
predicted figure 5) are found to differ in limb bone and perhaps also muscle
constraints. All having similar projections of the characteristic vector in
the $\lambda$-$\beta$ diagram, the limb bone stress indicated by the bone
slenderness for the African elephant is significantly distinct from that in
other elephants. According to the SSMM, in the Elephantidae and in particular
in Elephas maximus, the bone off-axial external muscle forces generated during
fast locomotion, broadly exceeding body weight and causing a complex bending-
torsion elastic bone stress, provide a relatively high level of limb
compliance conducted by the structurally adapted limb long bones. In contrast,
limb bones in Loxodonta africana generally adapted for axial bone compression,
are most likely tuned by limb muscles to employ better gravitation reaction
forces, in accord with relatively low bone slenderness, explained by the ESM.
Having long bones designed to maintain axial stress and avoiding bending and
torsion, African elephants can be expected to exhibit shorter stride lengths
and therefore to use higher stride frequencies than Asian elephants, at
increased locomotor speeds. From an energetic point of view, this implies the
higher energy cost of the Asian elephant locomotion, whereas higher duty
factors characteristic of African elephants (figure 5) indicate less bending
moments about the joints accommodated ground reaction forces. We infer that
Asian elephants, having more compliant limb bones than African elephants, are
broadly able to maintain higher speeds more easily that is statistically
supported by the observation (via $\beta<0.5$) of Asian elephants (figure 4A
by Hutchinson et al. 2006). Even the the limb duty factors in Asian and
African elephants may achieve those in mammals, the negative gradient for the
limb bone compliance limits the maximum stride length and therefore the
maximal running speed with respect to mammals.
It is traditionally believed that mechanical differences must be large to
produce differences in bone morphology (Frost 1990), but more recent studies
have demonstrated that temporal continuous stimulation three orders of
magnitude below the maximal peak forces characteristic of fast locomotion (see
Rubin & Lanyon 1982) is likely sufficient to produce significant changes in
bone morphology (Rubin et al. 2001). African elephants are typically found in
open environments and routinely undertake long-distance seasonal migrations at
leisurely paces, whereas Asian elephants are mostly found in topologically
more heterogeneous, forested environments and appear to undertake fewer and
shorter, if any, seasonal migrations (Sikes 1971; McKay 1973; Laws et al.
1975; Sukumar 1991, 1992). Potentially, this could imply subtle differences
even in low-force every-day locomotor mechanics imposed by the structure of
the environment. Thus, differences in ecology and migratory activity may
conceivably be responsible for the subtle differences in locomotor mechanics
between African and Asian elephants, as observed by Hutchinson et al. (2006).
In this study, more pronounced differences in allometric and mechanical
patterns are demonstrated for Elephas maximus, which appear more similar to
those in other mammals, and Loxodonta africana, which is divergent from both
and also from the Elephantidae.
On the other hand, neither allometric nor kinematic or kinetic studies deal
with forces and bone stresses directly. It remains therefore a challenge to
further analyze the reactive-force elastic-stress patterns revealed for limb
bones in extant elephants. Nevertheless, there is another difference between
the two species of extant elephants congruent with our findings. Because the
limb bone pattern for African elephants indicates axial bone stress, not
increasing with body mass (Rubin & Lanyon 1982, 1984) and therefore would
constitute non-critical stress (Kokshenev 2007), both the mean and maximal
body masses for Asian elephants are expected to be below of those for African
elephants. Indeed, African elephants appear to be larger on average than Asian
elephants; African elephant large bulls routinely weight $5-7$ tons, whereas
$4-5$ tons is more common for Asian elephant bulls (Wood 1976; Shoshani 1991).
Tentative maximal size of African elephant also appears to be distinctly
larger. World recorded bulls are as close to or even exceeding 4 m in standing
shoulder height, and estimated body masses of $10-12$ tons (Wood 1976;
McFarlan 1992), whereas Asian elephants are estimated at $3.3-3.4$ m in
shoulder height and around 8 tons (Pilla 1941; Wood 1976; Blashford-Snell &
Lenska 1996).
.
Acknowledgments
One of the authors (V.B.K.) acknowledges financial support by the CNPq.
.
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Figure Captions
.
Figure 1. Long bone articular lengths against diaphysial least circumferences
in extant elephants. A, humerus; B, ulna; C, femur; D, tibia. Closed squares
are Elephas maximus; open squares are Loxodonta africana. Regression
coefficients are shown in table 2.
.
Figure 2. A comparison of the predictions by the theory of similarity with the
bone slenderness exponents observed via the regression slopes in different
groups of proboscideans. Notations: open circles are McMahon’s predictions for
bones adapted for the influence of gravity ($\lambda_{0}=1,2/3$, and $1/2$ due
to GSM, ESM, and SSM); the closed circle shows mean data
$\lambda_{pred}=0.785$ predicted for the limb bone, which is primarily adapted
for resisting peak muscle forces during locomotion [discussed in equation (2)
]. The bars are the mean LS and RMA data for the limb bone characteristic of
Proboscidea, Elephantidae, and mammals taken from table 1 and these for
Elephas maximus and Loxodonta africana, taken from table 2. The error bars
show statistical variations between the means of regression data.
.
Figure 3. The observation of trends in limb bone mechanical adaptation in
different species. The bars show maximal variations of the mean exponents of
individual limb bones. The slenderness exponents in humerus (H), radius (R),
ulna (U), femur (F), and tibia (T) are analyzed in view of elastic similarity
models. The notations on the symmetry of bone compression follow from the
models described in Methods. Other notations correspond to those in figure 2.
.
Figure 4. Observation of elastic similarity in the effective forelimb bone
(humerus, radius, and ulna) and hindlimb bone (femur and tibia). The arrows
indicate deviations in the trends of adaptation for forelimb and hindlimb
mechanical functions. The bar shows statistical error. Other notations
correspond to those in fig. 3.
.
Figure 5. Limb bone scaling in mammals and elephants against limb kinematics
in fast walking. The vector positions and magnitudes indicate the slenderness
exponent and duty factor and the vector directions indicate their forelimb-
hindlimb gradients. The dashed vector position predicts the limb duty factor
$\beta_{pre}=0.58$ consistent with $\lambda_{\exp}=0.790$ for Elephantidae
obtained by liner interpolation between the kinematic data for Asian and
African elephants, as shown by the thin dashed line. The corresponding the
model duty factor gradient $\Delta\beta_{\operatorname{mod}}=0.043$ is found
in the inset, through the linear extrapolation (shown by the dashed line) of
the gradient known for Elephas maximus (blue point) and
$\Delta\lambda_{mod}=0.008$ adopted for the Loxodonta africana (green point).
Other data are provided above and/or taken from table 4 by Hutchinson et al.
2006.
|
arxiv-papers
| 2009-09-08T16:54:51 |
2024-09-04T02:49:05.139912
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Valery B. Kokshenev and Per Christiansen",
"submitter": "Valery B. Kokshenev",
"url": "https://arxiv.org/abs/0909.1516"
}
|
0909.1558
|
# Fast Winds and Mass Loss from Metal-Poor Field Giants111Data presented
herein were obtained at the W. M. Keck Observatory, which 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
A. K. Dupree Harvard-Smithsonian Center for Astrophysics, 60 Garden Street,
Cambridge, MA 02138 dupree@cfa.harvard.edu Graeme H. Smith University of
California Observatories/Lick Observatory, University of California, Santa
Cruz, CA 95064 graeme@ucolick.org Jay Strader222Hubble Fellow Harvard-
Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138
jstrader@cfa.harvard.edu
###### Abstract
Echelle spectra of the infrared He I $\lambda$10830 line were obtained with
NIRSPEC on the Keck 2 telescope for 41 metal-deficient field giant stars
including those on the red giant branch (RGB), asymptotic giant branch (AGB),
and red horizontal branch (RHB). The presence of this He I line is ubiquitous
in stars with $T_{eff}\gtrsim$ 4500K and $M_{V}$ fainter than $-$1.5, and
reveals the dynamics of the atmosphere. The line strength increases with
effective temperature for $T_{eff}\gtrsim$ 5300K in RHB stars. In AGB and RGB
stars, the line strength increases with luminosity. Fast outflows ($\gtrsim$
60 km s-1) are detected from the majority of the stars and about 40 percent of
the outflows have sufficient speed as to allow escape of material from the
star as well as from a globular cluster. Outflow speeds and line strengths do
not depend on metallicity for our sample ([Fe/H]= $-$0.7 to $-$3.0) suggesting
the driving mechanism for these winds derives from magnetic and/or
hydrodynamic processes. Gas outflows are present in every luminous giant, but
are not detected in all stars of lower luminosity indicating possible
variability. Mass loss rates ranging from $\sim 3\times 10^{-10}$ to $\sim
6\times 10^{-8}$ $M_{\sun}\ yr^{-1}$ estimated from the Sobolev approximation
for line formation represent values with evolutionary significance for red
giants and red horizontal branch stars. We estimate that 0.2 M☉ will be lost
on the red giant branch, and the torque of this wind can account for
observations of slowly rotating RHB stars in the field. About 0.1–0.2 M☉ will
be lost on the red horizontal branch itself. This first empirical
determination of mass loss on the RHB may contribute to the appearance of
extended horizontal branches in globular clusters. The spectra appear to
resolve the problem of missing intracluster material in globular clusters.
Opportunities exist for ’wind smothering’ of dwarf stars by winds from the
evolved population, possibly leading to surface pollution in regions of high
stellar density.
stars: chromospheres — stars: Population II — stars: winds, outflows
## 1 Introduction
The assumption of mass loss from stars evolving on the red giant branch of
globular clusters has yet to be tested through direct detection of winds. This
assumption remains one of the major concerns in the evolution of low mass
stars and may be related to the second-parameter problem (Sandage & Wildey
1967): differing horizontal branch morphology between globular clusters of the
same metallicity and age. Various explanations have been offered for
differences in the horizontal branch morphology including intrinsic
dispersions in the amount of stellar mass loss, rotation, or deep mixing,
environmental effects possibly correlated with cluster mass or central
density, heterogeneities in He abundance possibly as a result of cluster
pollution by intermediate-mass asymptotic giant branch (AGB) stars, or of the
infall of planets onto cluster stars (Buonanno et al. 1993; Buonanno et al.
1998; Catelan et al. 2001; Sills & Pinsonneault 2000; Recio-Blanco et al.
2006; Sandquist & Martel 2007; Soker et al. 2001b; Sneden et al. 2004;
Sweigart 1997; Peterson et al. 1995; Ventura & D’Antona 2005). Most recently
with the advent of infrared photometry discussed below, and the identification
of multiple populations on the main sequence of globular clusters with the
Hubble Space Telescope (Anderson 2002; Bedin et al. 2004; Piotto et al. 2007),
renewed attention is focussing on the mass loss process.
A related issue is the puzzling absence of the material lost from the red
giants in a globular cluster (Tayler & Wood 1975). This gas is expected to
accumulate in clusters between sweeps through the galactic plane, but
detection has proved elusive. Unconfirmed measures of 21-cm H-line emission
exist for NGC 2808 (Faulkner et al. 1991). The radio dispersion of milli-
second pulsars located in 47 Tuc and M15 hinted at an enhanced electron
density in the intracluster medium (Freire et al. 2001). Deep searches for H I
in 5 clusters gave one firm detection of H I emission, and possibly 2 others,
but the amount of mass inferred from this is a few orders of magnitude less
than expected (van Loon et al. 2006). A search for intracluster dust in 12
globular clusters with the Far-Infrared Surveyor on AKARI failed to detect
emission except possibly in one cluster (Matsunaga et al. 2008). There is a
notable lack of detections of intracluster material with the Spitzer Space
Telescope. Only 2 clusters of the many observed with Spitzer have ’possible’
detections of intracluster material (Boyer et al. 2006; Barmby et al. 2009).
One suggestion for removal of intracluster material is ram-pressure stripping
by the galactic halo (Frank & Gisler 1976, Okada et al. 2007), but Spitzer
observations can not confirm the expected relation between cluster kinematics
and the presence (or upper limit) of dust (Barmby et al. 2009).
Results from mid-infrared observations of metal-deficient field giants and
globular clusters with IRAS, the ISO satellite, the Spitzer Space Telescope,
and AKARI (Smith 1998; Origlia et al. 2002, 2007; Evans et al. 2003; Boyer et
al. 2006; Ita et al. 2007; Boyer et al. 2008) suggest that some giants have
produced circumstellar dust that could result from stellar winds. Since not
all giants display excess infrared emission, mass loss associated with dust
appears to be episodic (Origlia et al. 2007; Mészáros et al. 2008). Caloi and
D’Antona (2008) further suggest that the mass loss rate might be ‘sharply’
peaked at one value along the red giant branch. In the metal-rich open cluster
NGC 6791, the presence of low mass white dwarfs led Kalirai et al. (2007) to
conclude that mass loss is enhanced in high metallicity environments, although
that suggestion appears to be controversial (van Loon et al. 2008; Bedin et
al. 2008). Concurrently with the above results, the possibility of multiple
episodes of star-formation, and self-pollution in globular clusters is
receiving increased attention as a means to explain chemical variations and
multiple branches in the color-magnitude diagram of clusters (Lee et al. 1999;
Pancino et al. 2000; Anderson 2002; D’Antona et al. 2002; Piotto et al. 2007;
Kayser et al. 2008). This suggestion frequently resorts to AGB stars with mass
$>~{}$3M☉ ‘polluting’ either the surface of the cluster stars or the
environment in which a second generation of stars subsequently forms. The wind
velocities from the AGB stars must be slow so that material will not escape
the cluster. A long-standing suggestion envisions that pulsation near the top
of the AGB may degenerate into relaxation oscillations, during which mass loss
or envelope ejection occurs in rapid fashion, dubbed a ‘superwind’ (Renzini
1981; Bowen and Willson 1991). Another conjecture (Soker et al. 2001a) calls
for a superwind during the immediate post-RGB phase to explain gaps on the
horizontal branch. Clearly, both observations and theory currently allow a
great variety in both the presence and character of mass loss from cluster
stars. The spectra of He I $\lambda$10830 presented here can address these
questions.
## 2 Spectroscopic Diagnostics of Winds
Spectra of stars contain many features that can be used to detect winds
directly, but their diagnostic properties are related to the specific
conditions in a stellar atmosphere. In a cool giant star, atoms and ions form
at different levels in the chromosphere, loosely tied to the local electron
temperature which is increasing with height. Additionally, for strong lines
such as H$\alpha$, Ca II H and K, and Mg II, the profile is formed over an
extended range of atmospheric layers which results from differing opacities
across the line itself. Calculations for metal-poor giants demonstrate (Dupree
et al. 1992a; Mauas et al. 2006) the locations in the atmosphere where
commonly observed lines are formed. Mass flows can produce asymmetric profiles
of H$\alpha$ and Na D, as well as emission asymmetries and velocity shifts in
the reversed absorption core of Ca II and Mg II. These features have been
measured in red giants in many globular clusters (Cacciari et al. 2004;
McDonald and van Loon 2007; Mészáros et al. 2008, 2009b) and in metal-poor
field stars (Smith & Dupree 1988; Dupree & Smith 1995; Dupree et al. 2007).
However the velocities inferred from the optical profiles are generally less
than the escape velocity from the star, and can not truly be identified as
stellar winds.
We note the difference between the use of ’outflow’ and ’wind’ in this
context. By ’outflow’, we mean the presence of a velocity field within the
region of a chromosphere sampled by our spectral diagnostic in which a flow of
material occurs that is moving away from the stellar photosphere. The term
’wind’ is reserved for the particular case of an outflow in which the outflow
velocity exceeds the escape velocity within the region of the chromosphere
sampled by our spectral diagnostic.
A good diagnostic of winds is the near-infrared He I 10830Å line ($1s2s\
^{3}S\ -\ 1s2p\ ^{3}P$) which models show (Dupree et al. 1992a) is formed
higher in the metal-poor atmosphere than H$\alpha$ and Ca II K. Thus it might
be expected to trace out higher velocities, where the outflow becomes a wind,
than the optical diagnostics. Additionally, the lower level of this transition
is metastable, and is not closely linked to local physical conditions in the
wind, so it can absorb photospheric radiation and map out an expanding wind in
a luminous star.
The lower level of the $\lambda$10830 multiplet ($1s2s\ ^{3}S_{1}$) lies 19.7
eV above the ground state of He I and can be populated directly by collisions
from the ground state ($1s^{2}\ {}^{1}S$) although the rate is far less than
for an allowed transition. High temperatures ($\gtrsim$ 20,000 K) are
generally required. The ${}^{3}S$ level can also be populated by recombination
from the continuum. This latter pathway has long been studied especially in
the Sun and other cool stars (Goldberg 1939; Harvey & Sheeley 1979; Zirin
1975, 1982) because a source of EUV or X-ray radiation can photoionize He I
from the ground state or the ${}^{3}S$ level and the helium ion preferentially
recombines to the triplet state followed by subsequent cascade to the lower
level of the $\lambda$10830 transition. Thus stars with strong X-ray emission
display enhanced helium absorption in $\lambda$10830 (O’Brien & Lambert 1986;
Zarro & Zirin 1986; Sanz-Forcada & Dupree 2008). Depopulation of the ${}^{3}S$
state occurs at high densities with collisions to the ${}^{1}P$ level. Because
${}^{3}S$ has a long lifetime for decay to the ground state, (i.e. this level
is metastable), a significant population can build up, and provide an
opportunity for scattering of near-IR photons from the line itself or
continuum photons from the photosphere. If the chromosphere is expanding, this
transition can trace out the wind velocity as it scatters radiation while
being carried along in the expansion. An additional advantage of this
transition is that the profile can not be compromised by interstellar or
circumstellar absorption since it is not a resonance line.
The first detection of a wind in a metal-deficient star using the
$\lambda$10830 line was made (Dupree et al. 1992a) in the bright field giant,
HD 6833 where an outflow of 90 km s-1 was discovered $-$ a value comparable to
the chromospheric escape velocity. Subsequently, Smith et al. (2004)
identified He I absorption from one warm AGB star in the globular cluster M13
in addition to two other metal-poor field giants. The short wavelength
extension of the lines in these stars reached 90 to 140 km s-1 – again fast
enough to escape a chromosphere and also a globular cluster. A stellar
$T_{eff}$ greater than 4600K appeared required to populate the lower level of
the He I atom; thus Smith et al. (2004) suggested that the coolest red giants
can not produce this transition. Indeed, $\lambda$10830 was not detected in
the 5 coolest red giants observed in M13. While globular cluster stars
themselves remain ideal targets, the metal-deficient field giants are
brighter, more accessible to current instrumentation, and can act as
surrogates for cluster stars. We report here on the high-resolution
spectroscopy of the He I $\lambda$10830 line in 41 such field stars.
## 3 Observations and Reductions
The objective in this investigation was to study the systematics of the He I
line among evolved Population II stars in a variety of evolutionary states.
This goal suggests an observational program concentrating on halo field stars
rather than globular cluster stars, since in the latter only the upper regions
of the red giant and asymptotic giant branches can be studied at high signal-
to-noise, even with the NIRSPEC instrument on the Keck 2 telescope. Spectra of
the He I line for several red giants in the cluster M13 were published by
Smith et al. (2004). In our study, halo field stars were chosen from the lists
of Bond (1980) and Beers et al. (2000) with an effort to achieve a good
sampling in the red giant branch, red horizontal branch, and asymptotic giant
branch phases of evolution. No selection was made on the basis of metallicity
or proper motion, although certain radial velocities were avoided in order to
prevent overlap of the He I line with telluric absorption features. Some
chemically peculiar stars in the form of CH stars were included in the sample.
To facilitate high signal-to-noise spectroscopy, only stars with apparent
magnitudes of $V<11$ were observed, and most have $V<10$. Thus this sample
comprises relatively nearby halo stars, although most are beyond the limit at
which the Hipparcos satellite provided reliable parallaxes. In order to study
evolved stars, most of our targets have absolute magnitudes of $M_{V}<+2$, and
no subdwarfs were included in the program.
The spectra of 41 metal-poor halo field giants (Table 1) were obtained during
1.5 nights of observation in May 2005 using NIRSPEC (McLean et al. 1998, 2000)
on the Keck 2 telescope. Observations were made using the echelle cross-
dispersed mode of NIRSPEC with the NIRSPEC-1 order-sorting filter and a slit
of $0.43^{\arcsec}\times 12^{\arcsec}$ giving a nominal resolving power of
23,600. The long-wavelength blocking filter was not used in order to minimize
unwanted fringing. Total integration times accumulated for each star are
listed in Table 2; these times are generally broken into two shorter exposures
in the NOD-2 positions. Calibration exposures consisted of internal flat-field
lamps, NeArKr arcs, and dark frames. Spectra of rapidly rotating hot stars
obtained at airmasses similar to the target objects were used to identify, and
minimize or eliminate night sky emission lines.
Data reduction was performed using the REDSPEC package (McLean et al. 2003)
which was written specifically for NIRSPEC. Following dark subtraction and
flat fielding, the data frames were spatially rectified. Wavelength
calibration was performed using NeArKr arc lamp spectra taken after each
science observation. For this study, we extracted only order 70 (wavelength
coverage $\sim 1.079-1.095$ $\mu$m), which contains the He I line at 1.0830
$\mu$m. The strong Si I photospheric absorption line at 1.0827$\mu$m was used
to estimate the stellar radial velocity with an uncertainty of $\sim$5 km s-1.
Values of the radial velocity (RV) are given in Table 2.
The spectrum of each giant was normalized to a continuum determined by fitting
a 5th order cubic spline to the wavelength range 1.082 - 1.092 $\mu$m. The
wavelength scales of the spectra were then shifted onto the photospheric rest
frame of each star by applying a zero-point wavelength shift determined from
measuring the wavelength of the nearby photospheric 10827.09Å Si I line. The
equivalent width of the helium line was measured from the continuum normalized
spectra using the IRAF333Image Reduction and Analysis Facility (IRAF) written
and supported by the IRAF programming group at NOAO, operated by AURA under a
cooperative agreement with the NSF. (http://iraf.noao.edu/iraf/web) task
‘splot’ and measuring the width directly, or, if blended with the adjacent Si
I line, deconvolving the blend with Voigt profiles and dividing the total
equivalent width measured directly into appropriate fractions obtained from
the deconvolution. The photometric colors, extinction, and evolutionary state
as inferred for the target stars are included in Table 1. The footnotes to
this Table contain the references to the tabulated quantities. The assignment
of the evolutionary state of our targets is based on the Strömgren $c_{1}$ vs.
$b-y$ diagram and the color magnitude diagram ($M_{V}$ vs $B-V$). Parameters
of the He I line are given in Table 2. About half (21) of the 41 targets are
located on the red giant branch (RGB), and one (HD135148) is identified as a
CH star. Red horizontal branch (RHB) stars comprised 11 objects, and
asymptotic branch stars (AGB) made up 6 targets. Two subgiants and one semi-
regular red variable (TY Vir) completed the sample. The spectra of these stars
are shown in Figure 1 (RGB stars), Figure 2 (RHB stars), Figure 3 (AGB stars),
and Figure 4 (subgiant stars).
The stars in our sample show three basic types of He I line behavior: (i) a
helium line with a pure absorption profile, (ii) a P Cygni type profile in
which an absorption profile is paired with an emission feature to longer
wavelengths, and (iii) no helium line at all - neither in absorption or
emission. Inspection shows that the He I line is generally broader than the
neighboring Si I photospheric absorption line which is expected since He I
arises in the higher temperature chromosphere. In many cases where a He I
absorption profile is present, this profile extends to shorter wavelengths
providing evidence of a chromospheric outflow (see, for example, the spectra
of stars BD $+$30°2611 and HD 122956 in Figure 1, BD $-$03°5215, HD 119516 in
Figure 2, and HD 121135 and HD 107752 in Figure 3).
The star HD 135148, classified as a CH star, and identified as a spectroscopic
binary (Carney et al. 2003) exhibits (Figure 5) a substantial P Cygni profile
with deep absorption almost to zero flux at velocities $\sim-$60 km s-1, and
extending to $-$115 km s-1. The spectrum of the coolest target of our sample,
HD104207 (GK Com) is shown in Figure 6 where many photospheric absorption
features of neutral atoms appear in addition to a weak P Cygni feature of He
I. The value of [Fe/H]=$-$1.93 in this star is similar to many other stars in
our sample without such an array of neutral species in their spectra, thus
dramatically illustrating the effects of low effective temperature.
## 4 Discussion
The targeted sample includes $(B-V)$ colors ranging from 0.6 to 1.6, and spans
6 magnitudes, reaching from the tip of the RGB to several subgiant stars. Most
of the stars showed a He I $\lambda$10830 feature the presence of which is
shown as a function of position in the $M_{V}$ versus $(B-V)_{0}$ and Teff
diagrams in Figure 7\. The targets from M13 and the metal-deficient field
giants reported earlier (Dupree et al. 1992a; Smith et al. 2004) have been
added to this figure and their parameters are given in Table 3. Many of the
more luminous stars at $M_{V}=0$ and brighter, exhibit emission resulting from
scattering of the line above the stellar limb which is normal in large stars
with extended chromospheres. Since the He I line is formed at chromospheric
temperatures ($\sim$10,000$-$20,000K), it might be expected to vanish in the
coolest objects where the chromosphere does not attain sufficiently high
temperatures. A region exists in the color magnitude diagram, with magnitude
brighter than $-$1.5 and with $(B-V)_{0}>1.1$ where the $\lambda$10830 line
does not generally appear either in emission or absorption. Our earlier search
for He I along the red giant branch in M13 revealed absorption in an AGB star,
IV-15, near $(B-V)_{0}$ =1.02, but not in any of 5 cooler stars on the red
giant branch (Smith et al. 2004). The presence of a $\lambda$10830 line among
Population II giants as a function of position in a color-magnitude diagram
differs from Population I stars (O’Brien & Lambert 1986; Lambert 1987), where
the helium emission disappears near spectral type M1 in giants and supergiant
stars, corresponding to $T_{eff}$ of 3780 K (Tokunaga 2000). Based on our
earlier sample (Smith et al. 2004), we noted that Population II giants with
$T_{eff}$ less than 4600K did not show helium. However the larger sample
presented here contains several stars with $T_{eff}$ less than 4600K, and they
exhibit the $\lambda$10830 line. Two of these objects are somewhat anomalous
red giants. HD 104207 (GK Com) is the coolest star in the sample, and a semi-
regular variable. It is plausible that the atmosphere cycles through heating
and cooling phases, producing the helium line at certain times. The other
star, HD 135138, is a CH object, a spectroscopic binary with a degenerate
secondary star both of which could contribute to atmospheric conditions of
high excitation. Yet a handful of otherwise normal stars with T${}_{eff}<$
4600K remain: HD 6833; BD+30°2611; HD 141531; and HD 83212. This extended
survey of helium suggests that only the most luminous Population II stars
($M_{V}$ brighter than $-$1.5), with $T_{eff}\lesssim 4500$ K lack the helium
feature.
### 4.1 Equivalent Widths
The equivalent widths (EW) of the He I $\lambda$10830 absorption are shown in
Figure 8 and 9 as a function of $T_{eff}$ and values are listed in Table 2.
Repeated measurements suggest the error in measuring the equivalent width is
about 5%. The red giants may have an increasing equivalent width with
decreasing effective temperature; this is not unexpected as an extended
expanding atmosphere increases scattering in the line. In the coolest star, HD
104207, the He line is blended with Ti I absorption to shorter wavelengths.
Here, the measurement of the equivalent width is uncertain because of the
blend with both Ti I and Si I. A hint that the absorption is extended and the
equivalent width underestimated in HD 104207 comes from comparison of the
short wavelength side of another Si I line at $\lambda$10843.90 to that of the
line at $\lambda$10827.09 (Figure 6). There may be excess absorption on the
latter line arising from an extended He I profile.
The He I line is surprisingly strong in the RHB stars. In 4 out of 11 RHB
stars, the depth of the line extends 10 to 20 percent below the continuum and
reaches equivalent widths between 0.1 and 0.5Å. These stars also show a
dramatic increase in the helium equivalent width that sets in at
$T_{eff}\gtrsim 5300$K. The values of the equivalent widths are comparable to
those found for Population I stars including binaries which are well-known
X-ray sources (Zarro & Zirin 1986; Sanz-Forcada & Dupree 2008). RHB stars are
not known to be X-ray sources (which would enhance the ionization of He I,
populate the metastable level of He I by recombination and cascade, and create
a stronger line). There are no X-ray sources at the positions of the two stars
with strongest He absorption in the HEASARC Archives
[http://heasarc.gsfc.nasa.gov] indicating that X-ray illumination does not
appear to be present to strengthen the line. The RHB star, BD +17°3248 has a
chromosphere as documented also by the presence of Mg II ultraviolet emission
(Dupree et al. 2007).
The star HD 195636 displays an exceptionally strong He line. Preston (1997)
first noted that this red horizontal branch star is a rapid rotator, which is
confirmed by Carney et al. (2008) as a single star. A strong helium line is
present also in the RHB object, HD 119516 which is not rapidly rotating and
has not been identified as a binary (Carney et al. 2008). Thus rotation does
not appear related to the strength of the helium line, although our sample
consists only of 4 stars. It is interesting to note that these horizontal
branch stars are in a similar helium-burning evolutionary phase as Population
I clump giants. Clump stars, such as the well-studied Hyades giants, exhibit
magnetic activity cycles, ultraviolet emission, and X-rays (Baliunas et al.
1983, 1998).
The increasing strength of the He I line with higher effective
temperature—possibly connected to the development of a hotter
chromosphere—suggests that collisions might be effective in populating via the
forbidden transition from $1s^{2}\ {}^{1}S$ to $1s2s\ ^{3}S$. The strength of
the line in the CH star HD 135148, which has a degenerate companion,
demonstrates that ionization of He by a hot companion, with subsequent
recombination and cascade, can also be important in populating the $1s2s\
^{3}S$ level. Some fraction of stars will have undetected white dwarf
companions; this may be another parameter affecting the strength of the helium
line.
The absorption equivalent widths in this metal deficient sample are generally
lower than found in bright Population I stars of low magnetic activity. In
Figure 10, the absorption equivalent widths for single red giants (Luminosity
classes II-III, III, and III-IV) are taken from the high quality measures in
the sample of O’Brien and Lambert (1986). Many of the spectra show variability
in the equivalent widths, but the He I line is generally stronger in the stars
with roughly solar composition than in the metal-poor sample. A P Cygni
profile may suffer some filling in of the absorption by near-star scattering,
but not all lines display these profiles. The helium abundance, Y, increases
by $\sim$20% between metal-poor and solar models (Girardi et al. 2000), which
is not enough for a factor of $\sim$4 change in the equivalent width. For the
same energy input, the metal-poor chromospheres may be warmer since radiative
losses are less, however that would strengthen the helium line and not weaken
it. Perhaps the helium absorption is enhanced in the Population I stars. They
generally exhibit magnetic activity which leads to X-ray emission that
increases the lower-level population through photoionization followed by
recombination. Semi-empirical models of these chromospheres are needed.
The equivalent width of the $\lambda$10830 line as a function of metallicity
is shown in Figure 11 where no systematic dependence on [Fe/H] appears over
this range of lower [Fe/H]: $-$0.7 to $-$3.0. Red giants of solar metal
abundance show varying strengths of the He I line that tend to cluster between
100 and 200 mÅ, although several stars display values comparable to the metal-
deficient sample.444The stars that are X-ray sources among the Population I
giants show substantially increased strength in the helium lines (Zarro &
Zirin 1986; O’Brien & Lambert 1986; Sanz-Forcada & Dupree 2008), presumably
due to X-ray photoionization, followed by recombination contributing to the
population of the lower ${}^{3}S$ level. It is premature to speculate on the
helium abundance from the line strengths alone without modeling this and other
helium profiles, since they depend on chromospheric conditions. However, if
spectra could be obtained in a globular cluster, providing a larger sample of
similar stars, the relative abundance of helium might be assessed.
### 4.2 Line Profiles
About one-third of all the luminous stars in Figure 7 (AGB and RGB stars
brighter than $M_{V}=0.5$) show helium emission. In most of these stars, the
emission is accompanied by absorption. These classical P Cygni profiles by
their very nature mark an extended outflowing atmosphere. Absorption profiles
without emission can indicate atmospheric dynamics by their asymmetry. The
ratio of the short wavelength extent (at the continuum level) to the long
wavelength extent of an absorption feature relative to the photospheric rest
wavelength gives a measure of the line profile asymmetry. These values are
converted to velocity units and are shown in Figure 12 for the stars without
emission. The value of $B/R$ is given where $B$ denotes the blue (short
wavelength) extent and $R$ the red (long wavelength) extent. The majority of
the helium lines have $B/R>1$ signaling outflowing motions. Values of the
short wavelength extent of the He I absorption are taken as the terminal
velocity ($V_{term}$), and $B/R$ ratios are given in Table 2. It is generally
easy to see from the spectra that helium absorption can extend to the strong
Si I line at 10827.09Å. Such an extension implies an expansion velocity of at
least 90 km s-1. Many stars exhibit higher speeds with absorption evident in
the short wavelength wing of Si I. The outflow velocities measured by the
extent of the short wavelength wing are independent of [Fe/H] as shown in
Figure 13. Metallicity, naturally, is a factor determining the speeds of
radiatively-driven winds. And there is some evidence for dusty winds in OH/IR
sources in the low metallicity Magellanic clouds to have lower speeds as
compared to similar sources in the galactic center (Marshall et al. 2004).
However, the outflow speeds of gaseous winds detected here (presumably driven
by hydrodynamic or magnetic processes) do not depend on the [Fe/H] abundance,
and we conclude that these winds are not radiatively driven.
RHB stars have a convective core and a semi-convective envelope (Castellani et
al. 1971; Schwarzschild 1970) and so conditions exist for the acceleration of
a stellar wind by magnetic processes such as Alfvèn waves. In addition, if
high temperatures are produced in an extended chromosphere, these could
contribute to a thermally driven wind.
The extension of the 10830Å line to shorter wavelengths signals outflow that
in many stars is comparable in value to the escape velocity from the stellar
chromosphere:
$V_{esc}(\rm km\ \rm s^{-1})=620\left(\frac{{\it M/M_{\sun}}}{{\it
R/R_{\sun}}}\right)^{1/2},$ (1)
where $M$ is the stellar mass and $R$ is the distance from the star center to
some region in the chromosphere. We take the 10830Å line to be formed at 2R⋆
in the stellar chromosphere where R⋆ is the stellar photospheric radius. This
estimate does not require a helium model. Detailed calculations concur on the
location of the formation of H$\alpha$ in metal-poor stars. Observations of
higher outflow velocities as well as semi-empirical models confirm that the
10830Å line is formed above the H$\alpha$ core in luminous stars. Our
spherical models for metal-deficient giants (Dupree et al. 1984) have a
chromospheric extent of ’several’ stellar radii in order to produce the
H$\alpha$ line (1.2 R⋆ to 3.6R⋆). The recent (non-LTE, spherical expanding)
models of Mauas et al. (2006) note that the H$\alpha$ core is formed ’about 1
stellar radius above the photosphere’, similar to our spherical models.
Subsequent modeling of H$\alpha$ in M13, M15, M92 giants show the H$\alpha$
cores to be formed at 2R⋆ (Mészáros et al. 2009a). Hence, it appears
reasonable to assume the level of formation of the 10830Å line as 2R⋆. Table 2
contains an estimated stellar radius (Column 10) determined by evaluating the
bolometric correction for each star (Alonso et al. 1999) as a function of
$T_{eff}$ and [Fe/H]. The chromospheric escape velocity for each star is
tabulated in Column 11 of Table 2 using Equation (1). Here we have assumed
masses of a red giant (0.75M⊙), a red horizontal branch star (0.7M⊙), an AGB
star (0.6M⊙), a subgiant branch star (0.8M⊙), and a semi-regular variable
(0.6M⊙). Marked in boldface are values of $V_{term}$ when they are comparable
to or exceed the escape velocity at 2R⋆. These amount to 40% of our sample
where the helium line is detected.
Many of the remaining stars exhibit a short wavelength extension of
$\gtrsim$40 km s-1; this value exceeds the extension of the long wavelength
wing, and signals that outflow of material is present that has not yet reached
escape velocity. Where helium occurs in the luminous stars,
($M_{V}\lesssim-0.2$), a signature of outflow is found in each one. However, a
fraction of the lower luminosity objects do not show this signature,
suggesting that the gas outflow may be variable.555One star, BD+17°3248
($M_{V}=0.65$), was observed 3 years previously (Smith et al. 2004) and the
helium profile has not changed. Speeds greater than $\sim$ 12 km s-1 are
generally supersonic in fully ionized metal-deficient ([Fe/H]=$-$2) plasma at
chromospheric temperatures of 104K.
Three of the target stars (BD +17°3248, HD 122956, and HD 126587) have high
resolution Hubble Space Telescope spectra available of the Mg II line at
$\lambda$2800 (Dupree et al. 2007). Asymmetries of the line emissions indicate
motions in the chromosphere, and these were found in stars brighter than
MV=$-$0.8. Only the most luminous of the 3, HD 122956, shows a Mg II emission
asymmetry indicating outflow (short-wavelength emission peak less than the
long-wavelength emission peak). HD 122956 has a high value of the helium
terminal velocity, 110 km s-1 which exceeds the chromospheric escape velocity
of 78 km s-1. The RHB star, BD +17°3248, shows outflow in helium at an
intermediate velocity, whereas the helium line in HD 126587 appears symmetric.
However, models (Dupree et al. 1992a) suggest that Mg II is formed at lower
levels than the 10830Å line in a metal-deficient chromosphere.666Other cool
stars have observational signatures of this separation: Cepheids (Sasselov &
Lester 1994), a T Tauri star (Dupree et al. 2005), and the Sun (Avrett 1992).
Thus it is not surprising to find differing dynamical signatures in these line
diagnostics, in addition to possible time variations. There may be a
similarity here between the well-known changing asymmetries of H$\alpha$
emission wings in metal deficient giants (Smith & Dupree 1988; Cacciari et al.
2004; Mészáros et al. 2008, 2009b) and the Mg II emission.
The star HD 135148 deserves special mention. This RGB object is classified as
a CH star and Carney et al. (2003) obtained an orbital period of 1411 days for
the spectroscopic binary. Emission in the P Cygni profile arises from an
extended scattering atmosphere and the absorption extends to $\sim$$-$115 km
s-1. This value exceeds the escape velocity from the chromosphere,
$V_{esc}$=67 km s-1, where helium originates. Thus after the initial transfer
of material to the secondary star in the system, a substantial wind remains
from the cool star that is presently visible.
### 4.3 Estimate of the Mass Loss Rate
A rough estimate of the mass loss rate implied by the helium absorption in the
wind can be derived from the Sobolev optical depth. In an expanding
atmosphere, a photon emitted from the photosphere (or the line itself) can be
absorbed and then scattered when the absorption coefficient is “aligned” with
the photon. In this situation, a sufficient number of atoms occur at the
correct velocity to absorb and scatter photons. In a stellar wind, a narrow
interaction region will be present that depends on the velocity gradient in
the wind, and the width and strength of the line absorption coefficient. The
Sobolev approximation defines the interaction region to be very narrow for
simplification of the transfer equation, and this causes the absorption
parameters to be related only to local conditions (Lamers & Cassinelli 1999).
This approximation assumes that the density and velocity gradient do not
change significantly over the absorbing/scattering region.
The line optical depth at frequency, $\nu$, at star center, is given by
$\tau_{\nu}=\int_{0}^{\infty}\kappa_{\nu}(z)\rho(z)dz$ (2)
along the radial direction, $z$, where $\kappa_{\nu}$(cm2 g-1) is the line
absorption coefficient, and $\rho$(g cm-3) is the mass density in the lower
level of the transition. Taking the line profile function as a delta-function
(the Sobolev approximation) and inserting values for $\kappa_{\nu}$ [cf.
Equation (8.51) of Lamers & Cassinelli (1999) or Equation (8.8) of Hartmann
(1998)], we write the Sobolev optical depth, $\tau_{S}$ as:
$\tau_{S}=\frac{\pi e^{2}}{mc}\times
f\times\lambda_{0}\times\frac{N_{1}}{(dV/dz)}.$ (3)
where $dV/dz$ is the velocity gradient in the scattering region and $N_{1}$ is
the population in the lower level of the absorption line (the ${}^{3}S$ level
of He I). Conservation of mass gives
$\dot{M}=4\pi R^{2}V\mu m_{H}N_{H}(V)$ (4)
where $R$ is the radial distance (in units of $R_{\sun}$) at which the wind
has a velocity $V$(km s-1). NH is the hydrogen density at $V$, $\mu$ is the
mass per hydrogen nucleus, and mH is the mass of the hydrogen atom. Then
substituting for $N_{H}$ into Equation (4) [rewriting
$N_{H}=N_{1}\times(N_{H}/N_{1})$], and replacing the value of $N_{1}$ from the
expression for $\tau_{S}$ above, we find
$\dot{M}=\frac{4\pi R^{2}V\mu
m_{H}}{N_{1}/N_{H}}\times\frac{\tau_{S}}{\frac{\pi
e^{2}}{mc}f\lambda_{0}}\times\frac{dV}{dz}.$ (5)
We assume that $dV/dz\sim\Delta V/\Delta R=V/(R-R_{\star})$ where $\Delta V$
is the change in wind velocity, ($R-R_{\star}$) is the distance over which the
speed changes from zero at the stellar photosphere to a value of $V$ at
distance $R$. In our estimate, we adopt $R=2R_{\star}$ since the 10830Å line
is formed in the chromosphere (see discussion in Section 4.2) and $R$ is
measured from the center of the star. Then with $\mu=1.4$, and
$m_{H}=1.67\times 10^{-24}$ g, we have:
$\dot{M}\ {(\rm M_{\sun}\ \rm yr^{-1})}=\frac{1.22\times
10^{-18}\tau_{S}(R/R_{\sun})^{2}V^{2}}{(N_{1}/N_{H})\times
f\times\lambda(\AA)\times(R_{\star}/R_{\sun})}$ (6)
where $N_{1}/N_{H}$ is the ratio of the population in the lower ${}^{3}S$
level of the helium transition to the total hydrogen density. To evaluate the
mass loss rate from Equation (6) at a distance of 1 $R_{\star}$ above the
photosphere, we set $\tau_{S}$ =1 and the oscillator strength, $f=0.54$ for
the $\lambda$10830 multiplet. With these values, the mass loss rate becomes,
$\dot{M}\ (\rm M_{\sun}\ \rm yr^{-1})=\frac{8.37\times 10^{-22}{\it
R_{\star}V^{2}_{term}}}{{\it(N_{1}/N_{H})}}$ (7)
where $R_{\star}$ is the stellar (photospheric) radius (in units of
$R_{\sun}$), and $V_{term}$ is the observed terminal velocity (km s-1) in the
helium line.
Now, the value of $N_{1}/N_{H}$ can be estimated using our semi-empirical non-
LTE models (Avrett & Loeser 2008) of cool star chromospheres (see Appendix A).
The semi-empirical models suggest that an upper limit of the population ratio,
$N_{1}/N_{H}$ for typical line strengths found in the targets reported here
(line depth $\sim$0.9) corresponds to 6.3$\times$10-8 where
$N_{He}/N_{H}=0.1$. As a lower limit, we take the value derived from the solar
model: 1.0$\times$10-8. The mass loss rates are estimated using values for
$V_{term}$ (where $V_{term}>45$km s-1) and $R_{\star}$ contained in Table 2,
and are shown in Fig. 14. Two rates are plotted for each star corresponding to
the upper and lower limit on $N_{1}/N_{H}$. The mass loss rate generally
increases with increasing stellar bolometric magnitude. The uncertainty in
these estimates arises predominantly from the population of the lower
${}^{3}S$ level of the $\lambda$10830 transition. A discussion of the error
estimate is given in Appendix B. Equation (7) does not apply to the
exceptionally deep P Cygni profile of HD 135148 such as that shown in Fig. 5
because it likely overestimates the mass loss rate.
Undoubtedly there is variation in the gas mass loss rate for stars on the RGB
- and probably all stars considered here. The solar mass loss rate changes
over the solar cycle by about a factor of 1.5 (Wang 1998). Mészáros et al.
(2009a) found variations ranging between a factor of 2 to 6 in the mass loss
rate of individual globular cluster red giants. Accompanying the changes in
the mass loss rate may be variations in the size of the outflow velocities.
However several observations suggest that outflows occur continually as stars
evolve through the upper part of the RGB. The helium line profiles reported
here overwhelmingly display a signature of outflowing gas in stars brighter
than $M_{V}\sim 0$. H$\alpha$ line cores show generally increasing outflowing
velocities with luminosity (Mészáros et al. 2008, 2009b). Even though some
measured outflow velocities are less than the escape velocity, conservation of
mass suggests that the velocities will yield meaningful mass loss rates. We
emphasize the difference between the presence of mass loss from red giants
determined from diagnostics of the gas (from He I 10830Å and the H$\alpha$
line) and that derived from infrared detections of circumstellar dust.
Evidence suggests that dust formation is an episodic process (Origlia et al.
2007; Mészáros 2008), whereas the velocity measurements of the gas indicate
continuous outflow of material.
With regard to the presence of a wind directly indicated by the helium line
profile, a fraction of the stars in Table 2 and 3 show velocities exceeding
the chromospheric escape velocity. There are several possible interpretations
of these measurements: (1) Only some fraction of the stars have outflows that
develop into winds; (2) All stars develop winds for some fraction of their
evolutionary phase brighter than $M_{V}\sim 0$. (3) The outflow velocities
vary, probably along with the mass loss rates, and we need diagnostics formed
higher in the atmospheres to attempt the detection of escape velocities. The
absence of inflowing velocities both in H-$\alpha$ and the helium line would
suggest that the last option appears to be the most likely.
### 4.4 Evolutionary Effects of Mass Loss
Stars on the red giant branch exhibit mass loss rates that increase with
luminosity. At a magnitude comparable to the horizontal branch, $\dot{M}\sim
3.2\times 10^{-9}$ to $3.2\times 10^{-8}$ $M_{\sun}\ yr^{-1}$. At higher
luminosities on the RGB, the mass loss rates increase by about a factor of 2,
viz., $6.3\times 10^{-9}$ to $6.3\times 10^{-8}$$M_{\sun}\ yr^{-1}$. A low-
mass star spends about 50 Myr in the RGB phase, thus, taking the geometric
mean of the minimum and maximum values, ($4.5\times 10^{-9}$$M_{\sun}\
yr^{-1}$), the total mass lost as indicated by the helium line would amount to
$\sim$ 0.2 $M_{\sun}$. This is the amount generally demanded by stellar
evolution considerations which range from $\sim$0.15 to 0.22$M_{\odot}$ (Rood
1973; Lee et al. 1994; Caloi & D’Antona 2008; Dotter 2008). Moreover, this
mass loss rate is consistent with values derived from the H$\alpha$ profiles
of globular cluster giants (Mauas et al. 2006; Mészáros et al. 2009a).
Stars on the AGB have estimated mass loss rates of 0.25$-$1.6$\times$ 10-8
$M_{\sun}\ yr^{-1}$. For a 20 Myr lifetime on the AGB, additional mass loss of
$\sim$ 0.1M☉ could result for the AGB objects here (assuming a mean mass loss
of 6.3 $\times$ 10-9$M_{\sun}\ yr^{-1}$). Groenewegen & deJong (1993) find
about 0.16$-$0.2 $M_{\odot}$ is lost on the AGB. The sample of AGB stars
included here has low luminosities ($M_{bol}\approx$ 0 to $-$1.9), and it is
likely that the mass loss increases with luminosity which would increase the
empirical total mass loss.
Minimum and maximum values of the mass loss rate for the RHB stars in our
sample range from $4.5\times 10^{-10}$ to $2.2\times 10^{-8}$ $M_{\sun}\
yr^{-1}$(Fig. 14). Taking a median of $3\times 10^{-9}$$M_{\sun}\ yr^{-1}$,
and a lifetime of 75 Myr, this rate implies a total mass loss of
0.2$M_{\odot}$. Note that only half of the sample of RHB stars has an
asymmetric profile indicating outflow velocities, suggesting that the outflow
and hence the mass loss rate may be lower. Some evolutionary models have
considered mass loss on the RHB. Yong et al. (2000) and earlier, Demarque &
Eder (1985) evaluated models of horizontal branch stars with the ad hoc
assumption of mass loss to test its effects, and concluded that mass loss
rates between 10-10 and 10-9 $M_{\sun}\ yr^{-1}$ could produce the observed
extended blue HB. Koopmann et al. (1994) set an upper limit to the HB mass
loss rate of 10-9 $M_{\sun}\ yr^{-1}$, based on calculations for M4. These
values are not substantially discrepant from the mass loss rate inferred from
the He I line profiles.
Vink and Cassisi (2002) evaluated the effect of radiatively driven winds in
horizontal branch stars, but found the mass loss rates too low for
evolutionary effects. However, winds can be driven in several other ways which
is likely here, given the presence of convection layers in these stars
(Castellani et al. 1971; Schwarzschild 1970). It is worth noting that even
models of RGB and AGB stars currently consider sub-surface dynamo activity
(Busso et al. 2007, Nordhaus et al. 2008) which could lead to winds driven by
magnetic processes.
Carney et al. (2008) noted that red horizontal branch stars in the field are
rotating slower than expected considering the rotation observed in their
predecessors on the red giant branch. Angular momentum carried away by the
stellar winds could offer an explanation of this discrepancy. Here we estimate
the torque required to spin down a red giant with the mass loss rates
discussed above. To be effective, magnetic fields must be present and we
assume that the moment of inertia of the star does not change. The torque
required, $\tau_{\star}$ (dyne cm) under these conditions is
$\tau_{\star}=(I_{o}\omega_{o})/\Delta t$ where $I_{o}$ is the stellar moment
of inertia, $\omega_{o}$, the initial angular velocity, and $\Delta t$ the
time required to completely stop the rotation of the red giant. Taking, a
tangential rotation velocity of 2 km s-1 [from the Carney et al. (2008)
measures, and setting $sin\ i=0.3$] and R⋆=50R☉ for a 0.8M☉ red giant star, we
find that a torque of 7$\times$1035 dyne cm is required to spin down the star
in 20 Myr. Matt & Pudritz (2008) evaluated the torque created on a star by
winds of various mass loss rates in the presence of a stellar magnetic field
with several configurations and strengths. A simple parameterization of the
quantity: $(B_{\star}R_{\star})^{2}/(\dot{M}V_{esc})$ can predict the ’lever
arm’ to evaluate the torque exerted by the wind on the star. For a red giant
with the parameters above, a dipole magnetic field of 100 gauss, and
$\dot{M}$= 4.5 $\times$10-9 $M_{\sun}\ yr^{-1}$, the predicted wind torque
amounts to 3 $\times$ 1036 dyne cm, a value that exceeds the amount required
for spindown, making winds a plausible explanation of the low velocities of
RHB stars. However we caution that a change in the moment of inertia could
affect these results as well as the magnetic field strength and
configuration.777Y. C. Kim made available his calculations of the moment of
inertia for a 1 M☉ star with [Fe/H]=$-$0.25 as it evolves up the red giant
branch. These show that the moment of inertia increases by a factor of 6.5,
based on internal structural changes including an increase in radius, as the
star evolves from an effective temperature of 4500K to 4000K. Such a factor
would appear to allow spin-down to occur.
These results can be used to calculate a total budget of mass loss for post-
main sequence evolution. Taking values of $\dot{M}$ estimated above, we find a
star would lose 0.2 M☉ on the RGB, 0.1–0.2 M☉ on the RHB, and 0.1–0.2 M☉ on
the AGB and as a planetary nebulae (Bianchi et al. 1995), totaling 0.4–0.6 M☉.
This total is in harmony with recent studies of globular clusters that suggest
anywhere from 0.5 to over 1 $M_{\odot}$ is lost between the main sequence
turnoff and the white dwarf cooling sequence (Moehler et al. 2004; Hansen et
al. 2007; Richer et al. 2008). Our budget should not be strictly taken to
apply to current turnoff stars of mass $\sim 0.8$ M☉, since many of the white
dwarfs observed in clusters are remnants of stars of higher initial masses,
where more mass loss is demanded in post-main sequence evolution. In addition,
the Population II field giants that make up our sample potentially had a range
of initial masses, and thus some may have mass loss rates higher than expected
for globular clusters.
### 4.5 The Fate of Wind Material
If red giants in globular clusters have winds similar to those detected among
Population II field giants, it is worth considering whether the wind material
would have enough energy to escape not only from the stars themselves but also
from the parent cluster. Escape velocities from the cores of Milky Way
globular clusters have been evaluated by McLaughlin and van der Marel (2005).
Central escape velocities vary from $\sim$ 2 km s-1 (AM1, Pal 5 and Pal 14) to
$\sim$ 90 km s-1 for massive clusters (NGC 6388 and NGC 6441). A majority of
the stars brighter than MV = 0 in our sample exhibit terminal velocities in
excess of the stellar escape velocity from the chromosphere. These speeds,
ranging from 90 to 170 km s-1 also exceed many of the escape speeds from
clusters too. Since the cluster escape speed corresponds to the speed
necessary to remove material from the core, the value will obviously decrease
with distance from the core. One could envision a scenario in which material
does not have sufficient energy to escape the cluster core, but it could
escape if arising from a star located further out in the cluster.
Several authors have considered in detail the dynamics of wind material
deposited in the intracluster medium by the stellar population. VandenBerg and
Faulkner (1977) used hydrodynamic equations to construct time dependent models
of gas flow from a cluster assuming the stellar wind has a velocity $\sim$20
km s-1. Depending upon the initial assumptions of the energy available, they
found that both outflows and inflows of material could occur, the latter as a
result of radiative cooling. In some models material was retained in the
cluster core. Smith (1999) concluded that additional energy could be injected
into the gas by solar-like winds originating from cluster dwarf stars and this
would be sufficient to establish an outflow of material from a cluster even if
red giant winds are slow. If the giant winds are fast, comparable to the solar
wind with V $\sim$ 450 km s-1, steady state outflows would result, even from
clusters with the highest values of escape velocity (Faulkner & Freeman 1977).
The results presented here from helium lines suggest that the winds from red
giants and red horizontal branch stars can be much faster than generally
assumed, and these would naturally lead to a steady-state outflow.
Recent observations of the fast solar wind suggest that interactions between
colliding winds within the cluster itself may also be of some consequence.
Insight can be drawn from the Sun where spacecraft have located the
termination shock and characterized its physical parameters (Richardson et al.
2008). The shock occurs $\sim$100 AU distant from the Sun, and the wind speed
at that point is comparable to the speed in the corona, namely 400 km s-1.
While the solar wind is hot and driven by a combination of gas and wave
pressure, a cool wind can be driven by wave pressure alone (Cranmer 2008).
However, convective envelopes in red horizontal branch stars, and recent
conjectures of dynamo activity in RGB and AGB stars would allow MHD processes
to occur as well (Busso et al. 2007; Nordhaus et al. 2008). Thus, it appears
possible that the stellar winds will not be substantially decelerated up to
the termination shock, but maintain the values indicated by the helium line.
At large distances from the star, the pressure of the stellar wind will
eventually be balanced by the opposing pressure (both gas and magnetic)
presented by the surrounding intercluster medium, $P_{icm}$. When these
pressures are equal, a termination shock occurs at some distance, $R_{TS}$,
viz.:
$R^{2}_{TS}=\frac{\dot{M}_{\star}\times V_{wind}}{4\pi P_{icm}}.$ (8)
Rewriting this using astronomical units,
$R_{TS}(AU)=48.4\times\sqrt{\frac{\dot{M}_{\star}(M_{\sun}\ yr^{-1})\times
V_{wind}(km\ s^{-1})}{P_{icm}(dyne\ cm^{-2})}}$ (9)
and assuming $\dot{M}=$2$\times$10-9 M☉ yr-1 for an average RGB star, Vwind=
100 km s-1, and a (total) interstellar pressure of 4.0$\times$10-13 dyne cm-2
for a distance of 1.5 kpc above the Galactic plane (Cox 2005), we find the
transition shock distance is 34,000 AU or 0.2 pc from the star. The central
luminosity densities for globular clusters have median values of $\sim$ 3000
$L_{\sun}pc^{-3}$ (Harris 1996), and with a typical star of 0.1 to 0.2
$L_{\sun}$, the median stellar density in the core is greater than 104 stars
pc-3, giving an average separation between stars of less than 9500 AU -
smaller than the termination shock distance. While uncertainty exists in both
the pressure outside and within clusters888The pressure in the galactic halo
is not firmly known. O VI absorption suggests warm (3 $\times$105K) extended
low density regions are present at high galactic latitudes (Dixon and Sankrit
2008) with thermal pressures of 0.7 to 1 $\times$ 10-12 dyne cm-2. Electron
densities in the intercluster material of 47 Tuc derived from pulsar
dispersion measures (Freire et al. 2001) indicate $n_{e}=0.067\pm 0.015\
cm^{-3}$. They concluded ionized material was dominant, where for a
temperature of 104K, the thermal pressure equals 9$\times$10-14 dyne cm-2. van
Loon et al (2006) detected 0.3$M_{\sun}$ of material in the core of M15 using
the Arecibo telescope. Assuming that this material is evenly distributed in
the beam we find a hydrogen density of 2.29$\times$10-2 cm-3 for the diffuse
gas in the core. With a temperature of 100K, the pressure will be
3.2$\times$10-16 dynes cm-2. Faulkner and Freeman (1977) constructed time-
independent gas flow models for tightly bound clusters. These models suggest
pressures at the tidal radius ranging from 0.1 to 5.5 $\times$10-16 dyne cm-2.
At the sonic point in the flow which lies in the cluster interior where the
stellar density is down by a factor of $\sim$100 from the core density, the
gas pressures are much higher, ranging from 5$\times$10-15 to 3.5$\times$10-12
dynes cm-2. Our estimate of the transition shock distance varies inversely as
the square root of the external gas pressure, so that an order of magnitude
change in the gas pressure, changes the distance by a factor $\sim$ 3., these
estimates suggest that the central regions of clusters may well be filled with
expanding warm material, enveloping the surrounding stars. Since the majority
of cluster stars are still on the main sequence with winds of $\dot{M}\sim
2\times 10^{-14}$ $M_{\sun}\ yr^{-1}$ and $V_{wind}\sim 400$ km s-1 (adopting
solar parameters), these winds can not balance the pressure of the more
massive winds arising from the luminous stars, raising the possibility that
the dwarf winds are in fact smothered by the giant winds, allowing for
pollution of the surface layers of the dwarfs. As the stellar density
decreases towards the cluster edges, the effect of smothered winds and
resultant surface pollution would decrease, possibly leading to spatially
dependent self-pollution within a cluster. Even beyond the termination shock
crossing, the wind speed decreases by about a factor of 2 (based on the solar
termination shock measured by Richardson et al. 2008), but the temperature and
density increase. In many cases, even the decreased speed will allow escape of
these fast winds from the cluster.
Central escape velocities for the most massive clusters can reach $\sim$90 km
s-1 (McLaughlin & van der Marel 2005). While the helium line profile suggests
that some objects possess these high velocities, others do not (see Figure
13). The extent of velocity variability in the helium line is unknown at
present. The velocity of a red giant wind at levels higher than the formation
region of the helium line may (or may not) reach escape velocities.
Calculations of line forming regions for strong optical, IR, and near UV lines
have been made for metal deficient red giants (Dupree et al. 1992a; Mauas et
al. 2006; Mészáros et al. 2009a) but only through the low chromosphere and not
for higher levels of the atmosphere. In the high chromosphere, the most
straightforward diagnostics of winds lie in the ultraviolet region of the
spectrum, where Lyman-$\alpha$ and resonance lines of C II ($\lambda$1335) and
Si II ($\lambda$1800) might be observable.999UV and Far-uv spectra obtained
with IUE, HST, and FUSE of the brightest of the metal-deficient targets in the
field, HD 6833, are weak and can not definitively reveal emission. One
advantage of metal-poor targets is that their high velocities move the spectra
away from local interstellar absorption which could compromise the resonance
line profiles. Infrared transitions of hydrogen such as members of the
Paschen, Brackett, and Pfund series arise from higher levels of hydrogen and
hence are formed deeper in the atmosphere and can not help in detecting a
wind.
## 5 Conclusions
The He I 10830Å transition maps atmospheric dynamics to higher levels than
optical or near-uv diagnostics. The profile of this near-infrared line in
metal-poor stars gives evidence for outflow in most of the RGB, AGB, and RHB
stars showing helium. In many objects, the speeds are comparable to the escape
velocities from both the stellar chromosphere and globular clusters. The fact
that all the luminous stars with helium absorption exhibit expanding
chromospheres suggests that the mass outflows as traced by gas occurs
continually, with most probably variable mass loss rates.
Our estimate suggests that the mass loss observed directly on the RGB will
provide the requisite amount needed by stellar evolution calculations. Mass
loss detected in RHB stars appears at a rate sufficient to cause extension of
the horizontal branch. It will be very useful to obtain ultraviolet spectra of
some of these stars to identify higher temperature plasma, and to track the
acceleration in the chromosphere. Better estimates of mass loss rates can be
obtained with semi-empirical modeling of the line profiles.
These results demonstrate that chromospheric material has sufficient speed to
escape these stars and become a stellar wind. If the star were in a globular
cluster, the high-speed wind continues with little diminution, filling the
cluster with expanding warm gas. In the process, red giant winds could smother
the substantially less massive winds from dwarf stars possibly allowing for
surface pollution. Beyond the termination shock, velocities decrease but could
still escape the cluster. Thus fast winds observed in the helium line offer a
straightforward way to understand the absence of intracluster material in
globular clusters.
We are grateful to Steve Cranmer and Aad van Ballegooijen for insight into the
solar wind. Gene Avrett kindly made the helium calculations for the Sun
available before publication. We thank Y. C. Kim for providing us with his
pre-publication results for the evolution of the moment of inertia of red
giants. This research has made use of NASA’s Astrophysics Data System Abstract
Service and SIMBAD database (CDS, Strasbourg, France). We wish to extend
special thanks to those of Hawaiian ancestry from whose sacred mountain of
Mauna Kea we are privileged to conduct observations. Without their generous
hospitality, the Keck results presented in this paper would not have been
possible. Facilities: Keck 2 (NIRSPEC)
## Appendix A Estimation of Level Population
The level population for the ${}^{3}S$ level of He I, n(${}^{3}S$), is
estimated from non-LTE calculations using semi-empirical models for different
stars. A detailed discussion of helium excitation processes in the solar
atmosphere is given in Andretta & Jones (1997) and is not reviewed here. Our
estimates of the population levels derive from the PANDORA code (Avrett &
Loeser 2003) which was used in either plane-parallel or spherical form, with
an expanding wind in most models. A 13-level helium atom plus continuum was
generally used; some models had a 5-level helium atom plus continuum to
expedite calculation. The velocity field was introduced explicitly into the
source functions. Solutions for hydrogen populations and ionization are
iterated first, and then they are followed by similar iterations for helium.
The highest value n(${}^{3}S$)/n($He_{tot}$) in each model is shown in Fig. 15
as a function of the maximum depth of absorption in the helium 10830Å line.
The solar model represents the quiet sun for a static plane-parallel
atmosphere which is described in detail elsewhere (Avrett & Loeser 2008), and
E. Avrett kindly made an advance copy of the helium results available for use
here. High It may well be that the mass loss rates are variable. For instance
only one-half of the RHB stars exhibit outflow. And recent chromospheric
models calculated for varying H$\alpha$ profiles in globular cluster red
giants, reveal changes in mass loss rate by a factor of 6 (Meszaros et al.
2009a). Thus the total mass lost would seem to intrinsically span a range of
values. luminosity stars (similar to giants and supergiants) were also
calculated in plane-parallel and spherical models both in a static atmosphere
and also with an assumed mass outflow. The models were of cool stars (such as
$\beta$ Dra, $\alpha$ Aqr, and $\alpha$ Boo) with effective temperatures
approximately solar. The photospheric model has little or no influence on the
helium line profile since it is formed totally in the chromosphere. Results
for Mg II and helium lines in a supergiant were described elsewhere (Dupree et
al. 1992b). A cool dwarf star with an extended chromosphere (TW Hya) producing
a P Cygni profile was also modeled in helium, with a spherical approximation
and a substantial high velocity wind (Dupree et al. 2008).
Figure 15 shows the maximum depth of the He I 10830Å as a function of the
population ratio n(${}^{3}S$)/n($He_{tot}$) of helium. The three plane
parallel models (for the Sun and the supergiant and giant stars marked with
vertical lines in Figure 15) show generally lower values of the population
ratio. As one might expect, there is a correlation between the depth of the
line and the level population. Higher level populations result in increasing
the line depth. The models with high outflow velocities in the chromosphere
(110$-$200 km s-1) at the atmospheric level where the ${}^{3}S$ population
maximizes, create deeper absorption profiles, but the level populations are
remarkably similar. One of our target stars, HD 135148 has an extreme 10830Å
line depth ($\sim$0.1) likely caused by radiation from the hot companion
contributing to the photoionization-recombination processes populating the
lower ${}^{3}S$ level. Other stars observed in this paper have absorption
depths relative to the continuum between 0.8 and $\sim$0.95. Fig. 15 shows
that such absorption depths occur for a population ratio of log
n(${}^{3}S$)/n($He_{tot}$) $<\ -$6.2. This provides our estimate of an upper
limit to the population ratio, which in turn translates into a lower limit to
the inferred mass loss rate for the discussion in Section 4.3.
These models and the assumptions of the Sobolev approximation contain
uncertainties. The atomic physics (collisional excitation rates,
photoexcitation and photoionization cross-sections, and radiative and
dielectronic recombination rates) are continually updated, and these
calculations of populations were made more than a year ago. Only the giant
model for $\alpha$ Boo was constructed using a metal-poor abundance set. For
the same chromospheric input energy, one might expect the chromospheric
temperatures to be higher in a metal-poor environment because the radiative
losses are less. However the Alpha Boo (giant) model in which the abundances
are a factor of 3 less than solar appears to give results in harmony with
those from solar abundance models. Moreover, these are ’semi-empirical’
models, generally constructed to fit observed line profiles, so that the
required temperature/density structure accommodates a non-solar metallicity.
Our use of the Sobolev approximation introduces assumptions as well. We have
adopted a conservative value for dV/dz. By setting this gradient equal to
$V/(R-R_{\star})$ in section 4.3, our estimate basically averages the entire
velocity increase from 0 km s-1 to the observed He I outflow velocity over the
entire 1 stellar radius of the chromosphere between the surface and the radius
of He I formation. The wind might become more accelerated with increasing
distance from the photosphere, [and acceleration has been observed in red
giants in globular clusters (Mészáros et al. 2009b)], or else be
preferentially accelerated at higher altitudes. This would cause our estimate
of dV/dz (and hence the mass loss rate) to be an underestimate. We have taken
the region of helium line formation as 1 stellar radius above the photosphere,
based on several models of the level of formation of the H$\alpha$ line
placing it at 2R⋆ in globular cluster red giants (Mauas et al. 2006; Mészaáros
et al. 2009a). However, helium is formed above the H$\alpha$ line and so this
could also lead to an underestimate of the mass loss rate. We have set
$\tau_{S}$ equal to 1.0 in the Sobolev approximation, following other authors
(Hartmann 1998). It probably would be less in the chromosphere in a fully non-
LTE calculation, and thus decrease the mass loss rate, but other atmospheric
parameters would undoubtedly change.
As a next step, calculations of the level populations and the line profiles
should be carried out in non-LTE, with spherical coordinates, and include
velocity fields for a grid of temperatures and values of [Fe/H]. It would be
optimum also to have other chromospheric lines than helium, such as H$\alpha$,
Ca II K, and Mg II to constrain the chromospheric structure. While the
H$\alpha$ profiles are known to vary, currently we are ignorant of possible
changes in Ca II, Mg II, and the He I 10830Å lines. Of course, such diagnostic
lines must be acquired with a variety of instruments on the ground and from
space, and realistically may be difficult to achieve - much less achieve
simultaneously. No measure exists of X-rays from these sources; such a
measurement could also provide constraints on any X-ray illumination of the
chromosphere.
It is reassuring that the mass loss rates for stars on the red giant branch
span values of $\approx 3\times 10^{-10}$ $M_{\sun}\ yr^{-1}$ to $6.3\times
10^{-8}$$M_{\sun}\ yr^{-1}$, and these values are congruent with those
determined by non-LTE spherical models of the H$\alpha$ line. Mauas et al.
(2006) find values ranging from $1.1\times 10^{-10}$ to $3.9\times 10^{-9}$
$M_{\sun}\ yr^{-1}$ for 5 red giants in NGC 2808. Mészáros et al. (2009a) find
similar values of the mass loss rate: $5.7\times 10^{-10}$ to $4.8\times
10^{-9}$ $M_{\sun}\ yr^{-1}$ for 15 red giants in 3 globular clusters (M13,
M15, and M92) from the H$\alpha$ line too. At the highest luminosity, the
values from helium derived here appear to be higher than the H$\alpha$
modeling, but in agreement with the classical ’Reimers’ value (Reimers 1975).
There may well be wind variability too, and this could cause changes in the
mass loss rate by perhaps a factor of 6, based on the H$\alpha$ variability
(Mészáros et al. 2009b). Further work would be beneficial to establish mass
loss rates.
## Appendix B Propagation of Errors in the Mass Loss Rate
We estimate the propagated rms error $\sigma$($\dot{M}$) in the mass loss rate
by evaluating the contribution of 5 variables to $\dot{M}$: R, V, $N_{rel}$=
$N_{1}/N_{H}$, $\tau_{S}$, and $D_{V}$= $dV/dz$. Here, as defined in Section
4.3, $R$ is the distance from the center of the star with radius $R_{\star}$,
$V$ is the expansion velocity, $N_{rel}$ is the population of the lower level
of the He I 10830Å line relative to hydrogen, $\tau_{S}$ is the Sobolev
optical depth, and $D_{V}$ is the velocity gradient. The error is given by:
$\sigma^{2}(\dot{M})=\sum_{X=R,V,N_{rel},\tau_{S},D_{V}}\left(\frac{\partial\dot{M}}{\partial
X}\right)^{2}\sigma^{2}(X).$ (B1)
By evaluating $\left(\frac{\sigma(\dot{M})}{\dot{M}}\right)^{2}$,
$\sigma(\dot{M})$ can be estimated as a function of $\dot{M}$ to assess the
contribution of each quantity to the error. Writing Equation (B1) in terms of
the variables gives,
$\displaystyle\left(\frac{\sigma(\dot{M})}{\dot{M}}\right)^{2}$
$\displaystyle=$
$\displaystyle\frac{1}{\dot{M}^{2}}\left(\frac{\partial\dot{M}}{\partial
N_{rel}}\right)^{2}\sigma^{2}(N_{rel})+\frac{1}{\dot{M}^{2}}\left(\frac{\partial\dot{M}}{\partial\tau_{S}}\right)^{2}\sigma^{2}(\tau_{S})+\frac{1}{\dot{M}^{2}}\left(\frac{\partial\dot{M}}{\partial
V}\right)^{2}\sigma^{2}(V)$ (B2) $\displaystyle+$
$\displaystyle\frac{1}{\dot{M}^{2}}\left(\frac{\partial\dot{M}}{\partial
R}\right)^{2}\sigma^{2}(R)+\frac{1}{\dot{M}^{2}}\left(\frac{\partial\dot{M}}{\partial
D_{V}}\right)^{2}\sigma^{2}(D_{V})$
From Equation (5) in Section 4.3,
$\dot{M}=K\frac{R^{2}V\tau_{S}D_{V}}{N_{rel}}$ (B3)
where $K$ is a constant composed of atomic and scaling parameters. Taking the
partial derivatives of these variables, and reinserting the expression for
$\dot{M}$, we find,
Substituting these quantities into Equation (B2) yields,
$\left(\frac{\sigma(\dot{M})}{\dot{M}}\right)^{2}=\left(\frac{\sigma(N_{rel})}{N_{rel}}\right)^{2}+\left(\frac{\sigma(\tau_{S})}{\tau_{S}}\right)^{2}+\left(\frac{\sigma(V)}{V}\right)^{2}+\left(\frac{2\sigma(R)}{R}\right)^{2}+\left(\frac{\sigma(D_{V})}{D_{V}}\right)^{2}$
(B4)
Now supposing that $\frac{\sigma(N_{rel})}{N_{rel}}=3$,
$\frac{\sigma(\tau_{S})}{\tau_{S}}=1$, $\frac{\sigma(V)}{V}=\frac{1}{2}$,
$\frac{\sigma(R)}{R}=\frac{1}{2}$, and $\frac{\sigma(D_{V})}{D_{V}}=1$ and
substituting these values into Equation (B4), we find
$\left(\frac{\sigma(\dot{M})}{\dot{M}}\right)^{2}=9+1+\frac{1}{4}+1+1$ (B5)
so that the largest contribution to the error arises from the uncertainty in
$N_{1}$, and
$\sigma({\dot{M}})=3.5\dot{M}.$ (B6)
Thus, in this Sobolev approximation, the uncertainty in the mass loss rate
principally depends on the population of the lower level of the He I 10830Å
transition.
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Figure 1: Red giant (RGB) stars showing the region of the He I line. The Si I
photospheric line is the strong narrow absorption at 10827.09Å. The position
and extent of the He I triplet line spanning 10830.081-10830.341Å are marked.
The strongest member of the multiplet occurs at 10830.341Å. Weak narrow
emission features sometimes arise from the incomplete sky subtraction. A bad
pixel causes the narrow emission spike in BD+09°2870.
Figure 2: Red horizontal branch stars in the sample. See caption for Figure 1
Figure 3: The He I line in normalized spectra of the 6 AGB stars in our
sample. The position of the He I $\lambda$10830.341 transition is marked by a
broken line set to zero velocity. The continuum level is set at 1.0 for each
star, with each spectrum offset by a constant value; the extent of 0.1 in the
continuum level is shown. The strong Si I photospheric line at $-$90 km s-1
dominates this region of the spectrum. The AGB stars are arranged in order
(lower to upper spectra) of increasing $(B-V)_{0}$ values which are noted
below the stellar identification. The most luminous objects show weak emission
in He I. Evidence of absorption at high negative velocities is seen in the 4
coolest stars: BD+12°2547, HD 121135, HD 107752, BD+18°2757.
Figure 4: He I spectra for the two subgiants in the sample.
Figure 5: HD 135148, a CH star, shows a substantial deep P Cygni profile with
extent at least to the Si I line and beyond that indicates a wind velocity of
$\sim$ 115 km s-1.
Figure 6: The He spectral region of HD 104207 (GK Com), the coolest red giant
in this sample, where the spectrum is dominated by neutral lines of Si, Ti,
Fe, and Ca. In addition, the Ti I line at 10828.04Å occurs in very cool stars
near the Si I transition at 10827.09Å. The He I line at $\lambda$10830.34
appears to have weak emission longward of the absorption. The short wavelength
wing of the Si I 10827.09Å profile may have additional absorption when
compared to the other Si I transition at 10843.9Å, possibly caused by extended
helium absorption.
Figure 7: Location of target stars in a color-magnitude diagram (top panel)
and in a Teff-magnitude diagram (lower panel). Results for 6 giants in M13 and
5 metal-poor field giants reported previously (Dupree et al. 1992a; Smith et
al. 2004) have been added to this figure (see Table 3). The curves mark 12.1
Gyr isochrones (VandenBerg et al. 2006) for an abundance [Fe/H]=$-$2.01 (solid
line) and [Fe/H]=$-$1.53 (broken line). Plus signs mark both P Cygni profiles
and asymmetric profiles signaling mass outflow.
Figure 8: Equivalent width of the He I absorption line as a function of
$T_{eff}$ with evolutionary status of the stars indicated as follows: RGB=red
giant branch; AGB= asymptotic giant branch; SGB= subgiant branch. The plus
($+$) signs overplotted indicate stars with P Cygni profiles. Lower limits to
the equivalent widths occur when the helium absorption overlaps the Si I
photospheric line.
Figure 9: Equivalent width of $\lambda$10830 in the RHB sample as a function
of $T_{eff}$. Absorption becomes systematically stronger in stars with
$T_{eff}\gtrsim$ 5320K. The plus ($+$) sign overplotted indicates a star with
a P Cygni profile. Lower limits to the equivalent widths occur when the helium
absorption overlaps the Si I photospheric line. Errors in the equivalent width
amount to $\sim\pm$5%.
Figure 10: Equivalent widths of the He I absorption in red giants and
subgiants from our sample and from the red giants in Population I stars
reported by O’Brien & Lambert (1987). Many of the Population I objects were
observed several times and the lower and upper values of the equivalent widths
are connected by solid lines. The plus ($+$) sign overplotted indicates a P
Cygni profile. Some Population I giants with strong X-ray emission have
substantially larger equivalent widths (O’Brien & Lambert 1986; Sanz-Forcada &
Dupree 2008), and these are omitted from this figure.
Figure 11: Equivalent widths of He I absorption as a function of [Fe/H] with
evolutionary status of the stars indicated, as follows: RGB=red giant branch;
RHB=red horizontal branch; AGB= asymptotic giant branch; SGB= subgiant branch.
The CH star, HD 135148 with EW= 2390 mÅ and [Fe/H]=$-$1.9 has been omitted as
well as 3 stars (2 RGB and TY Vir) not showing helium. Stars displaying a P
Cygni profile are marked with a plus ($+$) sign; lower limits are shown where
the helium absorption extends into the neighboring Si I line. There is no
systematic dependence of the equivalent width on [Fe/H] between [Fe/H]=$-$0.7
and $-$3.0. Figure 12: The ratio of the short wavelength velocity extent
(BLUE) to the long wavelength velocity extent (RED) of the absorption profile
of the He I 10830Å line as a function of absolute visual magnitude. Only stars
with helium absorption lines and no emission are included here. A line arising
in a stationary atmosphere has a ratio of 1; outflow is indicated when
Blue/Red $>$1\. An ’up’ arrow marks stars for which the blue velocity wing
reached the Si I line, 90 km s-1 to shorter wavelengths, but the limiting
extent could not be determined because of overlap with the Si I feature. The
error bar marks the estimated 15% uncertainty in the measurement of the ratio.
The majority of these stars have asymmetric absorption profiles signalling
outflow.
Figure 13: Short wavelength extent of the helium 10830Å line as a function of
[Fe/H]. An extent of $\sim$40 km s-1 appears to correspond to the normal
thermal and turbulent width of the line (marked by the broken line). No
systematic dependence is present as a function of metallicity. Stars
displaying a P Cygni profile are marked with a plus ($+$) sign. Upward
pointing arrows denote lower limits because the helium absorption extends into
the neighboring Si I line and the extent of the helium profile can not be
determined.
Figure 14: The relation between the bolometric absolute magnitude, $M_{bol}$,
and the mass loss rate (Equation 7) as inferred from the He I 10830Å profile
in the stars with $V_{term}$ exceeding the thermal width of 45 km s-1. Upper
and lower limits are shown that derive from the limits on the the lower level
population ratio. Some stars exhibit absorption extending into the Si I line
located $-$90 km s-1 from He I. Since the termination of the He I profile is
difficult to establish, the figure shows a lower limit arrow on these rates.
Figure 15: The depth of the He I 10830Å absorption as a function of the ratio
of the lower level population, n(${}^{3}S$), in He I to the total helium
abundance, n(Hetotal), as calculated for various stellar chromospheres. Here,
Hetotal= He I + He II. The vertical bars on two stars indicate the span of
line-core depth values at different $\mu$ where $\mu$=cos $\theta$, the polar
angle with respect to the stellar photosphere in the plane-parallel
approximation that was used. With the exception also of the Sun, all of the
other models were calculated in a spherical approximation. The giant star
model has parameters: $T_{eff}$ = 4250K, log g = 1.6, and solar abundances
other than hydrogen and helium are decreased by a factor of 0.33. From these
results, we take an upper limit of the lower level population ratio,
n(${}^{3}S$)/n($He_{total}$)= $-$6.2 dex, and a lower limit of this ratio of
$-$7.0 dex. Table 1: Metal-Poor Field Giants Observed
Star | $V$ | $B-V$ | $U-B$ | Ref. | $E(b-y)$ | $M_{V}$ | $V$ | $b-y$ | $c_{1}$ | Ref. | E($B-V$) | ($B-V$)0 | [Fe/H] | Ref. | Evol.aaEvolutionary state: RGB = red giant branch; SGB = subgiant branch; SR= semiregular variable; AGB = asymptotic giant branch decided on the basis of (b$-$y, c1) diagram and $M_{V}$; RHB = red horizontal branch decided on the basis of (b$-$y, c1) diagram and $M_{V}$.
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
BD +01 3070 | 10.060 | 0.769 | $\cdots$ | 3 | 0.016 | 1.46 | 10.038 | 0.487 | 0.264 | 2 | 0.022 | 0.747 | $-$1.85 | 2 | RGB
BD +05 3098 | 10.530 | 0.780 | $\cdots$ | 3 | 0.028 | $-$0.04 | 10.537 | 0.542 | 0.380 | 2 | 0.038 | 0.742 | $-$2.4 | 2 | RGB
BD +09 2574 | 10.517 | 0.793 | $\cdots$ | 3 | 0.000 | 0.27 | 10.523 | 0.530 | 0.379 | 2 | 0.000 | 0.793 | $-$1.95 | 2 | RGB
BD +09 2860 | 10.830 | 0.710 | $\cdots$ | 3 | 0.003 | 0.73 | 10.83 | 0.440 | 0.430 | 2 | 0.004 | 0.706 | $-$1.6 | 2 | RHB
BD +09 2870 | 9.440 | 1.042 | $\cdots$ | 3 | 0.012 | $-$1.27 | 9.426 | 0.647 | 0.527 | 2 | 0.016 | 1.026 | $-$2.37 | 6 | RGB
BD +09 3223 | 9.260 | 0.670 | $\cdots$ | 3 | 0.041 | 0.58 | 9.253 | 0.469 | 0.481 | 2 | 0.056 | 0.614 | $-$2.26 | 6 | RHB
BD +10 2495 | 9.723 | 0.749 | $\cdots$ | 3 | 0.002 | 0.29 | 9.745 | 0.522 | 0.361 | 2 | 0.003 | 0.746 | $-$1.83 | 7 | RGB
BD +11 2998 | 9.067 | 0.679 | $\cdots$ | 3 | 0.035 | 0.77 | 9.058 | 0.453 | 0.505 | 2 | 0.048 | 0.631 | $-$1.17 | 6 | RHB
BD +12 2547 | 9.920 | 1.034 | $\cdots$ | 3 | 0.004 | $-$0.94 | 9.92 | 0.635 | 0.420 | 2 | 0.005 | 1.029 | $-$0.72 | 7 | AGB
BD +17 3248 | 9.37 | 0.66 | 0.08 | 1 | 0.040 | 0.65 | 9.352 | 0.486 | 0.445 | 2 | 0.055 | 0.605 | $-$2.02 | 6 | RHB
BD +18 2757 | 9.795 | 0.745 | 0.155 | 1 | 0.000 | $-$0.80 | 9.84 | 0.550 | 0.490 | 2 | 0.00 | 0.745 | $-$2.19 | 6 | AGB
BD +18 2976 | 9.850 | 1.051 | $\cdots$ | 3 | 0.005 | $-$1.32 | 9.835 | 0.655 | 0.527 | 2 | 0.006 | 1.045 | $-$2.4 | 2 | RGB
BD +30 2611 | 9.125 | 1.240 | 1.125 | 1 | 0.003 | $-$1.11 | 9.143 | 0.807 | 0.551 | 2 | 0.004 | 1.236 | $-$1.49 | 6 | RGB
BD +52 1601 | 8.800 | 0.901 | $\cdots$ | 3 | 0.000 | 0.13 | 8.80 | 0.555 | 0.445 | 2 | 0.00 | 0.901 | $-$1.58 | 6 | RGB
BD +54 1323 | 9.343 | 0.670 | $\cdots$ | 3 | 0.003 | 0.69 | 9.33 | 0.470 | 0.440 | 2 | 0.004 | 0.666 | $-$1.65 | 6 | RHB
BD +58 1218 | 9.960 | 0.835 | $\cdots$ | 3 | 0.000 | 0.39 | 9.96 | 0.515 | 0.360 | 2 | 0.00 | 0.835 | $-$2.66 | 7 | RGB
BD $-$03 5215 | 10.170 | 0.719 | $\cdots$ | 3 | 0.032 | 0.73 | 10.188 | 0.435 | 0.499 | 2 | 0.043 | 0.676 | $-$1.66 | 1 | RHB
CD $-$30 8626 | 9.703 | 0.759 | 0.250 | 1 | 0.034 | 0.27 | 9.719 | 0.521 | 0.479 | 2 | 0.047 | 0.712 | $-$1.7 | 2 | AGB
HD 83212 | 8.335 | 1.070 | 0.760 | 1 | 0.018 | $-$0.92 | 8.328 | 0.694 | 0.571 | 2 | 0.025 | 1.045 | $-$1.49 | 2 | RGB
HD 93529 | 9.300 | 0.881 | 0.370 | 1 | 0.048 | 1.05 | 9.306 | 0.582 | 0.409 | 2 | 0.066 | 0.815 | $-$1.67 | 6 | RGB
HD 101063 | 9.460 | 0.755 | 0.098 | 1 | 0.027 | 2.74 | 9.47 | 0.499 | 0.284 | 2 | 0.037 | 0.718 | $-$1.13 | 7 | SGB
HD 104207 | 6.984 | 1.574 | 1.565 | 1 | $\cdots$ | $-$2.48 | $\cdots$ | $\cdots$ | $\cdots$ | 4 | 0.04 | 1.534 | $-$1.93 | 4 | RGB
HD 105546 | 8.616 | 0.708 | $\cdots$ | 3 | 0.000 | 0.79 | 8.61 | 0.460 | 0.420 | 2 | 0.000 | 0.708 | $-$1.27 | 6 | RHB
HD 107752 | 10.05 | 0.75 | 0.18 | 1 | 0.000 | $-$0.68 | 9.994 | 0.578 | 0.463 | 2 | 0.000 | 0.75 | $-$2.88 | 7 | AGB
HD 108317 | 8.038 | 0.631 | $\cdots$ | 3 | 0.000 | 0.52 | 8.044 | 0.450 | 0.311 | 2 | 0.000 | 0.631 | $-$2.24 | 6 | RHB
HD 108577 | 9.597 | 0.694 | 0.134 | 1 | 0.015 | $-$0.57 | 9.581 | 0.506 | 0.500 | 2 | 0.021 | 0.673 | $-$2.28 | 6 | AGB
HD 110184 | 8.305 | 1.175 | 0.765 | 1 | 0.000 | $-$2.14 | 8.293 | 0.818 | 0.712 | 2 | 0.000 | 1.175 | $-$2.56 | 6 | RGB
HD 110885 | 9.180 | 0.672 | $\cdots$ | 3 | 0.000 | 0.74 | 9.18 | 0.423 | 0.492 | 2 | 0.000 | 0.672 | $-$1.44 | 8 | RHB
HD 111721 | 7.971 | 0.799 | 0.157 | 1 | 0.006 | 1.16 | 7.98 | 0.526 | 0.315 | 2 | 0.008 | 0.791 | $-$1.26 | 8 | RGB
HD 113002 | 8.745 | 0.747 | 0.209 | 1 | $\cdots$ | 2.95 | $\cdots$ | $\cdots$ | $\cdots$ | 5 | 0.020 | 0.727 | $-$1.08 | 5 | SGB
HD 115444 | 8.967 | 0.784 | 0.173 | 1 | 0.000 | $-$0.49 | 8.98 | 0.575 | 0.425 | 2 | 0.000 | 0.784 | $-$2.77 | 6 | RGB
HD 119516 | 9.090 | 0.661 | $\cdots$ | 3 | 0.000 | 0.56 | 9.09 | 0.410 | 0.525 | 2 | 0.000 | 0.661 | $-$2.5 | 2 | RHB
HD 121135 | 9.357 | 0.795 | $\cdots$ | 3 | 0.008 | $-$0.36 | 9.368 | 0.530 | 0.509 | 2 | 0.011 | 0.784 | $-$1.57 | 6 | AGB
HD 122956 | 7.22 | 1.01 | $\cdots$ | 1 | 0.042 | $-$0.69 | 7.251 | 0.667 | 0.480 | 2 | 0.058 | 0.95 | $-$1.78 | 6 | RGB
HD 126587 | 9.125 | 0.818 | 0.160 | 1 | 0.058 | $-$0.11 | 9.097 | 0.596 | 0.381 | 2 | 0.079 | 0.739 | $-$3.06 | 7 | RGB
HD 126778 | 8.168 | 0.916 | 0.666 | 1 | 0.000 | 1.05 | 8.15 | 0.596 | 0.449 | 2 | 0.000 | 0.916 | $-$0.7 | 2 | RGB
HD 135148 | 9.490 | 1.388 | $\cdots$ | 3 | 0.016 | $-$0.87 | 9.425 | 0.869 | 0.440 | 2 | 0.022 | 1.366 | $-$1.90 | 6 | RGBbbCH star.
HD 141531 | 9.130 | 1.240 | $\cdots$ | 3 | 0.012 | $-$1.46 | 9.145 | 0.765 | 0.603 | 2 | 0.016 | 1.224 | $-$1.62 | 8 | RGB
HD 161770 | 9.681 | 0.665 | $-$0.041 | 1 | 0.054 | 1.45 | 9.70 | 0.500 | 0.281 | 2 | 0.074 | 0.591 | $-$2.12 | 7 | RGB
HD 195636 | 9.540 | 0.645 | $-$0.005 | 1 | 0.044 | 0.51 | 9.552 | 0.467 | 0.481 | 2 | 0.060 | 0.585 | $-$2.83 | 10 | RHB
TY Vir | 8.100 | 1.28 | 1.00 | 1 | 0.012 | $-$1.17 | 8.165 | 0.938 | 0.711 | 2 | 0.016 | 1.26 | $-$1.78 | 9 | SR
References. — 1. Mermilliod et al. 1997; 2. Anthony-Twarog & Twarog 1994; 3.
HIPPARCOS Input Catalogue; 4. GK Com (Var.); M4 III; Beers et al. 2000 data
used to derive $M_{V}$; 5. Beers et al. (2000) data used to derive $M_{V}$; 6.
Pilachowski et al. (1996); 7. Pilachowski et al. (1993); 8\. Gratton et al.
2000; 9. Fulbright 2000; 10. Anthony-Twarog & Twarog 1998.
Table 2: Parameters of He I 10830Å Line
Star | Exp.aaTotal exposure time, usually divided into several nodded segments. | Evol.bbEvolutionary stage estimated from photometry: RGB= red giant branch; AGB=asymptotic giant branch; RHB= red horizontal branch; SGB= sub-giant branch; SR= semi-regular variable. | RV | Teff | LineccCode for the presence of He I $\lambda$10830; 0: no He line observed; 1: He absorption; 2: P Cygni profile or emission observed (2 stars). | EWddEquivalent Width: Positive values indicate equivalent width of absorption line below the local continuum; negative values indicate equivalent width of the emission line. | VtermeeFurthest extent of the short wavelength absorption edge. | $B/R$ff$B/R$ is the ratio of the short wavelength extent of helium absorption ($V_{term}$) to the long wavelength extent. | R⋆ | Vesc(2R⋆)ggStellar mass assumed: RGB=0.75M⊙; RHB = 0.7M⊙; AGB =0.6M⊙; SGB=0.8M⊙; and SR=0.6M⊙. | Ref.
---|---|---|---|---|---|---|---|---|---|---|---
| (s) | | (km s-1) | (K) | | (mÅ) | (km s-1) | | (R⊙) | (km s-1) | (for Teff)
BD +01 3070 | 400 | RGB | $-$329.9 | 5130 | 1 | 146.3 | $\gtrsim$90 | $\gtrsim$0.80 | 6.4 | 150 | 1
BD +05 3098 | 1440 | RGB | $-$160.5 | 4930 | 1 | 51.5 | 51 | 1.19 | 14.3 | 100 | 1
BD +09 2574 | 1200 | RGB | $-$49.8 | 4860 | 2 | 56.1 | 52 | 0.55 | 12.8 | 106 | 1
BD +09 2860 | 2160 | RHB | $-$20.7 | 5240 | 1 | 38.2 | 44 | 0.97 | 8.5 | 126 | 1
BD +09 2870 | 360 | RGB | $-$120.1 | 4600 | 2 | 15.9 | 60 | 0.84 | 30.7 | 69 | 2
BD +09 3223 | 720 | RHB | 67.3 | 5310 | 1 | 25.5 | 44 | 1.08 | 8.9 | 123 | 1
BD +10 2495 | 840 | RGB | 263.2 | 4920 | 1 | 25.2 | 46 | 1.11 | 12.3 | 108 | 1
BD +11 2998 | 480 | RHB | 50.7 | 5360 | 1 | 54.1 | 45 | 0.47 | 7.8 | 131 | 1
BD +12 2547 | 420 | AGB | 5.3 | 4610 | 2 | 8.8: | $\gtrsim$90 | $\gtrsim$0.86 | 26.0 | 67 | 1
BD +17 3248 | 720 | RHB | $-$147.4 | 5250 | 1 | 45.4 | 62 | 1.25 | 8.8 | 123 | 2
BD +18 2757 | 500 | AGB | $-$29.0 | 4840 | 1 | 86.3 | $\gtrsim$90 | $\gtrsim$2.76 | 21.3 | 74 | 1
BD +18 2976 | 500 | RGB | $-$167.4 | 4550 | 2 | 17.2: | 143 | 2.65 | 32.5 | 67 | 1
BD +30 2611 | 200 | RGB | $-$282.8 | 4275 | 1 | 92.6 | 143 | 1.78 | 35.9 | 63 | 2
BD +52 1601 | 400 | RGB | $-$47.4 | 4750 | 0 | 0. | $\cdots$ | $\cdots$ | 14.6 | 99 | 2
BD +54 1323 | 600 | RHB | $-$67.2 | 5300 | 1 | 45.0 | 36 | 0.48 | 8.4 | 126 | 2
BD +58 1218 | 600 | RGB | $-$305.2 | 4950 | 1 | 37. | 65 | 1.54 | 11.7 | 111 | 3
BD $-$03 5215 | 1200 | RHB | $-$294.5 | 5420 | 1 | 232.4 | $\gtrsim$90 | $\gtrsim$1.39 | 7.8 | 131 | 1
CD $-$30 8626 | 600 | AGB | 266.2 | 5000 | 1 | 25.6 | 41 | 1.08 | 11.9 | 99 | 1
HD 83212 | 500 | RGB | 108.0 | 4550 | 2hhEmission clearly present in HD 141531 but absorption not apparent. | $-$43.6 | $\gtrsim$90 | $\gtrsim$1.42 | 26.9 | 73 | 2
HD 93529 | 300 | RGB | 145.6 | 4650 | 1 | 39.9 | 38 | 0.79 | 10.2 | 119 | 2
HD 101063 | 400 | SGB | 182.6 | 5070 | 1 | 68.3 | 48 | 1.04 | 3.6 | 205 | 3
HD 104207 | 6 | RGB | 35.6 | 3916 | 2 | 23.8 | $\gtrsim$90 | $\gtrsim$8.69 | 95.0 | 39 | 4
HD 105546 | 240 | RHB | 18.1 | 5300 | 1 | 31.7 | 35 | 0.99 | 8.0 | 130 | 2
HD 107752 | 600 | AGB | 219.2 | 4750 | 1 | 71.7 | $\gtrsim$90 | $\gtrsim$0.92 | 21.4 | 73 | 1
HD 108317 | 150 | RHB | 4.4 | 5230 | 1 | 23.2 | 35 | 1.11 | 9.5 | 119 | 1
HD 108577 | 400 | AGB | $-$112.1 | 4975 | 1 | 20.6 | 42 | 1.11 | 17.8 | 80 | 2
HD 110184 | 120 | RGB | 138.5 | 4250 | 0 | 0. | $\cdots$ | $\cdots$ | 59.0 | 49 | 2
HD 110885 | 600 | RHB | $-$48.8 | 5330 | 1 | 14.1 | $\gtrsim$90 | $\gtrsim$0.85 | 8.1 | 129 | 1
HD 111721 | 240 | RGB | 20.5 | 5080 | 1 | 48.4 | 50 | 1.11 | 7.5 | 138 | 5
HD 113002 | 240 | SGB | $-$95.2 | 5007 | 1 | 30.1 | 57 | 1.46 | 3.4 | 212 | 4
HD 115444 | 300 | RGB | $-$27.7 | 4750 | 2 | 32.6 | 65 | 0.71 | 19.6 | 86 | 2
HD 119516 | 600 | RHB | $-$287.2 | 5440 | 1 | 466. | 121 | 1.29 | 8.6 | 125 | 1
HD 121135 | 600 | AGB | 125.0 | 4925 | 2 | 63.2 | 104 | 1.47 | 16.5 | 84 | 2
HD 122956 | 70 | RGB | 165.2 | 4600 | 1 | 84. | 110 | 2.99 | 23.4 | 78 | 2
HD 126587 | 480 | RGB | 149.3 | 4960 | 1 | 20.9 | 39 | 0.97 | 14.7 | 99 | 5
HD 126778 | 150 | RGB | $-$138.8 | 4847 | 2 | 33.8 | 41 | 0.56 | 8.9 | 127 | 6
HD 135148 | 320 | RGB | $-$85.4 | 4275 | 2 | 2389.6 | 117 | 1.51 | 32.2 | 67 | 2
HD 141531 | 440 | RGB | 2.3 | 4340 | 2ggStellar mass assumed: RGB=0.75M⊙; RHB = 0.7M⊙; AGB =0.6M⊙; SGB=0.8M⊙; and SR=0.6M⊙. | $-$12.8 | $\cdots$ | $\cdots$ | 40.1 | 60 | 1
HD 161770 | 1020 | RGB | $-$130.6 | 5406 | 1 | 30.5 | 32 | 0.91 | 5.7 | 159 | 4
HD 195636 | 600 | RHB | $-$257.7 | 5370 | 2 | 313.7 | 171 | 1.76 | 9.1 | 122 | 1
TY Vir | 70 | SR | 229.1 | 4350 | 0 | 0. | $\cdots$ | $\cdots$ | 34.8 | 58 | 8
References. — 1. Carney et al. 2003; 2. Pilachowski et al. 1996; 3\. Carney et
al. 2008; 4. Alonso et al. 1999; 5. Rossi et al. 2005; 6\. Cenarro et al. 2007
8. Andrievsky et al. 2007
Table 3: He I 10830Å Observations from Previous Publications
Star | MV | (B$-$V)0 | [Fe/H] | He 10830Å | VtermaaFurthest extent of the short wavelength absorption edge. | Teff | R⋆ | Vesc(2R⋆)bbStellar mass assumed: RGB=0.75M⊙; RHB=0.7M⊙; AGB =0.6M⊙. | Ref.
---|---|---|---|---|---|---|---|---|---
| | | | | (km s-1) | (K) | (R⊙) | (km s-1) |
Field Giants
BD +17°3248ccRHB star. | +0.65 | 0.605 | $-$2.1 | absorption | 60 | 4625 | 8.8 | 123 | 2
HD 6833 | $-$0.9 | 1.08 | $-$0.91 | absorption | $\gtrsim$90 | 4400 | 29.6 | 70 | 1,3
HD 122563 | $-$1.24 | 0.90 | $-$2.6 | absorption | 140 | 4625 | 29.6 | 70 | 2
HD 165195 | $-$2.14 | 1.07 | $-$2.4 | not detect. | $\cdots$ | 4450 | $\cdots$ | $\cdots$ | 2
HD 221170 | $-$1.67 | 1.09 | $-$2.0 | not detect. | $\cdots$ | 4425 | $\cdots$ | $\cdots$ | 2
Red Giants in M13
II-33 | $-$1.78 | 1.20 | $-$1.51 | not detect. | $\cdots$ | 4390 | $\cdots$ | $\cdots$ | 2
III-37 | $-$1.67 | 1.14 | $-$1.51 | not detect. | $\cdots$ | 4400 | $\cdots$ | $\cdots$ | 2
III-63 | $-$2.25 | 1.37 | $-$1.51 | not detect. | $\cdots$ | 4200 | $\cdots$ | $\cdots$ | 2
III-73 | $-$2.13 | 1.28 | $-$1.51 | not detect. | $\cdots$ | 4300 | $\cdots$ | $\cdots$ | 2
IV-15ddAGB star. | $-$1.49 | 1.02 | $-$1.51 | absorption | 30 | 4650 | 32.7 | 59 | 2
IV-25 | $-$2.36 | 1.52 | $-$1.51 | not detect. | $\cdots$ | 4000 | $\cdots$ | $\cdots$ | 2
References. — 1. Dupree et al. 1992a; 2. Smith et al. 2004 3. Smith & Dupree
1988
|
arxiv-papers
| 2009-09-08T20:02:02 |
2024-09-04T02:49:05.149580
|
{
"license": "Public Domain",
"authors": "A. K. Dupree (1), G. H. Smith (2), and J. Strader (1 and 3) ((1)\n Harvard-Smithsonian Center for Astrophysics, (2) University of California\n Observatories/Lick Observatory, (3) Hubble Fellow)",
"submitter": "Andrea Dupree",
"url": "https://arxiv.org/abs/0909.1558"
}
|
0909.1594
|
# Computation and Dynamics:
Classical and Quantum
Vladimir V. Kisil School of Mathematics
University of Leeds
Leeds LS2 9JT
UK kisilv@maths.leeds.ac.uk http://www.maths.leeds.ac.uk/~kisilv/
###### Abstract.
We discuss classical and quantum computations in terms of corresponding
Hamiltonian dynamics.
On leave from Odessa University.
The author acknowledges the support of the EPSRC Network on Semantics of
Quantum Computation (EP/E006833/2).
††copyright: ©:
## 1\. Introduction
It is well known that classical computations are modelled by abstract
“machines” first introduced in works of E. Post and A. Turing, see
[KnuthACP250]*§ 1.4.5 and § 2.6 for historical notes and further references.
We are going to demonstrate and exploit an explicit analogy between the
process of computation on such abstract machines and a Hamiltonian dynamics of
a particle in the phase space. We will use for this purpose the Post machine
since its description is simpler. In the following we will call it simply the
_machine_.
(a) (b)
Figure 1. (a) A symbolic representation of the classical Post machine (a
single tape).
(b) A symbolic representation of computations with a quantum superposition of
tapes. The programme contains new types of instructions but it is still
classical.
A state of the machine is described by two independent components: the tape
(data) and the instruction list (programme), see Fig. 1(a). The tape is
assumed to be an infinite sequence of cells with only a finite number of them
holding mark $1$, all others assumed to be “empty” (holding $0$). Another
important property of the tape is the _current cell_ for
observation/modification pointed by a reading head.
The second machine’s component— _programme_ —is a finite list of instructions
with the second pointer marking the current command. The statements are taken
from a very limited set and request modifications of the current tape’s cell
or respective movements of the reading head and the instruction pointer.
###### Remark 1.1.
The division into “data tape” and “programme” seems to be a fundamental one.
This duality is reflected in both—architectures of modern computers and the
computer science paradigm of “Algorithms and Data Structures”
[WirthAlgorithsDS].
A typical quantum computation [Shor94] [Grover96] can be modelled by a
_quantisation_ of the tape in a machine, see Fig. 1(b). This means that
instead of a classical tape holding a sequence of classical bits one considers
a quantum tape: a finite number of cells holding _qubits_. Qubits are assumed
to be able to store linear combinations (superpositions) of values $0$ and
$1$.
A quantisation of the other half—the programme—is rarely considered: it is
still a linear sequence of corresponding instructions, which are unitary
operators on qubits in this case. Thus a common quantum computer is strictly
speaking semi-classical or quantum-classical computer only. To get fully
quantised computer one can additionally request superpositions of computer
states and/or programmes. However a realisation of superposition for
instructions can be confusing.
In this paper we consider an alternative approach. Firstly we get unification
of the tape and the reading head position into a single coordinate. Computer’s
programme is linked to the another coordinate. Then we can quantise it in a
single move. Computational speed of such a computer cannot be directly
compared to a classical one, since it will not only process data in parallel
but also perform different computational stages at the same time.
## 2\. Phase Space Computations and Hamiltonian Dynamics
To obtain the dynamical description of computations we blend the state of the
tape and position of the reading head into a single parameter. We interpret
the finite sequence of $1$’s and enclosed among them $0$’s on the tape as a
dyadic rational number with the binary point at the immediate right to the
current cell. Then the standard actions of the reading head (the first column
of Tab. 1) can be translated into operations on the set $\mathbb{D}{}$ of
dyadic numbers (the second column of Tab. 1).
Head action | Arithmetic operation | Value of $\Delta_{p}H(q_{0},p_{0})$
---|---|---
Head to the left | Divide the fraction by $2$ | $-\frac{1}{2}q_{0}$
Head to the right | Multiply the fraction by $2$ | $q_{0}$
Replace $0$ by $1$ | Add $1$ to the fraction | $1$
Replace $1$ by $0$ | Subtract $1$ from the fraction | $-1$
Table 1. The first column lists actions of the reading head of a machine, the
second column translates them into dyadic arithmetic. The third column
provides values of a Hamiltonian which direct those transformations.
Similarly the set $\mathbb{Z}_{n}{}=\\{1,2,\ldots,n\\}$ can index a programme
of $n$ instructions. Thus a full state of a machine is described by a point
$(q,p)\in\mathbb{D}{}\times\mathbb{Z}_{n}{}$. Calculation is a dynamic on this
set with a discrete time parameter $t\in\mathbb{N}{}$. An iteration from a
current state $(q_{t},p_{t})$ to the next one $(q_{t+1},p_{t+1})$ is given by
the pair of finite differences equations:
(1) $\Delta_{t}q=\Delta_{p}H,\qquad\Delta_{t}p=-\Delta_{q}H.$
Here $\Delta_{q}H$ and $\Delta_{p}H$ is a pair of functions
$\mathbb{D}{}\times\mathbb{Z}_{n}{}\rightarrow\mathbb{Z}{}$ and
$\mathbb{D}{}\times\mathbb{Z}_{n}{}\rightarrow\mathbb{D}{}$ respectively. The
function $\Delta_{p}H$ defines transformations of the tape according to the
third column in Tab. 1. The programme flow is directed by $\Delta_{q}H$ as
described in Tab. 2.
###### Remark 2.1.
We intentionally use notations resembling Hamiltonian dynamics in order to
exploit the duality between data and algorithms mentioned in Rem. 1.1. However
the exact mathematical formalism for this duality is still missing. For
example, canonical transformations mixing data and programme can be related to
the philosophy behind Prolog and Lisp programming languages.
Instruction pointer | Value of $\Delta_{q}H(q_{0},p_{0})$
---|---
Next Instruction | $-1$
Go to $p_{1}$ | $p_{0}-p_{1}$
If cell is $1$ go to $p_{1}$ | $\left\\{\begin{array}[]{ll}p_{0}-p_{1},&\text{ if }[q_{0}]=1\mod 2;\\\ 1,&\text{ if }[q_{0}]\neq 1\mod 2.\end{array}\right.$
Table 2. Movements of the programme pointer (the first column) and the
corresponding values of a Hamiltonian (the second column).
###### Theorem 2.2.
Calculations of a Post machine is described by a discrete dynamics in the
phase space $\mathbb{D}{}\times\mathbb{Z}_{n}{}$ defined by the equations (1).
A programme corresponds to a Hamiltonian governing the dynamic.
Abstract computing | Hamilton dynamic
---|---
Tape state | Coordinate
Inner state | Momentum
Program | Hamiltonian
Execution | Dynamics
Inclusion-Exclusion | Wave superposition
Table 3. The correspondence between element of abstract calculations and
dynamics in the phase space.
Tab. 3 shows a correspondence between notions of computing and Hamilton
dynamics. Developing this approach we can define a _fully_ quantum computation
through quantisation of the classical discrete dynamics. This gives
simultaneous propagation along all possible paths, which means parallel
procession of data _and_ the programme similarly.
## 3\. Example: Polynomial Sequences of Binomial Type
Classical computations of many combinatorial quantities is based on the
inclusion-exclusion principle [StanleyI]*§ 2.1. Its quantum counterpart is the
superposition of wave functions: the resulting probability can be anything
from the sum (inclusion) to the difference (exclusion) of given probabilities.
Thus such combinatorial calculations are very suitable for quantum
computations.
For example, let $q_{n}(x)$ be a _token_ [Kisil01b] [Kisil97b] from
$\mathbb{N}{}$ to $\mathbb{R}{}$, i.e. the sequence of polynomials of $\deg
q_{n}=n$ satisfying to the identity:
$q_{n}(x+y)=\sum_{k=0}^{n}q_{k}(y)q_{n-k}(x),$
If $q_{n}(x)$ is such a token then a polynomial sequence $p_{n}(x)=n!q_{n}(x)$
is of _binomial type_ [RotaWay]*§ 4.3. Examples are provided by power
monomials, falling (rising) factorials, Abel, Laguerre and many other famous
polynomials.
A dynamics in a configurational space $Q$ can be described by the _propagator_
$K(q_{2},t_{2};q_{1},t_{1})$—a complex valued function defined on
$\mathbb{Q}{}\times\mathbb{R}{}\times\mathbb{Q}{}\times\mathbb{R}{}$. It is a
probability amplitude for a transition $q_{1}\rightarrow q_{2}$ from a state
$q_{1}$ at time $t_{1}$ to $q_{2}$ at time $t_{2}$. The fundamental assumption
about the quantum world is the _absence of trajectories_ for a system’s
evolution through the configurational space $\mathbb{Q}{}$: the system at any
time $t_{i}$ could be found at any point $q_{i}$.
R. Feynman developing ideas of A. Einstein, M.V. Smoluhovski and P.A.M. Dirac
proposed an expression for the propagator via the “integral over all possible
paths”:
$K(q_{2},t_{2};q_{1},t_{1})=\int\frac{\mathcal{D}q\,\mathcal{D}p}{h}\exp\left(\frac{i}{\hbar}\int\limits_{t_{1}}^{t_{2}}dt\left(p\dot{q}-H(p,q)\right)\right).$
Here $H(p,q)$ is the Hamiltonian of the system. The inner integral is over a
path in the phase space. The outer integral is taken over “all possible paths
between two given points with respect to a measure
$\mathcal{D}q\,\mathcal{D}p$ on paths in the phase space”.
###### Proposition 3.1.
Any quantum system is a quantum computer for an evaluation of its own
propagator $K$, computation is done simultaneously along all possible paths.
In this way we obtain the path computation formula for polynomials $q_{n}$
[Kisil98d]:
$q_{n}(x)=\int\\!\mathcal{D}k\mathcal{D}p\,\exp\\!\\!\int\limits_{0}^{x}(-ipk^{\prime}+h(p))\,dt,\qquad\text{
where }h(p)=\sum_{k=0}^{\infty}q_{k}^{\prime}(0)e^{ipk}.$
Thus a quantum system with the above Hamiltonian $h(p)$ allows to calculates
$q_{n}$ in a single operation (measurement). This looks unrealistically quick
and one can ask: how to compare speeds of quantum and classical computations
after all?
## 4\. Quantum Computers with Classical Terminals
A discussion of quantum computers is often limited to quantum algorithms.
However this an oversimplification, which does not include the process of
qubit preparation (input of data), building sequences of quantum gates
(programming) and reading of the final state (data output). Of course, in the
classical case these three processes can be done in a negligible time in
comparison with the actual computation. However, this is no longer true for
quantum computations.
###### Example 4.1.
Let us review two most known quantum algorithms.
1. (1)
Shor’s factorisation algorithm [Shor94] required the quantum circuit to be
reassembled accordingly every time a new random number was chosen for a test.
Thus the time of circuit assembling (programming) should be included in the
overall computational cost.
2. (2)
Grover’s database search algorithm [Grover96] requires several repeated
recalculations, each of which would destroy the database (the projection
postulate of quantum measurement [Mackey63] ). Thus the time for rebuilding a
database (data input) and measurement (data output) should be included in the
overall computational cost.
For more realistic consideration we have to add _classical interfaces_ for
input and output to make quantum computations really usable. At present even a
simple quantum step like two qbits swapping is done by a millions of classical
computational steps. Is it a _present day_ technological limitation or
_fundamental_ exchange rate between cost of a quantum and classical
computation? If _an application_ of an existent quantum gate is so expensive,
how expensive is _to built_ a case-specific quantum circuit for $f(x)=a^{x}$
[Grover96] or quantum Fourier transform [Shor94]? Such questions are already
hinted in [Shor94] but are rarely discussed in depth. Consequently we miss not
only clear answers but even the understanding of their importance.
In the first half of this paper we presented classical and quantum
computations as dynamics. Then a quantum computer with classical terminals
shall be represented by a dynamics of a quantum-classical aggregate system. Is
there a consistent theory to describe such a dynamics? This is a debated
topics with the majority of physicists believing that this is fundamentally
impossible [CaroSalcedo99] [Sahoo04]. If this is so, shall it be interpreted
as our inability (as a macroscopic and thus classical objects) to efficiently
interact with quantum computing devices even if they are to be built?
Quantum-classical dynamics is oftenly connected with an existence of special
quantum-classic bracket which shall unify (and replace) both quantum
commutator and Poisson brackets. A mathematical model for a classical system
attached as an input/output terminal to a quantum computer can be attempted
from the quantum-classical formalism proposed in [Kisil02e] [BrodlieKisil03a]
[Kisil05c] [Kisil09a]. Such a model would provide an opportunity for effective
estimation of the overall cost of quantum computing during the entire cycle:
preparation-computation-reading.
To stimulate an attention to this issue we wish to conclude by the following:
###### Conjecture 4.2 (“Golden rule” of quantum-classic information).
A gain in quantum algorithms is outweighed by losses in classical I/O and
programing.
## References
|
arxiv-papers
| 2009-09-08T23:24:15 |
2024-09-04T02:49:05.161894
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Vladimir V. Kisil",
"submitter": "Vladimir V Kisil",
"url": "https://arxiv.org/abs/0909.1594"
}
|
0909.1619
|
Strange effect of disorder on electron transport through a thin film
Santanu K. Maiti1,2,∗
1Theoretical Condensed Matter Physics Division, Saha Institute of Nuclear
Physics,
1/AF, Bidhannagar, Kolkata-700 064, India
2Department of Physics, Narasinha Dutt College, 129, Belilious Road,
Howrah-711 101, India
Abstract
A novel feature of electron transport is explored through a thin film of
varying impurity density with the distance from its surface. The film,
attached to two metallic electrodes, is described by simple tight-binding
model and its coupling to the electrodes is treated through Newns-Anderson
chemisorption theory. Quite interestingly it is observed that, in the strong
disorder regime the amplitude of the current passing through the film
increases with the increase of the disorder strength, while it decreases in
the weak disorder regime. This anomalous behavior is completely opposite to
that of conventional disordered systems. Our results also predict that the
electron transport is significantly influenced by the finite size of the thin
film.
PACS No.: 73.23.-b, 73.63.Rt, 85.65.+h
Keywords: Green’s function; Thin film; Disorder; Conductance; DOS.
∗Corresponding Author: Santanu K. Maiti
Electronic mail: santanu.maiti@saha.ac.in
## 1 Introduction
In the last few decades considerable attention has been paid to the
propagation of electrons through quantum devices with various geometric
structures where the electron transport is predominantly coherent [1, 2].
Recent progress in creating such quantum devices has enabled us to study the
electron transport through them in a very tunable environment. By using single
molecule or cluster of molecules it can be made possible to construct the
efficient quantum devices that provide a signature in the design of future
nano-electronic circuits. Based on the pioneering work of Aviram and Ratner
[3] in which a molecular electronic device has been predicted for the first
time, the development of a theoretical description of molecular electronic
devices has been pursued. Later, several experiments [4, 5, 6, 7, 8] have been
performed through different molecular bridge systems to understand the basic
mechanisms underlying such transport. Though electron transport properties
through several bridge systems have been described elaborately in lot of
theoretical as well as experimental papers, but yet the complete knowledge of
the conduction mechanism in this scale is not well understood even today. For
example, it is not so transparent how the molecular transport is affected by
its coupling with the side attached electrodes or by the geometry of the
molecule itself. Several significant factors are there which control the the
electron conduction across a bridge system and all these effects have to be
considered properly to study the electron transport. In a their work,
Ernzerhof et al. [9] have manifested a general design principle through some
model calculations, to show how the molecular structure plays a key role in
determining the electron transport. The molecular coupling with the electrodes
is also another important factor that controls the current in a bridge system.
In addition to these, the quantum interference of electron waves [10, 11, 12,
13, 14, 15, 16, 17, 18] and the other parameters of the Hamiltonian that
describe the system provide significant effects in the determination of the
current through the bridge system. Now in these small-scale devices, dynamical
fluctuations play an active role which can be manifested through the
measurement of “shot noise”, a direct consequence of the quantization of
charge. It can be used to obtain information on a system which is not
available directly through the conductance measurements, and is generally more
sensitive to the effects of electron-electron correlations than the average
conductance [19, 20].
In this present paper, we will describe quite a different aspect of quantum
transport than the above mentioned issues. Using the advanced nanoscience and
technology, it can be made possible to fabricate a nano-scale device where the
charge carriers are scattered mainly from its surface boundaries [21, 22, 23,
24, 25] and not from the inner core region. It is completely opposite to that
of a traditional doped system where the dopant atoms are distributed uniformly
along the system. For example, in shell-doped nanowires the dopant atoms are
spatially confined within a few atomic layers in the shell region of a
nanowire. In such a shell-doped nanowire, Zhong and Stocks [22] have shown
that the electron dynamics undergoes a localization to quasi-delocalization
transition beyond some critical doping. In other very recent work [24], Yang
et al. have also observed the localization to quasi-delocalization transition
in edge disordered graphene nanoribbons upon varying the strength of the edge
disorder. From the extensive studies of the electron transport in such systems
where the dopant atoms are not distributed uniformly along the system, it has
been suggested that the surface states [26], surface scattering [27] and the
surface reconstructions [28] may be responsible to exhibit several diverse
transport properties. Motivated by such kind of systems, in this article we
consider a special type of thin film where disorder strength varies smoothly
from layer to layer with the distance from the film surface. This system shows
a peculiar behavior of the electron transport where the current amplitude
increases with the increase of the disorder strength in the limit of strong
disorder, while the amplitude decreases in the weak disorder limit. On the
other hand, for the conventional disordered system i.e., the system subjected
to uniform disorder, the current amplitude always decreases with the increase
of the disorder strength. From our study it is also observed that the electron
transport through the film is significantly influenced by its size which
reveals the finite quantum size effects. Here we reproduce an analytic
approach based on the tight-binding model to investigate the electron
transport through the film system, and adopt the Newns-Anderson chemisorption
model [29, 30, 31] for the description of the electrodes and for the
interaction of the electrodes with the film.
We organize this paper as follows. In Section $2$, we describe the model and
the methodology for the calculation of the transmission probability ($T$) and
the current ($I$) through a thin film attached to two metallic electrodes by
the use of Green’s function technique. Section $3$ discusses the significant
results, and finally , we summarize our results in Section $4$.
## 2 Model and the theoretical description
This section describes the model and the methodology for the calculation of
the transmission probability ($T$), conductance ($g$) and the current ($I$)
through a thin film attached to two one-dimensional metallic electrodes
Figure 1: A thin film of four layers attached to two metallic electrodes
(source and drain), where the different layers are subjected to different
impurity strengths. The top most front layer (green color) is subjected to the
highest impurity strength and the strength of the impurity decreases smoothly
to-wards the bottom layer keeping the lowest bottom layer (light blue color)
as impurity free. The two electrodes are attached at the two extreme corners
of the bottom layer.
by using the Green’s function technique. The schematic view of such a bridge
system is illustrated in Fig. 1.
For low bias voltage and temperature, the conductance $g$ of the film is
determined by the Landauer conductance formula [32],
$g=\frac{2e^{2}}{h}T$ (1)
where the transmission probability $T$ becomes [32],
$T={\mbox{Tr}}\left[\Gamma_{S}G_{F}^{r}\Gamma_{D}G_{F}^{a}\right]$ (2)
Here $G_{F}^{r}$ and $G_{F}^{a}$ correspond to the retarded and advanced
Green’s functions of the film, and $\Gamma_{S}$ and $\Gamma_{D}$ describe its
coupling with the source and the drain, respectively. The Green’s function of
the film is written in this form,
$G_{F}=\left(E-H_{F}-\Sigma_{S}-\Sigma_{D}\right)^{-1}$ (3)
where $E$ is the energy of the injecting electron and $H_{F}$ represents the
Hamiltonian of the film which can be written in the tight-binding form within
the non-interacting picture like,
$H_{F}=\sum_{i}\epsilon_{i}c_{i}^{\dagger}c_{i}+\sum_{<ij>}t\left(c_{i}^{\dagger}c_{j}+c_{j}^{\dagger}c_{i}\right)$
(4)
In this expression, $\epsilon_{i}$’s are the site energies and $t$ corresponds
to the nearest-neighbor hopping strength. As an approximation, we set the
hopping strengths along the longitudinal and the transverse directions in each
layer of the film are identical with each other which is denoted by the
parameter $t$. Similar hopping strength $t$ is also taken between two
consecutive layers, for simplicity. Now in order to introduce the impurities
in the thin film where the different layers are subjected to different
impurity strengths, we choose the site energies ($\epsilon_{i}$’s) randomly
from a “Box” distribution function such that the top most front layer becomes
the highest disordered layer with strength $W$ and the strength gradually
decreases to-wards the bottom layer as a function of $W/N_{l}$ ($N_{l}$ be the
total number of layers in the film), keeping the lowest bottom layer as
impurity free. On the other hand, in the traditional disordered thin film all
the layers are subjected to the same disorder strength $W$. In our present
model we use the similar kind of tight-binding Hamiltonian as prescribed in
Eq.(4) to describe the side attached electrodes, where the site energy and the
nearest-neighbor hopping strength are represented by the symbols
$\epsilon_{i}^{\prime}$ and $v$, respectively. The parameters $\Sigma_{S}$ and
$\Sigma_{D}$ in Eq.(3) correspond to the self-energies due to coupling of the
film with the source and the drain, respectively, where all the informations
of this coupling are included into these two self-energies and are described
by the Newns-Anderson chemisorption model [29, 30, 31]. This Newns-Anderson
model permits us to describe the conductance in terms of the effective film
properties multiplied by the effective state densities involving the coupling,
and allows us to study directly the conductance as a function of the
properties of the electronic structure of the film between the electrodes.
The current passing through the film can be regarded as a single electron
scattering process between the two reservoirs of charge carriers. The current-
voltage relationship can be obtained from the expression [32],
$I(V)=\frac{e}{\pi\hbar}\int\limits_{-\infty}^{\infty}\left(f_{S}-f_{D}\right)T(E)dE$
(5)
where $f_{S(D)}=f\left(E-\mu_{S(D)}\right)$ gives the Fermi distribution
function with the electrochemical potential $\mu_{S(D)}=E_{F}\pm eV/2$.
Usually, the electric field inside the thin film, especially for small films,
seems to have a minimal effect on the $g$-$E$ characteristics. Thus it
introduces very little error if we assume that, the entire voltage is dropped
across the film-electrode interfaces. The $g$-$E$ characteristics are not
significantly altered. On the other hand, for larger system sizes and higher
bias voltage, the electric field inside the film may play a more significant
role depending on the size and the structure of the film [33], but yet the
effect is quite small.
In this article, we concentrate our study on the determination of the typical
current amplitude which can be expressed through the relation,
$I_{typ}=\sqrt{<I^{2}>_{W,V}}$ (6)
where $W$ and $V$ correspond to the impurity strength and the applied bias
voltage, respectively.
Throughout this article we study our results at absolute zero temperature, but
the qualitative behavior of all the results are invariant up to some finite
temperature ($\sim 300$ K). The reason for such an assumption is that the
broadening of the energy levels of the thin film due to its coupling with the
electrodes is much larger than that of the thermal broadening. For simplicity,
we take the unit $c=e=h=1$ in our present calculations.
## 3 Results and discussion
Here we focus the significant results and describe the strange effect of
impurity on electron transport through a thin film subjected to the smoothly
varying impurity density from its surface. These results provide the basic
conduction mechanisms and the essential principles for the control of electron
transport in a bridge system. An anomalous feature of the electron transport
through this system is observed, where the current amplitude increases with
the increase of the impurity strength in the strong impurity regime, while the
current amplitude decreases with the impurity strength in the weak impurity
regime. This peculiar behavior is completely opposite to that of the
traditional doped film in which the current amplitude always decreases with
the increase of the doping concentration. Throughout our discussion we choose
the values of the different parameters as follows: the coupling strengths of
the film to the electrodes $\tau_{S}=\tau_{D}=1.5$, the hopping strengths
$t=2$ and $v=4$ respectively in the film and and in the two electrodes. The
site energies ($\epsilon_{i}^{\prime}$’s) in the electrodes are set to zero,
for the sake of simplicity. In addition to these parameters, three other
parameters are also introduced those are represented by $N_{x}$, $N_{y}$ and
$N_{z}$, where the first two of them correspond to the total number of lattice
sites in each layer of the film along the $x$ and $y$ directions,
respectively, and the third one ($N_{z}$) represents the total number lattice
sites along the $z$ direction of the film. In the smoothly varying disordered
film, the different layers are subjected to the strengths $W_{l}=W/N_{l}$,
keeping the top most front layer as the highest disordered layer with strength
$W$ and the lowest bottom layer as disorder free. While, for the conventional
disordered film, all the layers are subjected to the identical strength $W$.
Now both for these two cases, we choose the site energies randomly from a
“Box”
Figure 2: Typical current amplitudes ($I_{typ}$) as a function of the impurity
strength ($W$) for the thin films with six layers ($N_{z}=6$), where the
system size of each layer is taken as: $N_{x}=3$ and $N_{y}=3$. The red and
the blue curves correspond to the results for the smoothly varying and the
conventional disordered films, respectively.
distribution function, and accordingly, we determine the typical current
amplitude ($I_{typ}$) by averaging over a large number ($50$) of random
disordered configurations in each case to achieve much more accurate result.
On the other hand, for the averaging over the bias voltage ($V$), we compute
the results considering the range of $V$ within $-16$ to $16$ in each case. In
this present study, we concentrate ourselves only on the smaller system sizes,
since all the qualitative behaviors remain invariant even for the larger
system sizes, and therefore, the numerical results can be computed in the low
cost of time. The variation of the typical current amplitudes ($I_{typ}$) as a
function of the impurity strength ($W$) for the thin films with system size
$N_{x}=3$, $N_{y}=3$ and $N_{z}=6$ is shown in Fig. 2. The red and the blue
curves correspond to the results for the smoothly varying and the conventional
disordered films, respectively. For the conventional disordered film, the
typical current amplitude decreases sharply with the increase of the impurity
strength ($W$). This behavior can be well understood from the theory of
Figure 3: (a) $g(E)$-$E$ (red color) and (b) $\rho(E)$-$E$ (blue color) curves
for an ordered ($W=0$) thin film with six layers ($N_{z}=6$), where the system
size of each layer is taken as: $N_{x}=3$ and $N_{y}=3$.
Anderson localization, where more localization is achieved with the increase
of the disorder strength [34]. Such a localization phenomenon is well
established in the transport community from a long back ago. A dramatic
feature is observed only when the disorder strength decreases smoothly from
the top most highest disordered layer, keeping the lowest bottom layer as
disorder free. In this particular system, the current amplitude initially
decreases with the increase of the impurity strength, while beyond some
critical value of the impurity strength $W=W_{c}$ (say) the amplitude
increases. This phenomenon is completely opposite in nature from the
traditional disordered system, as discussed above. Such an anomalous behavior
can be explained in this way. We can treat the smoothly varying disordered
film with ordered bottom layer as an order-disorder separated film. In such an
order-disorder separated film, a gradual
Figure 4: (a) $g(E)$-$E$ (red color) and (b) $\rho(E)$-$E$ (blue color) curves
for a smoothly varying disordered ($W=10$, weak disorder limit) thin film with
six layers ($N_{z}=6$), where the system size of each layer is taken as:
$N_{x}=3$ and $N_{y}=3$.
separation of the energy spectra of the disordered layers and the ordered
layer takes place with the increase of the disorder strength $W$. In the limit
of strong disorder, the energy spectrum of the order-disorder separated film
contains localized tail states with much small and central states with much
large values of localization length. Hence the central states gradually
separated from the tail states and delocalized with the increase of the
strength of the disorder. To understand it precisely, here we present the
behavior of the conductance for the three different cases considering the
disorder strengths $W=0$, $W=10$ and $W=30$. The results are shown in Fig. 3,
Fig. 4 and Fig. 5, respectively. In every case the pictures of the density of
states (DOS) are also given to show clearly that with the increase of the
disorder strength more energy eigenstates appear in the energy regimes for
which the conductance is zero. Thus the separation of the localized and the
delocalized eigenstates is clearly visible from these pictures. Hence for the
coupled order-disorder separated film, the coupling between the localized
states with the extended states is strongly
Figure 5: (a) $g(E)$-$E$ (red color) and (b) $\rho(E)$-$E$ (blue color) curves
for a smoothly varying disordered ($W=30$, strong disorder limit) thin film
with six layers ($N_{z}=6$), where the system size of each layer is taken as:
$N_{x}=3$ and $N_{y}=3$.
influenced by the strength of the disorder, and this coupling is inversely
proportional to the disorder strength $W$ which indicates that the influence
of the random scattering in the ordered layer due to the strong localization
in the disordered layers decreases. Therefore, in the limit of weak disorder
the coupling effect is strong, while the coupling effect becomes less
significant in the strong disorder regime. Accordingly, in the limit of weak
disorder the electron transport is strongly influenced by the impurities at
the disordered layers such that the electron states are scattered more and
hence the current amplitude decreases. On the other hand, for the strong
disorder limit the extended states are less influenced by the disordered
layers and the coupling effect gradually decreases with the increase of the
impurity strength which provide the larger current amplitude in the strong
disorder limit. For large enough impurity strength, the extended states are
almost unaffected by the impurities at the disordered layers and in that case
the current is carried only by these extended states in the ordered layer
which is the trivial limit. So the exciting limit is the intermediate limit of
$W$. In order to investigate the finite size effect on the electron transport,
we also calculate the typical current amplitude for the other two different
system sizes of the thin film those are plotted in Fig. 6 and
Figure 6: Typical current amplitudes ($I_{typ}$) as a function of the impurity
strength ($W$) for the thin films with seven layers ($N_{z}=7$), where the
system size of each layer is taken as: $N_{x}=3$ and $N_{y}=3$. The red and
the blue curves correspond to the identical meaning as in Fig. 2.
Fig. 7, respectively. In Fig. 6, we plot the typical current amplitudes for
the films with system size $N_{X}=3$, $N_{y}=3$ and $N_{z}=7$, while the
results for the films with system size $N_{x}=3$, $N_{y}=3$ and $N_{z}=8$ are
shown in Fig. 7. The red and the blue curves of these two figures correspond
to the identical meaning as in Fig. 2. Since both for these two films we will
get the similar behavior of the conductance and the density of states, we do
not describe these results further. The variation of the typical current
amplitudes for these films with the disorder strength shows quite similar
behavior as discussed earlier. But the significant point is that the typical
current amplitude where it goes to a minimum strongly depends on the system
size of the film which reveals the finite quantum size effects in the study of
electron transport phenomena. The underlying physics behind the location of
the minimum in the current versus disorder curve is very interesting. The
current amplitude is controlled by the two competing mechanisms. One is the
random scattering in the ordered
Figure 7: Typical current amplitudes ($I_{typ}$) as a function of the impurity
strength ($W$) for the thin films with eight layers ($N_{z}=8$), where the
system size of each layer is taken as: $N_{x}=3$ and $N_{y}=3$. The red and
the blue curves correspond to the identical meaning as in Fig. 2.
layer due to the localization in the disordered layers which tends to decrease
the current, and the other one is the vanishing influence of random scattering
in the ordered layer due to the strong localization in the disordered layers
which provides the enhancement of the current. Now depending on the ratio of
the atomic sites in the disordered region to the atomic sites in the ordered
region, the vanishing effect of random scattering from the ordered states
dominates over the non-vanishing effect of random scattering from these states
for a particular disorder strength $(W=W_{c})$, which provides the location of
the minimum in the current versus disorder curve.
## 4 Concluding Remarks
In conclusion, we have investigated a novel feature of disorder on electron
transport through a thin film of variable disorder strength from its surface
attached to two metallic electrodes by the Green’s function formalism. A
simple tight-binding model has been used to describe the film, where the
coupling to the electrodes has been treated through the use of Newns-Anderson
chemisorption theory. Our results have predicted that, in the smoothly varying
disordered film the typical current amplitude increases with the increase of
the disorder strength in the strong disorder regime, while the amplitude
decreases in the weak disorder regime. This behavior is completely opposite to
that of the conventional disordered film, where the current amplitude always
decreases with the disorder strength and such a strange phenomenon has not
been pointed out previously in the literature. In this context we have also
discussed the finite size effect on the electron transport by calculating the
typical current amplitude in different film sizes. From these results it has
been observed that, the typical current amplitude where it goes to a minimum
strongly depends on the size of the film which manifests the finite size
effect on the electron transport. Thus we can predict that, there exists a
strong correlation between the localized states at the disordered layers and
the extended states in the ordered layer which depends on the strength of the
disorder, and it provides a novel phenomenon in the transport community.
Similar type of anomalous quantum transport can also be observed in lower
dimensional systems like, edge disordered graphene sheets of single-atom-
thick, surface disordered finite width rings, nanowires, etc. Our study has
suggested that the carrier transport in an order-disorder separated mesoscopic
device may be tailored to desired properties through doping for different
applications.
Throughout our discussions we have used several realistic approximations by
neglecting the effects of the electron-electron interaction, all the inelastic
scattering processes, the Schottky effect, the static Stark effect, etc. More
studies are expected to take into account all these approximations for our
further investigations.
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|
arxiv-papers
| 2009-09-09T04:19:01 |
2024-09-04T02:49:05.166247
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Santanu K. Maiti",
"submitter": "Santanu Maiti Kumar",
"url": "https://arxiv.org/abs/0909.1619"
}
|
0909.1671
|
# Fluid Models of Many-server Queues with Abandonment
Jiheng Zhang
Department of Industrial Engineering and Logistic Management
The Hong Kong University of Science and Technology
jiheng@ust.hk
###### Abstract
We study many-server queues with abandonment in which customers have general
service and patience time distributions. The dynamics of the system are
modeled using measure-valued processes, to keep track of the residual service
and patience times of each customer. Deterministic fluid models are
established to provide first-order approximation for this model. The fluid
model solution, which is proved to uniquely exists, serves as the fluid limit
of the many-server queue, as the number of servers becomes large. Based on the
fluid model solution, first-order approximations for various performance
quantities are proposed.
_Key words and phrases:_ many-server queue, abandonment, measure valued
process, quality driven, efficiency driven, quality and efficiency driven.
## 1 Introduction
Recently, there has been a great interest in queues with a large number of
servers, motivated by applications to telephone call centers. Since a customer
can easily hang up after waiting for too long, abandonment is a non-negligible
aspect in the study of many-server queues. In our study, a customer can leave
the system (without getting service) once has been waiting in queue for more
than his patience time. Both patience and service times are modeled using
random variables. A recent statistical study by Brown et al. [2] suggests that
the exponential assumption on service time distribution, in many cases, is not
valid. In fact, the distribution of service times at call centers may be log-
normal in some cases as shown in [2]. This emphasizes the need to look at the
many-server model with generally distributed service and patience times.
In this paper, we study many-server queues with general patience and service
times. The queueing model is denoted by $G/GI/n$+$GI$. The $G$ represents a
general stationary arrival process. The first $GI$ indicates that service
times come from a sequences of independent and identically distributed (IID)
random variables with a general distribution. The $n$ denotes the number of
homogeneous servers. There is an unlimited waiting space, called buffer, where
customers wait and can choose to abandon if their patience times expires
before their service starts. Again, the patience times of each customer are
IID and with a general distribution (the $GI$ after the ‘+’ sign).
Useful insights can be obtained by considering a many-server queue in limit
regimes where the number $n$ of servers increases along with the arrival rate
$\lambda^{n}$ such that the traffic intensity
$\rho^{n}=\frac{\lambda^{n}}{n\mu}\to\rho\textrm{ as }n\to\infty,$
where $\mu$ is the service rate of a single server (in other words, the
reciprocal of the mean service time), and $\rho\in[0,\infty)$. Since the
abandonment ensures stability, the limit $\rho$ in the above need not to be
less than 1. In fact, according to $\rho$, the limit regimes can be divided
into three classes, i.e. _Efficiency-Driven_ (ED) regime when $\rho>1$,
_Quality-and-Efficiency-Driven_ (QED) regime when $\rho=1$ and _Quality-
Driven_ (QD) regime when $\rho<1$. The QED regime is also called _Halfin-
Whitt_ regime due to the seminal work Halfin and Whitt [11]. With this
motivation, we establish the fluid (also called law of large number) limit for
the $G/GI/n$+$GI$ queue in all the ED, QED and QD limit regimes.
We show that the fluid model has an equilibrium, which yields approximations
for various performance quantities. These fluid approximations work pretty
well in the ED and QD regime where $\rho$ is not that close to 1, as
demonstrated in the numerical experiments of Whitt [28]. However, when $\rho$
is very close (say within 5%) to 1, the fluid approximations lose their
accuracy and we shall look at a more refined limit, the diffusion limit, in
this case. Diffusion limit is not within the scope of the current paper.
One of the challenges in studying many-server queue with general service (as
well as patience) time is that Markovian analysis can not be used. In a system
where multiple customers are processed at the same time, such as the many-
server queue, how to describe the system becomes an important issue. The
number of customers in the system does not give much information since they
may all have large remaining service times or all have small remaining service
times, and this information can affect future evolution of the system. We
choose finite Borel measures on $(0,\infty)$ to describe the system. At any
time $t\geq 0$, instead of recording the total number of customers in service
(i.e. the number of busy servers), we record all the remaining patience times
using measure $\mathcal{Z}(t)$. For any Borel set $C\in(0,\infty)$,
$\mathcal{Z}(t)(C)$ indicates the number of customers in server with
_remaining service time_ belongs to $C$ at that time. Similar idea applies for
the remaining patience times. We first introduce the _virtual buffer_ , which
holds all the customers who have arrived but not yet scheduled to receive
service (assuming they are infinitely patient). We record all the remaining
patience times for those in the virtual buffer using finite Borel measure
$\mathcal{R}(t)$ on $\mathbb{R}=(-\infty,\infty)$. At time $t\geq 0$,
$\mathcal{R}(t)(C)$ indicates the number of customers in the virtual buffer
with _remaining patience time_ belongs to the Borel set $C$. The descriptor
$(\mathcal{R}(\cdot),\mathcal{Z}(\cdot))$ contains very rich information,
almost all information about the system can be recovered from it. Note that a
customer with negative remaining patience time has already abandoned. So the
actual number of customers in the buffer is
$Q(t)=\mathcal{R}(t)((0,\infty))\textrm{ for all }t\geq 0.$
More details will be discussed when we rigorously introduce the mathematical
model in Section 2. In literature, another descriptor that keeps track of the
ages of customers in service and the ages of customers in waiting have been
used, e.g. [15, 28]; The age proceses have the advantage of being observable,
without requiring future information, though their analysis is often more
complicated. Both age and residul descriptions of the system often results in
the same steady state insights. In this paper, we focus on residual processes
only.
The framework of using measure-valued process has been successfully applied to
study models where multiple customers are processed at the same time. Existing
works include Gromoll and Kruk [8], Gromoll, Puha and Williams [9] and
Gromoll, Robert and Zwart [10], to name a few. Most of these works are on the
processor sharing queue and related models where there is no waiting buffer.
Recently, Zhang, Dai and Zwart [31, 30] apply the measure-valued process to
study the limited processor sharing queue, where only limited number of
customers can be served at any time with extra customers waiting in a buffer.
Many techniques in this paper closely follows from those developed in [31].
There has been a huge literature on many-server queue and related models since
the seminal work by Halfin and Whitt [11]. But there are not many successes
with the case where the service time distribution is allowed to be non-
exponential. One exception is the work of Reed [25], in which fluid and
diffusion limits of the customer-count process of many server queues (without
abandonment) are established where few assumptions beyond a first moment are
placed on the service time distribution. Later, Puhalskii and Reed [23] extend
the aforementioned results to allow noncritical loading, generally distributed
service times, and general initial conditions. Jelenković et al. [13] study
the many-server queue with deterministic service times; Garmarnik and
Momčilović [6] study the model with lattice-valued service times; Puhalskii
and Reiman [24] study the model with phase-type service time distributions.
Mandelbaum and Momčilović [18] study the virtual waiting time processes, and
Kaspi and Ramanan [16] study the fluid limit of measure-valued processes for
many-server queues with general service times. For the many-server queue with
abandonment, a version of the fluid model have been established as a
conjecture in Whitt [27], where a lot of insight was demonstrated, which help
greatly in our work. Recently, Kang and Ramanan also worked on the same topic
and summarized their result in the technical report [15]. Although we focus on
the same topic, our work uses different methodology from that in [15] and
requires less assumptions on the service time distribution. In our work, the
only assumption on the service time distribution is continuity, while the
service time distribution in [15] is required to have a density and the hazard
rate function must be either bounded or lower lower-semicontinuous. From the
modeling aspect, our approach mainly based on tracking the “residual”
processes, while [15] tracks the “age” processes for studying the queueing
model. Also, we propose a quite simple fluid model, which facilitates the
analysis. The existence of solution to the fluid model in [15] is proved by
showing each fluid limit satisfies the fluid model equations. The current
paper proves the existence directly from the definition of the fluid model
without invoking fluid limits. In addition, we verify in the end of this paper
(c.f. Section 6) that our fluid model is consistent with the special case
where both service and patience times are exponentially distributed, as
established in Whitt [27] for the ED regime, Garnet et al. [7] for QED regime
and Pang and Whitt [21] and Puhalskii [22] for all three regimes. Additional
works on many-server queues with abandonment includes Dai, He and Tezcan [4]
for phase-type service time distributions and exponential patience time
distribution; Zeltyn and Mandelbaum [29] for exponential service time
distribution and general patience time distributions; Mandelbaum and
Momčilović [19] for both general service time distribution and general
patience time distribution. The difference between our work and [19] is that
we study the fluid limit of measure-valued processes in all three regimes, and
[19] studies the diffusion limit of customer count processes and virtual
waiting processes in the QED regime. By assuming a convenient initial
condition, [19] does not require a detailed fluid model analysis.
The paper is organized as follows: We begin in Section 2 by formulating the
mathematical model of the $G/GI/n$+$GI$ queue. The dynamics of the system are
clearly described by modeling with measure-valued processes; see (2.4) and
(2.5). The main results, including a characterization of the fluid model and
the convergence of the stochastic processes underlying the $G/GI/n$+$GI$ queue
to the fluid model solution are stated in Section 3. In Section 4, we explore
the fluid model and give proofs of all the results on the fluid model. Section
5 is devoted to establishing the convergence of stochastic processes, which
includes the proof of pre-compactness and the characterization of the limit as
the fluid model solution.
### 1.1 Notation
The following notation will be used throughout. Let $\mathbb{N}$, $\mathbb{Z}$
and $\mathbb{R}$ denote the set of natural numbers, integers and real numbers
respectively. Let $\mathbb{R}_{+}=[0,\infty)$. For $a,b\in\mathbb{R}$, write
$a^{+}$ for the positive part of $a$, $\lfloor{a}\rfloor$ for the integer
part, $\lceil{a}\rceil$ for $\lfloor{a}\rfloor+1$, $a\vee b$ for the maximum,
and $a\wedge b$ for the minimum. For any $A\subset\mathbb{R}$, denote
$\mathscr{B}(A)$ the collection of all Borel subsets which are subsets of $A$.
Let $\mathbf{M}$ denote the set of all non-negative finite Borel measures on
$\mathbb{R}$, and $\mathbf{M}_{+}$ denote the set of all non-negative finite
Borel measures on $(0,\infty)$. To simplify the notation, let us take the
convention that for any Borel set $A\subset\mathbb{R}$,
$\nu(A\cap(-\infty,0])=0$ for any $\nu\in\mathbf{M}_{+}$. Also, by this
convention, $\mathbf{M}_{+}$ is embedded as a subspace of $\mathbf{M}$. For
$\nu_{1},\nu_{2}\in\mathbf{M}$, the Prohorov metric is defined to be
$\begin{split}\mathbf{d}[\nu_{1},\nu_{2}]=\inf\Big{\\{}\epsilon>0:\nu_{1}(A)\leq\nu_{2}(A^{\epsilon})+\epsilon&\text{
and }\\\ \nu_{2}(A)\leq\nu_{1}(A^{\epsilon})+\epsilon&\text{ for all closed
Borel set }A\subset\mathbb{R}\Big{\\}},\end{split}$
where $A^{\epsilon}=\\{b\in\mathbb{R}:\inf_{a\in A}|a-b|<\epsilon\\}$. This is
the metric that induces the topology of weak convergence of finite Borel
measures. (See Section 6 in [1].) For any Borel measurable function
$g:\mathbb{R}\to\mathbb{R}$, the integration of this function with respect to
the measure $\nu\in\mathbf{M}$ is denoted by $\langle{g},{\nu}\rangle$.
Let $\mathbf{M}_{+}\times\mathbf{M}$ denote the Cartesian product. There are a
number of ways to define the metric on the product space. For convenience we
define the metric to be the maximum of the Prohorov metric between each
component. With a little abuse of notation, we still use $\mathbf{d}$ to
denote this metric.
Let $(\mathbf{E},\pi)$ be a general metric space. We consider the space
$\mathbf{D}$ of all right-continuous $\mathbf{E}$-valued functions with finite
left limits defined either on a finite interval $[0,T]$ or the infinite
interval $[0,\infty)$. We refer to the space as $\mathbf{D}([0,T],\mathbf{E})$
or $\mathbf{D}([0,\infty),\mathbf{E})$ depending upon the function domain. The
space $\mathbf{D}$ is also known as the space of càdlàg functions. For
$g(\cdot),g^{\prime}(\cdot)\in\mathbf{D}([0,T],\mathbf{E})$, the uniform
metric is defined as
$\upsilon_{T}[g,g^{\prime}]=\sup_{0\leq t\leq T}\pi[g(t),g^{\prime}(t)].$
(1.1)
However, a more useful metric we will use is the following Skorohod $J_{1}$
metric,
$\varrho_{T}[g,g^{\prime}]=\inf_{f\in\Lambda_{T}}(\|f\|^{\circ}_{T}\vee\upsilon_{T}[g,g^{\prime}\circ
f]),$ (1.2)
where $g\circ f(t)=g(f(t))$ for $t\geq 0$ and $\Lambda_{T}$ is the set of
strictly increasing and continuous mapping of $[0,T]$ onto itself and
$\|f\|^{\circ}_{T}=\sup_{0\leq s<t\leq
T}\big{|}\log\frac{f(t)-f(s)}{t-s}\big{|}.$
If $g(\cdot)$ and $g^{\prime}(\cdot)$ are in the space
$\mathbf{D}([0,\infty),\mathbf{E})$, the Skorohod $J_{1}$ metric is defined as
$\varrho[g,g^{\prime}]=\int_{0}^{\infty}e^{-T}(\varrho_{T}[g,g^{\prime}]\wedge
1)dT.$ (1.3)
By saying convergence in the space $\mathbf{D}$, we mean the convergence under
the Skorohod $J_{1}$ topology, which is the topology induced by the Skorohod
$J_{1}$ metric [5].
We use “$\to$” to denote the convergence in the metric space
$(\mathbf{E},\pi)$, and use “$\Rightarrow$” to denote the convergence in
distribution of random variables taking value in the metric space
$(\mathbf{E},\pi)$.
## 2 Stochastic Model
In this section, we first describe the $G/GI/n$+$GI$ queueing system and then
introduce a pair of measure-valued processes that capture the dynamics of the
system.
There are $n$ identical servers in the system. Customers arrive according to a
general stationary arrival process (the initial G) with arrival rate
$\lambda$. Let $a_{i}$ denote the arrival time of the $i$th arriving customer,
$i=1,2,\cdots$. An arriving customer enters service immediately upon arrival
if there is a server available. If all $n$ servers are busy, the arriving
customer waits in a buffer, which has infinite capacity. Customers are served
in the order of their arrival by the first available server. Waiting customers
may also elect to abandon. We assume that each customer has a random patience
time. A customer will abandon immediately when his waiting time in the buffer
exceeds his patience time. Once a customer starts his service, the customer
remains until the service is completed. There are no retrials; abandoning
customers leave without affecting future arrivals.
The two GIs in the notation mean that the service times and patience times
come from two independent sequences of iid random variables; these two
sequences are assumed to be independent of the arrival process. Let $u_{i}$
and $v_{i}$ denote the patience and service time of the $i$th arriving
customer, $i=1,2,\cdots$. In many applications such as telephone call centers,
customers cannot see the queue (the case of invisible queues, c.f. [20]), thus
do not know the experience of other customers. In such a case, it is natural
to assume that patience times are iid. Denote $F(\cdot)$ and $G(\cdot)$ the
distributions for the patience and service times, respectively.
To describe the system using measure-valued process, we first introduce the
notion of _virtual buffer_. The virtual buffer holds all customers in the real
buffer and some of the abandoned customers. An abandoned customer continues to
wait in the virtual buffer when he first abandons until it were his turn for
service had he not abandoned. At this time, he leaves the virtual buffer. At
any time $t\geq 0$, $\mathcal{R}(t)$ denotes a measure in $\mathbf{M}$ such
that $\mathcal{R}(t)(C)$ is the number of customers in the virtual buffer with
remaining patience time in $C\in\mathscr{B}(\mathbb{R})$. Please note that
this way of modeling requires $\mathcal{R}(\cdot)$ to be a measure on
$\mathbb{R}$, not just $(0,\infty)$. It is clear that
$Q(t)=\mathcal{R}(t)((0,\infty))\text{ and }R(t)=\mathcal{R}(t)(\mathbb{R})$
(2.1)
represent the number of customers waiting in the real buffer and number of
customers in the virtual buffer, respectively.
We also use a measure to describe the server. At any time $t\geq 0$,
$\mathcal{Z}(t)$ denotes a measure in $\mathbf{M}_{+}$ such that
$\mathcal{Z}(t)(C)$ is the number of customers in service with remaining
service time in $C\in\mathscr{B}((0,\infty))$. Different from the virtual
buffer, the servers only hold customers with positive remaining service times,
so we only care about the subsets in $(0,\infty)$. The quantity
$Z(t)=\mathcal{Z}(t)((0,\infty)),$ (2.2)
represents the number of customers in service at any time $t\geq 0$.
The measure-valued (taking value in $\mathbf{M}\times\mathbf{M}_{+}$)
stochastic process $(\mathcal{R}(\cdot),\mathcal{Z}(\cdot))$ serves as the
descriptor for the $G/GI/n$+$GI$ queueing model. Before we use it to describe
the dynamics of the system, let us first talk about the initial condition,
since the system is allowed to be non-empty initially. The initial state
specifies $R(0)$, the number of customers in the virtual buffer as well as
their remaining patience times $u_{i}$ and service times $v_{i}$,
$i=1-R(0),2-R(0),\cdots,0$. The initial state also specifies $Z(0)$, the
number of customers in service as well as their remaining service times
$v_{i}$, $i=1-R(0)-Z(0),\cdots,-R(0)$. Briefly, the initial customers are
given negative index, in order not to conflict with the index of arriving
customers. Those initial customers in the buffer are also assumed to have
i.i.d. service times with distribution $G(\cdot)$. For each $t\geq 0$, denote
$E(t)$ the number of customers that has arrived during time interval $(0,t]$.
Arriving customers are indexed by $1,2,\cdots$ according to the order of their
arrival. By this way of indexing customers, it is clear that the index of the
first customer in the virtual buffer at time $t\geq 0$ is $B(t)+1$, where
$B(t)=E(t)-R(t).$ (2.3)
Denote $w_{i}$ the waiting time of the $i$th customers; then
$\tau_{i}=a_{i}+w_{i}$ is the time when the $i$th job starts _service_ for all
$i\geq 1-R(0)$. For $i<0$, $a_{i}$ may be a negative number indicating how
long the $i$th customer had been there by time 0. We will impose some
conditions on $a_{i}$’s with $i<0$ later on. Let $\delta_{x}$ and
$\delta_{(x,y)}$ denote the Dirac point measure at $x\in\mathbb{R}$ and
$(x,y)\in\mathbb{R}^{2}$, respectively. Denote $C+x=\\{c+x:x\in C\\}$ for any
subset $C\subset\mathbb{R}$ and $C_{x}=(x,\infty)$. For any subsets
$C,C^{\prime}\subset\mathbb{R}$, let $C\times C^{\prime}$ denote the Cartesian
product. Using the Dirac measure and the above introduced notations, the
evolution of the system can be captured by the following _stochastic dynamic
equations_ :
$\displaystyle\mathcal{R}(t)(C)$
$\displaystyle=\sum_{i=1+B(t)}^{E(t)}\delta_{u_{i}}(C+t-a_{i}),\quad\textrm{for
all }C\in\mathscr{B}(\mathbb{R}),$ (2.4)
$\displaystyle\begin{split}\mathcal{Z}(t)(C)&=\sum_{i=1-R(0)-Z(0)}^{-R(0)}\delta_{v_{i}}(C+t)\\\
&\quad+\sum_{i=1-R(0)}^{B(t)}\delta_{(u_{i},v_{i})}(C_{0}+\tau_{i}-a_{i})\times(C+t-\tau_{i}),\end{split}\quad\textrm{for
all }C\in\mathscr{B}((0,\infty)),$ (2.5)
for all $t\geq 0$. Denote the total number of customers in the system by
$X(t)=Q(t)+Z(t)\quad\text{for all }t\geq 0.$
The following _policy constraints_ must be satisfied at any time $t\geq 0$,
$\displaystyle Q(t)$ $\displaystyle=(X(t)-n)^{+},$ (2.6) $\displaystyle Z(t)$
$\displaystyle=(X(t)\wedge n),$ (2.7)
where $n$, as introduced above, denotes the number of servers in the system.
## 3 Main Results
The main results of this paper contains two parts. The first part is a
characterization of the fluid model, including the existence and uniqueness of
the fluid model solution, and the equilibrium of the fluid model; these
results are summarized in Section 3.1. The second part is the convergence of
the stochastic processes to the fluid model solution; this result is stated in
Section 3.2.
### 3.1 Fluid Model
To study the stochastic model, we introduce a determinisitic fluid model. To
simplify notations, let $F^{c}(\cdot)$ denote the complement of the patience
time distribution $F(\cdot)$, i.e. $F^{c}(x)=1-F(x)$ for all $x\in\mathbb{R}$;
the complement of the service time distribution, denoted by $G^{c}(\cdot)$, is
defined in the same way. We introduce the following _fluid dynamic equations_
:
$\displaystyle\bar{\mathcal{R}}(t)(C_{x})$
$\displaystyle=\lambda\int_{t-\frac{\bar{R}(t)}{\lambda}}^{t}F^{c}(x+t-s)ds,\quad
t\geq 0,\quad x\in\mathbb{R},$ (3.1)
$\displaystyle\bar{\mathcal{Z}}(t)(C_{x})$
$\displaystyle=\bar{\mathcal{Z}}(0)(C_{x}+t)+\int_{0}^{t}F^{c}\left(\bar{R}(s)/\lambda\right)G^{c}(x+t-s)d\bar{B}(s),\quad
t\geq 0,\quad x\in(0,\infty),$ (3.2)
where $C_{x}=(x,\infty)$ and $\bar{B}(s)=\lambda s-\bar{R}(s)$. Here, all the
time dependent quanities are assumed to be right continuous on $[0,\infty)$
and to have left limits in $(0,\infty)$; furthermore, $\bar{B}(\cdot)$ is a
non-decreasing function, and the integral $\int_{0}^{t}g(s)\,d\bar{B}(s)$ is
interpreted as the Lebesgue-Stieltjes integral on the interval $(0,t]$. The
quantities $\bar{R}(\cdot)$, $\bar{Q}(\cdot)$, $\bar{Z}(\cdot)$ and
$\bar{X}(\cdot)$ are defined in the same way as their stochastic counterparts
in (2.1), (2.2) and (2). The following policy constraints must be satisfied
for all $t\geq 0$,
$\displaystyle\bar{Q}(t)$ $\displaystyle=(\bar{X}(t)-1)^{+},$ (3.3)
$\displaystyle\bar{Z}(t)$ $\displaystyle=(\bar{X}(t)\wedge 1).$ (3.4)
The fluid dynamic equations (3.1) and (3.2) and the policy constraints (3.3)
and (3.4) define a _fluid model_ , which is denoted by $(\lambda,F,G)$.
Denote
$(\bar{\mathcal{R}}_{0},\bar{\mathcal{Z}}_{0})=(\bar{\mathcal{R}}(0),\bar{\mathcal{Z}}(0))$
the initial condition of the fluid model. For the convenience of notations,
also denote $\bar{Q}_{0}=\bar{Q}(0)$, $\bar{Z}_{0}=\bar{Z}(0)$ and
$\bar{X}_{0}=\bar{Q}_{0}+\bar{Z}_{0}$. We need to require that the initial
condition satisfies the dynamic equations and the policy constraints, i.e.
$\displaystyle\bar{\mathcal{R}}_{0}(C_{x})$
$\displaystyle=\lambda\int_{0}^{\frac{\bar{R}_{0}}{\lambda}}F^{c}(x+s)ds,\quad
x\in\mathbb{R},$ (3.5) $\displaystyle\bar{Q}_{0}$
$\displaystyle=(\bar{X}_{0}-1)^{+},$ (3.6) $\displaystyle\bar{Z}_{0}$
$\displaystyle=(\bar{X}_{0}\wedge 1).$ (3.7)
We also require that
$\bar{\mathcal{Z}}_{0}(\\{0\\})=0,$ (3.8)
which means that nobody with remaining service time 0 stays in the server. We
call any element
$(\bar{\mathcal{R}}_{0},\bar{\mathcal{Z}}_{0})\in\mathbf{M}\times\mathbf{M}_{+}$
a _valid_ initial condition if it satisfies (3.5)–(3.8).
We call
$(\bar{\mathcal{R}}(\cdot),\bar{\mathcal{Z}}(\cdot))\in\mathbf{D}([0,\infty),\mathbf{M}\times\mathbf{M}_{+})$
a solution to the fluid model $(\lambda,F,G)$ with a valid initial condition
$(\bar{\mathcal{R}}_{0},\bar{\mathcal{Z}}_{0})$ if it satisfies the fluid
dynamic equations (3.1) and (3.2) and the policy constraints (3.3) and (3.4).
Denote $\mu$ the reciprocal of first moment of the service time distribution
$G(\cdot)$. Let
$M_{F}=\inf\\{x\geq 0:F(x)=1\\}.$ (3.9)
By the right continuity, it is clear that $F(x)<1$ for all $x<M_{F}$ and
$F(x)=1$ for all $x\geq M_{F}$. If the patience time distribution $F(\cdot)$
has a density $f(\cdot)$, then define the hazard rate $h_{F}(\cdot)$ of the
distribution $F(\cdot)$ by
$h_{F}(x)=\left\\{\begin{array}[]{ll}\frac{f(x)}{1-F(x)}&x<M_{F},\\\ 0&x\geq
M_{F}.\end{array}\right.$
###### Theorem 3.1 (Existence and Uniqueness).
Assume the service time distribution satisfies both that
$G(\cdot)\textrm{ is continuous,}$ (3.10)
and that
$0<\mu<\infty.$ (3.11)
Assume the patience time distribution satisfies either that
$F(\cdot)\textrm{ is Lipschitz continuous},$ (3.12)
or that $F(\cdot)$ has a density $f(\cdot)$ such that the hazard rate is
bounded, i.e.
$\sup_{x\in[0,\infty)}h_{F}(x)<\infty.$ (3.13)
There exists a unique solution to the fluid model $(\lambda,F,G)$ for any
valid initial condition $(\bar{\mathcal{R}}_{0},\bar{\mathcal{Z}}_{0})$.
The above theorem provides the foundation to further study the fluid model. A
key property is that the fluid model has an equilibrium state. An equilibrium
state is defined as the following:
###### Definition 3.1.
An element
$(\bar{\mathcal{R}}_{\infty},\bar{\mathcal{Z}}_{\infty})\in\mathbf{M}\times\mathbf{M}_{+}$
is called an _equilibrium state_ for the fluid model $(\lambda,F,G)$ if the
solution to the fluid model with initial condition
$(\bar{\mathcal{R}}_{\infty},\bar{\mathcal{Z}}_{\infty})$ satisfies
$(\bar{\mathcal{R}}(t),\bar{\mathcal{Z}}(t))=(\bar{\mathcal{R}}_{\infty},\bar{\mathcal{Z}}_{\infty})\quad\textrm{for
all }t\geq 0.$
This definition says that if a fluid model solution starts from an equilibrium
state, it will never change in the future. To present the result about
equilibrium state, we need to introduce some more notation. For the service
time distribution function $G(\cdot)$ on $\mathbb{R}_{+}$, the associated
_equilibrium_ distribution is given by
$G_{e}(x)=\mu\int_{0}^{x}G^{c}(y)dy,\quad\textrm{for all }x\geq 0.$
###### Theorem 3.2.
Assume the conditions in Theorem 3.1. The state
$(\bar{\mathcal{R}}_{\infty},\bar{\mathcal{Z}}_{\infty})$ is an equilibrium
state of the fluid model $(\lambda,F,G)$ if and only if it satisfies
$\displaystyle\bar{\mathcal{R}}_{\infty}(C_{x})$
$\displaystyle=\lambda\int_{0}^{w}F^{c}(x+s)ds,\quad x\in\mathbb{R},$ (3.14)
$\displaystyle\bar{\mathcal{Z}}_{\infty}(C_{x})$
$\displaystyle=\min\left(\rho,1\right)[1-G_{e}(x)],\quad x\in(0,\infty),$
(3.15)
where $w$ is a solution to the equation
$F(w)=\max\left(\frac{\rho-1}{\rho},0\right).$ (3.16)
###### Remark 3.1.
If equation (3.16) has multiple solutions, then the equilibrium is not unique
(any solution $w$ gives an equilibrium). If the equation has a unique solution
(for example when $F(\cdot)$ is strictly increasing), then the equilibrium
state is unique.
The quantity $w$ is interpreted to be the _offered_ waiting time for an
arriving customer. If his patience time exceeds $w$, he will not abandon.
Thus, the probabilty of his abandonment is given by $F(w)$, which is equal to
$(\rho-1)/\rho$ when $\rho>1$; the latter quantity is the fraction of traffic
that has to be discarded due to the overloading. From (3.14),
$\bar{\mathcal{R}}_{\infty}(C_{x})=\lambda w$ for $x\leq-w$. Thus, the average
number of customers in the virtual buffer is
$\bar{R}_{\infty}=\bar{\mathcal{R}}_{\infty}(\mathbb{R})=\lambda w,$
which is consistent with the Little’s law. From (3.15), the average number of
busy servers is
$\bar{Z}_{\infty}=\bar{\mathcal{Z}}_{\infty}((0,\infty))=\min(\rho,1),$
which is intuitively clear. These observations and interpretations were first
made by Whitt [28], where approximation formulas based on a conjectured fluid
model were also given, and were compared with extensive simulation results.
The approximation formulas derived from our fluid model is consistent with
those formulas in Whitt [28].
### 3.2 Convergence of Stochastic Models
We consider a sequence of queueing systems indexed by the number of servers
$n$, with $n\to\infty$. Each model is defined in the same way as in Section 2.
The arrival rate of each model is assumed be to proportional to $n$. To
distinguish models with different indices, quantities of the $n$th model are
accompanied with superscript $n$. Each model may be defined on a different
probability space $(\Omega^{n},\mathcal{F}^{n},\mathbb{P}^{n})$. Our results
concern the asymptotic behavior of the descriptors under the _fluid_ scaling,
which is defined by
$\bar{\mathcal{R}}^{n}(t)=\frac{1}{n}\mathcal{R}^{n}(t),\quad\bar{\mathcal{Z}}^{n}(t)=\frac{1}{n}\mathcal{Z}^{n}(t),$
(3.17)
for all $t\geq 0$. The fluid scaling for the arrival process $E^{n}(\cdot)$ is
defined in the same way, i.e.
$\bar{E}^{n}(t)=\frac{1}{n}E^{n}(t),$
for all $t\geq 0$. We assume that
$\bar{E}^{n}(\cdot)\Rightarrow\lambda\cdot\quad\text{as }n\to\infty.$ (3.18)
Since the limit is deterministic, the convergence in distribution in (3.18) is
equivalent to convergence in probability; namely, for each $T>0$ and each
$\epsilon>0$,
$\lim_{n\to\infty}\mathbb{P}^{n}\Bigl{(}\sup_{0\leq t\leq
T}|\bar{E}^{n}(t)-\lambda t|>\epsilon\Bigr{)}=0.$
Denote $\nu^{n}_{F}$ and $\nu^{n}_{G}$ the probability measures corresponding
to the patience time distribution $F^{n}$ and the service time distribution
$G^{n}$, respectively. Assume that as $n\to\infty$,
$\nu^{n}_{F}\to\nu_{F},\quad\nu^{n}_{G}\to\nu_{G},$ (3.19)
where $\nu_{F}$ and $\nu_{G}$ are some probability measures with associated
distribution functions $F$ and $G$. Also, the following initial condition will
be assumed:
$\displaystyle(\bar{\mathcal{R}}^{n}(0),\bar{\mathcal{Z}}^{n}(0))$
$\displaystyle\Rightarrow(\bar{\mathcal{R}}_{0},\bar{\mathcal{Z}}_{0})\quad\textrm{as
}n\to\infty,$ (3.20)
where, almost surely, $(\bar{\mathcal{R}}_{0},\bar{\mathcal{Z}}_{0})$ is a
valid initial condition and
$\displaystyle\bar{\mathcal{R}}_{0}\textrm{ and }\bar{\mathcal{Z}}_{0}\textrm{
has no atoms}.$ (3.21)
###### Theorem 3.3.
In addition to the assumptions (3.10)–(3.13) in Theorem 3.1, if the sequence
of many-server queues satisfies (3.18)–(3.21), then
$(\bar{\mathcal{R}}^{n}(\cdot),\bar{\mathcal{Z}}^{n}(\cdot))\Rightarrow(\bar{\mathcal{R}}(\cdot),\bar{\mathcal{Z}}(\cdot))\quad\textrm{
as }n\to\infty,$
where, almost surely, $(\bar{\mathcal{R}}(\cdot),\bar{\mathcal{Z}}(\cdot))$ is
the unique solution to the fluid model $(\lambda,F,G)$ with initial condition
$(\bar{\mathcal{R}}_{0},\bar{\mathcal{Z}}_{0})$.
## 4 Properties of the Fluid Model
In this section, we analyze the proposed fluid model and establish some basic
properties of the fluid model solution. The proof of Theorem 3.1 for existence
and uniqueness and the proof of Theorem 3.2 for characterization of the
equilibrium will be presented in Section 4.1 and Section 4.2, respectively.
### 4.1 Existence and Uniqueness of Fluid Model Solutions
We first present some calculus on the fluid dynamic equations (3.1) and (3.2),
which define the fluid model. It follows from (3.1) that
$\displaystyle\bar{Q}(t)=\bar{\mathcal{R}}(t)(C_{0})=\lambda\int_{t-\frac{\bar{R}(t)}{\lambda}}^{t}F^{c}(t-s)ds=\lambda\int_{0}^{\frac{\bar{R}(t)}{\lambda}}F^{c}(s)ds.$
Let
$F_{d}(x)=\int_{0}^{x}[1-F(y)]dy\quad\textrm{for all }x\geq 0.$
Please note that the density of $F_{d}(\cdot)$ is not scaled by the mean of
$F(\cdot)$. Thus, this is not exactly the equilibrium distribution associated
with $F(\cdot)$. In fact, we do not need the mean
$N_{F}=\int_{0}^{\infty}[1-F(y)]dy$ (4.1)
to be finite. Now we have
$\frac{\bar{Q}(t)}{\lambda}=F_{d}(\frac{\bar{R}(t)}{\lambda}).$ (4.2)
It follows from (3.2) that
$\displaystyle\bar{Z}(t)=\bar{\mathcal{Z}}(t)(C_{0})$
$\displaystyle=\bar{\mathcal{Z}}_{0}(C_{0}+t)+\lambda\int_{0}^{t}F^{c}(\frac{\bar{R}(s)}{\lambda})G^{c}(t-s)ds$
$\displaystyle\quad-\int_{0}^{t}F^{c}(\frac{\bar{R}(s)}{\lambda})G^{c}(t-s)d\bar{R}(s).$
Note that by (4.2),
$d\bar{Q}(s)=F^{c}(\frac{\bar{R}(s)}{\lambda})d\bar{R}(s)$. So
$\displaystyle\bar{Z}(t)$
$\displaystyle=\bar{\mathcal{Z}}_{0}(C_{0}+t)+\frac{\lambda}{\mu}\int_{0}^{t}F^{c}(\frac{\bar{R}(s)}{\lambda})dG_{e}(t-s)-\int_{0}^{t}G^{c}(t-s)d\bar{Q}(s).$
Performing change of variable and integration by parts, we have
$\begin{split}\bar{Z}(t)&=\bar{\mathcal{Z}}_{0}(C_{t})+\frac{\lambda}{\mu}\int_{0}^{t}F^{c}(\frac{\bar{R}(t-s)}{\lambda})dG_{e}(s)\\\
&\quad-\bar{Q}(t)G^{c}(0)+\bar{Q}(0)G^{c}(t)+\int_{0}^{t}\bar{Q}(t-s)dG(s).\end{split}$
(4.3)
We wish to represent the term $F^{c}(\frac{\bar{R}(\cdot)}{\lambda})$ using
$\bar{Q}(\cdot)$. Recall $M_{F}$ and $N_{F}$, which are defined in (3.9) and
(4.1), respectively. It is clear that $F_{d}(x)$ is strictly monotone for
$x\in[0,M_{F})$. Thus, $F_{d}^{-1}(y)$ is well defined for each
$y\in[0,N_{F})$. We define $F_{d}^{-1}(y)=M_{F}$ for all $y\geq N_{F}$. Thus,
(4.2) implies that
$F^{c}(\frac{\bar{R}(t)}{\lambda})=F^{c}\left(F_{d}^{-1}(\frac{\bar{Q}(t)}{\lambda})\right).$
(4.4)
Note that $G^{c}(0)=1$ by assumption (3.10). Combining (3.3), (3.4), (4.3),
and (4.4), we obtain
$\begin{split}\bar{X}(t)&=\bar{\mathcal{Z}}_{0}(C_{t})+\bar{Q}_{0}G^{c}(t)\\\
&\quad+\frac{\lambda}{\mu}\int_{0}^{t}F^{c}\Big{(}F_{d}^{-1}(\frac{(\bar{X}(t-s)-1)^{+}}{\lambda})\Big{)}dG_{e}(s)\\\
&\quad+\int_{0}^{t}(\bar{X}(t-s)-1)^{+}dG(s).\end{split}$
Now, introduce
$H(x)=\begin{cases}F^{c}(F_{d}^{-1}(\frac{x}{\lambda}))&\text{ if }0\leq
x<\lambda,\\\ 0&\text{ if }x\geq\lambda,\end{cases}$
and
$\zeta_{0}(\cdot)=\bar{\mathcal{Z}}_{0}(C_{0}+\cdot)+\bar{Q}_{0}G^{c}(\cdot)$.
It then follows that
$\begin{split}\bar{X}(t)&=\zeta_{0}(t)+\rho\int_{0}^{t}H\big{(}(\bar{X}(t-s)-1)^{+}\big{)}dG_{e}(s)+\int_{0}^{t}(\bar{X}(t-s)-1)^{+}dG(s).\end{split}$
(4.5)
Please note that $\zeta_{0}(\cdot)$ depends only on the initial condition and
$H(\cdot)$ is a function defined by the arrival rate $\lambda$ and the
patience time distribution $F(\cdot)$. The equation (4.5) serves as a key to
the analysis of the fluid model.
###### Proof of Theorem 3.1.
We first prove the existence. Given a valid initial condition
$(\bar{\mathcal{R}}_{0},\bar{\mathcal{Z}}_{0})$ (i.e. an element in
$\mathbf{M}\times\mathbf{M}_{+}$ that satisfies (3.5)– (3.8)), we now
construct a solution $(\bar{\mathcal{R}}(\cdot),\bar{\mathcal{Z}}(\cdot))$ to
the fluid model $(\lambda,F,G)$ with this initial condition. If the patience
time distribution $F(\cdot)$ is Lipschitz continuous, then it is clear that
$H(\cdot)$ is also Lipschitz continuous; if $F(\cdot)$ has a density, then the
function $H(\cdot)$ is differentiable and has derivative
$H^{\prime}(x)=-f(y)\frac{1}{F_{d}^{\prime}(y)}=\frac{-f(y)}{1-F(y)}=-h_{F}(y),$
on the interval $(0,\lambda N_{F})$ if $y=F_{d}^{-1}(x)$ and $H(x)=0$ for all
$x\geq\lambda N_{F}$. By condition (3.13),
$\sup_{0<x<\lambda N_{F}}|H^{\prime}(x)|=\sup_{y\in[0,M_{F})}h_{F}(y),$
which implies that $H(\cdot)$ is Lipschitz continuous. It follows from Lemma
A.1 that the equation (4.5) has a unique solution $\bar{X}(\cdot)$. Denote
$\bar{Q}(t)=(\bar{X}(t)-1)^{+}$. We first claim that $\bar{Q}(t)/\lambda\leq
N_{F}$ for all $t\geq 0$. The claim is automatically true if $N_{F}=\infty$.
Now, let us consider the case where $N_{F}<\infty$. Since
$(\bar{\mathcal{R}}_{0},\bar{\mathcal{Z}}_{0})$ is a valid initial condition,
$\bar{Q}(0)/\lambda\leq N_{F}$. Suppose there exists $t_{1}>0$ such that
$\bar{Q}(t_{1})/\lambda>N_{F}$. Let $t_{0}=\sup\\{s:\bar{Q}(s)/\lambda\leq
N_{F},s\leq t_{1}\\}$. So we have that $\lim_{t\to
t_{0}}\bar{Q}(t)/\lambda\leq N_{F}$, since $\bar{Q}(\cdot)$ has left limit.
Let $\delta=(Q(t_{1})/\lambda-N_{F})/4$ and pick
$t_{\delta}\in[t_{0}-\delta,t_{0}]$ such that $\bar{Q}(t_{\delta})/\lambda\leq
N_{F}+\delta$. By Lemma A.2,
$\frac{\bar{Q}(t^{\prime})}{\lambda}-\frac{\bar{Q}(t)}{\lambda}\leq\int_{t}^{t^{\prime}}F^{c}(F_{d}^{-1}(\frac{\bar{Q}(s)}{\lambda}))ds$
(4.6)
for any $t<t^{\prime}$. This gives that
$\displaystyle\frac{\bar{Q}(t_{1})}{\lambda}$
$\displaystyle\leq\frac{\bar{Q}(t_{\delta})}{\lambda}+\int_{t_{\delta}}^{t_{1}}[1-F(F_{d}^{-1}(\frac{\bar{Q}(s)}{\lambda}))]ds$
$\displaystyle\leq
N_{F}+\delta+\int_{t_{\delta}}^{t_{0}}1ds+\int_{t_{0}}^{t_{1}}0ds$
$\displaystyle\leq N_{F}+2\delta<\frac{\bar{Q}(t_{1})}{\lambda},$
which is a contradiction. This proves the claim. Let
$\displaystyle\bar{Z}(t)$ $\displaystyle=\min(\bar{X}(t),1),$
$\displaystyle\bar{R}(t)$ $\displaystyle=\lambda
F_{d}^{-1}(\frac{\bar{Q}(t)}{\lambda}),$ $\displaystyle\bar{B}(t)$
$\displaystyle=\lambda t-\bar{R}(t),$
for all $t\geq 0$. Next, we claim that the process $\bar{B}(\cdot)$ is non-
decreasing. To prove this claim, it is enough show that
$F_{d}^{-1}(\frac{\bar{Q}(t^{\prime})}{\lambda})-F_{d}^{-1}(\frac{\bar{Q}(t)}{\lambda})\leq
t^{\prime}-t$ (4.7)
for any $t\leq t^{\prime}$. Since $F_{d}^{-1}$ is a non-decreasing function,
the inequality holds trivially when $\bar{Q}(t^{\prime})\leq\bar{Q}(t)$. We
now focus on the case where $\bar{Q}(t^{\prime})>\bar{Q}(t)$. Note that the
function $F_{d}^{-1}(\cdot)$ is convex, since the derivative is non-
decreasing. This together with (4.6) implies that
$F_{d}^{-1}(\frac{\bar{Q}(t^{\prime})}{\lambda})\leq
F_{d}^{-1}(\frac{\bar{Q}(t)}{\lambda})+{F_{d}^{-1}}^{\prime}(\frac{\bar{Q}(t)}{\lambda})\int_{t}^{t^{\prime}}F^{c}(F_{d}^{-1}(\frac{\bar{Q}(s)}{\lambda}))ds.$
If $\bar{Q}(t)\leq\bar{Q}(s)$ for all $s\in[t,t^{\prime}]$, then due to the
fact that $F^{c}(F_{d}^{-1}(\cdot))$ is non-increasing, we have
$\displaystyle F_{d}^{-1}(\frac{\bar{Q}(t^{\prime})}{\lambda})$
$\displaystyle\leq
F_{d}^{-1}(\frac{\bar{Q}(t)}{\lambda})+\frac{1}{F^{c}(F_{d}^{-1}(\frac{\bar{Q}(t)}{\lambda}))}F^{c}(F_{d}^{-1}(\frac{\bar{Q}(t)}{\lambda}))(t^{\prime}-t)$
$\displaystyle\leq F_{d}^{-1}(\frac{\bar{Q}(t)}{\lambda})+t^{\prime}-t,$
which gives (4.7); otherwise, let $t^{*}\in(t,t^{\prime})$ be the point where
$\bar{Q}(\cdot)$ achieves minimum. Since $\bar{Q}(t)>\bar{Q}(t^{*})$, we have
$F_{d}^{-1}(\frac{\bar{Q}(t^{*})}{\lambda})-F_{d}^{-1}(\frac{\bar{Q}(t)}{\lambda})\leq
t^{*}-t.$
Since $\bar{Q}(t^{*})\leq\bar{Q}(s)$ for all $s\in[t^{*},t^{\prime}]$, by the
same reasoning in the above, we also have
$F_{d}^{-1}(\frac{\bar{Q}(t^{\prime})}{\lambda}\leq
F_{d}^{-1}(\frac{\bar{Q}(t^{*})}{\lambda})+t^{\prime}-t^{*}.$
The above two inequalities also leads to (4.7). So the claim is proved. We now
construct a fluid model solution by letting
$\displaystyle\bar{\mathcal{R}}(t)(C_{x})$
$\displaystyle=\lambda\int_{t-\frac{\bar{R}(t)}{\lambda}}^{t}F^{c}(x+t-s)ds,$
$\displaystyle\bar{\mathcal{Z}}(t)(C_{x})$
$\displaystyle=\bar{\mathcal{Z}}_{0}(C_{x}+t)+\int_{0}^{t}F^{c}(\frac{\bar{R}(s)}{\lambda})G^{c}(x+t-s)d\bar{B}(s),$
for all $t\geq 0$. It is clear that the above defined
$(\bar{\mathcal{R}}(\cdot),\bar{\mathcal{Z}}(\cdot))$ satisfies the fluid
dynamic equations (3.1) and (3.2) and constraints (3.3) and (3.4). So we
conclude that $(\bar{\mathcal{R}}(\cdot),\bar{\mathcal{Z}}(\cdot))$ is a fluid
model solution.
It now remains to show the uniqueness. Suppose there is another solution to
the fluid model $(\lambda,F,G)$ with initial condition
$(\bar{\mathcal{R}}_{0},\bar{\mathcal{Z}}_{0})$, denoted by
$(\bar{\mathcal{R}}^{\dagger}(\cdot),\bar{\mathcal{Z}}^{\dagger}(\cdot))$.
Similarly, denote
$\displaystyle\bar{R}^{\dagger}(t)$
$\displaystyle=\bar{\mathcal{R}}^{\dagger}(\mathbb{R}),$
$\displaystyle\bar{Z}^{\dagger}(t)$
$\displaystyle=\bar{\mathcal{Z}}^{\dagger}((0,\infty)),$
for all $t\geq 0$. It must satisfy the fluid dynamic equations (3.1) and (3.2)
and constraints (3.3) and (3.4). For all $t\geq 0$, let
$\bar{Q}^{\dagger}(t)=\lambda\bar{F}_{d}(\frac{\bar{R}^{\dagger}(t)}{\lambda}).$
According to the algebra at the beginning of Section 4.1,
$\bar{X}^{\dagger}(\cdot)$ must also satisfy equation (4.5). By the uniqueness
of the solution to the equation (4.5) in Lemma A.1,
$\bar{X}^{\dagger}(t)=\bar{X}(t)\quad\text{for all }t\geq 0.$
This implies that $\bar{R}^{\dagger}(t)=\bar{R}(t)$. By the dynamic equations
(3.1) and (3.2), we must have that
$(\bar{\mathcal{R}}^{\dagger}(t),\bar{\mathcal{Z}}^{\dagger}(t))=(\bar{\mathcal{R}}(t),\bar{\mathcal{Z}}(t))\quad\text{for
all }t\geq 0.$
This completes the proof. ∎
### 4.2 Equilibrium State of the Fluid Model Solution
In this section, we first intuitively explain what an equilibrium should be.
Then we rigorously prove it in Theorem 3.2. To provide some intuition, note
that in the equilibrium, by equation (3.1), one should have
$\bar{\mathcal{R}}_{\infty}(C_{x})={\lambda}\int_{0}^{\bar{R}_{\infty}/\lambda}F^{c}(x+s)ds,$
for the buffer. This immediately implies that
$\bar{\mathcal{R}}_{\infty}(C_{x})=\lambda[F_{d}(x+\frac{\bar{R}_{\infty}}{\lambda})-F_{d}(x)].$
So the rate at which customers leave the buffer due to abandonment is:
$\displaystyle\lim_{x\to
0}\frac{\bar{\mathcal{R}}_{\infty}(C_{0})-\bar{\mathcal{R}}_{\infty}(C_{x})}{x}=\lambda
F(\frac{\bar{R}_{\infty}}{\lambda}).$
In the equilibrium, intuitively, the number of customers in service should not
change and the distribution for the remaining service time should be the
equilibrium distribution $G_{e}(\cdot)$, i.e.
$\bar{\mathcal{Z}}_{\infty}(C_{x})=\bar{Z}_{\infty}[1-G_{e}(x)].$
The rate at which customers depart from the server is:
$\displaystyle\lim_{x\to
0}\frac{\bar{\mathcal{Z}}_{\infty}(C_{0})-\bar{\mathcal{Z}}_{\infty}(C_{x})}{x}=\bar{Z}_{\infty}\mu.$
The arrival rate must be equal to the summation of the departure rate from
server (due to service completion) and the one from buffer (due to
abandonment), i.e.
$\lambda=\lambda F(\frac{\bar{R}_{\infty}}{\lambda})+\bar{Z}_{\infty}\mu.$
(4.8)
It follows directly from (4.2) that
$\bar{Q}_{\infty}=\lambda F_{d}(\frac{\bar{R}_{\infty}}{\lambda}).$ (4.9)
If $\bar{R}_{\infty}>0$, then according to (4.9) we have $\bar{Q}_{\infty}>0$.
Thus $\bar{Z}_{\infty}=1$ according to policy constraints. By (4.8), $\rho>1$
and $\frac{\bar{R}_{\infty}}{\lambda}$ is a solution to the equation
$F(w)=\frac{\rho-1}{\rho}$. If $\bar{R}_{\infty}=0$, then according to (4.8)
we have $\rho=\bar{Z}_{\infty}\leq 1$. In summary, we have that
$\displaystyle\bar{Q}_{\infty}$ $\displaystyle=\lambda F_{d}(w),$
$\displaystyle\bar{Z}_{\infty}$ $\displaystyle=\min(\rho,1),$
where $w$ is a solution to the equation $F(w)=\max(\frac{\rho-1}{\rho},0)$.
This is consistent with the one in [28], which is derived from a conjecture of
a fluid model. Now, we rigorously prove this result.
###### Proof of Theorem 3.2.
If $(\bar{\mathcal{R}}_{\infty},\bar{\mathcal{Z}}_{\infty})$ is an equilibrium
state, then according to the definition, it must satisfies
$\displaystyle\bar{\mathcal{R}}_{\infty}(C_{x})$
$\displaystyle=\lambda\int_{t-\frac{\bar{R}_{\infty}}{\lambda}}^{t}F^{c}(x+t-s)ds,\quad
t\geq 0,$ (4.10) $\displaystyle\bar{\mathcal{Z}}_{\infty}(C_{x})$
$\displaystyle=\bar{\mathcal{Z}}_{\infty}(C_{x}+t)+\int_{0}^{t}F^{c}(\frac{\bar{R}_{\infty}}{\lambda})G^{c}(x+t-s)d\lambda
s,\quad t\geq 0.$ (4.11)
It follows from (4.11) that
$\displaystyle\bar{\mathcal{Z}}_{\infty}(C_{x})-\bar{\mathcal{Z}}_{\infty}(C_{x}+t)$
$\displaystyle=\rho
F^{c}(\frac{\bar{R}_{\infty}}{\lambda})\mu\int_{0}^{t}G^{c}(x+t-s)ds$
$\displaystyle=\rho
F^{c}(\frac{\bar{R}_{\infty}}{\lambda})[G_{e}(x+t)-G_{e}(x)],\quad t\geq 0.$
Taking $t\to\infty$, one has
$\bar{\mathcal{Z}}_{\infty}(C_{x})=\rho
F^{c}(\frac{\bar{R}_{\infty}}{\lambda})G_{e}^{c}(x).$ (4.12)
Thus $\bar{Z}_{\infty}=\rho F^{c}(\frac{\bar{R}_{\infty}}{\lambda})$.
According to (4.2), we have that
$\bar{Q}_{\infty}=\lambda F_{d}(\frac{\bar{R}_{\infty}}{\lambda}).$
First assume that $\bar{R}_{\infty}>0$. Then $\bar{Q}_{\infty}>0$, and thus
$\bar{Z}_{\infty}=1$ by the policy constraints (3.3) and (3.4). Therefore,
$\rho F^{c}(\frac{\bar{R}_{\infty}}{\lambda})=1$, which implies that
$F(\frac{\bar{R}_{\infty}}{\lambda})=\frac{\rho-1}{\rho}$ and $\rho>1$. Now
assume that $\bar{R}_{\infty}=0$. Then $\bar{Z}_{\infty}=\rho$, which must be
less than or equal to 1 by the policy constraints. Summarizing the cases where
$\rho>1$ and $\rho\leq 1$, we have that the equilibrium state must satisfy
(3.14)–(3.16).
If a state $(\bar{\mathcal{R}}_{\infty},\bar{\mathcal{Z}}_{\infty})$ satisfies
(3.14)–(3.16), then let
$\displaystyle(\bar{\mathcal{R}}(t),\bar{\mathcal{Z}}(t))=(\bar{\mathcal{R}}_{\infty},\bar{\mathcal{Z}}_{\infty}),$
for all $t\geq 0$. If $\rho\leq 1$, then
$\bar{\mathcal{R}}(\cdot)\equiv\mathbf{0}$ and $\bar{Z}(\cdot)\equiv\rho$; if
$\rho>1$, then $\bar{R}(\cdot)\equiv\lambda w$ and $\bar{Z}(\cdot)\equiv 1$,
where $w$ is a solution to equation (3.16). It is easy to check that
$(\bar{\mathcal{R}}(\cdot),\bar{\mathcal{Z}}(\cdot))$ is a fluid model
solution in both cases. So by definition, the state
$(\bar{\mathcal{R}}_{\infty},\bar{\mathcal{Z}}_{\infty})$ is a equilibrium
state. ∎
## 5 Fluid Approximation of the Stochastic Models
Similar to (2.3), let
$B^{n}(t)=E^{n}(t)-R^{n}(t).$ (5.1)
It follows from (2.4) and (2.5) that the dynamics for the fluid scaled
processes can be written as
$\displaystyle\bar{\mathcal{R}}^{n}(t)(C)$
$\displaystyle=\frac{1}{n}\sum_{i=B^{n}(t)+1}^{E^{n}(t)}\delta_{u^{n}_{i}}(C+t-a^{n}_{i}),\quad\textrm{for
all }C\in\mathscr{B}(\mathbb{R}),$ (5.2)
$\displaystyle\begin{split}\bar{\mathcal{Z}}^{n}(t)(C)&=\bar{\mathcal{Z}}^{n}(s)(C+t-s)\\\
&\quad+\frac{1}{n}\sum_{i=B^{n}(s)+1}^{B^{n}(t)}\delta_{(u^{n}_{i},v^{n}_{i})}(C_{0}+\tau^{n}_{i}-a^{n}_{i})\times(C+t-\tau^{n}_{i}),\end{split}\quad\textrm{for
all }C\in\mathscr{B}((0,\infty)),$ (5.3)
for all $0\leq s\leq t$.
### 5.1 Precompactness
We first establish the following precompactness for the sequence of fluid
scaled stochastic processes
$\\{(\bar{\mathcal{R}}^{n}(\cdot),\bar{\mathcal{Z}}^{n}(\cdot))\\}$.
###### Theorem 5.1.
Assume (3.18)–(3.21). The sequence of the fluid scaled stochastic processes
$\\{(\bar{\mathcal{R}}^{n}(\cdot),\bar{\mathcal{Z}}^{n}(\cdot))\\}_{N\in\mathbb{N}}$
is precompact as $n\to\infty$; namely, for each subsequence
$\\{(\bar{\mathcal{R}}^{n_{k}}(\cdot),\bar{\mathcal{Z}}^{n_{k}}(\cdot))\\}_{n_{k}}$
with $n_{k}\to\infty$, there exists a further subsequence
$\\{(\bar{\mathcal{R}}^{n_{k_{j}}}(\cdot),\bar{\mathcal{Z}}^{n_{k_{j}}}(\cdot))\\}_{n_{k_{j}}}$
such that
$(\bar{\mathcal{R}}^{n_{k_{j}}}(\cdot),\bar{\mathcal{Z}}^{n_{k_{j}}}(\cdot))\Rightarrow(\tilde{\mathcal{R}}(\cdot),\tilde{\mathcal{Z}}(\cdot))\quad\text{as
}j\to\infty,$
for some
$(\tilde{\mathcal{R}}(\cdot),\tilde{\mathcal{Z}}(\cdot))\in\mathbf{D}([0,\infty),\mathbf{M}\times\mathbf{M}_{+})$.
The remaining of this section is devoted to proving the above theorem. By
Theorem $3.7.2$ in [5], it suffices to verify ($a$) the compact containment
property, Lemma 5.1 and ($b$) the oscillation bound, Lemma 5.4 below.
#### 5.1.1 Compact Containment
A set $\mathbf{K}\subset\mathbf{M}$ is relatively compact if
$\sup_{\xi\in\mathbf{K}}\xi(\mathbb{R})<\infty$, and there exists a sequence
of nested compact sets $A_{j}\subset\mathbb{R}$ such that $\cup
A_{j}=\mathbb{R}$ and
$\lim_{j\to\infty}\sup_{\xi\in\mathbf{K}}\xi(A_{j}^{c})=0,$
where $A_{j}^{c}$ denotes the complement of $A_{j}$; see [14], Theorem A$7.5$.
The first major step to prove Theorem 5.1 is to establish the following
_compact containment_ property.
###### Lemma 5.1.
Assume (3.18)–(3.21). Fix $T>0$. For each $\eta>0$ there exists a compact set
$\mathbf{K}\subset\mathbf{M}$ such that
$\liminf_{n\to\infty}\mathbb{P}^{n}\Big{(}{(\bar{\mathcal{R}}^{n}(t),\bar{\mathcal{Z}}^{n}(t))\in\mathbf{K}\times\mathbf{K}\text{
for all }t\in[0,T]}\Big{)}\geq 1-\eta.$
To prove this result, we first need to establish some bound estimations. For
the convenience of notation, denote
$\bar{E}^{n}(s,t)=\bar{E}^{n}(t)-\bar{E}^{n}(s)$ for any $0\leq s\leq t$. Fix
$T>0$. It follows immediately from condition (3.18) that for each $\epsilon>0$
there exists an $n_{0}$ such that when $n>n_{0}$,
$\mathbb{P}^{n}\Big{(}{\sup_{0\leq s<t\leq
T}|\bar{E}^{n}(s,t)-\lambda(t-s)|<\epsilon}\Big{)}\geq 1-\epsilon.$ (5.4)
To facilitate some arguments later on, we derive the following result from the
above inequality.
###### Lemma 5.2.
Fix $T>0$. There exists a function $\epsilon_{E}(\cdot)$, with
$\lim_{n\to\infty}\epsilon_{E}(n)=0$ such that
$\mathbb{P}^{n}\Big{(}{\sup_{0\leq s<t\leq
T}|\bar{E}^{n}(s,t)-\lambda(t-s)|<\epsilon_{E}(n)}\Big{)}\geq
1-\epsilon_{E}(n),$
for each $n\geq 0$.
The derivation of the above lemma from (5.4) follows the same as the proof of
Lemma $5.1$ in [31]. We omit the proof for brevity. Based on the above lemma,
we construct the following event,
$\Omega^{n}_{E}=\\{\sup_{t\in[0,T]}|\bar{E}^{n}(s,t)-\lambda(t-s)|<\epsilon_{E}(n)\\}.$
(5.5)
We have that on this event, the arrival process is regular, i.e.
$\bar{E}^{n}(s,t)$ is “close” to $\lambda(t-s)$. And this event has “large”
probability, i.e.
$\lim_{n\to\infty}\mathbb{P}^{n}\Big{(}{\Omega^{n}_{E}}\Big{)}=1.$ (5.6)
###### Proof of Lemma 5.1.
By the convergence of the initial condition (3.20), for any $\epsilon>0$,
there exists a relatively compact set $\mathbf{K}_{0}\subset\mathbf{M}$ such
that
$\liminf_{n\to\infty}\mathbb{P}^{n}\Big{(}{\bar{\mathcal{R}}^{n}(0)\in\mathbf{K}_{0}\textrm{
and }\bar{\mathcal{Z}}^{n}(0)\in\mathbf{K}_{0}}\Big{)}>1-\epsilon.$ (5.7)
Denote the event in the above probability by $\Omega^{n}_{0}$. On this event,
by the definition of relatively compact set in the space $\mathbf{M}$, there
exists a function $\kappa_{0}(\cdot)$ with $\lim_{x\to\infty}\kappa_{0}(x)=0$
such that
$\bar{\mathcal{R}}^{n}(0)(C_{x})\leq\kappa_{0}(x),\quad\bar{\mathcal{Z}}^{n}(0)(C_{x})\leq\kappa_{0}(x),$
(5.8)
and
$\bar{\mathcal{R}}^{n}(0)(C^{-}_{x})\leq\kappa_{0}(x),$ (5.9)
for all $x\geq 0$, where $C^{-}_{x}=(-\infty,-x)$ for any $y\in\mathbb{R}$.
(Remember that $\bar{\mathcal{Z}}^{n}(0)$ is a measure on $(0,\infty)$, so we
do not need to consider its measure of $C^{-}_{x}$.) It is clear that on the
event $\Omega^{n}_{E}\cap\Omega^{n}_{0}$, for any $t\leq T$ and all large $n$,
$\displaystyle\bar{\mathcal{R}}^{n}(t)(\mathbb{R})$
$\displaystyle\leq\sup_{n}\bar{\mathcal{R}}^{n}(0)(\mathbb{R})+2\lambda T,$
$\displaystyle\bar{\mathcal{Z}}^{n}(t)((0,\infty))$ $\displaystyle\leq 1,$
where the last inequality is due to the fact that $Z^{n}(\cdot)\leq n$. Again,
by the definition of relative compact set in $\mathbf{M}$, we have that
$\sup_{n}\bar{\mathcal{R}}^{n}(0)(\mathbb{R})=M_{0}<\infty$. It follows from
the dynamic equation (5.2) and (5.3) that for all $x>0$,
$\displaystyle\bar{\mathcal{R}}^{n}(t)(C_{x})$
$\displaystyle\leq\bar{\mathcal{R}}^{n}(0)(C_{x})+\frac{1}{n}\sum_{i=1}^{E^{n}(t)}\delta_{u^{n}_{i}}(C_{x}),$
$\displaystyle\bar{\mathcal{Z}}^{n}(t)(C_{x})$
$\displaystyle\leq\bar{\mathcal{Z}}^{n}(0)(C_{x})+\frac{1}{n}\sum_{i=1}^{E^{n}(t)}\delta_{v^{n}_{i}}(C_{x}).$
Denote
$\bar{\mathcal{L}}^{n}_{1}(t)=\frac{1}{n}\sum_{i=1}^{E^{n}(t)}\delta_{u^{n}_{i}}$
and
$\bar{\mathcal{L}}^{n}_{2}(t)=\frac{1}{n}\sum_{i=1}^{E^{n}(t)}\delta_{v^{n}_{i}}$.
Let us first study these two terms. Recall the definition of the event
$\Omega^{n}_{\text{GC}}(M,L)$ and the envelope function $\bar{f}$ (which
increases to infinity) in (B.7). For the application here, it is enough to set
$M=1$ and $L=2\lambda T$. On the event
$\Omega^{n}_{E}\cap\Omega^{n}_{\text{GC}}(M,L)$, we have
$\displaystyle\langle{\bar{f}},{\bar{\mathcal{L}}^{n}_{1}(t)}\rangle\leq\langle{\bar{f}},{\frac{1}{n}\sum_{i=1}^{2\lambda
Tn}\delta_{u^{n}_{i}}}\rangle\leq 2\lambda
T\langle{\bar{f}},{\nu_{F}}\rangle+1,$
for all large enough $n$. Similarly, on the same event we have that
$\displaystyle\langle{\bar{f}},{\bar{\mathcal{L}}^{n}_{2}(t)}\rangle\leq\langle{\bar{f}},{\frac{1}{n}\sum_{i=1}^{2\lambda
Tn}\delta_{v^{n}_{i}}}\rangle\leq 2\lambda
T\langle{\bar{f}},{\nu_{G}}\rangle+1,$
for all large enough $n$. Denote $M_{B}=2\lambda
T\max(\langle{\bar{f}},{\nu_{F}}\rangle,\langle{\bar{f}},{\nu_{G}}\rangle)+1$.
By Markov’s inequality, for all $x>0$ (again, on the same event and for all
large $n$)
$\bar{\mathcal{L}}^{n}_{1}(t)(C_{x})<M_{b}/\bar{f}(x),\quad\bar{\mathcal{L}}^{n}_{2}(t)(C_{x})<M_{b}/\bar{f}(x).$
Unlike the measure $\mathcal{Z}(t)\in\mathbf{M}_{+}$, the measure
$\mathcal{R}(t)\in\mathbf{M}$. So we need to consider all the test set
$C^{-}_{x}=(-\infty,-x)$ for $x\geq 0$. The following inequality again follows
from (5.2),
$\bar{\mathcal{R}}^{n}(t)(C^{-}_{x})\leq\bar{\mathcal{R}}^{n}(0)(C^{-}_{x}+t)+\frac{1}{n}\sum_{i=1}^{E^{n}(t)}\delta_{u^{n}_{i}}(C^{-}_{x}+t).$
Note that if we take $x>T$, then $\delta_{u^{n}_{i}}(C^{-}_{x}+t)=0$. So we
have that
$\bar{\mathcal{R}}^{n}(t)(C^{-}_{x})\leq\bar{\mathcal{R}}^{n}(0)(C^{-}_{x}+T)=\bar{\mathcal{R}}^{n}(0)(C^{-}_{x-T}),\quad\textrm{for
all }t\leq T.$ (5.10)
Now, define the set $\mathbf{K}\subset\mathbf{M}$ by
$\begin{split}\mathbf{K}=\Big{\\{}\xi\in\mathbf{M}:&\xi(\mathbb{R})<1+M_{0}+2\lambda
T,\\\ &\xi(C_{x})<\kappa_{0}(x)+M_{b}/\bar{f}(x)\textrm{ for all }x>0,\\\
&\xi(C^{-}_{x})\leq\kappa_{0}(x-T)\textrm{ for all }x\geq
T\Big{\\}}.\end{split}$
It is clear that $\mathbf{K}$ is relatively compact and on the event
$\Omega^{n}_{E}\cap\Omega^{n}_{\text{GC}}(M,L)\cap\Omega^{n}_{0}$,
$(\bar{\mathcal{R}}^{n}(t),\bar{\mathcal{Z}}^{n}(t))\in\mathbf{K}\times\mathbf{K}\text{
for all }t\in[0,T].$
The result of this lemma then follows immediately from (5.6), (5.7) and (B.8).
∎
#### 5.1.2 Oscillation Bound
The second major step to prove precompactness is to obtain the oscillation
bound in Lemma 5.4 below. The oscillation of a càdlàg function $\zeta(\cdot)$
(taking values in a metric space $(\mathbf{E},\pi)$) on a fixed interval
$[0,T]$ is defined as
$\mathbf{w}_{T}({\zeta}(\cdot),{\delta})=\sup_{s,t\in[0,T],|s-t|<\delta}\pi[\zeta(s),\zeta(t)].$
If the metric space is $\mathbb{R}$, we just use the Euclidean metric; if the
space is $\mathbf{M}$ or $\mathbf{M}_{+}$, we use the Prohorov metric
$\mathbf{d}$ defined in Section 1.1. For the measure-valued processes in our
model, oscillations mainly result from sudden departures of a large number of
customers. To control the departure process, we show that
$\bar{\mathcal{Z}}^{n}(\cdot)$ and $\bar{\mathcal{R}}^{n}(\cdot)$ assign
arbitrarily small mass to small intervals.
###### Lemma 5.3.
Assume (3.10), (3.18)–(3.21). Fix $T>0$. For each $\epsilon,\eta>0$ there
exists a $\kappa>0$ (depending on $\epsilon$ and $\eta$) such that
$\displaystyle\liminf_{n\to\infty}\mathbb{P}^{n}\Big{(}{\sup_{t\in[0,T]}\sup_{x\in\mathbb{R}_{+}}\bar{\mathcal{Z}}^{n}(t)([x,x+\kappa])\leq\epsilon}\Big{)}\geq
1-\eta.$ (5.11)
###### Proof.
First, We have that for any $\epsilon,\eta>0$, there exists a $\kappa$ such
that
$\liminf_{n\to\infty}\mathbb{P}^{n}\Big{(}{\sup_{x\in\mathbb{R}_{+}}\bar{\mathcal{Z}}^{n}(0)([x,x+\kappa])\leq\epsilon/2}\Big{)}\geq
1-\eta.$ (5.12)
This inequality is derived from the initial condition. The derivation is
exactly the same as in the proof of (5.14) in [31], so we omit it here for
brevity.
Now we need to extend this result to the interval $[0,T]$. Denote the event in
(5.12) by $\Omega^{n}_{0}$, and the event in Lemma 5.1 by
$\Omega^{n}_{C}(\mathbf{K})$. Fix $M=1$ and $L=2\lambda T$, Let
$\Omega^{n}_{1}(M,L)=\Omega^{n}_{0}\cap\Omega^{n}_{C}(\mathbf{K})\cap\Omega^{n}_{E}\cap\Omega^{n}_{\text{GC}}(M,L).$
(5.13)
By (5.12), Lemma 5.1, (5.6) and (B.8), for any fixed $M,L>0$,
$\liminf_{n\to\infty}\mathbb{P}^{n}\Big{(}{\Omega^{n}_{1}(M,L)}\Big{)}\geq
1-\eta.$
In the remainder of the proof, all random objects are evaluated at a fixed
sample path in $\Omega^{n}_{1}(M,L)$.
It follows from the fluid scaled stochastic dynamic equation (5.3) that
$\displaystyle\bar{\mathcal{Z}}^{n}(t)([x,x+\kappa])$
$\displaystyle\leq\bar{\mathcal{Z}}^{n}(0)([x,x+\kappa]+t)$
$\displaystyle\quad+\frac{1}{n}\sum_{i=B(0)+1}^{B(t)}\delta_{v^{n}_{i}}([x,x+\kappa]+t-\tau^{n}_{i}),$
for each $x,\kappa\in\mathbb{R}_{+}$. By (5.12), the first term on the right
hand side of the above equation is always upper bounded by $\epsilon/2$. Let
$S$ denote the second term on the right hand side of the preceding equation.
Now it only remains to show that $S<\epsilon/2$.
Let $0=t_{0}<t_{1}<\cdots<t_{J}=t$ be a partition of the interval $[0,t]$ such
that $|t_{j+1}-t_{j}|<\delta$ for all $j=0,\cdots,J-1$, where $\delta$ and $N$
are to be chosen below. Write $S$ as the summation
$S=\sum_{j=0}^{J-1}\frac{1}{n}\sum_{i=B(t_{j})+1}^{B(t_{j+1})}\delta_{v^{n}_{i}}([x,x+\kappa]+t-\tau^{n}_{i}).$
Recall that $\tau^{n}_{i}$ is the time that the $i$th job starts service, so
on each sub-interval $[t_{j},t_{j+1}]$ those $i$’s to be summed must satisfy
$t_{j}\leq\tau^{n}_{i}\leq t_{j+1}$. This implies that
$t-t_{j+1}\leq t-\tau_{i}\leq t-t_{j}.$
Then
$S\leq\sum_{j=0}^{J-1}\frac{1}{n}\sum_{i=B(t_{j})+1}^{B(t_{j+1})}\delta_{v^{n}_{i}}([x+t-t_{j+1},x+t-t_{j}+\kappa]).$
By (5.1), we have for all $j=0,\cdots,J-1$
$\displaystyle-\bar{R}^{n}(0)\leq\bar{B}^{n}(t_{j})$
$\displaystyle\leq\bar{E}^{n}(T),$ $\displaystyle
0\leq\bar{B}^{n}(t_{j+1})-\bar{B}^{n}(t_{j})$
$\displaystyle\leq\bar{E}^{n}(T)+\bar{R}^{n}(0).$
By Lemmas 5.1 and 5.2, $\bar{R}^{n}(0)<M_{0}$ and $\bar{E}^{n}(T)\leq 2\lambda
T$ on $\Omega^{n}_{C}(\mathbf{K})\cap\Omega^{n}_{E}$ for some constant
$M_{0}$. Take $M=\max(M_{0},2\lambda T)$ and $L=M_{0}+2\lambda T$, it follows
from the Glevenko-Cantelli estimate (B.7) that
$\displaystyle\quad\frac{1}{n}\sum_{i=B^{n}(t_{j})+1}^{B^{n}(t_{j+1})}\delta_{v^{n}_{i}}([x+t-t_{j+1},x+t-t_{j}+\kappa])$
$\displaystyle\leq\Big{(}\bar{B}^{n}(t_{j+1})-\bar{B}^{n}(t_{j})\Big{)}\nu^{n}([x+t-t_{j+1},x+t-t_{j}+\kappa])+\frac{\epsilon}{4J},$
for each $j<J$. By condition (3.19), for any $\epsilon_{2}>0$,
$\mathbf{d}[\nu^{n}_{G},\nu_{G}]<\epsilon_{2},$
for all large $n$. By the definition of Prohorov metric, we have
$\nu^{n}_{G}([x+t-t_{j+1},x+t-t_{j}+\kappa])\leq\nu_{G}([x+t-t_{j+1}-\epsilon_{2},x+t-t_{j}+\kappa+\epsilon_{2}]),$
for all large $n$. Since
$[x+t-t_{j+1}-\epsilon_{2},x+t-t_{j}+\kappa+\epsilon_{2}]$ is a close interval
with length less than $\kappa+\delta+2\epsilon_{2}$, by condition (3.10), we
can choose $\kappa,\delta,\epsilon_{2}$ small enough such that
$\nu([x+t-t_{j+1}-\epsilon_{2},x+t-t_{j}+\kappa+\epsilon_{2}])\leq\frac{\epsilon}{4M}.$
Thus, we conclude that
$\displaystyle
S\leq\frac{\epsilon}{4J}[\bar{B}^{n}(T)-\bar{B}^{n}(0)]+\frac{\epsilon}{4}\leq\epsilon/2.$
This completes the proof. ∎
###### Lemma 5.4.
Assume (3.10), (3.18)–(3.21). Fix $T>0$. For each $\epsilon,\eta>0$ there
exists a $\delta>0$ (depending on $\epsilon$ and $\eta$) such that
$\liminf_{n\to\infty}\mathbb{P}^{n}\Big{(}{\mathbf{w}_{T}({(\bar{\mathcal{R}}^{n},\bar{\mathcal{Z}}^{n})}(\cdot),{\delta})\leq
3\epsilon}\Big{)}\geq 1-\eta.$ (5.14)
###### Proof.
Define
$\Omega^{n}_{\text{Reg}}(\epsilon,\kappa)=\Big{\\{}\sup_{t\in[0,T]}\sup_{x\in\mathbb{R}_{+}}\bar{\mathcal{Z}}^{n}(t)([x,x+\kappa])\leq\epsilon\Big{\\}}.$
By (5.6) and Lemma 5.3, for each $\epsilon,\eta>0$ there exists a $\kappa>0$
such that
$\liminf_{n\to\infty}\mathbb{P}^{n}\Big{(}{\Omega^{n}_{E}\cap\Omega^{n}_{\text{Reg}}(\epsilon,\kappa)}\Big{)}>1-\eta.$
(5.15)
On the event $\Omega^{n}_{E}\cap\Omega^{n}_{\text{Reg}}(\epsilon,\kappa)$, we
have some control over the dynamics of the system. First, note that the number
of customers (in the virtual buffer, including those who have abandoned but
ought to get service if they did not) that enter the server during time
interval $(s,t]$ can be upper bounded by
$\bar{B}^{n}(s,t)\leq\bar{E}^{n}(s,t)+\bar{\mathcal{Z}}^{n}(s)([0,t-s]).$
When $t-s\leq\min(\frac{\epsilon}{2\lambda},\kappa)$, by the definition of
$\Omega^{n}_{E}$ and $\Omega^{n}_{\text{Reg}}(\epsilon,\kappa)$, we have
$\displaystyle\bar{E}^{n}(s,t)$ $\displaystyle\leq\epsilon$ (5.16)
$\displaystyle\bar{B}^{n}(s,t)$ $\displaystyle\leq 2\epsilon.$ (5.17)
Second, by the dynamic equation (5.2), for any $s<t$ and any set
$C\in\mathscr{B}(\mathbb{R})$,
$\displaystyle\bar{\mathcal{R}}^{n}(t)(C)-\bar{\mathcal{R}}^{n}(s)(C^{3\epsilon}))$
$\displaystyle\leq\bar{B}^{n}(s,t)+\bar{E}^{n}(s,t)$
$\displaystyle\quad+\frac{1}{n}\sum_{1+B^{n}(t)}^{E^{n}(s)}[\delta_{u^{n}_{i}}(C+t-a^{n}_{i})-\delta_{u^{n}_{i}}(C^{3\epsilon}+s-a^{n}_{i})],$
where $C^{a}$ is the $a$-enlargement of the set $C$ as defined in Section 1.1.
Note that when $t-s\leq 3\epsilon$, $C+t-a^{n}_{i}\subseteq
C^{3\epsilon}+s-a^{n}_{i}$ for all $i\in\mathbb{Z}$, which implies that the
second term in the above inequality is less than zero. By (5.16) and (5.17),
$\bar{\mathcal{R}}^{n}(t)(C)-\bar{\mathcal{R}}^{n}(s)(C^{3\epsilon}))\leq
3\epsilon.$
By Property (ii) on page 72 in [1], we have
$\mathbf{d}[\bar{\mathcal{R}}^{n}(t),\bar{\mathcal{R}}^{n}(s)]\leq 3\epsilon.$
(5.18)
Finally, by the dynamic equation (5.3),
$\bar{\mathcal{Z}}^{n}(t)(C)\leq\bar{\mathcal{Z}}^{n}(s)(C+t-s))+\bar{B}^{n}(s,t).$
Note that when $t-s\leq 2\epsilon$, $C+t-s\subseteq C^{2\epsilon}$, where
$C^{a}$ is the $a$-enlargement of the set $C$ as defined in Section 1.1. By
(5.17), we have
$\bar{\mathcal{Z}}^{n}(t)(C)\leq\bar{\mathcal{Z}}^{n}(s)(C^{2\epsilon})+2\epsilon.$
By Property (ii) on page 72 in [1], we have
$\mathbf{d}[\bar{\mathcal{Z}}^{n}(s),\bar{\mathcal{Z}}^{n}(t)]\leq 2\epsilon.$
(5.19)
The result of this lemma follows immediately from (5.15), (5.18) and (5.19). ∎
### 5.2 Convergence to the Fluid Model Solution
We have established the precompactness in Theorem 5.1. So every subsequence of
the fluid scaled processes has a further subsequence which converges to some
limit. For simplicity of notations, we index the convergent subsequence again
by $n$. So we have that
$(\bar{\mathcal{R}}^{n}(\cdot),\bar{\mathcal{Z}}^{n}(\cdot))\Rightarrow(\tilde{\mathcal{R}}(\cdot),\tilde{\mathcal{Z}}(\cdot))\quad\text{as
}n\to\infty.$ (5.20)
By the oscillation bound in Lemma 5.4, the limit
$(\tilde{\mathcal{R}}(\cdot),\tilde{\mathcal{Z}}(\cdot))$ is almost surely
continuous. We have the following result that further characterizes the above
limit.
###### Lemma 5.5.
Assume (3.10)–(3.13) and (3.18)–(3.21). The limit
$(\tilde{\mathcal{R}}(\cdot),\tilde{\mathcal{Z}}(\cdot))$ in (5.20) is almost
surely the solution to the fluid model $(\lambda,F,G)$ with initial condition
$(\bar{\mathcal{R}}_{0},\bar{\mathcal{Z}}_{0})$.
The rest of this section is devoted to characterizing the limits. To better
structure the proof, we first provide some preliminary estimates based on the
dynamic equations (5.2) and (5.3).
###### Lemma 5.6.
Let $\\{t_{j}\\}_{j=0}^{J}$ be a partition of the interval $[s,t]$ such that
$s=t_{0}<t_{1}<\ldots<t_{J}=t$. We have for any $x\in\mathbb{R}$,
$\displaystyle\bar{\mathcal{R}}^{n}(t)(C_{x})$
$\displaystyle\leq\sum_{i=0}^{J-1}\frac{1}{n}\sum_{i=1+E^{n}(t_{j})}^{E^{n}(t_{j+1})}\delta_{u^{n}_{i}}(C_{x}+t-t_{j})+|\bar{E}^{n}(s)-\bar{B}^{n}(t)|,$
(5.21) $\displaystyle\bar{\mathcal{R}}^{n}(t)(C_{x})$
$\displaystyle\geq\sum_{i=0}^{J-1}\frac{1}{n}\sum_{i=1+E^{n}(t_{j})}^{E^{n}(t_{j+1})}\delta_{u^{n}_{i}}(C_{x}+t-t_{j+1})-|\bar{E}^{n}(s)-\bar{B}^{n}(t)|.$
(5.22)
If in addition that
$\sup_{\tau\in[s,t]}|\bar{E}^{n}(\tau)-\lambda\tau|<\epsilon$, then for any
$x>0$,
$\displaystyle\begin{split}\bar{\mathcal{Z}}^{n}(t)(C_{x})&\leq\bar{\mathcal{Z}}^{n}(s)(C_{x}+t-s)\\\
&\quad+\sum_{j=0}^{J-1}\frac{1}{n}\sum_{i=1+B^{n}(t_{j})}^{B^{n}(t_{j+1})}\delta_{u^{n}_{i}}(C_{0}+\frac{\bar{R}^{n}_{L,j}-2\epsilon}{\lambda})\delta_{v^{n}_{i}}(C_{x}+t-t_{j}),\end{split}$
(5.23)
$\displaystyle\begin{split}\bar{\mathcal{Z}}^{n}(t)(C_{x})&\geq\bar{\mathcal{Z}}^{n}(s)(C_{x}+t-s)\\\
&\quad+\sum_{j=0}^{J-1}\frac{1}{n}\sum_{i=1+B^{n}(t_{j})}^{B^{n}(t_{j+1})}\delta_{u^{n}_{i}}(C_{0}+\frac{\bar{R}^{n}_{U,j}+2\epsilon}{\lambda})\delta_{v^{n}_{i}}(C_{x}+t-t_{j+1}),\end{split}$
(5.24)
where $\bar{R}^{n}_{L,j}=\inf_{t\in[t_{j},t_{j+1}]}\bar{R}^{n}(t)$ and
$\bar{R}^{n}_{U,j}=\sup_{t\in[t_{j},t_{j+1}]}\bar{R}^{n}(t)$.
###### Proof.
Note that $0\leq\delta_{u^{n}_{i}}(C)\leq 1$ for any Borel set $C$ and any
random variable $u^{n}_{i}$. So by the dynamic equation (5.2), we have
$\Big{|}\bar{\mathcal{R}}^{n}(t)(C)-\frac{1}{n}\sum_{i=E^{n}(s)+1}^{E^{n}(t)}\delta_{u^{n}_{i}}(C+t-a^{n}_{i})\Big{|}\leq|\bar{E}^{n}(s)-\bar{B}^{n}(t)|.$
For those $i$’s such that $E^{n}(t_{j})<i\leq E^{n}(t_{j+1})$, we have that
$t_{j}<a^{n}_{i}\leq t_{j+1}.$ (5.25)
This implies that $C_{x}+t-a_{i}\subseteq C_{x}+t-t_{j}$. So we have
$\displaystyle\sum_{i=1+E^{n}(t_{j})}^{E^{n}(t_{j+1})}\delta_{u^{n}_{i}}(C_{x}+t-a_{i})$
$\displaystyle\leq\sum_{i=1+E^{n}(t_{j})}^{E^{n}(t_{j+1})}\delta_{u^{n}_{i}}(C_{x}+t-t_{j}).$
This establishes (5.21). Also, (5.25) implies $C_{x}+t-t_{j+1}\subseteq
C_{x}+t-a_{i}$. So (5.22) follows in the same way.
For those $i$’s such that $B^{n}(t_{j})<i\leq B^{n}(t_{j+1})$, we have that
$t_{j}<\tau^{n}_{j}\leq t_{j+1}.$
Note that
$\bar{R}^{n}(\tau^{n}_{i})=\bar{E}^{n}(\tau^{n}_{i})-\bar{E}^{n}(a^{n}_{i})$
for each $i$. So, by the closeness between $\bar{E}^{n}(\cdot)$ and
$\lambda\cdot$, we have
$\displaystyle\quad|\bar{R}^{n}(\tau^{n}_{i})-\lambda(\tau^{n}_{i}-a^{n}_{i})|$
$\displaystyle\leq|\bar{R}^{n}(\tau^{n}_{i})-\bar{E}^{n}(\tau^{n}_{i})+\bar{E}^{n}(a^{n}_{i})|+|\bar{E}^{n}(\tau^{n}_{i})-\bar{E}^{n}(a^{n}_{i})-\lambda(\tau^{n}_{i}-a^{n}_{i})|$
$\displaystyle\leq 2\epsilon.$
So
$\bar{R}^{n}_{L,j}-2\epsilon\leq\lambda(\tau^{n}_{i}-a^{n}_{i})\leq\bar{R}^{n}_{U,j}+2\epsilon,$
for all $i$’s such that $B^{n}(t_{j})<i\leq B^{n}(t_{j+1})$. Thus,
$\sum_{i=1+B^{n}(t_{j})}^{B^{n}(t_{j+1})}\delta_{u^{n}_{i}}(C_{0}+\tau^{n}_{i}-a^{n}_{i})\delta_{v^{n}_{i}}(C_{x}+t-\tau^{n}_{j})\leq\sum_{i=1+B^{n}(t_{j})}^{B^{n}(t_{j+1})}\delta_{u^{n}_{i}}(C_{0}+\frac{\bar{R}^{n}_{L,j}-2\epsilon}{\lambda})\delta_{v^{n}_{i}}(C_{x}+t-t_{j}).$
This implies (5.23). And (5.24) can be proved in the same way. ∎
Recall the notations
$\bar{\mathcal{L}}^{n}(m,l),\bar{\mathcal{L}}^{n}_{p}(m,l)$ and
$\bar{\mathcal{L}}^{n}_{S}(m,l)$ are defined in (B.1)–(B.3) in the appendix.
Using these notations, Lemma 5.6 can be written as the following:
###### Lemma 5.7.
Let $\\{t_{j}\\}_{j=0}^{J}$ be a partition of the interval $[s,t]$ such that
$s=t_{0}<t_{1}<\ldots<t_{J}=t$. We have for any $x\in\mathbb{R}$,
$\displaystyle\bar{\mathcal{R}}^{n}(t)(C_{x})$
$\displaystyle\leq\sum_{i=0}^{J-1}\langle{1_{(C_{x}+t-t_{j})}},{\bar{\mathcal{L}}^{n}_{p}(E^{n}(t_{j}),\bar{E}^{n}(t_{j},t_{j+1})}\rangle+|\bar{E}^{n}(s)-\bar{B}^{n}(t)|,$
(5.26) $\displaystyle\bar{\mathcal{R}}^{n}(t)(C_{x})$
$\displaystyle\geq\sum_{i=0}^{J-1}\langle{1_{(C_{x}+t-t_{j+1})}},{\bar{\mathcal{L}}^{n}_{p}(E^{n}(t_{j}),\bar{E}^{n}(t_{j},t_{j+1})}\rangle-|\bar{E}^{n}(s)-\bar{B}^{n}(t)|.$
(5.27)
If in addition that
$\sup_{\tau\in[s,t]}|\bar{E}^{n}(\tau)-\lambda\tau|<\epsilon$, then for any
$x>0$,
$\displaystyle\begin{split}\bar{\mathcal{Z}}^{n}(t)(C_{x})&\leq\bar{\mathcal{Z}}^{n}(s)(C_{x}+t-s)\\\
&\quad+\sum_{j=0}^{J-1}\langle{1_{(C_{0}+\frac{\bar{R}^{n}_{L,j}-2\epsilon}{\lambda})\times(C_{x}+t-t_{j})}},{\bar{\mathcal{L}}^{n}(B^{n}(t_{j}),\bar{B}^{n}(t_{j},t_{j+1}))}\rangle,\end{split}$
(5.28)
$\displaystyle\begin{split}\bar{\mathcal{Z}}^{n}(t)(C_{x})&\geq\bar{\mathcal{Z}}^{n}(s)(C_{x}+t-s)\\\
&\quad+\sum_{j=0}^{J-1}\langle{1_{(C_{0}+\frac{\bar{R}^{n}_{U,j}+2\epsilon}{\lambda})\times(C_{x}+t-t_{j+1})}},{\bar{\mathcal{L}}^{n}(B^{n}(t_{j}),\bar{B}^{n}(t_{j},t_{j+1}))}\rangle.\end{split}$
(5.29)
Fix a constant $T>0$ and let $M=1$ and $L=2\lambda T$. Denote the random
variable
$\bar{V}^{n}_{M,L}=\max_{-nM<m<nM}\sup_{l\in[0,L]}\sup_{x,y\in\mathbb{R}}\left\\{\begin{array}[]{l}\big{|}\bar{\mathcal{L}}^{n}(m,l)(C_{x}\times
C_{y})-l\nu_{F}^{n}(C_{x})\nu_{G}^{n}(C_{y})\big{|}\\\
+\big{|}\bar{\mathcal{L}}^{n}_{F}(m,l)(C_{x})-l\nu_{F}^{n}(C_{x})\big{|}\\\
+\big{|}\bar{\mathcal{L}}^{n}_{G}(m,l)(C_{x})-l\nu_{G}^{n}(C_{x})\big{|}\end{array}\right\\}.$
(5.30)
By Lemma B.1, for any fixed constants $M,L>0$,
$\bar{V}^{n}_{M,L}\Rightarrow 0\quad\textrm{as }n\to\infty.$
By the assumption (3.18), we have
$\bar{E}^{n}(\cdot)\Rightarrow\lambda\cdot\quad\text{as }n\to\infty.$
Since both the above two limits are deterministic, those convergences are
joint with the convergence of
$(\bar{\mathcal{R}}^{n}(\cdot),\bar{\mathcal{Z}}^{n}(\cdot))$. Now, for each
$n\geq 1$, we can view
$(\bar{E}^{n}(\cdot),\bar{\mathcal{R}}^{n}(\cdot),\bar{\mathcal{Z}}^{n}(\cdot),V_{M,L})$
as a random variable in the space $\mathbf{E}_{1}$, which is the product space
of three $\mathbf{D}([0,\infty),\mathbb{R})$ spaces and the space
$\mathbb{R}$. And
$(\bar{\mathcal{L}}^{n}(m,\cdot),\bar{\mathcal{L}}^{n}_{F}(m,\cdot),\bar{\mathcal{L}}^{n}_{G}(m,\cdot):m\in\mathbb{Z})$
in the product space $\mathbf{E}_{2}$ of countable many
$\mathbf{D}([0,\infty),\mathbf{M})$ spaces. It is clear that both
$\mathbf{E}_{1}$ and $\mathbf{E}_{2}$ are complete and separable metric
spaces. Using the extension of Skorohod representation Theorem, Lemma C.1, we
assume without loss of generality that
$\bar{E}^{n}(\cdot),\bar{\mathcal{R}}^{n}(\cdot),\bar{\mathcal{Z}}^{n}(\cdot),\bar{V}^{n}_{M,L},\bar{\mathcal{L}}^{n}(m,\cdot),\bar{\mathcal{L}}^{n}_{F}(m,\cdot),\bar{\mathcal{L}}^{n}_{G}(m,\cdot),m\in\mathbb{Z}$,
and $(\tilde{\mathcal{R}}(\cdot),\tilde{\mathcal{Z}}(\cdot))$ are defined on a
common probability space
$(\tilde{\Omega},\tilde{\mathcal{F}},\tilde{\mathbb{P}})$ such that, almost
surely,
$\Big{(}(\bar{\mathcal{R}}^{n}(\cdot),\bar{\mathcal{Z}}^{n}(\cdot)),\bar{V}^{n}_{M,L},\bar{E}^{n}(\cdot)\Big{)}\to\Big{(}(\tilde{\mathcal{R}}(\cdot),\tilde{\mathcal{Z}}(\cdot)),0,\lambda\cdot\Big{)}\textrm{\quad
as }n\to\infty,$ (5.31)
and inequalities (5.26)–(5.29) and equation (5.30) also hold almost surely.
Note that the convergence of each function component in the above is in the
Skorohod $J_{1}$ topology. Since the limit is continuous, the convergence is
equivalent to the convergence in the uniform norm on compact intervals. Thus
as $n\to\infty$,
$\displaystyle\sup_{t\in[0,T]}\mathbf{d}[\bar{\mathcal{R}}^{n}(t),\tilde{\mathcal{R}}(t)]\to
0,$ (5.32)
$\displaystyle\sup_{t\in[0,T]}\mathbf{d}[\bar{\mathcal{Z}}^{n}(t),\tilde{\mathcal{Z}}(t)]\to
0,$ (5.33) $\displaystyle\sup_{t\in[0,T]}\big{|}\bar{E}^{n}(t)-\lambda
t\big{|}\to 0,$ (5.34)
where $\mathbf{d}$ is the Skorohod metric defined in Section 1.1. Same as on
the original probability space, let
$\displaystyle\bar{R}^{n}(\cdot)=\langle{1},{\bar{\mathcal{R}}^{n}(\cdot)}\rangle,$
$\displaystyle\quad\bar{Q}^{n}(\cdot)=\langle{1_{(0,\infty)}},{\bar{\mathcal{R}}^{n}(\cdot)}\rangle,$
$\displaystyle\bar{Z}^{n}(\cdot)=\langle{1},{\bar{\mathcal{Z}}^{n}(\cdot)}\rangle,$
$\displaystyle\quad\bar{X}^{n}(\cdot)=\bar{Q}^{n}(\cdot)+\bar{Z}^{n}(\cdot),$
and
$\bar{B}^{n}(\cdot)=\bar{E}^{n}(\cdot)-\bar{R}^{n}(\cdot).$
According to (5.32) and (5.34), we have
$\sup_{t\in[0,T]}\big{|}\bar{B}^{n}(t)-\tilde{B}(t)\big{|}\to 0.$ (5.35)
For each $n$, let $\tilde{\Omega}_{n,2}$ be an event of probability one on
which the stochastic dynamic equations (5.2) and (5.3) and the policy
constraints (2.6) and (2.7) hold. Define
$\tilde{\Omega}_{0}=\tilde{\Omega}_{1}\cap(\cap_{n=0}^{\infty}\tilde{\Omega}^{n}_{n,2})$,
where $\tilde{\Omega}_{1}$ is the event of probability one on which (5.31)
holds. Then $\tilde{\Omega}_{0}$ also has probability one. Based on Lemma 5.6
and the above argument using Skorohod Representation theorem, we can now prove
Lemma 5.5.
###### Proof of Lemma 5.5.
For any $t\geq 0$, fix a constant $T>t$. Let us now study
$(\tilde{\mathcal{R}}(\cdot),\tilde{\mathcal{Z}}(\cdot))$ on the time interval
$[0,T]$. It is enough to show that on the event $\tilde{\Omega}_{0}$,
$(\tilde{\mathcal{R}}(t),\tilde{\mathcal{Z}}(t))$ satisfies the fluid model
equation (3.1)–(3.2) and the constraints (3.3)–(3.4). Assume for the remainder
of this proof that all random objects are evaluated at a sample path in the
event $\tilde{\Omega}_{0}$.
We first verify (3.1). For any $\epsilon>0$, consider the difference
$\displaystyle\quad\tilde{\mathcal{R}}(t)(C_{x})-\int_{t-\frac{\tilde{R}(t)}{\lambda}}^{t}F^{c}(x+t-s)d\lambda
s$
$\displaystyle=\tilde{\mathcal{R}}(t)(C_{x})-\bar{\mathcal{R}}^{n}(t)(C^{\epsilon}_{x})+\bar{\mathcal{R}}^{n}(t)(C^{\epsilon}_{x})-\int_{t-\frac{\tilde{R}(t)}{\lambda}}^{t}F^{c}(x+t-s)d\lambda
s,$
where $C^{\epsilon}_{x}$ is the $\epsilon$-enlargement of the set $C_{x}$ as
defined in Section 1.1, which is essentially $C_{x-\epsilon}$. Let
$t_{0}=t-\tilde{R}(t)/\lambda$. According to (5.26), we have that
$\begin{split}&\quad\tilde{\mathcal{R}}(t)(C_{x})-\int_{t-\frac{\tilde{R}(t)}{\lambda}}^{t}F^{c}(x+t-s)d\lambda
s\\\
&\leq\tilde{\mathcal{R}}(t)(C_{x})-\bar{\mathcal{R}}^{n}(t)(C^{\epsilon}_{x})+|\bar{E}^{n}(t_{0})-\bar{B}^{n}(t)|\\\
&\quad\sum_{i=0}^{J-1}\langle{1_{(C^{\epsilon}_{x}+t-t_{j})}},{\bar{\mathcal{L}}^{n}_{p}(E^{n}(t_{j}),\bar{E}^{n}(t_{j},t_{j+1})}\rangle-\int_{t_{0}}^{t}F^{c}(x+t-s)d\lambda
s,\end{split}$ (5.36)
where $\\{t_{j}\\}_{j=0}^{J}$ is a partition of the interval $[t_{0},t]$ such
that $t_{0}<t_{1}<\ldots<t_{J}=t$ and $\max_{j}(t_{j+1}-t_{j})<\delta$ for
some $\delta>0$. By the definition of Prohorov metric and the convergence in
(5.32), the first term on the right hand side of (5.36) is bounded by
$\epsilon$ for all large $n$. By (5.32) and (5.34)
$\displaystyle|\bar{B}^{n}(t)-\bar{E}^{n}(t_{0})|$
$\displaystyle=|\bar{E}^{n}(t)-\bar{R}^{n}(t)-\bar{E}^{n}(t_{0})|$
$\displaystyle\leq|\bar{E}^{n}(t)-\lambda
t|+|\bar{R}^{n}(t)-\tilde{R}(t)|+|\bar{E}^{n}(t_{0})-\lambda
t_{0}|<3\epsilon,$
for all large $n$. So
$\begin{split}&\quad\tilde{\mathcal{R}}(t)(C_{x})-\int_{t-\frac{\tilde{R}(t)}{\lambda}}^{t}F^{c}(x+t-s)d\lambda
s\\\ &\leq
4\epsilon+\sum_{i=0}^{J-1}\langle{1_{(C^{\epsilon}_{x}+t-t_{j})}},{\bar{\mathcal{L}}^{n}_{p}(E^{n}(t_{j}),\bar{E}^{n}(t_{j},t_{j+1})}\rangle-\int_{t_{0}}^{t}F^{c}(x+t-s)d\lambda
s,\end{split}$ (5.37)
for all large $n$. Similarly, according to (5.27), we have
$\begin{split}&\quad\tilde{\mathcal{R}}(t)(C_{x})-\int_{t-\frac{\tilde{R}(t)}{\lambda}}^{t}F^{c}(x+t-s)d\lambda
s\\\
&\geq-4\epsilon+\sum_{i=0}^{J-1}\langle{1_{(C^{\epsilon}_{x}+t-t_{j+1})}},{\bar{\mathcal{L}}^{n}_{p}(E^{n}(t_{j}),\bar{E}^{n}(t_{j},t_{j+1})}\rangle-\int_{t_{0}}^{t}F^{c}(x+t-s)d\lambda
s,\end{split}$ (5.38)
for all large $n$. Note that for each $j$, we have
$\displaystyle\quad\langle{1_{(C_{x}+t-t_{j})}},{\bar{\mathcal{L}}^{n}_{p}(E^{n}(t_{j}),\bar{E}^{n}(t_{j},t_{j+1})}\rangle$
$\displaystyle\leq\langle{1_{(C_{x}+t-t_{j})}},{\bar{\mathcal{L}}^{n}_{p}(E^{n}(t_{j}),\lambda(t_{j+1}-t_{j})+2\epsilon}\rangle$
$\displaystyle\leq[\lambda(t_{j+1}-t_{j})+2\epsilon]\nu^{n}_{F}(C^{\epsilon}_{x}+t-t_{j})+\epsilon$
$\displaystyle\leq[\lambda(t_{j+1}-t_{j})+2\epsilon][\nu_{F}(C_{x}+t-t_{j})+\epsilon]+\epsilon$
$\displaystyle\leq\lambda(t_{j+1}-t_{j})\nu_{F}(C_{x}+t-t_{j})+(3+\lambda\delta)\epsilon$
for all large $n$, where the first inequality is due to (5.34), the second one
is due to (5.31) (the component of $\bar{V}^{n}_{M,L}$), the third one is due
to (3.19), and the last one is due to algebra. Similarly, we can show that
$\displaystyle\quad\langle{1_{(C_{x}+t-t_{j+1})}},{\bar{\mathcal{L}}^{n}_{p}(E^{n}(t_{j}),\bar{E}^{n}(t_{j},t_{j+1})}\rangle$
$\displaystyle\geq\lambda(t_{j+1}-t_{j})\nu_{F}(C_{x}+t-t_{j+1})-(3+\lambda\delta)\epsilon$
for all large $n$. Note that
$\sum_{j=0}^{J-1}\lambda(t_{j+1}-t_{j})F^{c}(x+t-t_{j})$ and
$\sum_{j=0}^{J-1}\lambda(t_{j+1}-t_{j})F^{c}(x+t-t_{j+1})$ serve as the upper
and lower Reimann sum of the integral $\int_{t_{0}}^{t}F^{c}(x+t-s)d\lambda
s$, which converge to the integration as $n\to\infty$. So by (5.37) and
(5.38), we have that for all large $n$,
$\big{|}\tilde{\mathcal{R}}(t)(C_{x})-\int_{t-\frac{\tilde{R}(t)}{\lambda}}^{t}F^{c}(x+t-s)d\lambda
s\big{|}\leq(3+\lambda\delta)J\epsilon+5\epsilon.$
We conclude that
$\tilde{\mathcal{R}}(t)(C_{x})-\int_{t-\frac{\tilde{R}(t)}{\lambda}}^{t}F^{c}(x+t-s)d\lambda
s=0$ since $\epsilon$ in the above can be arbitrary. This verifies (3.1).
Next, we verify (3.2). For any $\epsilon>0$, consider the difference
$\begin{split}&\quad\Big{|}\tilde{\mathcal{Z}}(t)(C_{x})-\bar{\mathcal{Z}}_{0}(C_{x}+t)-\int_{0}^{t}F^{c}(\frac{\tilde{R}(s)}{\lambda})G^{c}(x+t-s)d[\lambda
s-\tilde{R}(s)]\Big{|}\\\
&\leq|\tilde{\mathcal{Z}}(t)(C_{x})-\bar{\mathcal{Z}}^{n}(t)(C_{x}^{\epsilon})|+|\tilde{\mathcal{Z}}_{0}(C_{x}+t)-\bar{\mathcal{Z}}^{n}(0)(C_{x}^{\epsilon}+t)|\\\
&\quad+\Big{|}\bar{\mathcal{Z}}^{n}(t)(C_{x}^{\epsilon})-\bar{\mathcal{Z}}^{n}(0)(C_{x}^{\epsilon}+t)-\int_{0}^{t}F^{c}(\frac{\tilde{R}(s)}{\lambda})G^{c}(x+t-s)d[\lambda
s-\tilde{R}(s)]\Big{|},\end{split}$ (5.39)
where the above inequality is due to the fluid scaled stochastic dynamic
equation (5.3). Again, by the definition of Prohorov metric and the
convergence in (5.33), each of the first two terms on the right hand side in
the above inequality is less than $\epsilon$ for all large $n$. Let
$\\{t_{j}\\}_{j=0}^{J}$ be a partition of the interval $[0,t]$ such that
$0=t_{0}<t_{1}<\ldots<t_{J}=t$ and $\max_{j}(t_{j+1}-t_{j})<\delta$ for some
$\delta>0$. Let
$\tilde{R}_{U,j}=\sup_{t\in[t_{j},t_{j+1}]}\tilde{R}(t),\quad\tilde{R}_{L,j}=\inf_{t\in[t_{j},t_{j+1}]}\tilde{R}(t).$
By (5.32), we have that
$|\bar{R}^{n}_{U,j}-\tilde{R}_{U,j}|\leq\epsilon,\quad|\bar{R}^{n}_{L,j}-\tilde{R}_{L,j}|\leq\epsilon,$
for all large $n$. So for each $j$, we have
$\displaystyle\quad\langle{1_{(C_{0}+\frac{\bar{R}^{n}_{L,j}-2\epsilon}{\lambda})\times(C^{\epsilon}_{x}+t-t_{j})}},{\bar{\mathcal{L}}^{n}(B^{n}(t_{j}),\bar{B}^{n}(t_{j},t_{j+1}))}\rangle$
$\displaystyle\leq\langle{1_{(C_{0}+\frac{\tilde{R}_{L,j}-3\epsilon}{\lambda})\times(C^{\epsilon}_{x}+t-t_{j})}},{\bar{\mathcal{L}}^{n}(B^{n}(t_{j}),\tilde{B}(t_{j+1})-\tilde{B}(t_{j})+2\epsilon)}\rangle$
$\displaystyle\leq[\tilde{B}(t_{j+1})-\tilde{B}(t_{j})+2\epsilon]\nu^{n}_{F}(C_{0}+\frac{\tilde{R}_{L,j}-3\epsilon}{\lambda})\nu^{n}_{G}(C^{\epsilon}_{x}+t-t_{j})+\epsilon$
$\displaystyle\leq[\tilde{B}(t_{j+1})-\tilde{B}(t_{j})+2\epsilon][\nu_{F}(C_{0}+\frac{\tilde{R}_{L,j}}{\lambda})+\frac{3\epsilon}{\lambda}][\nu_{G}(C_{x}+t-t_{j})+\epsilon]+\epsilon$
for all large $n$, where the first inequality is due to (5.35), the second one
is due to (5.31) (the component of $\bar{V}^{n}_{M,L}$), the third one is due
to (3.19). Let $M_{B}$ be a finite upper bound of
$\tilde{B}(t_{J})-\tilde{B}(t_{0})$, the above inequality can be further
bounded by
$\displaystyle[\tilde{B}(t_{j+1})-\tilde{B}(t_{j})]\nu_{F}(C_{0}+\frac{\tilde{R}_{L,j}}{\lambda})\nu_{G}(C_{x}+t-t_{j})+(\frac{3}{\lambda}+2)M_{B}\epsilon+3\epsilon.$
Similarly, we can show that
$\displaystyle\quad\langle{1_{(C_{0}+\frac{\bar{R}^{n}_{U,j}+2\epsilon}{\lambda})\times(C_{x}+t-t_{j+1})}},{\bar{\mathcal{L}}^{n}(B^{n}(t_{j}),\bar{B}^{n}(t_{j},t_{j+1}))}\rangle$
$\displaystyle\geq[\tilde{B}(t_{j+1})-\tilde{B}(t_{j})]\nu_{F}(C_{0}+\frac{\tilde{R}_{L,j}}{\lambda})\nu_{G}(C_{x}+t-t_{j})-(\frac{3}{\lambda}+2)M_{B}\epsilon-3\epsilon.$
Note that
$\sum_{j=0}^{J-1}[\tilde{B}(t_{j+1})-\tilde{B}(t_{j})]F^{c}(\frac{\tilde{R}_{U,j}}{\lambda})G^{c}(x+t-t_{j})$
and
$\sum_{j=0}^{J-1}[\tilde{B}(t_{j+1})-\tilde{B}(t_{j})]F^{c}(\frac{\tilde{R}_{L,j}}{\lambda})G^{c}(x+t-t_{j+1})$
serve as the upper and lower Reimann sum of the integral
$\int_{t_{0}}^{t}F^{c}(\frac{\tilde{R}(s)}{\lambda})G^{c}(x+t-s)d\tilde{B}(s)$,
which converge to the integration as $n\to\infty$. So, by (5.28) and (5.29),
we have that for all large $n$,
$\Big{|}\bar{\mathcal{Z}}^{n}(t)(C_{x}^{\epsilon})-\bar{\mathcal{Z}}^{n}(0)(C_{x}^{\epsilon}+t)-\int_{t_{0}}^{t}F^{c}(\frac{\tilde{R}(s)}{\lambda})G^{c}(x+t-s)d\tilde{B}(s)\Big{|}\leq(\frac{3}{\lambda}+2)M_{B}\epsilon+3\epsilon+\epsilon.$
In summary, the right hand side of (5.39) can be bounded by a finite multiple
of $\epsilon$. We conclude that the left hand side of (5.39) must be 0 since
it does not depend on $\epsilon$, which can be arbitrary. This verifies (3.2).
The verification of fluid constrains (3.3) and (3.4) is quite straightforward.
Basically, it is just passing the fluid scaled stochastic constraints
$\displaystyle\bar{Q}^{n}(t)$ $\displaystyle=(\bar{X}^{n}(t)-1)^{+},$
$\displaystyle\bar{Z}^{n}(t)$ $\displaystyle=(\bar{X}^{n}(t)\wedge 1),$
to $n\to\infty$. We omit it for brevity. ∎
## 6 The Special Case with Exponential Distribution
In this section, we verify that the fluid model developed in this paper for
the general patience and service time distributions is consistent with the one
in [27], that was obtained in the special case where both distributions are
assumed to be exponential.
Our fluid model equations implies the key relationship (4.5). Now, we
specialize in the case with exponential distribution, i.e.
$F(t)=F_{e}(t)=1-e^{-\alpha t},\quad G(t)=G_{e}(t)=1-e^{-\mu t},\quad\textrm{
for all }t\geq 0.$
Now (4.5) becomes
$\begin{split}\bar{X}(t)&=\zeta_{0}(t)+\rho\int_{0}^{t}\big{[}1-\frac{\alpha}{\lambda}\big{(}(\bar{X}(t-s)-1)^{+}\big{)}\big{]}\mu
e^{-\mu s}ds+\int_{0}^{t}(\bar{X}(t-s)-1)^{+}\mu e^{-\mu s}ds.\end{split}$
In the case of exponential service time distribution, the remaining service
time of those initially in service and the service times of those initially
waiting in queue are also assumed to be exponentially distributed. So we have
$\zeta_{0}(t)=\bar{\mathcal{Z}}_{0}(C_{0}+t)+\bar{Q}_{0}e^{-\mu
t}=\bar{X}_{0}e^{-\mu t},$
where $\bar{X}_{0}=\bar{Z}_{0}+\bar{Q}_{0}$ is the initial number of customers
in the system. By some algebra, the above two equations can be simplified as
the following,
$\bar{X}(t)=\bar{X}_{0}e^{-\mu t}+\rho[1-e^{-\mu
t}]+(\mu-\alpha)\int_{0}^{t}(\bar{X}(t-s)-1)^{+}e^{-\mu s}ds.$ (6.1)
By the change of variable $t-s\to s$, the above integration can be written as
$\int_{0}^{t}(\bar{X}(t-s)-1)^{+}e^{-\mu s}ds=e^{-\mu
t}\int_{0}^{t}(\bar{X}(s)-1)^{+}e^{\mu s}ds.$
Taking the derivative on both sides of (6.1) yields
$\displaystyle\bar{X}^{\prime}(t)$ $\displaystyle=-\mu X_{0}e^{-\mu t}+\mu\rho
e^{\mu t}$ $\displaystyle\quad+(\mu-\alpha)[-\mu e^{-\mu
t}\int_{0}^{t}(\bar{X}(s)-1)^{+}e^{\mu s}ds+e^{-\mu t}(\bar{X}(t)-1)^{+}e^{\mu
t}]$ $\displaystyle=-\mu X_{0}e^{-\mu t}-\mu\rho[1-e^{\mu t}]+\mu\rho$
$\displaystyle\quad-\mu(\mu-\alpha)e^{-\mu
t}\int_{0}^{t}(\bar{X}(s)-1)^{+}e^{\mu s}ds+(\mu-\alpha)(\bar{X}(t)-1)^{+}$
$\displaystyle=-\mu\bar{X}(t)+\mu\rho+(\mu-\alpha)(\bar{X}(t)-1)^{+}.$
Using the notation in [27], $a^{-}=-\min(0,a)$ for any $a\in\mathbb{R}$. Note
that $a=\min(a,1)+(a-1)^{+}=1-(a-1)^{-}+(a-1)^{+}$. So the above equation
further implies
$\bar{X}^{\prime}(t)=\mu(\rho-1)-\alpha(\bar{X}(t)-1)^{+}+\mu(\bar{X}(t)-1)^{-},\quad\textrm{for
all }t\geq 0.$
This equation is consistent with Theorem 2.2 in [27] ($\mu$ is assumed to be 1
in that paper).
## Acknowledgements
The author would like to express the gratitude to his Ph.D supervisors,
Professor Jim Dai and Professor Bert Zwart, for many useful discussions. The
author is grateful to Professor Christian Gromoll from the department of
mathematics at University of Virginia for suggesting a nice method on using
Skorohod representation theorem to make the presentation in Section 5.2
rigorous. This research is supported in part by National Science Foundation
grants CMMI-0727400 and CNS-0718701.
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## Appendix A A Convolution Equation
###### Lemma A.1.
Assume that $G(\cdot)$ is a distribution function with $G(0)<1$,
$\zeta(\cdot)\in\mathbf{D}([0,T],\mathbb{R})$, $H(\cdot)$ is a Lipschitz
continuous function, and $\rho\in\mathbb{R}$. There exists a unique solution
$x^{*}(\cdot)\in\mathbf{D}([0,T],\mathbb{R})$ to the following equation:
$x(t)=\zeta(t)+\rho\int_{0}^{t}H\big{(}(x(t-s)-1)^{+}\big{)}dG_{e}(s)+\int_{0}^{t}(x(t-s)-1)^{+}dG(s),$
(A.1)
where, $G_{e}$ is the equilibrium distribution of $G$ as defined in Section
3.1.
###### Proof.
Suppose $H(\cdot)$ is Lipschitz continuous with constant $L$. The equilibruim
distribution has density $\mu[1-G(\cdot)]$, so
$|G_{e}(t)-G_{e}(s)|\leq\mu|t-s|$ for any $s,t\in\mathbb{R}$. Since $G(0)<1$,
there exists $b>0$ such that
$\kappa:=\rho L[G_{e}(b)-G_{e}(0)]+[G(b)-G(0)]<1.$
Now consider the space $\mathbf{D}([0,b],\mathbb{R})$ (all real valued càdlàg
functions on $[0,b]$, c.f. Section 1.1) is a subset of the Banach space of
bounded, measurable functions on $[0,b]$, equipped with the sup norm. One can
check that this subset is closed in the Banach space. Thus, the space
$\mathbf{D}([0,b],\mathbb{R})$ itself, equipped with the uniform metric
$\upsilon_{T}$ (defined in Section 1.1), is complete.
For any $y\in\mathbf{D}([0,b],\mathbb{R})$, define $\Psi(y)$ by
$\Psi(y)(t)=\zeta(t)+\rho\int_{0}^{t}H\left((y(t-s)-1)^{+}\right)dG_{e}(s)+\int_{0}^{t}(y(t-s)-1)^{+}dG(s),$
for any $t\in[0,b]$. By convention, the integration $\int_{0}^{t}y(t-s)dF(s)$
is interpreted to be $\int_{(0,t]}y(t-s)dF(s)$ (c.f. Page 43 in [3]). We prove
the existence and uniqueness of the solution to equation (A.1) by showing that
$\Psi$ is a contraction mapping on $\mathbf{D}([0,b],\mathbb{R})$. According
to the proof of Lemma A.1 in [31], the convolution of a càdlàg function with a
distribution function is still a càdlàg function. So $\Psi$ is a mapping from
$\mathbf{D}([0,b],\mathbb{R})$ to $\mathbf{D}([0,b],\mathbb{R})$. Next, we
show that the mapping $\Psi$ is a contraction. For any
$y,y^{\prime}\in\mathbf{D}([0,b],\mathbb{R})$, we have that
$\displaystyle\upsilon_{b}[\Psi(y),\Psi(y^{\prime})]$
$\displaystyle\leq\sup_{t\in[0,b]}\rho\int_{0}^{t}L\big{|}(y(t-s)-1)^{+}-(y^{\prime}(u-v)-1)^{+}\big{|}dG_{e}(s)$
$\displaystyle\quad+\sup_{t\in[0,b]}\int_{0}^{t}\big{|}(y(t-s)-1)^{+}-(y^{\prime}(t-s)-1)^{+}\big{|}dG(s)$
$\displaystyle\leq\rho
L\int_{0}^{b}\upsilon_{b}[y,y^{\prime}]dG_{e}(s)+\int_{0}^{b}\upsilon_{b}[y,y^{\prime}]dG(s)$
$\displaystyle\leq\kappa\upsilon_{b}[y,y^{\prime}].$
Since $\kappa<1$, the mapping $\Psi$ is a contraction. By the contraction
mapping theorem (c.f. Theorem $3.2$ in [12]), $\Psi$ has a unique fixed point
$x$, i.e. $x=\psi(x)$. This implies that $x\in\mathbf{D}([0,b],\mathbb{R})$ is
the unique solution to equation (A.1) on $[0,b]$.
It now remains to extend the existence and uniqueness result from $[0,b]$ to
$[0,T]$. Denote $x_{b}(t)=x(b+t)$,
$\zeta_{b}(t)=\zeta(b+t)+\rho\int_{t}^{b+t}H\left((x(b+t-s)-1)^{+}\right)dG_{e}(s)+\int_{t}^{b+t}(x(b+t-s)-1)^{+}dG(s)$,
then we have for $t\in[0,T-b]$,
$x_{b}(t)=\zeta_{b}(t)+\rho\int_{0}^{t}H\left((x_{b}(t-s)-1)^{+}\right)dG_{e}(s)+\int_{0}^{t}(x_{b}(t-s)-1)^{+}dG(s).$
(A.2)
It follows from the previous argument that there is unique solution
$x_{b}(\cdot)$ to the above equation. Thus, we obtain a unique extension of
the solution to (A.1) on the interval $[0,2b]$. Repeating this approach for
$N$ time with $N\geq\lceil{T/b}\rceil$ gives a unique solution on the interval
$[0,T]$. ∎
###### Lemma A.2.
Assume the same condition as in Lemma A.1. Let
$x(\cdot)\in\mathbf{D}([0,T],\mathbb{R})$ be the solution to equation (A.1).
If $\rho=\lambda/\mu$ with $\lambda,\mu>0$ ($\mu$ is the mean of $G$) ,
$H(x)\geq 0$ for all $x\geq 0$, and $\zeta(\cdot)$ satisfies the following
condition
$\zeta(t)=h(t)+(\zeta(0)-1)^{+}[1-G(t)],$ (A.3)
where $h(\cdot)$ is a non-increasing function, then the function
$(x(t)-1)^{+}-\lambda\int_{0}^{t}H\left((x(s)-1)^{+}\right)ds$
is non-increasing on the interval $[0,T]$.
###### Proof.
To simplify the notation, let $Q(t)=(x(t)-1)^{+}$ and
$D(t)=Q(t)-\lambda\int_{0}^{t}H\left(Q(s)\right)ds$ (A.4)
for all $t\in[0,T]$. Since $G_{e}(\cdot)$ is the equilibrium distribution, we
have
$\displaystyle x(t)$
$\displaystyle=\zeta(t)+\rho\int_{0}^{t}H\left(Q(t-s)\right)\mu[1-G(s)]ds+\int_{0}^{t}Q(t-s)dG(s)$
$\displaystyle=\zeta(t)+\lambda\int_{0}^{t}H\left(Q(s)\right)ds-\lambda\int_{0}^{t}H\left(Q(s)\right)G(t-s)ds+\int_{0}^{t}Q(t-s)dG(s).$
Applying Fubini’s Theorem (c.f. Theorem 8.4 in [17]) to the second to the last
integral in the above, we have
$\displaystyle\int_{0}^{t}H\left(Q(s)\right)G(t-s)ds$
$\displaystyle=\int_{0}^{t}\int_{0}^{t-s}H\left(Q(s)\right)dG(\tau)ds$
$\displaystyle=\int_{0}^{t}\int_{0}^{t-\tau}H\left(Q(s)\right)dsdG(\tau).$
So we obtain
$x(t)-\lambda\int_{0}^{t}H\left(Q(s)\right)ds=\zeta(t)+\int_{0}^{t}\left[Q(t-s)-\lambda\int_{0}^{t-s}H\left(Q(\tau)\right)d\tau\right]dG(s).$
According to the above definition of $D(\cdot)$, we have
$\left(x(t)\wedge 1\right)+D(t)=\zeta(t)+\int_{0}^{t}D(t-s)dG(s).$ (A.5)
It now remains to use (A.5) to show that $D(\cdot)$ is non-increasing, i.e.
for any $t,t^{\prime}\in[0,T]$ with $t\leq t^{\prime}$, we have $D(t)\geq
D(t^{\prime})$. Since $G(0)<1$, there exists $a>0$ such that $G(a)<1$. We
first show that $D(\cdot)$ is non-increasing on the interval $[0,a]$. Let
$D^{*}=\sup_{\\{(t,t^{\prime})\in[0,a]\times[0,a]:t\leq
t^{\prime}\\}}D(t^{\prime})-D(t).$
Since $D(\cdot)$ is càdlàg, according to Theorem 6.2.2 in the supplement of
[26], it is bounded on the interval $[0,a]$. Thus, $D^{*}$ is finite. We will
prove by contradiction that $D^{*}\leq 0$, which shows that $D(\cdot)$ is non-
increasing on $[0,a]$. Assume on the contrary that $D^{*}>0$. Applying (A.5),
we have
$\displaystyle D(t^{\prime})-D(t)$ $\displaystyle=(x(t)\wedge
1)-(x(t^{\prime})\wedge 1)+\zeta(t^{\prime})-\zeta(t)$
$\displaystyle\quad+\int_{0}^{t^{\prime}}D(t^{\prime}-s)dG(s)-\int_{0}^{t}D(t-s)dG(s)$
$\displaystyle=(x(t)\wedge 1)-(x(t^{\prime})\wedge
1)+\zeta(t^{\prime})-\zeta(t)$
$\displaystyle\quad+\int_{t}^{t^{\prime}}D(t^{\prime}-s)dG(s)+\int_{0}^{t}[D(t^{\prime}-s)-D(t-s)]dG(s).$
It follows from (A.1) and (A.4) that $D(0)=(\zeta(0)-1)^{+}$. This together
with condition (A.3) implies that
$\zeta(t^{\prime})-\zeta(t)=h(t^{\prime})-h(t)+D(0)[G(t)-G(t^{\prime})].$
(A.6)
So
$\begin{split}D(t^{\prime})-D(t)&=(x(t)\wedge 1)-(x(t^{\prime})\wedge
1)+h(t^{\prime})-h(t)\\\
&\quad+\int_{t}^{t^{\prime}}[D(t^{\prime}-s)-D(0)]dG(s)+\int_{0}^{t}[D(t^{\prime}-s)-D(t-s)]dG(s).\end{split}$
(A.7)
If $x(t^{\prime})<1$, by (A.4),
$\displaystyle
D(t^{\prime})-D(t)=-\lambda\int_{t}^{t^{\prime}}H\left(Q(s)\right)ds-Q(t),$
which is always non-positive; if $x(t^{\prime})\geq 1$, then $(x(t)\wedge
1)-(x(t^{\prime})\wedge 1)\leq 0$. So it follows from (A.7) and $h(\cdot)$
being non-increasing that
$\displaystyle D(t^{\prime})-D(t)$
$\displaystyle\leq\int_{t}^{t^{\prime}}[D(t^{\prime}-s)-D(0)]dG(s)+\int_{0}^{t}[D(t^{\prime}-s)-D(t-s)]dG(s)$
$\displaystyle\leq\int_{0}^{t^{\prime}}D^{*}dG(s)=D^{*}G(t^{\prime})\leq
D^{*}G(a),$
where the last inequality follows from the assumption that $D^{*}$ is non-
negative. Summarizing both cases of $x(t^{\prime})$, we have
$D(t^{\prime})-D(t)\leq\max(0,D^{*}G(a))$
for all $t,t^{\prime}\in[0,a]>0$ with $t\leq t^{\prime}$. Taking the supremum
on both sides over the set $\\{(t,t^{\prime})\in[0,a]\times[0,a]:t\leq
t^{\prime}\\}$ gives $D^{*}\geq F(a)D^{*}$. This implies that
$[1-G(a)]D^{*}\leq 0$. Since $G(a)<1$, it contradicts the assumption that
$D^{*}>0$. So we must have $D^{*}\leq 0$, this implies that $D(\cdot)$ is non-
increasing on $[0,a]$. We next extend this property to the interval $[0,T]$
using induction. Suppose we can show that $D(\cdot)$ is non-decreasing on the
interval $[0,na]$ for some $n\in\mathbb{N}$. Introduce $D_{na}(t)=D(na+t)$,
$x_{na}(t)=x(na+t)$ and
$\zeta_{na}(t)=\zeta(na+t)+\int_{0}^{na}D(na-s)dG(t+s).$ (A.8)
It is clear that the shifted functions satisfy
$\left(x_{na}(t)\wedge
1\right)+D_{na}(t)=\zeta_{na}(t)+\int_{0}^{t}D_{na}(t-s)dG(s).$ (A.9)
To show that $D(\cdot)$ is non-increasing on $[na,(n+1)a]$ is the same as to
show that $D_{na}(\cdot)$ is non-increasing on $[0,a]$. For this purpose, it
is enough to verify that $\zeta_{na}(\cdot)$ satisfy the condition (A.6).
Performing integration by parts on (A.8) gives
$\displaystyle\zeta_{na}(t)$
$\displaystyle=h(na+t)+(\zeta(0)-1)^{+}[1-G(na+t)]+\int_{0}^{na}D(na-s)dG(t+s)$
$\displaystyle=h(na+t)+(\zeta(0)-1)^{+}[1-G(na+t)]$
$\displaystyle\quad+D(0)G(na+t)-D(na)G(t)-\int_{0}^{na}G(t+s)dD(na-s).$
It follows from (A.1) and (A.4) that $D(0)=(\zeta(0)-1)^{+}$, so we can write
$\zeta_{na}(\cdot)$ as
$\displaystyle\zeta_{na}(t)=h_{na}(t)+D_{na}(0)[1-G(t)],$
where
$h_{na}(t)=h(na+t)+(\zeta(0)-1)^{+}-D_{na}(0)-\int_{0}^{na}G(t+s)dD(na-s)$.
Since $G(\cdot)$ is non-decreasing and $D(\cdot)$ is non-increasing, the
integral $-\int_{0}^{na}G(t+s)dD(na-s)$ is non-increasing as a function of
$t$. So we can conclude that $h_{na}(\cdot)$ is non-increasing, i.e.
$\zeta_{na}(\cdot)$ satisfies condition (A.6). Thus, we extend the non-
increasing interval to $[0,(n+1)a]$. By induction, the function $D(\cdot)$ is
non-increasing on the interval $[0,T]$. ∎
## Appendix B Glivenko-Cantelli Estimates
An important preliminary result is the following Glivenko-Cantelli estimate.
It is used in Section 5. It is convenient to state it as a general result,
since the Glivenko-Cantelli estimate requires weaker conditions and gives
stronger results than those in this paper.
For each $n$, let $\\{u^{n}_{i}\\}_{i\in\mathbb{Z}}$ be a sequence of i.i.d.
random variables with probability measure $\nu_{F}^{n}(\cdot)$, let
$\\{u^{n}_{i}\\}_{i\in\mathbb{Z}}$ be a sequence of i.i.d. random variables
with probability measure $\nu_{G}^{n}(\cdot)$. For any $n,m\in\mathbb{Z}$ and
$l\in\mathbb{R}_{+}$, define
$\displaystyle\bar{\mathcal{L}}^{n}_{F}(m,l)$
$\displaystyle=\frac{1}{n}\sum_{i=m+1}^{m+\lfloor{nl}\rfloor}\delta_{u^{n}_{i}},$
(B.1) $\displaystyle\bar{\mathcal{L}}^{n}_{G}(m,l)$
$\displaystyle=\frac{1}{n}\sum_{i=m+1}^{m+\lfloor{nl}\rfloor}\delta_{v^{n}_{i}},$
(B.2) $\displaystyle\bar{\mathcal{L}}^{n}(m,l)$
$\displaystyle=\frac{1}{n}\sum_{i=m+1}^{m+\lfloor{nl}\rfloor}\delta_{(u^{n}_{i},v^{n}_{i})},$
(B.3)
where $\delta_{x}$ denotes the Dirac measure of point $x$ on $\mathbb{R}$ and
$\delta_{(x,y)}$ denotes the Dirac measure of point $(x,y)$ on
$\mathbb{R}\times\mathbb{R}$. So $\bar{\mathcal{L}}^{n}_{F}(m,l)$ and
$\bar{\mathcal{L}}^{n}_{G}(m,l)$ are measures on $\mathbb{R}$ and
$\bar{\mathcal{L}}^{n}(m,l)$ is a measure on $\mathbb{R}\times\mathbb{R}$.
Denote $C_{x}=(x,\infty)$, for all $x\in\mathbb{R}$. We define two classes of
testing functions by
$\displaystyle\mathscr{V}$
$\displaystyle=\left\\{1_{C_{x}}(\cdot):x\in\mathbb{R}\right\\},$
$\displaystyle\mathscr{V}_{2}$ $\displaystyle=\left\\{1_{C_{x}\times
C_{y}}(\cdot,\cdot):x,y\in\mathbb{R}\right\\}.$
It is clear that $\mathscr{V}$ is a set of functions on $\mathbb{R}$ and
$\mathscr{V}_{2}$ is a set of functions on $\mathbb{R}\times\mathbb{R}$.
Define an envelop function for $\mathscr{V}$ as follows. Since
$\nu_{F}^{n}\to\nu_{F}$, by Skorohod representation theorem, there exists
random variables $X^{n}$ (with law $\nu_{F}^{n}$) and $X$ (with law
$\nu_{F}$), such that $X^{n}\to X$ almost surely as $r\to\infty$. Thus there
exists a random variable $X^{*}$ such that almost surely,
$X^{*}=\sup_{r}X^{n}.$
Let $\nu_{F}^{*}$ be the law of $X^{*}$. Since $L_{2}(\nu_{F}^{*})$ (the space
of square integrable functions with respect to the measure $\nu_{F}^{*}$)
contains continuous unbounded functions, there exists a continuous unbounded
function $f_{\nu_{F}}:\mathbb{R}_{+}\to\mathbb{R}$ that is increasing,
satisfies $f_{\nu_{F}}\geq 1$ and
$\langle{f_{\nu_{F}}^{2}},{\nu_{F}}\rangle<\infty$. Similarly, based on the
weak convergence $\nu_{G}^{n}\to\nu_{G}$, we can construct a function
$f_{\nu_{G}}$ that is increasing, satisfies $f_{\nu_{G}}\geq 1$ and
$\langle{f_{\nu_{G}}^{2}},{\nu_{G}}\rangle<\infty$. Now, define function
$\bar{f}:\mathbb{R}_{+}\to\mathbb{R}$ by
$\bar{f}(x)=\min\left(f_{\nu_{F}}(x),f_{\nu_{G}}(x)\right)$ and function
$\bar{f}_{2}:\mathbb{R}_{+}\times\mathbb{R}_{+}\to\mathbb{R}$ by
$\bar{f}_{2}(x,y)=\min\left(f_{\nu_{F}}(x),f_{\nu_{G}}(y)\right)$ for all
$x,y\in\mathbb{R}_{+}$. Note that we have to following properties,
$\displaystyle\bar{f}\textrm{ is increasing and unbounded},$ (B.4)
$\displaystyle f\leq\bar{f}\textrm{ for all }f\in\mathscr{V},$ (B.5)
$\displaystyle f\leq\bar{f}_{2}\textrm{ for all }f\in\mathscr{V}_{2}.$ (B.6)
So we call $\bar{f}$ and $\bar{f}_{2}$ the envelop function for $\mathscr{V}$
and $\mathscr{V}_{2}$ respectively. Finally, let
$\bar{\mathscr{V}}=\\{\bar{f}\\}\cup\mathscr{V}$ and
$\bar{\mathscr{V}}_{2}=\\{\bar{f}_{2}\\}\cup\mathscr{V}_{2}$.
###### Lemma B.1.
Assume that
$\nu^{n}_{F}\to\nu_{F},\quad\nu^{n}_{G}\to\nu_{G}\textrm{ as }n\to\infty.$
Fix constants $M,L>0$. For all $\epsilon,\eta>0$,
$\displaystyle\limsup_{n\to\infty}\mathbb{P}^{n}\Big{(}{\max_{-nM<m<nM}\sup_{l\in[0,L]}\sup_{f\in\bar{\mathscr{V}}}\Big{|}\langle{f},{\bar{\mathcal{L}}^{n}_{F}(m,l)}\rangle-l\langle{f},{\nu_{F}^{n}}\rangle\Big{|}>\epsilon}\Big{)}<\eta,$
$\displaystyle\limsup_{n\to\infty}\mathbb{P}^{n}\Big{(}{\max_{-nM<m<nM}\sup_{l\in[0,L]}\sup_{f\in\bar{\mathscr{V}}}\Big{|}\langle{f},{\bar{\mathcal{L}}^{n}_{G}(m,l)}\rangle-l\langle{f},{\nu_{G}^{n}}\rangle\Big{|}>\epsilon}\Big{)}<\eta,$
$\displaystyle\limsup_{n\to\infty}\mathbb{P}^{n}\Big{(}{\max_{-nM<m<nM}\sup_{l\in[0,L]}\sup_{f\in\bar{\mathscr{V}}_{2}}\Big{|}\langle{f},{\bar{\mathcal{L}}^{n}(m,l)}\rangle-l\langle{f},{(\nu_{F}^{n},\nu_{G}^{n})}\rangle\Big{|}>\epsilon}\Big{)}<\eta.$
This kind of results have been widely used in the study of measure valued
processes, see [8, 10, 31]. The proof of the first two inequalities in the
above lemma follows exactly the same way as the one for Lemma $B.1$ in [31],
and the proof of the third inequality in the above lemma follows exactly the
same as the one for Lemma $5.1$ in [10]. We omit the proof for brevity. By the
same reasoning as for Lemma 5.2, there exists a function
$\epsilon_{\text{GC}}(\cdot)$, which vanishes at infinity such that the
$\epsilon$ and $\eta$ in the above lemma can be replaced by the function
$\epsilon_{\text{GC}}(n)$ for each index $n$. Based on this, we construct the
following event,
$\begin{split}\Omega^{n}_{\text{GC}}(M,L)&=\Big{\\{}\max_{-nM<m<nM}\sup_{l\in[0,L]}\sup_{f\in\bar{\mathscr{V}}}\Big{|}\langle{f},{\bar{\mathcal{L}}^{n}_{F}(m,l)}\rangle-l\langle{f},{\nu_{F}^{n}}\rangle\Big{|}\leq\epsilon_{\text{GC}}(n)\Big{\\}}\\\
&\quad\cap\Big{\\{}\max_{-nM<m<nM}\sup_{l\in[0,L]}\sup_{f\in\bar{\mathscr{V}}}\Big{|}\langle{f},{\bar{\mathcal{L}}^{n}_{G}(m,l)}\rangle-l\langle{f},{\nu_{G}^{n}}\rangle\Big{|}\leq\epsilon_{\text{GC}}(n)\Big{\\}}\\\
&\quad\cap\Big{\\{}\max_{-nM<m<nM}\sup_{l\in[0,L]}\sup_{f\in\bar{\mathscr{V}}_{2}}\Big{|}\langle{f},{\bar{\mathcal{L}}^{n}(m,l)}\rangle-l\langle{f},{(\nu_{F}^{n},\nu_{G}^{n})}\rangle\Big{|}\leq\epsilon_{\text{GC}}(n)\Big{\\}}.\end{split}$
(B.7)
It is clear that for any fixed $M,L>0$,
$\lim_{n\to\infty}\mathbb{P}^{n}\Big{(}{\Omega^{n}_{\text{GC}}(M,L)}\Big{)}=1.$
(B.8)
Intuitively, on the event $\Omega^{n}_{\text{GC}}(M,L)$ (whose probability
goes to 1 as $n\to\infty$ for any fixed constants $M,L$), the measures
$\bar{\mathcal{L}}^{n}_{F}(m,l)$, $\bar{\mathcal{L}}^{n}_{G}(m,l)$ and
$\bar{\mathcal{L}}^{n}(m,l)$ are very “close” to $l\nu_{F}^{n}$,
$l\nu_{G}^{n}$ and $l(\nu_{F}^{n},\nu_{G}^{n})$, respectively.
## Appendix C An Extension of Skorohod Representation Theorem
In this section, we present a slight extension, Lemma C.1 below, of the
Skorohod Representation Theorem (c.f. Theorem 3.2.2 in [26]). The proof of
Lemma C.1 is built on the proof of Theorem 3.2.2 provided in the supplement of
[26], with slight extension to deal with the product of two matric spaces.
Let $(\mathbf{E}_{1},\pi_{1})$ and $(\mathbf{E}_{2},\pi_{2})$ be two complete
and separable metric spaces. Let $(\mathbf{E}_{1}\times\mathbf{E}_{2},\pi)$
denote the product space of them, with the product metric $\pi$ obtained by
the maximum metric.
###### Lemma C.1.
Consider a sequence of random variables $\\{(X_{n},Y_{n}),n\geq 1\\}$ in the
product space $\mathbf{E}_{1}\times\mathbf{E}_{2}$. If $X_{n}\Rightarrow X$,
then there exists other random elements of
$\mathbf{E}_{1}\times\mathbf{E}_{2}$, $\\{(\tilde{X}_{n},\tilde{Y}_{n}),n\geq
1\\}$, and $\tilde{X}$, defined on a common underlying probability space, such
that
$(\tilde{X}_{n},\tilde{Y}_{n})\stackrel{{\scriptstyle
d}}{{=}}(X_{n},Y_{n}),n\geq 1,\quad\tilde{X}\stackrel{{\scriptstyle d}}{{=}}X$
and almost surely,
$\tilde{X}_{n}\to\tilde{X}\quad\textrm{as }n\to\infty.$
###### Proof.
In order to present the proof, we first need some preliminaries. A nested
family of countably partitions of a set $A$ is a collection of subsets
$A_{i_{1},\ldots,i_{k}}$ indexed by $k$-tuples of positive integers such that
$\\{A_{i}:i\geq 1\\}$ is a partition of $A$ and
$\\{A_{i_{1},\ldots,i_{k+1}}:i_{k+1}\geq 1\\}$ is a partition of
$A_{i_{1},\ldots,i_{k}}$ for all $k\geq 1$ and
$(i_{1},\ldots,i_{k})\in\mathbb{N}_{+}^{k}$. Let $\mathbb{P}_{1}$ denote the
probability measure on the space where $X$ lives on. Since the space
$(\mathbf{E}_{1},\pi_{1})$ is separable, according to Lemma 1.9 in the
supplement of [26], there exists a nested family of countably partitions
$\\{E^{1}_{i_{1},\ldots,i_{k}}\\}$ of $(\mathbf{E}_{1},\pi_{1})$ that
satisfies
$\displaystyle\text{rad}(E^{1}_{i_{1},\ldots,i_{k}})<2^{-k},$ (C.1)
$\displaystyle\mathbb{P}_{1}(\partial E^{1}_{i_{1},\ldots,i_{k}})=0,$ (C.2)
where $\text{rad}(A)$ denotes the radius of the set $A$ in a metric space, and
$\partial(A)$ denote the boundary of the set $A$. Since the space
$(\mathbf{E}_{2},\pi_{2})$ is separable, by the same lemma, there exists a
nested sequence of countably partitions
$\\{E^{2}_{i^{\prime}_{1},\ldots,i^{\prime}_{k^{\prime}}}\\}$ of
$(\mathbf{E}_{2},\pi_{2})$ that satisfies
$\displaystyle\text{rad}(E^{2}_{i^{\prime}_{1},\ldots,i^{\prime}_{k^{\prime}}})<2^{-k^{\prime}}.$
(C.3)
Note that for space $(\mathbf{E}_{2},\pi_{2})$, we only need a weaker version
of Lemma 1.9 in the supplement of [26].
The first step is to use this nested sequence of countably partitions to
construct random variables $\\{(\tilde{X}_{n},\tilde{Y}_{n}),n\geq 1\\}$ with
the same distribution for each $n$. For $n\geq 1$, we first construct
subintervals $I^{n}_{i_{1},\ldots,i_{k}}\subseteq[0,1)$ corresponding to the
marginal probability of $X_{n}$. Let
$I^{n}_{1}=[0,\mathbb{P}^{n}(E^{1}_{1}\times\mathbf{E}_{2}))$ and
$I^{n}_{i}=\Big{[}\sum_{j=1}^{i-1}\mathbb{P}^{n}(E^{1}_{j}\times\mathbf{E}_{2}),\sum_{j=1}^{i}\mathbb{P}^{n}(E^{1}_{j}\times\mathbf{E}_{2})\Big{)},\quad
i>1,$
where $\mathbb{P}^{n}$ is the probability measure on the space where
$(X_{n},Y_{n})$ lives. Let $\\{I^{n}_{i_{1},\ldots,i_{k+1}}:i_{k+1}\geq 1\\}$
be a countable partition of subintervals of $I^{n}_{i_{1},\ldots,i_{k}}$. If
$I^{n}_{i_{1},\ldots,i_{k}}=[a_{n},b_{n})$, then
$I^{n}_{i_{1},\ldots,i_{k+1}}=\Big{[}a_{n}+\sum_{j=1}^{i_{k+1}-1}\mathbb{P}^{n}(E^{1}_{i_{1},\ldots,i_{k},j}\times\mathbf{E}_{2}),a_{n}+\sum_{j=1}^{i_{k+1}}\mathbb{P}^{n}(E^{1}_{i_{1},\ldots,i_{k},j}\times\mathbf{E}_{2})\Big{)}.$
The length of each subinterval $I^{n}_{i_{1},\ldots,i_{k}}$ is the probability
$\mathbb{P}^{n}(E^{1}_{i_{1},\ldots,i_{k}}\times\mathbf{E}_{2})$. We then
construct further subintervals
$I^{n}_{i_{1},\ldots,i_{k};i^{\prime}_{1},\ldots,i^{\prime}_{k^{\prime}}}\subseteq
I^{n}_{i_{1},\ldots,i_{k}}$ corresponding to $(X_{n},Y_{n})$. If
$I^{n}_{i_{1},\ldots,i_{k}}=[a_{n},b_{n})$, then let
$I^{n}_{i_{1},\ldots,i_{k};1}=[a_{n},a_{n}+\mathbb{P}^{n}(E^{1}_{i_{1},\ldots,i_{k}}\times
E^{2}_{1}))$ and
$I^{n}_{i_{1},\ldots,i_{k};i^{\prime}}=\Big{[}a_{n}+\sum_{j^{\prime}=1}^{i^{\prime}-1}\mathbb{P}^{n}(E^{1}_{i_{1},\ldots,i_{k}}\times
E^{2}_{j^{\prime}}),a_{n}+\sum_{j^{\prime}=1}^{i^{\prime}}\mathbb{P}^{n}(E^{1}_{i_{1},\ldots,i_{k}}\times
E^{2}_{j^{\prime}})\Big{)},\quad i^{\prime}>1.$
Let
$\\{I^{n}_{i_{1},\ldots,i_{k};i^{\prime}_{1},\ldots,i^{\prime}_{k^{\prime}+1}}:i^{\prime}_{k^{\prime}+1}\geq
1\\}$ be countable partition of
$I^{n}_{i_{1},\ldots,i_{k};i^{\prime}_{1},\ldots,i^{\prime}_{k^{\prime}}}$. If
$I^{n}_{i_{1},\ldots,i_{k};i^{\prime}_{1},\ldots,i^{\prime}_{k^{\prime}}}=[a_{n},b_{n})$,
then
$\begin{split}&\quad
I^{n}_{i_{1},\ldots,i_{k};i^{\prime}_{1},\ldots,i^{\prime}_{k^{\prime}+1}}\\\
&=\Big{[}a_{n}+\sum_{j^{\prime}=1}^{i^{\prime}_{k^{\prime}+1}-1}\mathbb{P}^{n}(E^{1}_{i_{1},\ldots,i_{k}}\times
E^{2}_{i^{\prime}_{1},\ldots,i^{\prime}_{k},j^{\prime}}),a_{n}+\sum_{j^{\prime}=1}^{i^{\prime}_{k^{\prime}+1}}\mathbb{P}^{n}(E^{1}_{i_{1},\ldots,i_{k}}\times
E^{2}_{i^{\prime}_{1},\ldots,i^{\prime}_{k},j^{\prime}})\Big{)}.\end{split}$
The length of each subinterval
$I^{n}_{i_{1},\ldots,i_{k};i^{\prime}_{1},\ldots,i^{\prime}_{k^{\prime}}}$ is
the probability $\mathbb{P}^{n}(E^{1}_{i_{1},\ldots,i_{k}}\times
E^{2}_{i^{\prime}_{1},\ldots,i^{\prime}_{k^{\prime}}})$. Now from each
nonempty subset $E^{1}_{i_{1},\ldots,i_{k}}\times
E^{2}_{i^{\prime}_{1},\ldots,i^{\prime}_{k}}$ we choose one point
$(x_{i_{1},\ldots,i_{k}},y_{i^{\prime}_{1},\ldots,i^{\prime}_{k}})$. For each
$n\geq 1$ and $k\geq 1$, we define functions
$(x^{k}_{n},y^{k}_{n}):[0,1)\to\mathbf{E}_{1}\times\mathbf{E}_{2}$ by letting
$x^{k}_{n}(w)=x_{i_{1},\ldots,i_{k}}$ and
$y^{k}_{n}(w)=y_{i^{\prime}_{1},\ldots,i^{\prime}_{k}}$ for $\omega\in
I^{n}_{i_{1},\ldots,i_{k};i^{\prime}_{1},\ldots,i^{\prime}_{k}}$. By the
nested partition property and inequalities C.1 and C.3,
$\pi\big{(}(x^{k}_{n}(\omega),x^{k}_{n}(\omega)),(x^{k+j}_{n}(\omega),x^{k+j}_{n}(\omega))\big{)}<2^{-k}\quad\textrm{for
all }j,k,n$
and $\omega\in[0,1)$. Since $(\mathbf{E}_{1}\times\mathbf{E}_{2},\pi)$ is a
complete metric space, the above implies that there is
$(x_{n}(\omega),y_{n}(\omega))\in\mathbf{E}_{1}\times\mathbf{E}_{2}$ such that
$\pi\big{(}(x^{k}_{n}(\omega),x^{k}_{n}(\omega)),(x_{n}(\omega),x_{n}(\omega))\big{)}\to
0\quad\textrm{as }k\to\infty.$
We let $(\tilde{X}_{n},\tilde{Y}_{n})=(x_{n},y_{n})$ on $[0,1)$ for $n\geq 0$.
The next step is to construct $\tilde{X}$ and show that
$\tilde{X}_{n}\to\tilde{X}$ almost surely. For each $n\geq 1$, let
$\mathbb{P}^{n}_{1}$ denote the marginal probability of $X^{n}$. It is clear
that $I^{n}_{i_{1},\ldots,i_{k}}$ is the probability
$\mathbb{P}^{n}_{1}(E^{1}_{i_{1},\ldots,i_{k}})$. By (C.2), we have that
$\mathbb{P}^{n}_{1}(E^{1}_{i_{1},\ldots,i_{k}})\to\mathbb{P}_{1}(E^{1}_{i_{1},\ldots,i_{k}})$,
as $n\to\infty$. Consequently, the length of the interval
$I^{n}_{i_{1},\ldots,i_{k}}$ converges to the length of the interval
$I_{i_{1},\ldots,i_{k}}$, which is defined in a similar way as for
$I^{n}_{i_{1},\ldots,i_{k}}$ by letting
$I_{i_{1},\ldots,i_{k+1}}=\Big{[}a_{n}+\sum_{j=1}^{i_{k+1}-1}\mathbb{P}_{1}(E_{i_{1},\ldots,i_{k},j}),a_{n}+\sum_{j=1}^{i_{k+1}}\mathbb{P}_{1}(E_{i_{1},\ldots,i_{k},j})\Big{)},$
if $I_{i_{1},\ldots,i_{k}}=[a_{n},b_{n})$. Now from each nonempty subset
$E_{i_{1},\ldots,i_{k}}$ we choose one point $x_{i_{1},\ldots,i_{k}}$. For
each $k\geq 1$, we define functions $x^{k}:[0,1)\to\mathbf{E}_{1}$ by letting
$x^{k}(\omega)=x_{i_{1},\ldots,i_{k}}$ for $\omega\in
I^{n}_{i_{1},\ldots,i_{k}}$. By the nested partition property and inequalities
C.1,
$\pi_{1}(x^{k}(\omega),x^{k+j}(\omega))<2^{-k}\quad\textrm{for all }j,k$
and $\omega\in[0,1)$. Since $(\mathbf{E}_{1},\pi_{1})$ is a complete metric
space, the above implies that there is $x(\omega)\in\mathbf{E}_{1}$ such that
$\pi_{1}(x^{k}(\omega),x(\omega))\to 0\quad\textrm{as }k\to\infty.$
We let $\tilde{X}=x$ on $[0,1)$. Since
$\displaystyle\pi_{1}(\tilde{X}_{n}(\omega),\tilde{X}(\omega))$
$\displaystyle\leq\pi_{1}(\tilde{X}_{n}(\omega),\tilde{X}^{k}_{n}(\omega))+\pi_{1}(\tilde{X}^{k}_{n}(\omega),\tilde{X}^{k}(\omega))+\pi_{1}(\tilde{X}^{k}(\omega),\tilde{X}(\omega))$
$\displaystyle\leq 3\times 2^{-k},$
for all $\omega$ in the interior of $I_{i_{1},\ldots,i_{k}}$,
$\lim_{n\to\infty}\pi_{1}(\tilde{X}_{n}(\omega),\tilde{X}(\omega))\leq 3\times
2^{-k}.$
Since $k$ is arbitrary, we must have
$\tilde{X}_{n}(\omega)\to\tilde{X}(\omega)$ as $n\to\infty$ for all but at
most countably many $\omega\in[0,1)$.
It remains to show that $(\tilde{X}_{n},\tilde{Y}_{n})$ has the probability
laws $\mathbb{P}^{n}$. Let $\tilde{\mathbb{P}}$ denote the Lebesque measure on
$[0,1)$. It suffices to show that
$\tilde{\mathbb{P}}((\tilde{X}_{n},\tilde{Y}_{n})\in A)=\mathbb{P}^{n}(A)$ for
each $A$ such that $\mathbb{P}^{n}(\partial A)=0$. Let $A$ be such a set. Let
$A^{k}$ be the union of the sets $E^{1}_{i_{1},\ldots,i_{k}}\times
E^{2}_{i^{\prime}_{1},\ldots,i^{\prime}_{k}}$ such that
$E^{1}_{i_{1},\ldots,i_{k}}\times
E^{2}_{i^{\prime}_{1},\ldots,i^{\prime}_{k}}\subseteq A$ and let
${A^{\prime}}^{k}$ be the union of the sets $E^{1}_{i_{1},\ldots,i_{k}}\times
E^{2}_{i^{\prime}_{1},\ldots,i^{\prime}_{k}}$ such that
$E^{1}_{i_{1},\ldots,i_{k}}\times
E^{2}_{i^{\prime}_{1},\ldots,i^{\prime}_{k}}\cap A\neq\emptyset$. Then
$A^{k}\subseteq A\subseteq{A^{\prime}}^{k}$ and, by the construction above,
$\tilde{\mathbb{P}}((\tilde{X}_{n},\tilde{Y}_{n})\in
A^{k})=\mathbb{P}^{n}(A^{k})\textrm{ and
}\tilde{\mathbb{P}}((\tilde{X}_{n},\tilde{Y}_{n})\in{A^{\prime}}^{k})=\mathbb{P}^{n}({A^{\prime}}^{k})$
Now let $C^{k}=\\{s\in\mathbf{E}_{1}\times\mathbf{E}_{2}:\pi(s,\partial A)\leq
2^{-k}\\}$. Then ${A^{\prime}}^{k}-A^{k}\downarrow\partial A$ as $k\to\infty$.
Since $\mathbb{P}^{n}(\partial A)=0$ by assumption,
$\mathbb{P}^{n}(C^{k})\downarrow 0$ as $k\to\infty$. Hence
$\tilde{\mathbb{P}}((\tilde{X}_{n},\tilde{Y}_{n})\in
A)=\lim_{k\to\infty}\tilde{\mathbb{P}}((\tilde{X}_{n},\tilde{Y}_{n})\in
A^{k})=\lim_{k\to\infty}\mathbb{P}^{n}(A^{k})=\mathbb{P}^{n}(A).$
Following the same way, we can show that $\tilde{X}$ has probability law
$\mathbb{P}_{1}$. ∎
|
arxiv-papers
| 2009-09-09T15:39:38 |
2024-09-04T02:49:05.174174
|
{
"license": "Public Domain",
"authors": "Jiheng Zhang",
"submitter": "Jiheng Zhang",
"url": "https://arxiv.org/abs/0909.1671"
}
|
0909.1677
|
# Structure and energetics of Si(111)-(5$\times$2)-Au
Steven C. Erwin Center for Computational Materials Science, Naval Research
Laboratory, Washington, DC 20375, USA Ingo Barke Institut für Physik,
Universität Rostock, D-18051 Rostock, Germany F. J. Himpsel Department of
Physics, University of Wisconsin-Madison, Madison, WI 53706, USA
###### Abstract
We propose a new structural model for the Si(111)-(5$\times$2)-Au
reconstruction. The model incorporates a new experimental value of 0.6
monolayer for the coverage of gold atoms, equivalent to six gold atoms per
5$\times$2 cell. Five main theoretical results, obtained from first-principles
total-energy calculations, support the model. (1) In the presence of silicon
adatoms the periodicity of the gold rows spontaneously doubles, in agreement
with experiment. (2) The dependence of the surface energy on the adatom
coverage indicates that a uniformly covered phase is unstable and will phase-
separate into empty and covered regions, as observed experimentally. (3)
Theoretical scanning tunneling microscopy images are in excellent agreement
with experiment. (4) The calculated band structure is consistent with angle-
resolved photoemission spectra; analysis of their correspondence allows the
straightforward assignment of observed surface states to specific atoms. (5)
The calculated activation barrier for diffusion of silicon adatoms along the
row direction is in excellent agreement with the experimentally measured
barrier.
###### pacs:
68.43.Bc,73.20.At,68.37.Ef,68.43.Jk
## I Introduction
Forty years ago Bishop and Riviere first observed the (111) surface of silicon
to reconstruct, with fivefold periodicity, in the presence of gold.
bishop_j_phys_d_appl_physics_1969a Since that time the
Si(111)-(5$\times$2)-Au reconstruction has been widely studied, in several
hundred publications, as a prototype linear metallic chain system in which the
physics of one-dimensional metals is approximately
realized.hasegawa_j_phys_condens_matter_2000a ;
matsuda_j_phys_condens_matter_2007a
These investigations have provided very substantial insights into many aspects
of Si(111)-(5$\times$2)-Au.barke_solid_state_comm_2007a ;
barke_appl_surf_sci_2007a Less successful have been the many attempts to use
the clues provided by experiment to construct a complete structural model for
this complicated reconstruction. Beginning with the early work of LeLay over a
dozen different models have been proposed.lelay_surf_sci_1977a ;
berman_phys_rev_b_1988a ;
hasegawa_journal_of_vacuum_science__technology_a_1990a ; bauer_surf_sci_1991a
; schamper_phys_rev_b_1991a ; seehofer_surf_sci_1995a ;
marks_phys_rev_lett_1995a ; plass_surf_sci_1997a ; omahony_phys_rev_b_1994a ;
omahony_surf_sci_1992a ; shibata_phys_rev_b_1998b ; hasegawa_surf_sci_1996a ;
hasegawa_phys_rev_b_1996a ; hasegawa_surf_sci_1996b ;
erwin_phys_rev_lett_2003a ; kang_surf_sci_2003a ; riikonen_phys_rev_b_2005a ;
ren_phys_rev_b_2007a ; chuang_phys_rev_b_2008a All were eventually found to
be inconsistent with the results of scanning tunneling microscopy (STM),
angle-resolved photoemission spectroscopy (ARPES), or both.
Figure 1: (color online). Proposed structure of Si(111)-(5$\times$2)-Au with
gold coverage equal to 0.6 monolayer. Large yellow circles are gold, small
circles are silicon. The surface layer consists of a gold single row (S), gold
double row (D), and silicon honeycomb chain (HC). The surface energy is
minimized when this surface is decorated by silicon adatoms (dark blue) with
5$\times$4 periodicity, as shown. In the presence of adatoms the 5$\times$1
periodicity of the underlying substrate spontaneously doubles to 5$\times$2
(gray outline) due to dimerization within the gold double row.
In this paper we propose a new structural model for Si(111)-(5$\times$2)-Au
that is fully consistent with all experimental data to which we have compared.
The model is similar to one proposed by Erwin in
2003,erwin_phys_rev_lett_2003a but is modified to be consistent with a
recently revised value of the gold coverage.barke:155301 The modifications,
although seemingly minor, for the first time bring the predictions of the
model—for STM and ARPES as well as other phenomena—into excellent agreement
with experiment. More importantly, the new model opens the door to a more
fundamental physical understanding of Si(111)-(5$\times$2)-Au based on its
detailed atomic structure.
## II Structural model
The model proposed here is shown in Fig. 1. The basic structure is similar to
one proposed several years ago in Ref. erwin_phys_rev_lett_2003a, but there
are several important differences. For this reason it is useful to discuss
both the similarities and differences between the older model and the one
proposed here (hereafter the “2003 model” and the “2009 model”).
The starting point for the 2003 model was the experimental observation that
0.4 monolayer (ML) of gold induces a stable reconstruction of the Si(111)
surface. Using this coverage value the 2003 model was constructed with four
gold atoms per 5$\times$2 cell. As shown in Fig. 1 of Ref.
erwin_phys_rev_lett_2003a, , the gold atoms substitute for silicon atoms in
the topmost surface layer, forming two Au-Si chains oriented along the
[1$\bar{1}$0] direction. Adjacent to these two chains is a silicon “honeycomb
chain,” a thin graphitic strip of silicon that owes its stability to a Si=Si
double bond. The basic reconstruction just described has 5$\times$1
periodicity. The experimentally observed 5$\times$2 substrate periodicity was
argued to arise from a row of silicon rebonding atoms that bridges a channel
between one of the Au-Si rows and the silicon honeycomb chain. A full row of
rebonding atoms overcoordinates some atoms, while a half-occupied row, with
5$\times$2 periodicity, leaves some dangling bonds unsaturated. The latter
arrangement was argued to be energetically preferred when the system is doped
with extra electrons. These were supplied by silicon adatoms adsorbed on top
of the Au-Si chains.
The recently revised experimental determination of the gold coverage as 0.6 ML
obviously calls for a revised structural model as well.barke:155301 The 2009
model accommodates the additional two gold atoms per 5$\times$2 cell by
replacing the half-occupied rebonding row of silicon atoms from the 2003 model
with a full row of gold atoms. The other two building blocks of the 2003
model—the silicon honeycomb chain and the adsorbed silicon adatoms—are
unchanged in the new model. The total coverage of silicon atoms in the top
layer of the 2009 model varies, according to the coverage of adatoms, between
1.20 ML (for the undecorated surface) and 1.25 ML (for saturation coverage).
These values are consistent with the range of experimentally determined
silicon coverage, 1.1 to 1.3 ML (Ref. tanishiro1990a, ), 1.23$\pm$0.003 ML
(Ref. seifert_dissertation_2006a, ), and 1.3$\pm$0.1 ML (Ref. chin2007a, ).
The 2009 model is energetically more favorable than the 2003 model: the
theoretical surface energy is 17.1 meV/Å2 lower in the Au-rich limit, in which
the chemical potential is taken as the bulk energy per atom.chempotAu
Much of the rest of this paper will focus on the interesting role played by
the new row of gold atoms. But it is worth pausing briefly to place
Si(111)-(5$\times$2)-Au in the larger context of other metal-induced
reconstructions of Si(111). It is now known that a great variety of metal
adsorbates induce closely related reconstructions based on the silicon
honeycomb chain. But the details differ, sometimes with unexpected
consequences. Easily ionized adsorbates such as alkalis, alkaline earths, and
some rare earths form a family of “honeycomb-chain channel” (HCC)
reconstructions in which the adsorbates occupy channels between adjacent
silicon honeycomb chains.collazo-davila_phys_rev_lett_1998a ;
lottermoser_phys_rev_lett_1998a ; erwin_phys_rev_lett_1998a Each adsorbate is
three-fold coordinated by silicon atoms in the honeycomb chain, which are too
far away (3.0 Å) to form covalent bonds, consistent with a picture of ionic
charge donation and electrostatic attraction. The interactions between
adsorbates are also electrostatic, but repulsive.erwin_surf_sci_2005a Within
this family of HCC reconstructions the adsorbate coverage is determined by a
simple electron-counting rule first proposed by Lee et al.lee2001a ; lee2003a
and later generalized by Battaglia et al.battaglia2007a ; battaglia:075409
Adsorbates that are less ionic, such as silver and gold, also form
reconstructions based on the silicon honeycomb chain but the role of the
adsorbate is more interesting. Silver adsorbates occupy the channel of a HCC-
like reconstruction,collazo-davila_phys_rev_lett_1998a but the stronger
interaction between silver and silicon leads to a preference for two-fold
coordination of silver by silicon, at a much reduced distance of 2.6 Å. The
two-fold coordination brings silver adsorbates sufficiently close to each
other to allow pairing into silver dimers. Chuang et al.chuang2008a and
Urbieta et al.urbieta2009a showed that the phase of this pairing alternates
between adjacent channels, modulating the periodicity of the basic 3$\times$1
HCC reconstruction to a $c(12\times 2)$ variant.
Returning now to Si(111)-(5$\times$2)-Au, it appears that the role of the
adsorbate is still more complex. Gold is very reactive on silicon
surfaces.doremus2001a This reactivity is already evident in the Au-Si rows of
Fig. 1. Within these rows gold completely substitutes for the top layer of the
surface silicon bilayer, with each gold atom covalently bonded to three
silicon atoms. Likewise, each silicon atom at the edge of the silicon
honeycomb chain forms a covalent bond to a gold atom. The Au-Si bond lengths,
both within the Au-Si rows and bonded to the silicon honeycomb chain, are 2.5
Å, smaller than for any other adsorbate studied.
The most interesting aspect of the 2009 model is the behavior of the gold
double row labeled “Au D” in Fig. 1. As shown in the figure, the equilibrium
geometry of this double row is dimerized. Period doubling was also part of the
2003 model, but its origin—the half-occupied rebonding row—was simpler. In the
2009 model the dimerization occurs only in the presence of silicon adatoms, or
when the surface is doped with extra electrons. In Sec. IV we show that these
two scenarios are largely equivalent. We further demonstrate that in the
presence of adatoms the dimerization is driven by an unusual “double” Peierls
mechanism, in which the distortion opens two gaps in the band structure of the
undistorted 5$\times$1 substrate.
## III Methods
### III.1 Theoretical methods
First-principles total-energy calculations were used to determine equilibrium
geometries and relative energies of the basic model and its variants. The
calculations were performed in a slab geometry with four layers of Si plus the
reconstructed top surface layer and a vacuum region of 8 Å. All atomic
positions were relaxed, except the bottom Si layer and its passivating
hydrogen layer, until the largest force component on every atom was below 0.01
eV/Å. Total energies and forces were calculated within the PBE generalized-
gradient approximation to density-functional theory (DFT) using projector-
augmented-wave potentials, as implemented in vasp. kresse_phys_rev_b_1993a ;
kresse_phys_rev_b_1996a The plane-wave cutoff for all calculations was 250
eV.
The sampling of the surface Brillouin zone was chosen according to the size of
the surface unit cell and the relevant precision requirements. For example,
the dependence of the total energy on dimerization (Fig. 2) was calculated
using a 5$\times$4 unit cell and 2$\times$2 zone sampling, with convergence
checks using 4$\times$4 sampling. The dependence of the relative surface
energy on silicon adatom coverage (Fig. 3) requires greater precision because
the energy variations are smaller. Hence these surface energies were
calculated using a 5$\times$8 unit cell (to allow an adatom coverage of 1/8)
and 8$\times$4 sampling, with convergence checks using 12$\times$6 sampling.
Finally, the potential-energy surface for adatom diffusion (Fig. 7) was
calculated using a 5$\times$4 unit cell and 2$\times$2 zone sampling.
Simulated STM images (Fig. 4) were calculated using the method of Tersoff and
Hamann.tersoff_phys_rev_b_1985a For the filled-state image we integrated the
local density of states (LDOS) over a chosen energy window of occupied states
up to the Fermi level; for the empty-state image the integration was over
unoccupied states starting at the Fermi level. The simulated STM topography
under constant-current conditions was obtained by plotting the height at which
the integrated LDOS is constant.
### III.2 Experimental methods
Silicon wafers (from Virginia Semiconductors) were degassed for several hours
at 700 ∘C before flashing at 1250 ∘C for a few seconds. A rapid cool-down to
850 ∘C was followed by slow cooling to room temperature. An important
prerequisite for obtaining well-defined row structures was a flexible mount of
the samples that prevented strain from building up during high-temperature
flashing. Gold was evaporated from a Mo wire basket. The pressure was kept
below 5$\times$10−10 mbar throughout the sample preparation. All STM
measurements were carried out at room temperature with low tunneling currents
($\leq$50 pA).
Band dispersions were obtained using a Scienta 200 spectrometer with
$E,\theta$ multidetection and an energy resolution of 20 meV for electrons and
7 meV for photons. We used $p$-polarized synchrotron radiation at a photon
energy $h\nu$ = 34 eV, where the cross section of silicon surface states has a
maximum relative to the bulk states. The data are for a sample temperature
below 100 K.
## IV Energetics
This section addresses three issues related to the energetics of
Si(111)-(5$\times$2)-Au. We consider first the energetics of dimerization, and
show that dimerization occurs naturally within the 2009 model when silicon
adatoms or extra electrons are present. Next we examine how the surface energy
varies as a function of silicon adatom coverage; we find that the energy is
minimized for the coverage 1/4 shown in Fig. 1. Finally, we show that the
model explains the existence of a localized defect structure seen even on
carefully prepared surfaces.
### IV.1 Dimerization of the substrate
The model shown in Fig. 1 has 5$\times$4 periodicity, because silicon adatoms
decorate the surface in a 5$\times$4 arrangement. But it is clear from the
figure that the underlying substrate (that is, ignoring the adatoms) can be
understood more simply as a 5$\times$1 reconstruction whose periodicity is
doubled, along the chain direction, to 5$\times$2 by dimerization within the
Au double row. Understanding the nature and origin of this dimerization is
important for explaining many experimental aspects of Si(111)-(5$\times$2)-Au,
including the fine details of STM imagery, data from ARPES, the observation of
nanoscale phase separation,kirakosian_surf_sci_2003a ;
kirakosian_phys_rev_b_2003a ; mcchesney_phys_rev_b_2004a ;
yoon_phys_rev_b_2005a ; choi_phys_rev_lett_2008a the diffusion of silicon
adatoms on the surface,hasegawa_phys_rev_b_1996a ;
bussmann_phys_rev_lett_2008a and the existence and motion of domain walls
within the Au-Si rows.kang_phys_rev_lett_2008a
Figure 2: (color online). Variation of the total energy as a function of
dimerization along the chain direction. The dimensionless dimerization
parameter is $d=(a_{1}-a_{0})/a_{0}$, where $a_{1}$ is shown in the inset and
$a_{0}$ is the surface lattice constant. Total energies were calculated with
full relaxation for each constrained value of $d$.
We begin by defining more precisely the nature of the dimerization. The inset
to Fig. 2 shows a detail of the gold double row near a silicon adatom. Within
this double row the gold atoms have a ladder-like arrangement, with Au-Au
bonds as the rungs. In the absence of adatoms (or extra electrons) these rungs
are all parallel to each other, with a spacing equal to the silicon surface
lattice constant $a_{0}$. When adatoms decorate the surface in the 5$\times$4
arrangement shown in Fig. 1 the rungs rotate away from their parallel
alignment. This rotation occurs almost completely within the (111) surface
plane, and the rungs are quite rigid: the Au-Au bond length (2.94 Å) changes
by less than 1%. This rigidity is not surprising, because the Au-Au bond
length is already very close to the bulk gold bond length (2.88 Å). The sign
of the rotation alternates along the chain direction. Hence the dimerization
can be viewed as an antiferrodistortive instability.
To quantify the dimerization one could, of course, use the angle of rotation
of the Au-Au rungs. We choose instead a more physically transparent measure:
the distance $a_{1}$ between gold atoms on the side of the double row adjacent
to the silicon honeycomb chain, as labeled in Fig. 2. A dimensionless
dimerization parameter can then be defined, $d=(a_{1}-a_{0})/a_{0}$. For the
5$\times$4 arrangement of adatoms shown in Fig. 1, the equilibrium
dimerization parameter is $d_{\rm eq}=0.14$.
Some insight into the origin of the dimerization may be obtained by computing
the DFT total energy $E_{t}$ as a function of $d$ while relaxing all other
degrees of freedom. The results are shown in Fig. 2 for three variants of the
basic model: the undecorated and undoped surface (without adatoms or extra
electrons); the adatom-doped 5$\times$4 surface of Fig. 1; and the undecorated
surface doped with two extra electrons per 5$\times$4 cell. For the
undecorated undoped surface the minimum is at $d=0$, indicating that
dimerization is not stable. For the adatom-doped surface there is a single
minimum in the energy, as expected, at $d=+0.14$. For the electron-doped
surface the behavior of $E_{t}(d)$ is nearly indistinguishable, for positive
$d$, from that of the adatom-doped surface. This similarity strongly suggests
that each silicon adatom dopes two electrons to surface states. More
substantive evidence for this conjecture is found in the electronic structure
of these variants, as we show below in Sec. VI.
The behavior of $E_{t}(d)$ for negative $d$, however, is very different. This
is because adatom-doping strongly breaks the 5$\times$2 symmetry of the
surface, while electron-doping does not. This suggests another role for the
silicon adatoms: to pin the phase of the dimerization such that $d>0$ at the
position of the adatom. We will return to this role in Sec. VII when
considering the diffusion of adatoms.
Figure 3: Theoretical surface energy as a function of silicon adatom coverage.
Energies were calculated at the labeled coverages; the interpolating curve is
a guide to the eye. Adatoms occupy their preferred binding sites as shown in
Fig. 1. The surface energy is lowest for coverage 1/4, corresponding to the
fully saturated 5$\times$4 arrangement of adatoms. The dotted line highlights
a local maximum of the energy, and shows that a surface with coverage less
than 1/4 will phase separate into a mixture of empty and 1/4-covered regions,
as found experimentally.
### IV.2 Adatom coverage and phase separation
The preceding discussion and calculations were based on the assumption that
silicon adatoms decorate the surface in a 5$\times$4 arrangement, equivalent
to a coverage of 1/4 adatom per 5$\times$1 cell. This phase can be achieved
experimentally by depositing silicon onto the surface at temperatures around
300 ∘C until saturation is reached.bennewitz_nanotechnology_2002a But this
saturated phase is metastable: annealing causes half of the silicon adatoms to
diffuse away. The resulting equilibrium phase is not uniform, but instead
exhibits patches with local adatom coverage of 1/4 interspersed with patches
of undecorated surface; the global average adatom coverage is close to
1/8.kirakosian_surf_sci_2003a ; kirakosian_phys_rev_b_2003a ;
mcchesney_phys_rev_b_2004a ; yoon_phys_rev_b_2005a ; choi_phys_rev_lett_2008a
This experimental result poses two questions for theory. (1) What coverage of
silicon adatoms minimizes the calculated surface energy? (2) Does the observed
patchiness of the adatom distribution arise from an instability toward phase
separation?
To compare the energies of phases with different silicon adatom coverage, we
compute the relative surface energy
$E_{s}=E_{t}(N_{\rm Si})-N_{\rm Si}\;\mu_{\rm Si},$ (1)
where $E_{t}(N_{\rm Si})$ is the total energy of a surface unit cell
containing $N_{\rm Si}$ silicon atoms (including adatoms), and $\mu_{\rm Si}$
is the silicon chemical potential. Thermodynamic equilibrium between the
surface and the bulk requires $\mu_{\rm Si}$ to be the energy per atom in bulk
silicon. Although we do not explicitly consider the free energy at finite
temperature, configurational entropy effects may indeed play a role; see
below.
Within this formalism, adatoms are thermodynamically stable only if their
presence lowers the surface energy of the undecorated surface. Equation 1
shows that this requirement implies that the adsorption energy per adatom must
be greater than the silicon cohesive energy (5.4 eV within DFT/PBE).
Adsorption energies below this threshold imply that adatoms may be temporarily
metastable, but not thermodynamically stable. This is a very stringent
requirement, and one that is rarely satisfied in our experience (it was not in
the 2003 model).
Figure 4: Comparison of experimental and simulated (inset) STM images for
Si(111)-(5$\times$4)-Au. (a) Filled states, $V=-$0.7 eV. (b) Empty states,
$V=+$1.0 eV. The simulated images were obtained using slightly different
voltages: $V=-$1.0 and $+$0.5 eV, respectively. Projected positions are shown
for silicon adatoms and top-layer silicon and gold atoms; the size of the
circles indicates the height of the atoms.
We calculated the relative surface energies for five different silicon adatom
coverages. To minimize numerical uncertainties the same 5$\times$8 supercell
was used for each coverage, and full relaxation performed in every case. The
results are shown in Fig. 3. The surface energy is minimized for a coverage of
1/4. This is the coverage depicted in Fig. 1, and agrees well with the
experimentally observed local coverage within adatom-covered patches. In Sec.
VI we show why the value 1/4 is special: at this adatom coverage—but at no
other—the surface band structure is fully gapped and hence favored because
occupied states can move down in energy.
We turn now to the second question posed above. Of the five coverages
considered here, the second most-favorable is not the adjacent 1/8 or 3/8
phase, but rather the undecorated “empty” surface. In other words, the surface
energy of the intermediate 1/8 phase is higher than the average of the two
endpoint phases, empty and 1/4. This result suggests that a range of such
intermediate coverages between 0 and 1/4 may have energies above the tie line,
shown in Fig. 3, connecting the two endpoint phases. This implies that a
surface prepared with adatom coverage 1/8 (and perhaps any intermediate
coverage between 0 and 1/4) will phase separate into a mixture of empty and
1/4-covered regions.
Of course, this conclusion does not answer the related question of why the
observed global average coverage is close to 1/8. We hypothesize that the 1/8
phase may be stabilized at finite temperature by the entropy gained from
occupying only half the adatom sites of the 1/4 phase. We also do not address
here the characteristic size of the phase-separated regions; this would
require analyzing the energy cost of forming the phase boundary between the 0
and 1/4 phases. Finally, we leave for future analysis the possibility that the
phase-separated state benefits electrostatically from the charge transfer
proposed by Yoon et al. to take place between the 0 and 1/4
phases.yoon_phys_rev_b_2005a
### IV.3 Low-energy defect structure
In addition to the phase separation just discussed, STM images of
Si(111)-(5$\times$2)-Au sometimes reveal the presence of an occasional
“mirror-domain” defect.yoon_phys_rev_b_2005a ; seifert_dissertation_2006a The
signature of this defect is a slight shift, by roughly 3 Å to the right in
Fig. 1, of the bright protrusions that dominate the STM topography. The
intensity of the protrusions and the position and shape of the underlying
features are not affected.
A simple variation of the basic structural model accounts for the observed
defects. Starting from the structure shown in Fig. 1, one can construct a
mirror-image variant by reflecting the entire top layer of atoms through the
vertical plane bisecting the Si=Si double bonds of the honeycomb chain. (This
mirror plane was discussed in Ref. erwin_phys_rev_lett_1998a, and is shown in
Fig. 1 therein.) The reflection leaves the honeycomb chain unaffected but
reverses the ordering and orientation of the Au-Si rows. The geometry of the
rigidly reflected structure is already extremely close (within a few hundreths
of an Ångstrom) to the fully relaxed geometry. The reflection shifts the
silicon adatoms by 2.2 Å to the right, consistent with the observed shift.
The surface energy of the relaxed defect reconstruction is only slightly
higher, by 1.2 meV/Å2, than that of the original reconstruction. From
inspection of the atomic positions of the original and defect models, we judge
the energy cost of forming an interface between the two phases to be quite
small. Thus it is plausible that short sequences of this defect
structure—manifested as a small rightward shift of the silicon adatoms—should
appear even on carefully prepared surfaces, with minimal disruption to either
the primary reconstruction or the local adatom coverage.
## V Comparison with STM
Two decades of studies probing the Si(111)-(5$\times$2)-Au surface with STM
have created a richly detailed and consistent picture of its topography and
bias dependence. baski_phys_rev_b_1990a ; omahony_surf_sci_1992a ;
omahony_phys_rev_b_1994a ; shibata_phys_rev_b_1998b ; hasegawa_surf_sci_1996a
; hasegawa_phys_rev_b_1996a ; hasegawa_surf_sci_1996b ;
hasegawa_journal_of_vacuum_science__technology_a_1990a These data make
possible a very stringent test for any proposed structural model by comparing
theoretically simulated STM images to atomic-resolution experimental images.
In this section we show that simulated images based on the 2009 model are in
excellent and detailed agreement with recent experimental images. Moreover,
the new model resolves a small but irritating discrepancy found in comparisons
based on the earlier 2003 model.
The most pronounced features in STM imagery of the Si(111)-(5$\times$2)-Au
surface are the bright protrusions now well established as originating from
the silicon adatoms. In topographic maps these are generally the highest
features in both filled- and empty-state images. As discussed elsewhere
mcchesney_phys_rev_b_2004a and in the previous section, the adatoms occupy a
5$\times$4 lattice. Much detailed information is contained in the imagery in
between these lattice sites; the topography of this substrate has 5$\times$2
periodicity. Of special interest is the registry of the 5$\times$4 adatom
lattice and the 5$\times$2 substrate; the relative alignment of these two
lattices provides an important test for structural models. (We do not address
here the lack of correlation between the adatoms in different rows, previously
discussed in Ref. kirakosian_phys_rev_b_2003a, .)
Figure 4 shows experimental and theoretical simulated STM images for filled
and empty states of the Si(111)-(5$\times$2)-Au surface, in a region where the
silicon adatom coverage is 1/4. The agreement between experiment and theory is
excellent, and allows all of the main experimental features to be easily
identified. (1) The dark vertical channels separating the three main rows
shown in Fig. 4 arise from the doubly-bonded rungs of the silicon honeycomb
chains. (2) The bright protrusions are from the silicon adatoms. (3) The
triangular features with $\times$2 periodicity, constituting the right edge of
the rows, arise from the combination of two contributions: a pair of outer
gold atoms brought close by dimerization (the apex of the triangle) and two
silicon atoms at the left edge of the honeycomb chain (the base of the
triangle). (4) In the filled states, the left edge of the rows is not well
resolved, and appears to have either $\times$1 or very weak $\times$2
periodicity; these spots arise from the silicon atoms at the right edge of the
honeycomb chain. (5) In the empty states, the $\times$2 triangular features on
the right edge of the rows alternate with V-shaped features that open to the
left. These are a combination of two contributions: a pair of dimerized inner
gold atoms (the apex of the V) and Si-Au bonds that are slightly dimerized by
their proximity to the gold double row (the arms of the V).
In the experimental images the alignment of the 5$\times$4 adatom lattice and
the 5$\times$2 substrate topography is as follows. The bright protrusions are
located slightly to the left side within the main rows—except in the defect
regions discussed in the previous section, where the protrusions are shifted
to an equivalent location on the right side of the row. This off-center
location is accurately reproduced in the simulated images based on both the
original model and the defect model (not shown). Along the rows of the
experimental images, the 5$\times$4 protrusions are symmetrically straddled by
the 5$\times$2 triangular and V-shaped features of the substrate. Fig. 4 shows
that this symmetric registry is properly reproduced in the simulated
images—resolving a problem with the 2003 model first pointed out by Seifert
whereby the alignment of the two lattices was
asymmetric.seifert_dissertation_2006a
## VI Electronic structure
Photoemission studies of Si(111)-(5$\times$2)-Au began in the mid-1990s and
continue today.collins_surf_sci_1995a ;
okuda_j_electron_spectroscopy_and_rel_phenom_1996a ; okuda_appl_surf_sci_1997a
; hill_phys_rev_b_1997a ; hill_appl_surf_sci_1998a ; losio_phys_rev_lett_2000a
; altmann_phys_rev_b_2001a ;
himpsel_journal_of_electron_spectroscopy_and_related_phenomena_2002a ;
zhang_phys_rev_b_2002b ; mcchesney_phys_rev_b_2004a ; choi_phys_rev_lett_2008a
These studies have led to important insights into the electronic structure of
this complicated surface and provide tests complementary to STM for evaluating
structural models. In this section we address three aspects of the electronic
structure of Si(111)-(5$\times$2)-Au: the detailed mechanism that drives the
period doubling discussed in Sec. IV; a comparison of the theoretical band
structure to angle-resolved photoemission data; and an explanation of which
specific surface orbitals give rise to the observed band structure.
Figure 5: Calculated band structure of Si(111)-Au in four different scenarios.
(a) 5$\times$1 undecorated undoped surface. (b) 5$\times$2 electron-doped
surface (2 electrons per 5$\times$4 cell). (c) 5$\times$4 adatom-doped surface
(1 adatom per 5$\times$4 cell); (d) 5$\times$4 adatom-doped surface with spin-
orbit coupling included. In each panel the bands are plotted in the same
5$\times$4 surface Brillouin zone. The Fermi levels in panels (b), (c), and
(d) are set to zero, and the zone-boundary degeneracies in (a) and (b) are
aligned in order to highlight the evolution of the bands with doping.
Projected bulk silicon bands are shown in gray. The formation of a
hybridization gap at a band crossing (circled) and the opening of a gap at the
zone boundary (arrows) are indicated. The observed surface is phase-separated
into undecorated 5$\times$2 regions and adatom-doped 5$\times$4 regions in
equal proportion.
### VI.1 Origin of substrate period doubling
In earlier sections the geometry and energetics of the period doubling was
explored without any consideration of the underlying mechanism. Here we
suggest an explanation, based on the electronic structure, for why the
5$\times$1 substrate dimerizes to 5$\times$2 in the presence of silicon
adatoms or extra electrons.
We begin by considering a variant of the full 5$\times$4 model of Fig. 1 in
which the silicon adatoms are absent. When this surface is relaxed within DFT
the dimerization vanishes and hence the periodicity reverts to 5$\times$1\.
The theoretical band structure for this surface is shown in Fig. 5(a) for wave
vectors parallel to the chain direction and energies near the projected band
gap. The bands are plotted in the Brillouin zone of the full 5$\times$4 model
so that comparison with the bands of the full model can later be made. The
folding of the 5$\times$1 bands into the 5$\times$4 zone creates degeneracies
at the zone center $\Gamma$ and the 5$\times$4 zone boundary A4; the most
important zone-boundary degeneracy is marked by an arrow. The folding also
creates band crossings, the most important of which (circled) is inside the
band gap, about one-fourth of the way from $\Gamma$ to A4. The Fermi level is
very close to the top of the valence band and the system is metallic.
Next we consider how this band structure changes when we dope the surface with
extra electrons. It was demonstrated in Fig. 2 that a doping level of 2
electrons per 5$\times$4 cell leads to stable 5$\times$2 dimerization, with
distortion energetics nearly identical to that from silicon adatoms at 1/4
coverage. Figure 5(b) shows the band structure from this electron-doped
surface. The electron doping has two important consequences: the
antiferrodistortive dimerization creates a large hybridization gap at the band
crossing of the the undoped surface, and the Fermi level is pushed into this
gap. The band degeneracy at the 5$\times$4 zone boundary remains intact (see
arrow), because the electron-doped surface has perfect 5$\times$2 periodicity.
In the presence of silicon adatoms at 1/4 coverage this last degeneracy is
lifted. The last two panels of Fig. 5 show the bands calculated at two levels
of theory: (c) scalar relativistic; (d) fully relativistic with spin-orbit
coupling. (We will show in the next section that the gold character of these
surface bands is substantial, hence the spin-orbit splitting is large, about
0.2 eV.) Panel (d) shows that the system now develops a full gap. At 1/4
adatom coverage the Fermi level falls just inside this gap, making the system
insulating.
To summarize, we find that silicon adatoms at coverage 1/4 create a multiband
metal-insulator transition on Si(111)-(5$\times$2)-Au. The first
(electronically induced) gap arises from band hybridization originating from
dimerization and the resultant lowering of symmetry. The second (adatom
induced) gap arises from the 5$\times$4 potential of the adatoms, which lifts
the degeneracy at the 5$\times$4 zone boundary A4.
Figure 6: (color online) Comparison of angle-resolved photoemission data to
the theoretical band structure. (a) Photoemission-derived band dispersion
reproduced from Ref. mcchesney_phys_rev_b_2004a, . Band ${\bf 1^{\prime}}$
(red) has strong 5$\times$2 periodicity, band ${\bf 2}$ (yellow) has mainly
5$\times$1 periodicity, and band ${\bf 1^{\prime\prime}}$ (gray) has
5$\times$4 periodicity and becomes more intense with increasing silicon adatom
coverage.choi_phys_rev_lett_2008a Inset: Brillouin zones for 5$\times$1,
5$\times$2, and 5$\times$4 surface unit cells. (b) Theoretical band structure
of the 5$\times$2 electron-doped surface. The diameter of each circle is
proportional to the contribution from surface atoms in the Au-Si chains of
Fig. 1. (c) Theoretical band structure of the hypothetical undecorated undoped
5$\times$1 surface; these are to be compared with the folded bands in panels
(a) and (b). The labeled surface bands originate from gold and silicon
orbitals in the single (S) and double (D) rows marked in Fig. 1. The double
row leads to two bands, consisting of bonding (Db) and antibonding (Da)
combinations of orbitals.
### VI.2 Comparison with angle-resolved photoemission
The above discussion focused on bands above the Fermi level. Now we turn to
the occupied states, where we can make direct comparison to experimental data.
The experimental band structure is highly one-dimensional near the Fermi
level, and becomes gradually more two-dimensional for lower energies; here we
limit our discussion to the one-dimensional dispersion along the chain
direction. The results of our ARPES studies of Si(111)-(5$\times$2)-Au are
summarized in Fig. 6(a), which is reproduced from an earlier publication.
mcchesney_phys_rev_b_2004a By combining momentum- and energy-distribution
curves from several Brillouin zones, three bands can be identified: ${\bf
1^{\prime}}$, ${\bf 1^{\prime\prime}}$, and ${\bf 2}$. (These labels are used
by analogy to comparable bands at stepped Si(111)
surfaces.crain_phys_rev_b_2004a ) Figure 6(a) shows the bands in the repeated-
zone scheme of a surface with 5$\times$1 periodicity, for which A1 is the zone
boundary. The 5$\times$2 period doubling introduces backfolded replicas of
these bands, shown as dashed lines. The backfolded band ${\bf 1^{\prime}}$ is
indeed found where expected, indicating that this band has strong 5$\times$2
character. The backfolded replica of band ${\bf 2}$ is too weak to be
observed, indicated that it has mainly 5$\times$1 character. Band ${\bf
1^{\prime\prime}}$ becomes more intense as the silicon adatom coverage is
increased, indicating that it has strong 5$\times$4
character.choi_phys_rev_lett_2008a
It is difficult to compare directly the theoretical bands of the adatom-doped
surface to these experimental results, because the 5$\times$2 bands of Fig.
6(a) must be folded once more into the Brillouin zone of the 5$\times$4 cell.
This folding creates many bands in a small energy interval and overly
complicates the comparison between theory and experiment. We choose instead a
simpler and clearer approximate approach: to compare the experimental
5$\times$2 bands of Fig. 6(a) to the theoretical 5$\times$2 bands of the
electron-doped surface with no silicon adatoms. In doing so it must be kept in
mind that the 5$\times$4 potential of the adatoms, already seen to play an
important role for the unoccupied bands, will be absent.
Figure 6(b) shows the calculated bands for the 5$\times$2 electron-doped
surface, plotted in the 5$\times$1 repeated-zone scheme used in panel (a).
(The projected bulk bands, however, are shown for reasons of clarity in the
extended-zone scheme.) The diameter of each circle is proportional to the
summed projections of the state onto gold and silicon atoms in the single and
double Au-Si chains of Fig. 1. The solid colored curves represent our best
effort to match the three strongest surface bands to the ARPES bands. The
overall agreement for bands ${\bf 1^{\prime}}$ and ${\bf 2}$ is excellent,
despite the complexity of the calculated bands even for this simplified
surface. Note that in our interpretation, ${\bf 1^{\prime}}$ and ${\bf 2}$ are
not simple parabolic bands as depicted in Fig. 6(a). Instead, each comprises
two or more bands and exhibits several avoided crossings. Further support for
this interpretation will be presented in the next subsection, where we discuss
the orbital origin of the bands.
The agreement appears less satisfactory for band ${\bf 1^{\prime\prime}}$.
Although this band is correctly centered at the A2 point, and the shallow
dispersion near that point reasonable, it is shifted rigidly up in energy by
0.2 eV compared to experiment. We believe that this shift is a spurious effect
arising from the omission of silicon adatoms in the calculation. Indeed, Choi
et al. have shown experimentally that the ${\bf 1^{\prime\prime}}$ band shifts
upward as the adatom coverage is reduced. choi_phys_rev_lett_2008a For the
range of coverages studied (between 37 and 97% of the saturation 1/4 coverage)
the shift in energy was linear. Extrapolating this result to a surface free of
adatoms, one naively expects an upward shift in energy of the ${\bf
1^{\prime\prime}}$ band by 0.21 eV. Such a shift would bring the results of
ARPES and theory into excellent agreement for all three bands.
### VI.3 Orbital origin of the bands
We now make one last theoretical simplification by eliminating the extra
electrons. Upon relaxation the undoped surface is no longer dimerized, and the
periodicity reverts to 5$\times$1\. This simplification is justified if the
perturbation of the bands from the dimerization and the extra electrons is
small. In the limiting case of zero dimerization the exact 5$\times$2 bands
can be obtained from the 5$\times$1 bands by simple zone folding. Figure 6(c)
shows the calculated band structure for the simplified surface, plotted in the
conventional 5$\times$1 reduced-zone scheme. It is readily apparent that the
bands in (b) can indeed be accurately obtained from those in (c) by first
folding the 5$\times$1 bands about the zone midpoint A2, and then shifting the
Fermi level upward by the appropriate amount (0.3 eV) for electron doping of
rigid bands.
The advantage of the simplified 5$\times$1 band structure is that its orbital
origin is easy to analyze, because the bands fall almost entirely in the
projected gap and rarely cross. By examining individual states in real space,
we find that the middle band S originates primarily from gold and silicon
orbitals in the single chain “Au S” in Fig. 1. The other two bands are more
complicated. They arise from the double chain labeled “Au D.” The two bands
are the low-lying bonding (Db) and higher-lying antibonding (Da) combinations
of orbitals of the gold atoms that constitute each rung of the ladder
(combined with orbitals on the connecting silicon atoms).
By comparing the three panels of Fig. 6 we can now understand the orbital
origin of the ARPES bands. Band ${\bf 1^{\prime}}$ is the simplest and
corresponds to the antibonding Da band. Band ${\bf 2}$ is a superposition of
bands S and Db, which are energetically for energies below the Fermi level.
Band ${\bf 1^{\prime\prime}}$ has more complicated origin. It arises from two
effects: strong rehybridization (related to the dimerization) around A2 of
bands S and Db, and the energy shift discussed above from the adatom
potential.
We can also now better understand the two gaps created in Fig. 5 by adatom
doping. The orbital origin of the those bands can now be assigned by noting,
first, that the bands in Fig. 5(a) are identical to those in Fig. 6(c), folded
twice along the labeled vertical lines. It should be clear that the
electronically induced gap illustrated in Fig. 5(b) arises from hybridization
of bands S and Db, which cross near $\Gamma$ when folded into the 5$\times$4
zone. Likewise, the adatom-induced gap illustrated in Fig. 5(c) is created
entirely within the Da band at the second A4 point, which folds into the A4
zone boundary of the 5$\times$4 zone.
Figure 7: (color online). Theoretical potential energy surfaces for the
diffusion of a silicon adatom along the chain direction. Two approximations
were considered for determining the diffusion pathway. (1) Substrate atoms
were fixed rigidly at their equilibrium positions attained when adatoms are in
their lowest energy site (circles). (2) Substrate atoms were allowed to
conform to their “instantaneous” equilibrium positions as the adatom diffused
(squares). These two limiting cases give estimates for the activation barrier
of 1.6 and 1.1 eV, respectively. The experimentally measured barrier is 1.24
$\pm$ 0.08 eV.bussmann_phys_rev_lett_2008a
## VII Diffusion of silicon adatoms
The silicon adatoms decorating Si(111)-(5$\times$2)-Au are not immobile.
hasegawa_phys_rev_b_1996a Sequences of STM images of surfaces prepared with
adatom coverage near the equilibrium value 1/8 show evidence for diffusion of
adatoms along the Au-Si chain direction, by a defect-mediated mechanism of
unknown origin.bussmann_phys_rev_lett_2008a From measurements of the mean-
square displacement at different temperatures an effective activation barrier
of 1.24$\pm$0.08 eV was extracted.bussmann_phys_rev_lett_2008a In this
section we examine whether the structural model proposed in Fig. 1 can account
for this diffusion barrier.
Although it is usually straightforward to calculate a barrier for the
diffusion of an atom on a well-defined surface, Si(111)-(5$\times$2)-Au
presents an interesting complication. Normally one studies the diffusion of a
single atom in a supercell representing the clean surface, and relaxation of
the surface around the diffusing atom is allowed. But for
Si(111)-(5$\times$2)-Au the dimerization of the substrate is determined by the
location (and coverage) of the adatoms whose very motion is under study. Thus,
if the supercell is taken to be 5$\times$4 and the substrate is allowed to
relax, then the phase of the dimerization will track the position of the
adatom. This “conforming substrate” is physically unrealistic and will
underestimate the true diffusion barrier.
A more plausible scenario, in which the phase of the dimerization remains
fixed while the adatom diffuses, is difficult to implement without constraints
or a larger supercell. We take a simpler approach and adopt a perfectly rigid
5$\times$4 substrate, which will overestimate the true barrier. The true
barrier must then be between those of the conforming and the rigid substrate.
These two potential energy surfaces were calculated using a 5$\times$4 unit
cell in which the projected position of the adatom along the chain direction
was constrained. For the conforming scenario, all other degrees of freedom
were relaxed. For the rigid scenario, the two remaining adatom degrees of
freedom were relaxed while all other atoms were were fixed at their original
5$\times$4 positions.
Figure 7 shows the two resulting potential-energy surfaces. For the conforming
substrate the potential-energy surface has 5$\times$2 periodicity and an
activation barrier of 1.1 eV, while for the rigid substrate the periodicity is
5$\times$4 and the activation barrier is 1.6 eV. These two barriers nicely
bracket the experimental barrier of 1.24 eV. More detailed studies using
realistic boundary conditions will likely play an important role in unraveling
the nature of adatom diffusion on this surface.
## VIII Outlook
The structural model proposed here for Si(111)-(5$\times$2)-Au resolves one of
the longest-standing unsolved reconstructions of the silicon surface.
Predictions based on this model—for the dimerization of the underlying
substrate, for the saturation coverage of silicon adatoms, for phase
separation into adatom-covered and empty regions, for detailed STM imagery,
for electronic band structure, and for the diffusion of adatoms—are in
excellent agreement with experiments. Moreover, the physical mechanisms
underlying many widely studied phenomena in this system have now been
elucidated.
A number of other issues awaiting theoretical study can now be directly
addressed. These include, for example, the suggestion that phase separation is
accompanied by charge separation and the formation of a Schottky barrier at
the interface between adatom-covered and undecorated regions;
yoon_phys_rev_lett_2004a the structure and motion of the domain walls, within
the Au-Si rows, that form when the spacing between two neighboring adatoms is
not commensurate with the 5$\times$2 substrate; kang_phys_rev_lett_2008a and
the exploration of the fundamental limits on using Si(111)-(5$\times$2)-Au to
store and manipulate digital information at densities comparable to that of
DNA.bennewitz_nanotechnology_2002a ; kirakosian_surf_sci_2003a
Finally, we anticipate a renewal of theoretical interest in
Si(111)-(5$\times$2)-Au as a physical realization of a nearly one-dimensional
metal. Most studies to date have used parametrized models because a definitive
structural model has not been available.liu_nanotechnol_2008a With a model in
hand, the door is now open to insights from new theoretical investigations as
well.
## IX Acknowledgements
This work was supported by the Office of Naval Research, and by the NSF under
awards No. DMR-0705145 and DMR-0084402 (SRC). S.C.E. gratefully acknowledges
many helpful discussions with Christoph Seifert. I. B. acknowledges support
from the DAAD. Computations were performed at the DoD Major Shared Resource
Center at AFRL.
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|
arxiv-papers
| 2009-09-09T13:11:48 |
2024-09-04T02:49:05.185808
|
{
"license": "Public Domain",
"authors": "Steven C. Erwin, Ingo Barke, and F.J. Himpsel",
"submitter": "Steven C. Erwin",
"url": "https://arxiv.org/abs/0909.1677"
}
|
0909.1679
|
# The dangers of deprojection of proper motions
Paul J. McMillan, and James J. Binney
Rudolf Peierls Centre for Theoretical Physics, 1 Keble Road, Oxford, OX1 3NP,
UK E-mail: p.mcmillan1@physics.ox.ac.uk
###### Abstract
We re-examine the method of deprojection of proper motions, which has been
used for finding the velocity ellipsoid of stars in the nearby Galaxy. This
method is only legitimate if the lines of sight to the individual stars are
uncorrelated with the stars’ velocities. Very simple models are used to show
that spurious results similar to ones recently reported are obtained when
velocity dispersion decreases with galactocentric radius in the expected way.
A scheme that compensates for this bias is proposed.
###### keywords:
Galaxy: fundamental parameters – methods: statistical – Galaxy: kinematics and
dynamics
## 1 Introduction
Dehnen & Binney (1998, hereafter DB98) introduced a method for deprojecting
proper-motion data, which allowed them to explore the velocity distribution of
nearby stars in the Hipparcos catalogue (ESA, 1997), without knowing their
radial velocities. This works by taking a weighted ensemble average of the
proper motions of stars found in different parts of the sky, under the
assumption that the velocity distribution is uncorrelated with position on the
sky. This assumption was legitimate in the case of the sample studied by DB98
because all its stars lay within $\sim 100\,\mathrm{pc}$ of the Sun, so it was
reasonable to approximate the full phase space distribution function by the
velocity-space distribution at the Sun: $f(\mathbf{x},\mathbf{v})\simeq
f(\mathbf{x}_{\odot},\mathbf{v})$.
Recently Fuchs et al. (2009, hereafter F09) used the DB98 technique to study a
sample of stars taken from the Sloan Digital Sky Survey (SDSS: Abazajian et
al., 2009). This data set contains stars that extend up to $\sim
800\,\mathrm{pc}$ above the plane and span a range of galactocentric radii
$\sim 2\,\mathrm{kpc}$ wide. Since the velocity dispersion of stars varies
with both radius and distance from the plane, the validity of the assumption
that the velocity distribution is uncorrelated with sky position is
questionable for this spatially extended sample. In this paper we demonstrate
that applying the DB98 technique leads to erroneous results, particularly with
regard to the tilt of the velocity ellipsoid with respect to the Galactic
plane.
In Section 2 we briefly explain the DB98 method, and in Section 3 we
demonstrate that for a sample like that of F09 it gives a biased estimate of
the tilt of the velocity ellipsoid. Section 3.1 explains the origin of this
bias physically. Section 3.2 proposes a technique for removing the bias. In
Section 4 we discuss biases in the DB98 technique more generally.
## 2 Deprojection
The deprojection equations are stated and explained by DB98, and written out
in full by F09. We repeat them here for clarity.
We work in a Cartesian coordinate system, centred on the Sun, in which the
$x$-axis points towards the Galactic centre, the $y$-axis points in the
direction of Galactic rotation, and the $z$-axis points towards the north
Galactic pole. Given a star moving with heliocentric velocity
$\mathbf{v}\equiv(U,V,W)$, the observed proper-motion velocity is
$\mathbf{p}=\mathbf{v}-v_{\parallel}\mathbf{\hat{s}},$ (1)
where $\mathbf{\hat{s}}$ is the unit vector pointing from the Sun to the star,
and $v_{\parallel}$ is the component of $\mathbf{v}$ parallel to
$\mathbf{\hat{s}}$. This can be written in matrix form as
$\mathbf{p}=\mbox{\boldmath{$\mathsf{A}$}}\cdot\mathbf{v},\quad\mbox{where}\quad
A_{ij}=\delta_{ij}-\hat{s}_{i}\hat{s}_{j}.$ (2)
The velocity ellipsoid is defined by both the mean velocity and the velocity
dispersion. To determine the velocity dispersion tensor we use the equation
$\displaystyle p_{i}p_{j}$ $\displaystyle=$
$\displaystyle\sum_{k,l}A_{ik}v_{k}\,A_{jl}v_{l}$ (3) $\displaystyle=$
$\displaystyle\sum_{kl}B_{ijkl}v_{k}v_{l},$
where
$B_{ijkl}\equiv\frac{1}{2}(A_{ik}\,A_{jl}+A_{il}\,A_{jk})$ (4)
is the part of $\mathsf{A}$$\mathsf{A}$ that is symmetric in its last pair of
indices.
We are interested in situations in which we know $\mathbf{p}$ and
$\mathbf{\hat{s}}$ (and therefore $\mathsf{A}$ and $\mathsf{B}$) but do not
know $\mathbf{v}$. It is clear from the definition of $\mathbf{p}$ (equation
1) that in this case we cannot find $\mathbf{v}$ for an individual star
because we do not know $v_{\parallel}$. This is reflected in the fact that
$\mathsf{A}$ is singular.
We average equations (2) and (3) over a sample of stars. If the velocities
$\mathbf{v}$ of these stars are uncorrelated with their sky positions
$\mathbf{\hat{s}}$, they will be uncorrelated with $\mathsf{A}$ and
$\mathsf{B}$, and the expectation value of a product such as
$\mbox{\boldmath{$\mathsf{A}$}}\cdot\mathbf{v}$ will equal the expectation of
$\mathsf{A}$ times the expectation of $\mathbf{v}$. That is, when the
velocities are not correlated with $\mathbf{\hat{s}}$
$\langle\mathbf{p}\rangle=\langle\mbox{\boldmath{$\mathsf{A}$}}\cdot\mathbf{v}\rangle=\langle\mbox{\boldmath{$\mathsf{A}$}}\rangle\cdot\langle\mathbf{v}\rangle.$
(5)
Provided the stars are sufficiently widely spread on the sky, the matrix
$\langle\mbox{\boldmath{$\mathsf{A}$}}\rangle$ is not singular, so we can
write
$\langle\mathbf{v}\rangle=\langle\mbox{\boldmath{$\mathsf{A}$}}\rangle^{-1}\cdot\langle\mathbf{p}\rangle.$
(6)
Similarly,
$\langle\mathbf{vv}\rangle=\langle\mbox{\boldmath{$\mathsf{B}$}}\rangle^{-1}\cdot\langle\mathbf{pp}\rangle.$
(7)
## 3 Tests
Figure 1: Position of the 8 counting volumes, shaded black or grey
alternately as we look further from the plane, that make up a cone with its
vertex at the Sun, and its axis perpendicular to the Galactic plane.
In this section we demonstrate the danger of using the DB98 technique when the
key assumption of uncorrelated $\mathbf{v}$ and $\mathbf{\hat{s}}$ does not
hold. We do this by considering a simplified form of the problem addressed by
F09, which was to find the velocity ellipsoid of stars in the SDSS survey
volume. Fig. 1 shows our idealisation of the F09 counting volumes – we take
them to be slices of a cone with the Sun as its apex. In our usual Cartesian
coordinate system centred on the Sun, the cone is defined by
$\sqrt{X^{2}+Y^{2}}<0.8Z$ and $Z<800\,\mathrm{pc}$. It is split into eight
counting volumes that are each $100\,\mathrm{pc}$ thick (cf Fig. 5 of F09).
F09 validated their use of the DB98 method by drawing a velocity for every
star in their sample from a Schwarzschild distribution with a velocity
ellipsoid that was everywhere aligned with the $X,Y$ and $Z$ axes, and had
constant axis lengths $\sigma_{U}$, $\sigma_{V}$ and $\sigma_{W}$. This
velocity distribution does not vary with position, so $\mathbf{v}$ and
$\mathbf{\hat{s}}$ will be uncorrelated. In reality the velocity ellipsoid
will vary from point to point, both in the orientation of its principal axes,
and in the lengths of these axes. The lengths of these axes are expected to
vary with galactocentric radius $R$ roughly as
$\mbox{\boldmath$\sigma$}\propto\exp(-R/R_{\sigma})$, where $R_{\sigma}$ is of
order twice the disc’s scale length $R_{\rm d}$ (e.g. Binney & Tremaine,
2008). In the Milky Way, $R_{\rm d}\simeq 2.5\,\mathrm{kpc}$ (e.g. Jurić et
al., 2008), so $R_{\sigma}\sim 5\,\mathrm{kpc}$.
To illustrate the difficulty we adopt the distribution function
$\displaystyle f$ $\displaystyle\propto$
$\displaystyle\exp\left\\{-\frac{1}{2}\left[\left(\frac{v_{R}}{\sigma_{R}(R,z)}\right)^{2}\right.\right.+$
(9) $\displaystyle\left.\left.\left(\frac{v_{\phi}-v_{c}(R)-\langle
v_{\phi}(R,z)\rangle}{\sigma_{\phi}(R,z)}\right)^{2}+\left(\frac{v_{z}}{\sigma_{z}(R,z)}\right)^{2}\right]\right\\}.$
where again $(R,\phi,z)$ are cylindrical coordinates centred on the Galactic
Centre, $v_{c}(R)$ is the circular speed, and $\langle v_{\phi}(R,z)\rangle$
is the asymmetric drift. The velocity ellipsoid for this distribution function
is aligned with the cylindrical coordinate axes. We assume that we can
correctly compensate for the circular velocity using Oort’s constants (e.g.
Feast & Whitelock, 1997). In all cases we take constant $\langle
v_{\phi}(R,z)\rangle=-26\,\mathrm{km\,s}^{-1}$. In practice $\langle
v_{\phi}\rangle$ varies with $\sigma$, but this makes virtually no difference
to these results, so we ignore it for simplicity.
We consider the following three forms for $\sigma$:
1. 1.
Constant
$\mbox{\boldmath$\sigma$}\equiv(\sigma_{R},\sigma_{\phi},\sigma_{z})=(45,32,24)\,\mathrm{km\,s}^{-1}$.
This is nearly the same distribution function used by F09, except with the
velocity ellipsoid aligned with the cylindrical rather than Cartesian axes.
2. 2.
Radially varying $\sigma$
$\mbox{\boldmath$\sigma$}(R)=\mbox{\boldmath$\sigma$}(R_{0})\,\exp[(R_{0}-R)/R_{\sigma}]$
(10)
with $R_{\sigma}=5\,\mathrm{kpc}$, $R_{0}=8\,\mathrm{kpc}$, and
$\mbox{\boldmath$\sigma$}(R_{0})$ taking the same value as in case 1.
3. 3.
A form that varies both radially and vertically so as to provide reasonable
fits to the dispersions reported by F09:
$\displaystyle\mbox{\boldmath$\sigma$}(R,z)=(34+20z,23$ $\displaystyle+$
$\displaystyle 20z,19+30z)\,\mathrm{km\,s}^{-1}$ (11) $\displaystyle\times$
$\displaystyle\exp[(R_{0}-R)/R_{\sigma}],$
where $R_{\sigma}=5\,\mathrm{kpc}$ and $z$ is expressed in $\mathrm{kpc}$.
In each counting volume, we place 100,000 stars drawn randomly from a uniform
probability distribution over the entire volume. We assign each star a
velocity randomly chosen from the distribution function. We then “observe”
this star, and find its proper motion. This allows us to compare the values of
$\mathbf{v}$ and $\mathbf{v}\mathbf{v}$ we determine from deprojection
(equations 6 & 7) to the real values.
Since we consider everything with respect to the Cartesian axes defined in
Section 2, this yields values for $\langle U\rangle$, $\langle UU\rangle$,
$\langle UV\rangle$, etc. We can use these values (and the fact that the
centre of each counting volume lies at $X=Y=0$) to find the velocity
dispersions parallel to the cylindrical axes, $\sigma_{R},\,\sigma_{\phi}$ and
$\sigma_{z}$ and the mixed moments $\sigma_{R\phi}^{2},\,\sigma_{Rz}^{2}$ and
$\sigma_{\phi z}^{2}$.
Note that the mixed moments may be either positive or negative. In Figs. 2, 3
& 4 we plot $\sigma_{R\phi}$, $\sigma_{Rz}$ and $\sigma_{\phi z}$, which we
define by
$\sigma_{ij}\equiv\mathrm{sign}(\sigma^{2}_{ij})\sqrt{|\sigma^{2}_{ij}|}.$
(12)
The two vertex deviations, which describe the orientation of the velocity
ellipsoid with respect to the cylindrical axes, can be found from these values
as
$\Psi=-\frac{1}{2}\arctan{\frac{2\sigma^{2}_{R\phi}}{\sigma^{2}_{R}-\sigma^{2}_{\phi}}};$
(13)
$\alpha=-\frac{1}{2}\arctan{\frac{2\sigma^{2}_{Rz}}{\sigma^{2}_{R}-\sigma^{2}_{z}}}.$
(14)
Figure 2: Components of the velocity dispersion tensor $\sigma$ (bottom) and
the velocity ellipsoid tilt angle with respect to the plane, $\alpha$ (top) as
a function of height above the plane. The figure shows the true values from
the sample in each counting volume (dotted) and the values found by
deprojection (solid). The true velocity ellipsoid has principal axes aligned
with the cylindrical coordinate directions and axis lengths that are
independent of position (i.e. $\mbox{\boldmath$\sigma$}=\mathrm{const}$).
Figure 3: Similar to Fig. 2, except for
$\mbox{\boldmath$\sigma$}\propto\exp(-R/R_{\sigma})$, with
$R_{\sigma}=5\,\mathrm{kpc}$. Again, dotted lines show the true values, and
solid lines show those found by deprojection.
Figure 4: Similar to Figs. 2 and 3, with
$\mbox{\boldmath$\sigma$}\propto\exp(-R/R_{\sigma})$, with
$R_{\sigma}=5\,\mathrm{kpc}$, and with $\sigma$ varying with $z$ (as can be
seen in the figure). Again, dotted lines show the true values, and solid lines
show those found by deprojection. The value of $\sigma_{Rz}$ found here is
similar to that found in Figure 3, with the angle $\alpha$ being larger at
large $z$ because the velocity ellipsoid is rounder, due to $\sigma_{z}$
increasing more than $\sigma_{R}$ (c.f. equation 14).
In each of these cases, the values of $\langle U\rangle$, $\langle V\rangle$
and $\langle W\rangle$ determined from equation (6) are consistent with the
true values at $R_{0}$.
The lower panels of Figs. 2, 3 & 4 show the values of these velocity
dispersions and the mixed moments as functions of distance from the plane for
the three distribution functions described above: true values are shown by
dotted lines, while solid lines show values recovered by deprojection. We see
that deprojection yields reasonably accurate values of $\sigma_{R}$,
$\sigma_{\phi}$, $\sigma_{z}$, $\sigma_{R\phi}$ and $\sigma_{\phi z}$ even
when $\sigma$ varies significantly through the counting volumes, so the DB98
procedure is not strictly valid.
However, the value of $\sigma_{Rz}$ found by deprojection is materially
incorrect in all cases, being slightly negative when $\sigma$ does not vary
with $R$, and positive otherwise. The upper panels of Figs. 2, 3 & 4 show that
these incorrect values of $\sigma_{Rz}$ yield values of the tilt angle as
large as $\alpha\simeq-20\degr$. A tilt of the long axis of the ellipsoid
towards the plane implied by $\alpha\simeq-20\degr$ is similar to that seen by
F09. Thus our experiments demonstrate that the F09 tilt could be an artifact
that arises because the velocity dispersion increases inwards.
### 3.1 Physical interpretation
To understand why a radial gradient in $\sigma$ leads to an apparent tilt of
the velocity ellipsoid towards the plane, consider a simplified case in which
there are two fields, both at Galactic coordinate $b=90-\theta$. One is at
$l=0$ and the other is at $l=180$. The velocity measured by the proper motion,
$v_{\mu}$, is then
$v_{\mu}=\left\\{\begin{array}[]{ll}v_{R}\cos\theta+v_{z}\sin\theta,&\mbox{ at
}l=0;\\\ v_{R}\cos\theta-v_{z}\sin\theta,&\mbox{ at }l=180.\\\
\end{array}\right.$ (15)
Since $0<\theta<90$, both $\sin\theta$ and $\cos\theta$ are positive.
Therefore, in the field at $l=0$, $v_{\mu}$ is large when $v_{R}$ and $v_{z}$
have the same sign, while in the field at $l=180$ it is large when they take
opposite signs. In the absence of a radial gradient, the signature of a tilt
_towards_ the plane is therefore larger values of $v_{\mu}$ at $l=0$ than at
$l=180$. Clearly a radial gradient in $\sigma$ mimics this signature in the
absence of a tilt. Hence if one deprojects under the assumption that there is
no radial gradient, the algorithm will account for the data by reporting a
tilt towards the plane.
### 3.2 A workaround
Given that good sky coverage is essential to the success of the DB09 method,
one simply cannot assume that the velocity distribution is the same at the
locations of all the stars in a sample that reaches out to $\ga
1\,\mathrm{kpc}$ from the Sun. A remedy that can be considered is to adopt a
functional form for the radial variation of $\sigma$ and to use this form to
correct the observed proper-motion velocities to the values they would have
had if $\sigma$ had been independent of position. For example, for each star
we could calculate a “corrected” proper-motion velocity
$\mathbf{p}^{\prime}=(\mathbf{p}-\mbox{\boldmath{$\mathsf{A}$}}\cdot\mathbf{v}_{\mathrm{corr}})\exp[(R-R_{0})/R_{\sigma}^{\prime}]$
(16)
with $R_{\sigma}^{\prime}$ an estimate for the true value of the parameter
$R_{\sigma}$ that controls the radial variation of $\sigma$ (eq. 10) and
$\mathbf{v}_{\mathrm{corr}}=\mathbf{v_{\odot}}+\langle
v_{\phi}\rangle\mathbf{e}_{\phi}$ an adjustment for the Solar motion and
asymmetric drift. Thus defined, $\mathbf{p}^{\prime}$ would be expected to
average to zero over all directions and to be the proper-motion velocity if
there were no variation in $\sigma$ with radius.
Figure 5: Tilt angle with respect to the Galactic plane ($\alpha$) as a
function of height above the plane $z$. The true velocity ellipsoid is
oriented parallel to the cylindrical axes, with
$\mbox{\boldmath$\sigma$}\propto\exp(-R/(5\,\mathrm{kpc}))$. The figure shows
the true value of $\alpha$ for the sample (dotted), the value found from the
proper motions without applying any correction (solid), and values found
applying a correction (equation 16) with $R_{\sigma}^{\prime}=4\,\mathrm{kpc}$
(short dashed), $5\,\mathrm{kpc}$ (long dashed) and $6\,\mathrm{kpc}$ (dot
dashed). The “true” correction, $R_{\sigma}^{\prime}=5\,\mathrm{kpc}$ does not
return the true value of $\alpha$ because it does not correct for the fact
that the velocity ellipsoid is not aligned with the Cartesian axes.
We test this correction by applying it to simulated data generated as in case
(ii) above. We know the true value of $\mathbf{v}_{\mathrm{corr}}$ in this
case, so we ignore the relatively minor uncertainties which are caused by not
estimating this correctly.
The dashed lines in Fig. 5 show the tilt angle $\alpha$ found from corrected
proper-motion velocities for three values of $R_{\sigma}^{\prime}$:
$4\,\mathrm{kpc}$ (short-dashed), $5\,\mathrm{kpc}$ (long-dashed) and
$6\,\mathrm{kpc}$ (dot-dashed). In all three cases the corrected data give
much more accurate results than the uncorrected data (full curve), but the
most accurate results are obtained with $R_{\sigma}^{\prime}=6\,\mathrm{kpc}$
rather than the true value, $5\,\mathrm{kpc}$; with
$R_{\sigma}^{\prime}=5\,\mathrm{kpc}$ we find $\alpha\sim 1\degr$ at the
largest values of $z$ because the correction does not address the problem that
the axes of the velocity ellipsoid are aligned with the cylindrical rather
than Cartesian axes. A closely related bias is seen when $\sigma$ is constant
(Fig. 2). Using a value of $R_{\sigma}^{\prime}=6\,\mathrm{kpc}$ for the
correction gives $\alpha\simeq 0$ because it _under_ -compensates for the bias
due to the variation in $\sigma$, which inadvertently compensates for the bias
due to the alignment of the velocity ellipsoid’s axes.
If we considered the value $\alpha$ well established, we could use corrected
proper-motion velocities to determine $R_{\sigma}$ from the data.
## 4 Discussion
In this paper we have focused on the tilt of the velocity ellipsoid towards
the plane, and may have left the reader with the impression that, for example,
the tilt in the plane or the non-mixed terms ($\sigma_{R}$ etc.) are correctly
recovered by the DB98 technique. While this is true to a good approximation in
the cases shown here, it is not always true.
For example, consider the situation described in Section 3.2, in which we need
to know the value of $R_{\sigma}$ so we can compensate for the variation in
$\sigma$ across the counting volume. In an approach to the determination of
$R_{\sigma}$ we might split the data into two sets, for $|l|<90\deg$ and
$|l>90\deg$, and find $\sigma$ separately for each set – this gives us enough
information to find $R_{\sigma}$. However, if the data are split in this way,
they produce a bias in the values of the _non_ -mixed components of $\sigma$
(as well as the mixed components). This bias is in opposite directions for the
two data sets, so strongly affects the derived value of $R_{\sigma}$, but
cancels out when the two sets are considered together (hence the lack of bias
in the non-mixed components in Figs. 3 and 4).
Similar biases must _always_ be considered when using deprojection. In the
tests described above, the symmetry of the counting volumes cancelled out the
bias in most components of $\sigma$, effectively restricting it to
$\sigma_{Rz}$. The counting volumes of real data sets will not enjoy the high
degree of symmetry characteristic of our model sets, with the result that
biases in the values returned by the DB98 method will not be confined to
$\sigma_{Rz}$.
## 5 Conclusions
In this paper we have demonstrated that the statistical deprojection of proper
motions cannot be applied straightforwardly to data spanning a significant
volume of the Galaxy. This is primarily because the dependence of the velocity
dispersion $\sigma$ on position violates the central assumption of the method.
Using a simple model we have demonstrated that applying this method can
suggest a large tilt of the velocity ellipsoid towards the plane, even if the
actual tilt is zero. It seems very likely that this effect is responsible for
the remarkably large tilt, $\alpha=-20\degr$, reported by F09. Correcting for
this effect in the manner discussed in Section 3.2 would probably bring this
result much closer to the smaller tilt angles obtained using radial velocities
(e.g. Siebert et al., 2008; Bond et al., 2009).
We note, however, that all components of $\sigma$ other than $\sigma_{Rz}$
were nearly unaffected by this bias in our tests. For a realistic survey
volume such as that used by F09, these biases are likely to be larger than in
our tests and in some circumstances may materially affect the results.
## Acknowledgments
This research was supported by a grant from the Science and Technology
Facilities Council.
## References
* Abazajian et al. (2009) Abazajian K. N., Adelman-McCarthy J. K., Agüeros M. A. et al., 2009, ApJS, 182, 543
* Binney & Tremaine (2008) Binney J., Tremaine S., 2008, Galactic Dynamics: Second Edition. Princeton University Press
* Bond et al. (2009) Bond N. A., Ivezic Z., Sesar B. et al., 2009, ApJ, submitted (arXiv:0909.0013)
* Dehnen & Binney (1998) Dehnen W., Binney J. J., 1998, MNRAS, 298, 387 (DB98)
* ESA (1997) ESA, 1997, VizieR Online Data Catalog, 1239, 0
* Feast & Whitelock (1997) Feast M., Whitelock P., 1997, MNRAS, 291, 683
* Fuchs et al. (2009) Fuchs B., Dettbarn C., Rix H.-W. et al., 2009, AJ, 137, 4149 (F09)
* Jurić et al. (2008) Jurić M., Ivezić Ž., Brooks A. et al., 2008, ApJ, 673, 864
* Siebert et al. (2008) Siebert A., Bienaymé O., Binney J. et al., 2008, MNRAS, 391, 793
|
arxiv-papers
| 2009-09-09T11:14:08 |
2024-09-04T02:49:05.193596
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Paul J. McMillan and James J. Binney",
"submitter": "Paul McMillan",
"url": "https://arxiv.org/abs/0909.1679"
}
|
0909.1716
|
# Understanding the Clean Interface Between Covalent Si and Ionic Al2O3
H. J. Xiang National Renewable Energy Laboratory, Golden, Colorado 80401, USA
Juarez L. F. Da Silva National Renewable Energy Laboratory, Golden, Colorado
80401, USA Howard M. Branz National Renewable Energy Laboratory, Golden,
Colorado 80401, USA Su-Huai Wei National Renewable Energy Laboratory,
Golden, Colorado 80401, USA
###### Abstract
The atomic and electronic structures of the (001)-Si/(001)-$\gamma$-Al2O3
heterointerface are investigated by first principles total energy calculations
combined with a newly developed “modified basin hopping” method. It is found
that all interface Si atoms are 4-fold coordinated due to the formation of
Si-O and unexpected covalent Si-Al bonds in the new abrupt interface model.
And the interface has perfect electronic properties in that the unpassivated
interface has a large LDA band gap and no gap levels. These results show that
it is possible to have clean semiconductor/oxide interfaces.
###### pacs:
68.35.-p,73.20.-r,71.15.Nc,02.70.Uu
Interfaces between semiconductors and metal oxides are playing increasingly
important roles in advanced material science Forst2004 ; McKee1998 ; McKee2003
. In order to continue scaling electronic devices, a change from SiO2 (with a
dielectric constant $k$ about $3.9$) to high-$k$ oxides has been proposed for
the gate dielectric in future generation metal oxide semiconductor (MOS)
technologies. The key considerations for high-$k$ gate dielectrics include
high dielectric constant, high band offsets (at least 1 eV) with respect to
silicon, thermal stability, and minimization of electrical defects in the
interface. In particular, the quality of the interface is important for both
carrier mobility and device stability. However, control of the interface to
the Si substrate remains a stubborn outstanding problem. For example, hafnium-
based amorphous oxides has a bulk dielectric constant of $k\sim 22$
Ceresoli2006 , but its integration into the MOS gate stack poses substantial
technological challenges Kingon2000 . Epitaxial growth of oxides could lead to
more abrupt oxide-Si interfaces and consequently could offer solutions for the
end of the roadmap. Indeed, single crystal $\gamma$-Al2O3 ($k\sim 11$) thin
films have been epitaxially grown by molecular beam epitaxy on Si(001)
substrates Merckling2006 . Hence, Al2O3 could be a good candidate to be used
directly as a gate oxide or as a thin buffer barrier when combined with
high-$k$ amorphous or epitaxial oxides.
In another context, there have been some efforts in developing high quality
crystalline silicon (c-Si) film on inexpensive foreign substrates such as
oxides to reduce the Si material cost for terrestrial photovoltaic (PV) cells
Teplin2006 . Previous attempts to grow single crystal Si on some oxides such
as CeO2 failed due to the formation of SiO2 Teplin2006 . Recently, Findikoglu
et al. Findikoglu2005 demonstrated the growth of well-oriented Si thin films
with high carrier mobility on $\gamma$-Al2O3 substrate. In addition, Al2O3 has
been shown to passivate the c-Si surface efficiently for PV applications
Hoex2006 . These results suggest $\gamma$-Al2O3 could be a good substrate for
c-Si solar cell growth. Therefore, detailed knowledge of the Si/Al2O3
interface are vital.
Although many experimental studies have examined the growth of (001)
$\gamma$-Al2O3 on the Si (001) surface ($\gamma$-Al2O3/Si), the detailed
interface structure remains unclear. Theoretically, Boulenc and Devos
Boulenc2006 proposed an interface model for (001)-$\gamma$-Al2O3 grown on
(001)-Si surface by incorporating a defective spinel model Menendez2005 of
$\gamma$-Al2O3. To obtain an interface without gap states, they introduced
passivating O atoms to replace Si-Al and Si-Si bonds with Si-O and Al-O bonds.
However, it is not clear if their proposed substrate and interface Boulenc2006
have the lowest total energies because only a few models were tested. It is
also not clear if a sharp gap-states free interface can exist because the
large chemical and size difference between the semiconductor and the oxide.
Therefore, it is desirable to obtain an improved microscopic understanding of
the atomic and electronic structures of this important Si/Al2O3 interface.
In this Letter, we develop a new modified basin-hopping (BH) method to search
for the lowest energy structure of the (001)-Si/(001)-$\gamma$-Al2O3
interface. It is found that the new interface structure presents not only Si-O
bonds, but also Si-Al bonds, with all Si atoms 4-fold coordinated. Our density
functional calculation shows that the interface is semiconducting with a
type-I band alignment. Our results support the use of $\gamma$-Al2O3 as a gate
oxide or a substrate for the c-Si growth.
Our density functional theory calculations employed the frozen-core projector
augmented wave method (PAW) PAW encoded in the Vienna ab initio simulation
package VASP , and the local density approximation (LDA). We use a plane-wave
cutoff energy of 400 eV, except for the search of interface structures by the
BH method, where we use a soft O PAW pseudopotential with a cutoff energy of
212 eV.
As a prerequisite to build the Si/$\gamma$-Al2O3 interface, understanding the
$\gamma$-Al2O3 structure is necessary. Here, we adopt the bulk model
constructed by Krokidis et al. Krokidis2001 (We hereafter refer to this bulk
model as Krokidis model), which has a lower energy Digne2004 than traditional
defective spinel models Menendez2005 ; Pinto2004 , and is consistent with
experimental NMR and XRD results Zhou1991 . Our test calculations also confirm
the stability of the Krokidis model. The Krokidis model has a centrosymmetric
monoclinic structure (P21/m, No 11) with 1/4 four- and 3/4 six-fold
coordinated Al atoms. Our LDA optimization results in the following
parameters: $a=5.479$ Å, $b=8.255$ Å, $c=7.961$ Å, and $\beta=90.645^{\circ}$.
The lowest energy (001) $\gamma$-Al2O3 surface Digne2004 based on the
Krokidis model has many inequivalent surface sites (Fig. 1). Here, oxygen
atoms are indexed with capital letters and aluminum atoms with numbers
Valero2006 . It is noted that there is a mirror symmetry plane which is
perpendicular to $b$ and crosses Al 3 and 4. Therefore, the O at C (D) is
equivalent to the O at E (F), and the Al at 2 has the same environment as the
Al at 5. All surface Al atoms are pentacoordinated, except that Al atom 1 is
tetracoordinated and in a position slightly below the surface plane. All
surface oxygens are tricoordinated if only Al neighbors within 2 Å are
counted. However, C and E oxygen atoms have an additional nearby Al atom
besides the bonding Al atoms: i.e., the distance between Al 2 and oxygen C is
2.19 Å. In this sense, C and E oxygen atoms are quasi-four-fold coordinated,
as suggested in Fig. 1(a) by the dashed lines. Here, a four-layer (not
counting the tetrahedral Al atoms) symmetric slab model is adopted. After
relaxation, oxygen D (and F), oxygen A, and oxygen B move outward from the
surface by about 0.3 Å, 0.2 Å, and 0.1 Å, respectively. In contrast, oxygen C
and E stay in the surface due to the strong binding with four neighboring Al
atoms. It is noted that the surface is insulating due to the charge transfer
from surface Al atoms to O dangling bonds.
We first examine the thermodynamic stability of the interface by calculating
the enthalpy of two possible reactions Schlom2002 :
$\begin{array}[]{ll}\frac{3}{2}\mathrm{Si}+\mathrm{Al}_{2}\mathrm{O}_{3}\rightarrow
2\mathrm{Al}+\frac{3}{2}\mathrm{Si}\mathrm{O}_{2},\Delta
H=2.88\quad\mathrm{eV}\\\
\mathrm{Si}+\frac{5}{3}\mathrm{Al}_{2}\mathrm{O}_{3}\rightarrow\frac{4}{3}\mathrm{Al}+\mathrm{Si}\mathrm{Al}_{2}{\mathrm{O}}_{5},\Delta
H=0.99\quad\mathrm{eV}.\end{array}$ (1)
These positive reaction enthalpies indicate that the Si/Al2O3 interface is
thermodynamically stable, i.e., the formation of SiO2 and silicate is
unfavorable.
The construction of the interface model is a nontrivial task. Usually,
molecular dynamics simulations Broqvist2009 or intuition were employed for
this purpose. It should be noted that molecular dynamics simulations gives
different interface structures depending on initial conditions, and it is
almost impossible to guarantee that the constructed interface structure has
the lowest interface energy. And it is very hard to design a good interface
structure between two totally dissimilar materials just from chemical
intuition. Therefore, we develop a new modified BH method Wales1999 to
determine the most stable interface structure. In conventional BH method, each
BH run starts with a randomly chosen atomic configuration and is composed of a
given number of Monte Carlo steps. In each of these, the starting
configuration is first locally optimized to obtain an energy $E_{1}$. Then,
each atom is subjected to a random displacement in each of its Cartesian
coordinates, and a new locally optimized structure is obtained with energy
$E_{2}$. Here, $E_{1}$ and $E_{2}$ are the total energies from DFT
calculations. If $\exp[(E_{1}-E_{2})/k_{B}T]>r$, where $r$ is a random number
between 0 and 1 (Metropolis criterion), the new configuration is accepted
(otherwise the old configuration is kept), and the process is iterated. The BH
method has been widely used to search the global minimal structure of clusters
Wales1999 ; Yoo2003 ; Pei2008 ; Barcaro2007 . However, to our best knowledge,
the BH method has not been employed to search for the interface structure
between two surfaces.
In our newly developed modified BH method for finding lowest energy interface
structures, we name the two slabs as “top” and “bottom”, respectively (see
Fig. 2). In the case of the Si/Al2O3 interface, the Si (Al2O3) (001) slab is
the top (bottom) one. The Si slab has seven Si layers. The top Si layer forms
Si dimers and is passivated by H atoms. For the top Si slab, we have a rigid
layer, a buffer layer, and a hopping layer. The atoms in the rigid layer can
translate as a rigid body but the internal structure is fixed fix . The fixed
layer is held in place and the buffer layer is allowed to relax during the
optimization. In contrast, the atoms of the hopping layer move as in the usual
BH method but are restricted to the interface region. The typical value of the
hopping distance of the BH simulation is about $1.5$ Å. Our test calculations
indicate that swapping a Si for an Al atom is energetically unfavorable by
about 2 eV. Thus, the bottom Al2O3 slab is divided into two parts: a fixed
layer and a buffer layer. We note that the our modified BH method is rather
general and can be used to search for the lowest energy structure of other
interfaces.
Considering the lateral lattice contants of the (001) Si surface
[$a(\mathrm{Si})=5.404$ Å] and (001) $\gamma$-Al2O3 surface
[$a(\mathrm{Al}_{2}\mathrm{O}_{3})=5.479$ Å,
$b(\mathrm{Al}_{2}\mathrm{O}_{3})=8.255$ Å], the best lattice matching is
achieved by connecting the ($1\times 3$) Si (001) surface with the ($1\times
2$) (001) $\gamma$-Al2O3 surface. In this structure, the calculated lattice
mismatch is about 1.6%. Here, the in-plane lattice constants of the supercell
are fixed to be the theoretical lattice constants of bulk $\gamma$-Al2O3
because $\gamma$-Al2O3 has a large Young’s modulus. We perform several BH
simulations for 200 steps with different initial coordinates (the relative
position between the Si surface and the Al2O3 surface, and the atomic
positions of the atoms of the hopping layer). Finally, the lowest energy
interface structure found from the BH simulations is refined by performing a
full atomic relaxation of the whole system, including all atoms of the “fixed”
Al2O3 layer and “rigid” Si layer.
The lowest energy interface structure that we find is shown in Fig. 3. We can
see that the dimer structure at the Si (001) surfaces is preserved as a result
of the strong covalent Si-Si bond. We note that there is no dimer in the
interface of the initial structure, while dimers are formed during the
relaxation. At the interface, one Si atom of each dimer bonds with a three-
fold coordinated O atom of the Al2O3 surface, whereas the other Si atom forms
a bond with a four-fold coordinated Al atom. The Si-O and Si-Al bond lengths
are about 1.8 Å and 2.4 Å, respectively. The O atoms bonded with Si move
outward from the $\gamma$-Al2O3 surface in order to form bonds with Si atoms.
We find that oxygen C and E do not bond with Si atoms because it is
unfavorable for them to move outward due to the strong binding with the fourth
neighboring Al atom below the surface. The binding energy between the Si
surface and Al2O3 surface is calculated to be 2.96 eV/supercell, which
indicating the strong binding between the two surfaces. It should be noted
that there are some other nearly degenerate (within 20 meV/cell) interface
structures with feature similar to that shown in Fig. 3. In these metastable
structures, other Al and O atoms are bonded with Si dimers.
The DOS for the interface is shown in Fig. 3(c). We can see that the system is
semiconducting with an indirect band gap of 0.46 eV. Remarkably, this value is
larger than the LDA band gap (0.45 eV) of bulk Si. The presence of the band
gap is also consistent with the stability of the interface. The DOS plot shows
a type-I band alignment between Si and Al2O3. To compute an accurate band
offset, we align the energy levels using the core levels Wei1998 . The
calculated valence band offset is 2.40 eV. The measured value between Si and
$\alpha$-Al2O3 ranges from 2.90 eV to 3.75 eV Bersch2008 . The experimental
valence band offset between Si and $\gamma$-Al2O3 is expected to have a
similar value. The discrepancy between the experimental result and our
theoretical value are due to the different LDA error for the covalent Si and
ionic Al2O3 but the result is qualitatively correct Alkauskas2008 . To gain
insight into the electronic properties of the interface, we show the partial
charge densities of the topmost three HOMOs and bottommost three LUMOs of the
interface in Fig. 3(a) and (b). It is clear that the HOMOs are mainly
contributed by the directional covalent Si-Al bonds, and the LUMOs by the
antibonding Si-O bonds.
It is well known that each Si atom of the symmetric Si dimer of the Si (001)
surface has one dangling bond. On the free Si (001) surface, the tilt of the
Si dimer lifts the degeneracy of the Si dangling bonds and a band gap opens
because of the charge transfer from the inward Si atom to outward Si atom. In
the case of the Si/Al2O3 interface, the band gap opening mechanism is totally
different and much more efficient. As shown in Fig. 4, the lone pair electrons
of the surface O atom interact with the dangling bond of the nearest neighbor
Si atom, raising the level of the dangling bond. In contrast, the high-lying
empty Al orbital hybridizes with the dangling bond of the neighboring Si atom,
lowering the energy level of the Si orbital. As a consequence, the Si atom
bonded with the O atom transfers its dangling bond electron to the covalent
Si-Al bond, and the interface has a large band gap. This bonding mechanism
between Si and Al2O3 is consistent with the Bader charge analysis Bader : the
Si atom bonded with Al gains about 0.25 electrons, whereas the Si atom bonded
with O loses about 0.40 electrons. As a result, there is some small net charge
transfer (0.15 e/Si-dimer) from Si to Al2O3.
To investigate the kinetic stability of the interface, we calculate the energy
barrier of the sliding of the Si surface on the Al2O3 surface. To find the
transition state and energy barrier, we use the “climbing image nudged elastic
band” method Henkelman2000 . We consider the sliding of the Si surface along
the $b$ axis because the barrier of the sliding along other directions are
expected to be larger due to the need to break all Si-O and Si-Al bonds. The
final interface structure is obtained from the initial structure by sliding
the Si surface along the $b$ axis by $b$(Al2O3)$/3$; the final state is almost
degenerate with the initial state. In the transition state, there is some
remaining bonding between the Si surface and Al2O3 surface: one Si-O bond and
two Si-Al bonds. The energy barrier of the sliding is about 2.0 eV/supercell,
which makes the Si/Al2O3 interface kinetically stable.
In conclusion, we develop a general modified BH method to search for the
lowest energy structure of (001)-Si/$\gamma$-(001)-Al2O3 interface. It is
found that the interface Si dimers have a favorable 4-fold coordination due to
the formation of not only Si-O bonds, but also unexpected covalent Si-Al
bonds. Our study reveals that the Si/Al2O3 interface has the following
attractive properties: (i) The interface is sharp and is semiconducting with a
large LDA band gap; (ii) The band alignment between Si and $\gamma$-Al2O3 is
type-I with both valence band offset and conduction band offset larger than
1.5 eV; (iii) The interface is thermodynamically and kineticly stable. Our
results suggest that $\gamma$-Al2O3 can be used as a gate dielectric in future
MOS technologies or a substrate for the growth of c-Si for solar cells.
Work at NREL was supported by the U.S. Department of Energy, under Contract
No. DE-AC36-08GO28308.
## References
* (1) C. J. Först et al., Nature (London) 427, 53 (2004).
* (2) R. A. McKee, F. J. Walker, and M. F. Chisholm, Phys. Rev. Lett. 81, 3014 (1998).
* (3) R. A. McKee et al., Science 300, 1726 (2003).
* (4) D. Ceresoli and D. Vanderbilt, Phys. Rev. B 74, 125108 (2006).
* (5) A. I. Kingon, J. P. Maria, and S. K. Streiffer, Nature (London) 406, 1032 (2000).
* (6) C. Merckling et al., Appl. Phys. Lett. 89, 232907 (2006)
* (7) C. W. Teplin, D. S. Ginley and H. M. Branz, J. Non-Cryst. Solids 352, 984 (2006).
* (8) Alp T. Findikoglu et al., Adv. Mater. 17, 1527 (2005).
* (9) B. Hoex et al., Appl. Phys. Lett. 89, 042112 (2006).
* (10) P. Boulenc and I. Devos, Mater. Sci. Semicond. Process. 9, 949 (2006).
* (11) E. Menéndez-Proupin and G. Gutiérrez, Phys. Rev. B 72, 035116 (2005).
* (12) P. E. Blöchl, Phys. Rev. B 50, 17953 (1994); G. Kresse and D. Joubert, ibid 59, 1758 (1999).
* (13) G. Kresse and J. Furthmüller, Comput. Mater. Sci. 6, 15 (1996); Phys. Rev. B 54, 11169 (1996).
* (14) X. Krokidis et al., J. Phys. Chem. B 105, 5121 (2001).
* (15) M. Digne et al., J. Catal. 226, 54 (2004).
* (16) H. P. Pinto, R. M. Nieminen, and S. D. Elliott, Phys. Rev. B 70, 125402 (2004).
* (17) R.S. Zhou, R.L. Snyder, Acta Crystallogr. B 47, 617 (1991).
* (18) M. C. Valero, P. Raybaud, and P. Sautet, J. Phys. Chem. B 110, 1759 (2006).
* (19) D. G. Schlom and J. H. Haeni, MRS Bull. 27, 198 (2002).
* (20) P. Broqvist, J. F. Binder, and A. Pasquarello, Appl. Phys. Lett. 94, 141911 (2009).
* (21) D. J. Wales and H. A. Scheraga, Science 285, 1368 (1999); D. J. Wales and J. P. K. Doye, J. Phys. Chem. A 101, 5111 (1997).
* (22) S. Yoo and X. C. Zeng, J. Chem. Phys. 119, 1442 (2003).
* (23) Y. Pei and X. C. Zeng, J. Am. Chem. Soc. 130, 2580 (2008).
* (24) G. Barcaro and A. Fortunelli, Phys. Rev. B 76, 165412 (2007).
* (25) We modify the VASP code to relax a rigid body.
* (26) S.-H. Wei and A. Zunger, Appl. Phys. Lett. 72, 2011 (1998).
* (27) E. Bersch et al., Phys. Rev. B 78, 085114 (2008) and references therein.
* (28) A. Alkauskas et al., Phys. Rev. Lett. 101, 106802 (2008).
* (29) R. F. W. Bader, Atoms in Molecules-A Quantum Theory (Oxford University Press, Oxford, 1990); G. Henkelman, A. Arnaldsson, and H. Jńsson, Comput. Mater. Sci. 36, 254 (2006).
* (30) G. Henkelman, B. P. Uberuaga, and H. Jńsson, J. Chem. Phys. 113, 9901 (2000).
Figure 1: (Color online) (a) Top and (b) side view of the ($1\times 1$) (001)
$\gamma$-Al2O3 surface. Oxygen (small) atoms are indexed with capital letters
and aluminum atoms (large) are indexed with numbers.
Figure 2: (Color online) The definition of various layers of the Si/Al2O3
interface in our modified BH method.
Figure 3: (Color online) Interface structure and isosurface plots of the
partial charge density of (a) the topmost three HOMOs and (b) bottommost three
LUMOs of the Si/Al2O3 interface. (c) DOS plot for the Si/Al2O3 interface,
calculated with 0.1 eV broadening. The vertical dashed line denotes the top of
the valence band. The partial DOSs of the Si and Al2O3 surfaces are also
shown.
Figure 4: (Color online) Schematic illustration of the Si-Al and Si-O bond
formation and gap opening in the Si/Al2O3 interface. The valence-band maximum
(VBM) and conduction-band minimum (CBM) of bulk Si are also shown
schematically.
|
arxiv-papers
| 2009-09-09T14:42:38 |
2024-09-04T02:49:05.199709
|
{
"license": "Public Domain",
"authors": "H. J. Xiang, Juarez L. F. Da Silva, Howard M. Branz, and Su-Huai Wei",
"submitter": "H. J. Xiang",
"url": "https://arxiv.org/abs/0909.1716"
}
|
0909.1833
|
10.1080/0003681YYxxxxxxxx 1563-504X 0003-6811 00 00 2009 July
# Traveling Waves of Discrete Nonlinear Schrödinger Equations with Nonlocal
Interactions
Michal Fečkan†∗ and Vassilis M. Rothos††
†Department of Mathematical Analysis and Numerical Mathematics, Comenius
University, Mlynská dolina, 842 48 Bratislava, Slovakia, and Mathematical
Institute of Slovak Academy of Sciences, Štefánikova 49, 814 73 Bratislava,
Slovakia;
††School of Mathematics, Physics and Computational Sciences, Faculty of
Engineering, Aristotle University of Thessaloniki, Thessaloniki 54124, Greece
∗Corresponding author. Email: Michal.Feckan@fmph.uniba.sk
(v3.3 released July 2009)
###### Abstract
Existence and bifurcation results are derived for quasi periodic traveling
waves of discrete nonlinear Schrödinger equations with nonlocal interactions
and with polynomial type potentials. Variational tools are used. Several
concrete nonlocal interactions are studied as well.
###### keywords:
nonlocal interactions, discrete Schödinger equation, traveling wave, symmetry
34K14, 37K60, 37L60
## 1 Introduction
One of the most exciting areas in applied mathematics is the study of the
dynamics associated with the propagation of information. Coherent structures
like solitons, kinks, vortices, etc., play a central role, as carriers of
energy, in many nonlinear physical systems [17]. Solitons represent a rare
example of a (relatively) recently arisen mathematical object which has found
successful high-technology applications [24]. The nature of the system
dictates that the relevant and important effects occur along one axial
direction. Interplay between nonlinearity and periodicity is the focus of
recent studies in different branches of modern applied mathematics and
nonlinear physics. Applications range from nonlinear optics, in the dynamics
of guided waves in inhomogeneous optical structures and photonic crystal
lattices, to atomic physics, in the dynamics of Bose-Einstein condensate (BEC)
droplets in periodic potentials, and from condensed matter, in Josephson-
junction ladders, to biophysics, in various models of the DNA double strand.
Analysis and modeling of these physical situations are based on nonlinear
evolution equations derived from underlying physics equations, such as
nonlinear Maxwell equations with periodic coefficients [37]. In particular,
the systems of 2nd-order NLS equations, both continuous and discrete, were
applied in nonlinear physics to study a number of experimental and theoretical
problems. Spatial non-locality of the nonlinear response is also naturally
present in the description of BECs where it represents the finite range of the
bosonic interaction. Demands on the mathematics for techniques to analyze
these models may best be served by developing methods tailored to determining
the local behavior of solutions near these structures. The discreteness of
space i.e., the existence of an underlying spatial lattice is crucial to the
structural stability of these spatially localized nonlinear excitations.
During the early years, studies of intrinsic localized modes were mostly of a
mathematical nature, but the ideas of localized modes soon spread to
theoretical models of many different physical systems, and the discrete
breather concept has been recently applied to experiments in several different
physics subdisciplines. Most nonlinear lattice systems are not integrable even
if the partial differential equation (PDE) model in the continuum limit is.
While for many years spatially continuous nonlinear PDE’s and their localized
solutions have received a great deal of attention, there has been increasing
interest in spatially discrete nonlinear systems. Namely, the dynamical
properties of nonlinear systems based on the interplay between discreteness,
nonlinearity and dispersion (or diffraction) can find wide applications in
various physical, biological and technological problems. Examples are coupled
optical fibres (self-trapping of light) [18, 1, 19, 26], arrays of coupled
Josephson junctions [39], nonlinear charge and excitation transport in
biological macromolecules, charge transport in organic semiconductors [40].
Prototype models for such nonlinear lattices take the form of various
nonlinear lattices [6], a particularly important class of solutions of which
are so called discrete breathers which are homoclinic in space and oscillatory
in time. Other questions involve the existence and propagation of topological
defects or kinks which mathematically are heteroclinic connections between a
ground and an excited steady state. Prototype models here are discrete version
of sine-Gordon equations, also known as Frenkel-Kontorova (FK) models, e.g.
[4]. There are many outstanding issues for such systems relating to the global
existence and dynamics of localized modes for general nonlinearities, away
from either continuum or anti-continuum limits.
In the main part of the previous studies of the discrete NLS models the
dispersive interaction was assumed to be short-ranged and a nearest-neighbor
approximation was used. However, there exist physical situations that
definitely can not be described in the framework of this approximation. The
DNA molecule contains charged groups, with long-range Coulomb interaction
$1/r$ between them. The excitation transfer in molecular crystals [16] and the
vibron energy transport in biopolymers [35] are due to transition dipole-
dipole interaction with $1/r^{3}$ dependence on the distance, $r$. The
nonlocal (long-range) dispersive interaction in these systems provides the
existence of additional length-scale: the radius of the dispersive
interaction. We will show that it leads to the bifurcating properties of the
system due to both the competition between nonlinearity and dispersion, and
the interplay of long-range interactions and lattice discreteness.
In some approximation the equation of motion is the nonlocal discrete NLS
$\imath\dot{u}_{n}=\sum\limits_{m\neq
n}J_{n-m}(u_{n}-u_{m})+|u_{n}|^{2}u_{n},\quad n\in\mathbb{Z}\,,$ (1)
where the long-range dispersive coupling is taken to be either exponentially
$J_{n}=J{\rm e}^{-\beta|n|}$ with $\beta>0$, or algebraically
$J_{n}=J|n|^{-s}$ with $s>0$, decreasing with the distance $n$ between lattice
sites. In both cases the constant $J$ is normalized such that
$\sum_{n=1}^{\infty}J_{n}=1$, for all $\beta$ or $s$. The parameters $\beta$
and $s$ are introduced to cover different physical situations from the
nearest-neighbor approximation $(\beta\to\infty,s\to\infty)$ to the
quadrupole-quadrupole $(s=5)$ and dipole-dipole $(s=3)$ interactions. The
Hamiltonian $H$ and the number of excitations $N$
$H=\frac{1}{2}\sum_{n,m\in\mathbb{Z}}J_{n-m}|u_{n}-u_{m}|^{2}-\frac{1}{2}\sum_{n\in\mathbb{Z}}|u_{n}|^{4},\quad{\rm
and}\quad N=\sum_{n\in\mathbb{Z}}|u_{n}|^{2}$ (2)
are conserved quantities corresponding to the set of (1).
It should be also noted that the derivation of a discrete equation from the
Gross-Pitaevskii equation produces at the intermediate step a fully nonlocal
discrete NLS equation for the coefficients of the wave function expansion over
the complete set of the Wannier functions. Further reduction to the case of
the only band with the strong localization of the Wannier functions (the
tight-binding approximation) leads to the standard local DNLS equation.
Recently Abdullaev et al. [2] extended this approach to the case of periodic
nonlinearities and derived a number of nonintegrable lattices with different
nearest-neighbor nonlinearities.
In this paper, we study the discrete nonlinear Schrödinger equations on the
lattice $\mathbb{Z}$ (DNLS) with nonlocal interactions of forms
$\imath\dot{u}_{n}=\sum\limits_{j\in\mathbb{N}}a_{j}\Delta_{j}u_{n}+f(|u_{n}|^{2})u_{n},\quad
n\in\mathbb{Z}$ (3)
where $u_{n}\in\mathbb{C}$, $\Delta_{j}u_{n}:=u_{n+j}+u_{n-j}-2u_{n}$ are
$1$-dimensional discrete Laplacians and it holds
* (H1)
$f\in C(\mathbb{R}_{+},\mathbb{R})$ for $\mathbb{R}_{+}:=[0,\infty)$, $f(0)=0$
and $a_{j}\in\mathbb{R}$ with $\sum\limits_{j\in\mathbb{N}}|a_{j}|<\infty$.
Moreover, there are constants $s>0$, $\mu>1$, $c_{1}>0$, $c_{2}>0$ and
$\bar{r}>0$ such that
$\begin{gathered}|f(w)|\leq c_{1}(w^{s}+1),\quad c_{2}(w^{s+1}-1)\leq
F(w),\quad\mu F(w)-\bar{r}<f(w)w\end{gathered}$
for any $w\geq 0$, where $F(w)=\int\limits_{0}^{w}f(z)dz$. Furthermore,
$\limsup_{w\to 0_{+}}f(w)/w^{\widetilde{s}}<\infty$ for a constant
$\widetilde{s}>0$.
Of course we suppose that not all $a_{j}$ are zero. Note any polynomial
$f(w)=p_{1}w+\cdots+p_{s}w^{s}$, $s\in\mathbb{N}$ with $p_{s}>0$ satisfies
(H1). Furthermore, (3) can be rewritten into a standard form
$\imath\dot{u}_{n}=\sum\limits_{m\neq
n}a_{|m-n|}\left(u_{m}-u_{n}\right)+f(|u_{n}|^{2})u_{n},\quad n\in\mathbb{Z}.$
(4)
It is well known that (4) conserves two dynamical invariants
$\begin{gathered}\sum\limits_{n\in\mathbb{Z}}|u_{n}|^{2}\quad-\textrm{the
norm},\\\ \sum\limits_{n\in\mathbb{Z}}\left[-\frac{1}{2}\sum\limits_{m\neq
n}a_{|m-n|}\left|u_{m}-u_{n}\right|^{2}+F(|u_{n}|^{2})\right]\quad-\textrm{the
energy}.\end{gathered}$
Differential equations with nonlocal interactions on lattices have been
studied in [3, 5, 7, 8, 9, 13, 14, 25, 30], while DNLS (discrete nonlinear
Schrödinger) in [10, 11, 13, 22, 28]. Nowadays it is clear that a large number
of important models of various fields of physics are based on DNLS type
equations with several forms of polynomial nonlinearities starting with the
simplest self-focusing cubic (Kerr) nonlinearity, then following with the
cubic onsite nonlinearity relevant for Bose-Einstein condensates, then with
more general discrete cubic nonlinearity in Salerno model up to cubic-quintic
ones (see [11] for more references).
We are interested in the existence of traveling wave solutions
$u_{n}(t)=U(n-\nu t)$ of (3) with a quasi periodic function $U(z)$, $z=n-\nu
t$ and some $\nu\neq 0$.
First, we introduce a function
$\Phi(x):=\frac{4}{x}\sum\limits_{j\in\mathbb{N}}a_{j}\sin^{2}\left[\frac{x}{2}j\right]\,.$
###### Remark 1.1.
Clearly $\Phi\in C(\mathbb{R}\setminus\\{0\\},\mathbb{R})$, $\Phi$ is odd,
$\Phi(2\pi k)=0$ for any $k\in\mathbb{Z}\setminus\\{0\\}$, and $\Phi(x)\to 0$
as $|x|\to\infty$. If $\sum\limits_{j\in\mathbb{N}}j|a_{j}|<\infty$ then
$\Phi\in C(\mathbb{R},\mathbb{R})$ and if
$\sum\limits_{j\in\mathbb{N}}j^{2}|a_{j}|<\infty$ then $\Phi\in
C^{1}(\mathbb{R},\mathbb{R})$. Consequently the range
${\mathcal{R}}\Phi:=\Phi(\mathbb{R}\setminus\\{0\\})$ is either an interval
$[-\bar{R},\bar{R}]$ or $(-\bar{R},\bar{R})$ here with possibility
$\bar{R}=\infty$ (see Section 2.4 for concrete examples).
Now we can state the following existence result.
###### Theorem 1.2.
Let (H1) hold and $T>0$. Then for almost each
$\nu\in\mathbb{R}\setminus\\{0\\}$ and any rational $r\in\mathbb{Q}\cap(0,1)$,
there is a nonzero periodic traveling wave solution $u_{n}(t)=U(n-\nu t)$ of
(3) with $U\in C^{1}(\mathbb{R},\mathbb{C})$ and such that
$U(z+T)={\,\textrm{\rm e}}^{2\pi r\imath}U(z),\,\forall z\in\mathbb{R}\,.$ (5)
Moreover, for any $\nu\in\mathbb{R}\setminus\\{0\\}$ there is at most a finite
number of $\bar{r}_{1},\bar{r}_{2},\cdots,\bar{r}_{m}\in(0,1)$ such that
equation
$-\nu=\Phi\left(\frac{2\pi}{T}(\bar{r}_{j}+k)\right)$
has a solution $k\in\mathbb{Z}$. Then for any
$r\in(0,1)\setminus\\{\bar{r}_{1},\bar{r}_{2},\cdots,\bar{r}_{m}\\}$ there is
a nonzero quasi periodic traveling wave solution $u_{n}(t)=U(n-\nu t)$ with
the above properties. In particular, for any $|\nu|>\bar{R}$ and $r\in(0,1)$,
there is such a nonzero quasi periodic traveling wave solution.
When a nonresonance condition of Theorem 1.2 fails, then we have the following
bifurcation results.
###### Theorem 1.3.
Suppose $f\in C^{2}(\mathbb{R}_{+},\mathbb{R})$ with $f(0)=0$. If there are
$\bar{r}_{1}\in(0,1)$, $\nu\in{\mathcal{R}}\Phi\setminus\\{0\\}$ and $T>0$
such that all solutions $k_{1},k_{2},\cdots,k_{m_{1}}\in\mathbb{Z}$ of
equation
$-\nu=\Phi\left(\frac{2\pi}{T}(\bar{r}_{1}+k)\right)$
are either nonnegative or negative, and $m_{1}>0$. Then for any
$\varepsilon>0$ small there are $m_{1}$ branches of nonzero quasi periodic
traveling wave solutions
$u_{n,j,\varepsilon}(t)=U_{j,\varepsilon}(n-\nu_{\varepsilon}t)$ of (3) with
$U_{j,\varepsilon}\in C^{1}(\mathbb{R},\mathbb{C})$, $j=1,2,\cdots,m_{1}$, and
nonzero velocity $\nu_{\varepsilon}$ satisfying
$U_{j,\varepsilon}(z+T)={\,\textrm{\rm
e}}^{2\pi\bar{r}_{1}\imath}U_{j,\varepsilon}(z)$, $\forall z\in\mathbb{R}$
along with $\nu_{\varepsilon}\to\nu$ and $U_{j,\varepsilon}\rightrightarrows
0$ uniformly on $\mathbb{R}$ as $\varepsilon\to 0$.
###### Remark 1.4.
If $a_{j}\geq 0$ for all $j\in\mathbb{N}$, then the assumptions of Theorem 1.3
are satisfies for any $\nu\in{\mathcal{R}}\Phi\setminus\\{0\\}$ such that
$\frac{T}{2\pi}\Phi^{-1}(-\nu)\setminus\mathbb{Z}\neq\emptyset$, and so there
are bifurcations of quasi periodic traveling waves in the generic resonant
cases. On the other hand, if $\nu\in{\mathcal{R}}\Phi\setminus\\{0\\}$ with
$\frac{T}{2\pi}\Phi^{-1}(-\nu)\subset\mathbb{Z}$ then Theorem 1.2 is
applicable for any $r\in(0,1)$.
Theorem 1.3 is a Lyapunov center theorem for traveling wave solutions. Similar
results are derived in [23] for Fermi-Pasta-Ulam lattices.
We also discuss in Section 4 the extension of these results of (3) on the
lattices $\mathbb{Z}^{2}$ and $\mathbb{Z}^{3}$ [10, 11, 22, 28]. The final
Section 5 is devoted to traveling wave solutions of more general forms than
above [32].
## 2 Existence of Traveling Wave Solutions
In this section, we study the existence of traveling wave solutions of the
form $u_{n}(t)=U(n-\nu t)$, i.e. we are interested in the equation
$-\nu\imath
U^{\prime}(z)=\sum\limits_{j\in\mathbb{N}}a_{j}\partial_{j}U(z)+f(|U(z)|^{2})U(z)\,,$
(6)
where $z=n-\nu t$, $\nu\neq 0$ and $\partial_{j}U(z):=U(z+j)+U(z-j)-2U(z)$. We
are interested in the existence of quasi periodic solutions $U(z)$ of (6)
stated in Theorem 1.2.
### 2.1 Preliminaries
In this subsection we recall some results from critical point theory of [27].
Let $H$ be a Hilbert space and let $J\in C^{1}(H,\mathbb{R})$. Suppose
$H=H_{1}\oplus H_{2}$ for closed linear subspaces, and let
$e_{1},e_{2},\cdots$ be the orthonormal basis of $H_{1}$. Let us put
$H_{n}^{1}:=\textrm{span}\,\\{e_{1},e_{2},\cdots,e_{n}\\}$ and
$H_{n}:=H_{n}^{1}\oplus H_{2}$. Let $P_{n}$ be the orthogonal projection of
$H$ onto $H_{n}$. Set $J_{n}:=J/H_{n}$ \- the restriction of functional $J$ on
subspace $H_{n}$ \- and so $\nabla J_{n}(x)=P_{n}\nabla J(x)$ if $x\in H_{n}$.
###### Definition 2.1.
If there are two positive constants $\alpha$ and $\beta$ such that
$\begin{gathered}J(x)\geq 0\quad\forall x\in\\{x\in
H_{1}\mid\|x\|\leq\beta\\}\,,\\\ J(x)\geq\alpha\quad\forall x\in\\{x\in
H_{1}\mid\|x\|=\beta\\}\,,\\\ J(x)\leq 0\quad\forall x\in\\{x\in
H_{2}\mid\|x\|\leq\beta\\}\,,\\\ J(x)\leq-\alpha\quad\forall x\in\\{x\in
H_{2}\mid\|x\|=\beta\\}\,,\end{gathered}$
then $J$ is said to satisfy the local linking condition at $0$.
###### Definition 2.2.
We shall say that $J$ satisfies the Palais-Smale (PS)∗-condition if any
sequence $\\{x_{n}\\}_{n\in\mathbb{N}}$ in $H$ such that $x_{n}\in H_{n}$,
$J(x_{n})\leq c<\infty$ and $P_{n}\nabla J(x_{n})=\nabla J_{n}(x_{n})\to 0$ as
$n\to\infty$, possesses a convergent subsequence.
Now we can state the following theorem of [27] which we apply.
###### Theorem 2.3.
Suppose
* $(I_{1})$
$J\in C^{1}(H,\mathbb{R})$ satisfies (PS)∗-condition.
* $(I_{2})$
$J$ satisfies the local linking condition at $0$.
* $(I_{3})$
$\forall n$, $J_{n}(x)\to-\infty$ as $\|x\|\to\infty$ and $x\in H_{n}$.
* $(I_{4})$
$\nabla J=A+C$ for a bounded linear self-adjoint operator $A$ such that
$AH_{n}\subset H_{n}$, $\forall n\in\mathbb{N}$ and $C$ is a compact mapping.
Then $J$ possesses a critical point $\bar{x}$ with $|J(\bar{x})|\geq\alpha$.
###### Remark 2.4.
If $0$ is an indefinite nondegenerate critical point of $J$, then $J$
satisfies the local linking condition at $0$.
### 2.2 Proof of Theorem 1.2
In this section, we use Theorem 2.3 to prove Theorem 1.2. Without loss of
generality, we set $T=2\pi$. We suppose $\nu>0$, the case $\nu<0$ can be
handled similarly. First, we identify $\mathbb{C}$ with $\mathbb{R}^{2}$ in
this section. Let $r\in(0,1)$ be fixed. Next, we consider real Banach spaces
$L_{r}^{\widetilde{s}}:=\left\\{U\in
L^{\widetilde{s}}_{loc}(\mathbb{R},\mathbb{C})\mid U(z+2\pi)={\,\textrm{\rm
e}}^{2\pi r\imath}U(z),\,\forall z\in\mathbb{R}\right\\}$
for $\widetilde{s}\geq 1$. Clearly $U\in L_{r}^{\widetilde{s}}$ if and only if
$U(z)={\,\textrm{\rm e}}^{rz\imath}V(z)$ for some $V\in
L^{\widetilde{s}}:=L^{\widetilde{s}}(S^{2\pi},\mathbb{C})$. Consequently
$U_{1}(z+c_{1})\overline{U_{2}(z+c_{2})}$ is $2\pi$-periodic for any
$c_{1},c_{2}\in\mathbb{R}$ and $U_{1},U_{2}\in L_{r}^{\widetilde{s}}$, hence
$|U(z)|$ is $2\pi$-periodic. So we consider the norm on
$L_{r}^{\widetilde{s}}$ like on $L^{\widetilde{s}}$. In particular, we have
$V\in L_{r}^{2}\Leftrightarrow
V(z)=\sum\limits_{k\in\mathbb{Z}}V_{k}{\,\textrm{\rm
e}}^{(r+k)z\imath},\,V_{k}\in\mathbb{C},\,\sum\limits_{k\in\mathbb{Z}}|V_{k}|^{2}<\infty\,.$
Let
$\begin{gathered}X_{r}:=W_{r}^{1/2,2}(S^{2\pi},\mathbb{C})=\left\\{V\in
L^{2}_{r}\mid V(z)=\sum\limits_{k\in\mathbb{Z}}V_{k}{\,\textrm{\rm
e}}^{(r+k)z\imath},\,\sum\limits_{k\in\mathbb{Z}}|V_{k}|^{2}|r+k|<\infty\right\\}\,,\\\
Y_{r}:=W_{r}^{1,2}(S^{2\pi},\mathbb{C})=\left\\{V\in L^{2}_{r}\mid
V(z)=\sum\limits_{k\in\mathbb{Z}}V_{k}{\,\textrm{\rm
e}}^{(r+k)z\imath},\,\sum\limits_{k\in\mathbb{Z}}|V_{k}|^{2}(r+k)^{2}<\infty\right\\}\,.\end{gathered}$
Note $r+k\neq 0$ for any $k\in\mathbb{Z}$. Clearly $Y_{r}\subset X_{r}\subset
L_{r}^{2}$. We consider $L_{r}^{2}$, $X_{r}$ and $Y_{r}$ as real Hilbert
spaces with inner products
$\begin{gathered}\langle
V,W\rangle_{L_{r}^{2}}:=2\pi\Re\sum\limits_{k\in\mathbb{Z}}V_{k}\overline{W_{k}}=\Re\int_{0}^{2\pi}V(z)\overline{W(z)}dz\,,\\\
\langle
V,W\rangle_{X_{r}}:=2\pi\Re\sum\limits_{k\in\mathbb{Z}}V_{k}\overline{W_{k}}|r+k|\\\
\langle
V,W\rangle_{Y_{r}}:=2\pi\Re\sum\limits_{k\in\mathbb{Z}}V_{k}\overline{W_{k}}(r+k)^{2}\end{gathered}$
for $V(z)=\sum\limits_{k\in\mathbb{Z}}V_{k}{\,\textrm{\rm e}}^{(r+k)z\imath}$
and $W(z)=\sum\limits_{k\in\mathbb{Z}}W_{k}{\,\textrm{\rm e}}^{(r+k)z\imath}$.
Clearly $\|U\|_{L^{2}}=\|U\|_{L_{r}^{2}}\leq r_{1}\|U\|_{X_{r}}$, $\forall
U\in X_{r}$ and $\|U\|_{X_{r}}\leq
r_{1}\|U\|_{Y_{r}}=r_{1}\|U^{\prime}\|_{L^{2}}$, $\forall U\in Y_{r}$ for
$r_{1}:=\min\left\\{\sqrt{r},\sqrt{1-r}\right\\}$. The following result is
well-known [27, 34].
###### Lemma 2.5.
For each $\widetilde{s}\geq 1$, $X_{r}$ is compactly embedded into
$L_{r}^{\widetilde{s}}$.
On $X_{r}$, we consider a continuous symmetric bilinear form
$B_{r}(U,V):=4\pi\Re\sum\limits_{k\in\mathbb{Z}}V_{k}\overline{W_{k}}(r+k)\,.$
Note, if $U\in X_{r}$ and $V\in Y_{r}$, then
$2\Re\int_{0}^{2\pi}\imath U(z)\overline{V(z)}\,^{\prime}dz=B_{r}(U,V)\,.$
Now we consider a real functional
$\begin{gathered}I_{r}(U):=\frac{\nu}{2}B_{r}(U,U)+\int_{0}^{2\pi}\left\\{\sum\limits_{j\in\mathbb{Z}}\frac{a_{|j|}}{2}|U(z+j)-U(z)|^{2}-F(|U(z)|^{2})\right\\}dz\\\
=\frac{\nu}{2}B_{r}(U,U)+\int_{0}^{2\pi}\left\\{\sum\limits_{j\in\mathbb{N}}a_{j}|U(z+j)-U(z)|^{2}-F(|U(z)|^{2})\right\\}dz\end{gathered}$
on $X_{r}$. Then $I_{r}\in C^{1}(X_{r},\mathbb{R})$ and for $U\in X_{r}$,
$V\in Y_{r}$, we derive
$\begin{gathered}DI_{r}(U)V=\\\ 2\Re\left\\{\int_{0}^{2\pi}\left(\nu\imath
U(z)\overline{V(z)}\,^{\prime}-\left(\sum\limits_{j\in\mathbb{N}}a_{j}\partial_{j}U(z)+f(|U(z)|^{2})U(z)\right)\overline{V(z)}\right)dz\right\\}.\end{gathered}$
If $U\in X_{r}$ is a critical point of $I_{r}$ then
$\Re\left\\{\int_{0}^{2\pi}\left(\nu\imath
U(z)\overline{V(z)}\,^{\prime}-\left(\sum\limits_{j\in\mathbb{N}}a_{j}\partial_{j}U(z)+f(|U(z)|^{2})U(z)\right)\overline{V(z)}\right)dz\right\\}=0$
(7)
for any $V\in Y_{r}$. Replacing $V$ with $\imath V$ in (7), we obtain
$\int_{0}^{2\pi}\left(\nu\imath
U(z)\overline{V(z)}\,^{\prime}-\left(\sum\limits_{j\in\mathbb{N}}a_{j}\partial_{j}U(z)+f(|U(z)|^{2})U(z)\right)\overline{V(z)}\right)dz=0$
for any $V\in Y_{r}$. This means that $U$ is a weak solution of (6). Then a
standard regularity method shows [34] that $U$ is a $C^{1}$-smooth solution of
(6).
Now we split $X_{r}=X_{+}\oplus X_{-}$ for
$X_{-}:=\left\\{V(z)=\sum\limits_{k=-\infty}^{-1}V_{k}{\,\textrm{\rm
e}}^{(r+k)z\imath}\right\\},\quad
X_{+}:=\left\\{V(z)=\sum\limits_{k=0}^{\infty}V_{k}{\,\textrm{\rm
e}}^{(r+k)z\imath}\right\\}\,.$
Clearly if $U=U_{+}+U_{-}$ then
$B_{r}(U,U)=2\left(\|U_{+}\|^{2}_{X_{r}}-\|U_{-}\|^{2}_{X_{r}}\right)$.
Next, let us define $\widetilde{K}_{r}:L^{2}_{r}\to X_{r}$ as
$\langle\widetilde{K}_{r}H,V\rangle_{X_{r}}:=2\Re\int_{0}^{2\pi}H(z)\overline{V(z)}dz,\,\forall
V\in X_{r}\,.$ (8)
Then
$\widetilde{K}_{r}H=\sum_{k\in\mathbb{Z}}\frac{2H_{k}}{|r+k|}{\,\textrm{\rm
e}}^{(r+k)\imath z}$
and so $\widetilde{K}_{r}$ is compact. To study $\nabla I_{r}(u)$, we
introduce the mapping $\Psi_{r}:X_{r}\to X_{r}$ defined by
$\langle\Psi_{r}(U),V\rangle_{X_{r}}:=2\Re\int_{0}^{2\pi}f(|U(z)|^{2})U(z)\overline{V(z)}dz$
for any $V\in X_{r}$. By Lemma 2.5, the Nemytskij operator $U\to
f(|U(z)|^{2})U(z)$ from $X_{r}$ to $L_{r}^{2}$ is continuous. Using (8), we
get
$\Psi_{r}(U)=\widetilde{K}_{r}f(|U|^{2})U\,.$
Hence $\Psi_{r}:X_{r}\to X_{r}$ is compact and continuous.
###### Lemma 2.6.
Under (H1) it hods $D\Psi_{r}(0)=0$.
###### Proof 2.7.
There is a constant $c_{3}$ such that
$|f(w)|\leq c_{3}(w+w^{s})$
for any $w\geq 0$. Then by Lemma 2.5, we derive
$\begin{gathered}|f(|U|^{2})U|_{L_{r}^{2}}^{2}=\int_{0}^{2\pi}f(|U(z)|^{2})^{2}|U(z)|^{2}dz\\\
\leq 2c^{2}_{3}\int_{0}^{2\pi}\left(|U(z)|^{6}+|U(z)|^{2(2s+1)}\right)dz\leq
c^{2}_{4}\left(\|U\|^{3}_{X_{r}}+\|U\|^{2s+1}_{X_{r}}\right)^{2}\end{gathered}$
for a constant $c_{4}>0$. Hence
$\left|\langle\Psi_{r}(U),V\rangle_{X_{r}}\right|\leq
2\|f(|U|^{2})U\|_{L_{r}^{2}}\|V\|_{L_{r}^{2}}\leq
c_{5}\left(\|U\|^{3}_{X_{r}}+\|U\|^{2s+1}_{X_{r}}\right)\|V\|_{X_{r}}$
for a constant $c_{5}>0$. This implies
$\|\Psi_{r}(U)\|_{X_{r}}\leq
c_{5}\left(\|U\|^{3}_{X_{r}}+\|U\|^{2s+1}_{X_{r}}\right),\,\forall U\in
X_{r}\,.$
Since $\Psi_{r}(0)=0$ and $s>0$, we get $D\Psi_{r}(0)=0$. The proof is
finished.
Finally, define ${\mathcal{L}}_{r}:L_{r}^{2}\to L_{r}^{2}$ as
${\mathcal{L}}_{r}U:=\sum_{j\in\mathbb{N}}a_{j}\partial_{j}U(z)\,.$
Then
$\nabla
I_{r}(U)=\left(2\nu\bm{I}_{+}-2\nu\bm{I}_{-}-\widetilde{K}_{r}{\mathcal{L}}_{r}-\Psi_{r}\right)(U)$
(9)
for the identities $\bm{I}_{\pm}:X_{\pm}\to X_{\pm}$. Clearly
$A_{r}:=2\nu\bm{I}_{+}-2\nu\bm{I}_{-}-\widetilde{K}_{r}{\mathcal{L}}_{r}$
is a self-adjoint bounded operator $A_{r}:X_{r}\to X_{r}$ satisfying
$A_{r}U=2\sum_{k\in\mathbb{Z}}\left(\nu\,\textrm{sgn}\,(r+k)+\frac{4}{|r+k|}\sum_{j\in\mathbb{N}}a_{j}\sin^{2}\left[\frac{r+k}{2}j\right]\right)U_{k}{\,\textrm{\rm
e}}^{(r+k)\imath z}\,.$
Consequently, the spectrum $\sigma(A_{r})$ of $A_{r}$ is given by
$\sigma(A_{r})=\left\\{2\,\textrm{sgn}\,(r+k)\left(\nu+\Phi(r+k)\right)\mid
k\in\mathbb{Z}\right\\}\,.$
By Lemma 2.6, we get that under the assumption
$-\nu\neq\Phi(r+k)\,\forall k\in\mathbb{Z}\,,$ (10)
$0$ is an indefinite nondegenerate critical point of $I_{r}$: $\nabla
I_{r}(0)=0$ and $\textrm{Hess}\,I_{r}(0)=A_{r}$ with $0\notin\sigma(A_{r})$
and $X_{r}=X_{1,r}\oplus X_{2,r}$ with
$\sigma(A_{r}/X_{1,r})\subset(0,\infty)$ and
$\sigma(A_{r}/X_{2,r})\subset(-\infty,0)$ where $X_{1,r}$, $X_{2,r}$ are
suitable closed linear subspaces of $X_{r}$. Note $X_{1,r}$ and $X_{2,r}$ are
infinite dimensional, since $\Phi(r+k)\to 0$ as $|k|\to\infty$. Consequently
by Remark 2.4, under (10), $I_{r}$ satisfies the local linking condition at
$0$ in the sense of Definition 2.1, i.e. condition $(I_{2})$ of Theorem 2.3 is
verified.
We consider an equivalent scalar product $\langle\cdot,\cdot\rangle_{r}$ on
$X_{r}$ such that
$\langle A_{r}U,U\rangle_{r}=\|U_{1}\|_{r}^{2}-\|U_{2}\|_{r}^{2},\quad
U_{1}\in X_{1,r},\,U_{2}\in X_{2,r}\,.$
Note there is a linear isomorphism $K_{r}:X_{r}\to X_{r}$ such that
$\langle U,V\rangle_{X_{r}}=\langle K_{r}U,V\rangle_{r},\quad\forall U,\forall
V\in X_{r}\,.$
Clearly $K_{r}$ is self-adjoint and positive definite. Then
$\begin{gathered}I_{r}(U)=\frac{\nu}{2}\|U_{1}\|_{r}^{2}-\frac{\nu}{2}\|U_{2}\|_{r}^{2}-\int_{0}^{2\pi}F(|U(z)|^{2})dz\,,\\\
\nabla I_{r}(U)=\nu\bm{I}_{1}-\nu\bm{I}_{2}-K_{r}\Psi_{r}\,,\\\ \langle\nabla
I_{r}(U),V\rangle_{r}=DI_{r}(U)V=\nu\|V_{1}\|_{r}^{2}-\nu\|V_{2}\|_{r}^{2}\\\
-2\Re\int_{0}^{2\pi}f(|U(z)|^{2})U(z)\overline{V(z)}dz\,.\end{gathered}$
Let $X_{2,r}=\textrm{span}\,\\{e_{1},e_{2},\cdots\\}$ and $e_{i}$ are
eigenvectors of $A_{r}$. Then we take
$X_{n}=\textrm{span}\,\\{e_{1},e_{2},\cdots,e_{n}\\}\oplus X_{1,r}$ for $n\geq
3$. So clearly $AX_{n}\subset X_{n}$, i.e. condition $(I_{4})$ of Theorem 2.3
is verified. Let $P_{n}:X_{r}\to X_{n}$ be the orthogonal projection with
respect $\langle\cdot,\cdot\rangle_{r}$.
We suppose there is a sequence $\\{U_{m}\\}_{m\in\mathbb{N}}\subset X_{r}$,
$U_{m}\in X_{m}$ and a constant $c$ such that
$I_{r}(U_{m})\leq c\quad\textrm{and}\quad P_{m}\nabla I_{r}(U_{m})\to 0\,.$
Then for $m$ large we get,
$\begin{gathered}c+\|U_{m}\|_{r}\geq I_{r}(U_{m})-\frac{1}{2}\langle
P_{m}\nabla I_{r}(U_{m}),U_{m}\rangle_{r}\\\
=\int_{0}^{2\pi}\left[f(|U_{m}(z)|^{2})|U_{m}(z)|^{2}-F(|U_{m}(z)|^{2})\right]dz\\\
\geq\int_{0}^{2\pi}(\mu-1)F(|U_{m}(z)|^{2})dz-2\pi\bar{r}\\\
\geq(\mu-1)c_{2}\int_{0}^{2\pi}\left(|U_{m}(z)|^{2(s+1)}-1\right)dz-2\pi\bar{r}\\\
\geq(\mu-1)c_{2}\left(\|U_{m}\|^{2(s+1)}_{L^{2(s+1)}}-c_{6}\right)\end{gathered}$
(11)
for a constant $c_{6}>0$.
By following the same arguments, we derive
$\begin{gathered}\nu\|U_{1,m}\|^{2}_{r}\leq\|P_{m}\nabla
I_{m}(U_{m})\|\cdot\|U_{1,m}\|_{r}+2\int_{0}^{2\pi}f(|U_{m}(z)|^{2})|U_{m}(z)||U_{1,m}(z)|dz\\\
\leq\|U_{1,m}\|_{r}+2c_{7}\int_{0}^{2\pi}\left(|U_{m}(z)|^{2s+1}+1\right)|U_{1,m}(z)|dz\\\
\leq\|U_{1,m}\|_{r}+2c_{7}\left\||U_{m}|^{2s+1}+1\right\|_{L^{\frac{2(s+1)}{2s+1}}}\|U_{1,m}\|_{L^{2(s+1)}}\\\
\leq\|U_{1,m}\|_{r}+2c_{7}\left(\|U_{m}\|^{2s+1}_{L^{2(s+1)}}+1\right)\|U_{1,m}\|_{r}\end{gathered}$
and hence
$\|U_{1,m}\|_{r}\leq c_{8}\left(\|U_{m}\|^{2s+1}_{L^{2(s+1)}}+1\right)\,.$
Similarly we obtain
$\|U_{2,m}\|_{r}\leq c_{8}\left(\|U_{m}\|^{2s+1}_{L^{2(s+1)}}+1\right)$
and consequently by (11), we obtain
$\|U_{m}\|_{r}\leq 2c_{8}\left(\|U_{m}\|^{2s+1}_{L^{2(s+1)}}+1\right)\leq
c_{9}\left(\|U_{m}\|^{\frac{2s+1}{2(s+1)}}_{r}+1\right)$
for positive constants $c_{7}$, $c_{8}$ and $c_{9}$. Thus
$\\{U_{m}\\}_{m\in\mathbb{N}}\subset X_{r}$ is bounded. Since
$P_{m}\nabla I_{r}(U_{m})=\nu U_{1,m}-\nu U_{2,m}-K_{r}\Psi_{r}(U_{m})\to 0$
and $K_{r}\Psi_{r}$ is compact, there is a convergent subsequence of
$\\{U_{m}\\}_{m\in\mathbb{N}}$ in $X_{r}$. Summarizing, (PS)∗-condition is
verified for $I_{r}$, i.e. condition $(I_{1})$ of Theorem 2.3 is verified.
Next, let $U\in X_{n}$. Then using
$U_{1}\in\textrm{span}\,\\{e_{1},e_{2},\cdots,e_{n}\\}$, we derive
$\begin{gathered}I_{r}(U)=\frac{\nu}{2}\left(\|U_{1}\|_{r}^{2}-\|U_{2}\|_{r}^{2}\right)-\int_{0}^{2\pi}F(|U(z)|^{2})dz\\\
\leq\frac{\nu}{2}\left(\|U_{1}\|_{r}^{2}-\|U_{2}\|_{r}^{2}\right)-c_{2}\int_{0}^{2\pi}\left(|U(z)|^{2(s+1)}-1\right)dz\\\
\leq\frac{\nu}{2}\left(\|U_{1}\|_{r}^{2}-\|U_{2}\|_{r}^{2}\right)-c_{2}\|U\|^{2(s+1)}_{L^{2(s+1)}}+c_{10}\\\
\leq\frac{\nu}{2}\left(\|U_{1}\|_{r}^{2}-\|U_{2}\|_{r}^{2}\right)-c_{11}\left(\|U_{1}\|^{2(s+1)}_{L^{2}}+\|U_{2}\|^{2(s+1)}_{L^{2}}\right)+c_{10}\\\
\leq\frac{\nu}{2}\|U_{1}\|_{r}^{2}\left(1-c_{12}\|U_{1}\|^{2s}_{r}\right)-\frac{\nu}{2}\|U_{2}\|_{r}^{2}+c_{10}\end{gathered}$
for positive constants $c_{10}$, $c_{11}$ and $c_{12}$. Now it is clear that
$I_{r}(U)\to-\infty$ as $\|U\|_{r}\to\infty$, i.e. condition $(I_{3})$ of
Theorem 2.3 is verified.
Summarizing, under assumptions (H1) and (10), all conditions
$(I_{1})$-$(I_{4})$ of Theorem 2.3 are verified for $I_{r}$. Hence there is a
nonzero critical point $U_{r}\in X_{r}$ of $I_{r}$, which we already know to
be a $C^{1}$-smooth solution of (6) satisfying (5). Note (10) certainly holds
for any $|\nu|>\bar{R}$ and $r\in(0,1)$. Hence the proof of the second part of
Theorem 1.2 is finished. To prove the first part, it enough to observe that
the set
$\left\\{\Phi(r+k)\mid r\in\mathbb{Q}\cap(0,1),\quad k\in\mathbb{Z}\right\\}$
is countable, and thus for almost each $\nu\in\mathbb{R}\setminus\\{0\\}$ and
any $r\in\mathbb{Q}\cap(0,1)$, condition (10) holds.
### 2.3 Remarks
###### Remark 2.8.
When $r$ is rational in Theorem 1.2 then we get periodic $U(z)$ with
arbitrarily large minimal periods. If $r$ is irrational then clearly
$U(z)={\,\textrm{\rm e}}^{\frac{2\pi}{T}rz\imath}V(z)$ for a $T$-periodic
$V(z)=U(z){\,\textrm{\rm e}}^{-\frac{2\pi}{T}rz\imath}$. So $U(z)$ is quasi
periodic and its orbit in $\mathbb{C}\simeq\mathbb{R}^{2}$ is dense either in
a compact annulus or in a compact disc. But $|U(z)|$ is $T$-periodic in the
both cases.
###### Remark 2.9.
Changing $t\leftrightarrow-t$, we can also handle DNLS
$-\imath\dot{u}_{n}=\sum\limits_{j\in\mathbb{N}}a_{j}\Delta_{j}u_{n}+f(|u_{n}|^{2})u_{n},\quad
n\in\mathbb{Z}$ (12)
under (H1) and (10) becomes
$\nu\neq\Phi(r+k)\,\forall
k\in\mathbb{Z}\quad\textrm{and}\quad\nu\in(0,\bar{R})\,.$ (13)
###### Remark 2.10.
Assume that $U\in Y_{r}$ is a weak solution of (6), then
$\begin{gathered}|U(z)|\leq\sum_{k\in\mathbb{Z}}|U_{k}|\leq\sqrt{\sum_{k\in\mathbb{Z}}|U_{k}|^{2}(r+k)^{2}}\sqrt{\sum_{k\in\mathbb{Z}}(r+k)^{-2}}\\\
=\sqrt{\frac{\pi}{2}}{\rm cosec}\,\pi r\,\|U\|_{Y_{r}}.\end{gathered}$
Let $\widetilde{R}:=\max_{x\in\mathbb{R}_{+}}x\Phi(x)$. Then
$\begin{gathered}|\nu|\|U^{\prime}\|_{L^{2}}=|\nu|\|U\|_{Y_{r}}\leq\|{\mathcal{L}}_{r}U\|_{L^{2}}+\|f(|U|^{2}U)\|_{L^{2}}\\\
\leq\widetilde{R}\|U\|_{L^{2}}+c_{1}\left\||U|^{2s+1}+|U|\right\|_{L^{2}}\\\
\leq\left(\widetilde{R}r_{1}^{2}+c_{1}r_{1}^{2}+c_{1}\frac{\pi^{s+1}}{2^{s}}{\rm
cosec}^{2s+1}\,\pi r\,\|U\|_{Y_{r}}^{2s}\right)\|U\|_{Y_{r}}.\end{gathered}$
So if $U\neq 0$ then we obtain
$|\nu|\leq\widetilde{R}r_{1}^{2}+c_{1}r_{1}^{2}+c_{1}\frac{\pi^{s+1}}{2^{s}}{\rm
cosec}^{2s+1}\,\pi r\,\|U\|_{Y_{r}}^{2s},$
i.e.
$\|U\|_{Y_{r}}\geq\sqrt{2}\sqrt[2s]{\frac{|\nu|-\widetilde{R}r_{1}^{2}-c_{1}r_{1}^{2}}{c_{1}\pi^{s+1}{\rm
cosec}^{2s+1}\,\pi r}}$
for
$|\nu|\geq\widetilde{R}r_{1}^{2}+c_{1}r_{1}^{2}.$
Hence $\|U\|_{Y_{r}}\to\infty$ as $|\nu|\to\infty$ for a possible nonzero
solution $U\in Y_{r}$ of (6).
### 2.4 Examples
We first note
$\Phi(x)=\frac{2}{x}\sum_{j\in\mathbb{N}}a_{j}(1-\cos
xj)=\frac{2}{x}\left[\sum_{j\in\mathbb{N}}a_{j}-\Re\sum_{j\in\mathbb{N}}a_{j}{\,\textrm{\rm
e}}^{xj\imath}\right].$ (14)
Now we turn the the following concrete examples.
###### Example 2.11.
First we suppose that $a_{j}$ is decaying rapidly to $0$. Let
$a_{j}=\frac{1}{j!}$. Then
$\begin{gathered}\sum_{j\in\mathbb{N}}\frac{1}{j!}{\,\textrm{\rm
e}}^{xj\imath}={\,\textrm{\rm e}}^{{\,\textrm{\rm
e}}^{x\imath}}-1={\,\textrm{\rm e}}^{\cos x+\imath\sin x}-1\\\ ={\,\textrm{\rm
e}}^{\cos x}\left[\cos\sin x+\imath\sin\sin x\right]-1.\end{gathered}$
So by (14) we derive
$\Phi(x)=\frac{2}{x}\left[\sum_{j\in\mathbb{N}}\frac{1}{j!}-{\,\textrm{\rm
e}}^{\cos x}\cos\sin x+1\right]=\frac{2}{x}\left[{\,\textrm{\rm
e}}-{\,\textrm{\rm e}}^{\cos x}\cos\sin x\right].$
By Remark 1.1, $\Phi\in C^{1}(\mathbb{R},\mathbb{R})$ with the graph on
$[-4\pi,4\pi]$:
A numerical solution shows that $\Phi$ has a maximum
$\bar{R}=\Phi(x_{0})\doteq 3.15177$ at $x_{0}\doteq 1.03665$.
###### Example 2.12.
Now we suppose that $a_{j}$ is decaying exponentially to $0$. Let
$a_{j}={\,\textrm{\rm e}}^{-j}$, hence we have the discrete Kac-Baker
interaction kernel [13, 14]. Then
$\begin{gathered}\sum_{j\in\mathbb{N}}{\,\textrm{\rm e}}^{-j}{\,\textrm{\rm
e}}^{xj\imath}=\sum_{j\in\mathbb{N}}{\,\textrm{\rm
e}}^{(x\imath-1)j}=\frac{{\,\textrm{\rm e}}^{x\imath-1}}{1-{\,\textrm{\rm
e}}^{x\imath-1}}\\\ =\frac{\cos x+\imath\sin x}{{\,\textrm{\rm e}}-\cos
x-\imath\sin x}=\frac{{\,\textrm{\rm e}}\cos x-1+{\,\textrm{\rm e}}\imath\sin
x}{{\,\textrm{\rm e}}^{2}+1-2{\,\textrm{\rm e}}\cos x}.\end{gathered}$
So by (14) we derive
$\Phi(x)=\frac{2}{x}\left[\sum_{j\in\mathbb{N}}{\,\textrm{\rm
e}}^{-j}-\frac{{\,\textrm{\rm e}}\cos x-1}{{\,\textrm{\rm
e}}^{2}+1-2{\,\textrm{\rm e}}\cos x}\right]=\frac{2{\,\textrm{\rm
e}}({\,\textrm{\rm e}}+1)(1-\cos x)}{({\,\textrm{\rm e}}-1)x({\,\textrm{\rm
e}}^{2}+1-2{\,\textrm{\rm e}}\cos x)}.$
By Remark 1.1, $\Phi\in C^{1}(\mathbb{R},\mathbb{R})$ with the graph on
$[-4\pi,4\pi]$:
A numerical solution shows that $\Phi$ has a maximum
$\bar{R}=\Phi(x_{0})\doteq 0.992045$ at $x_{0}\doteq 0.991541$.
###### Example 2.13.
In this example, we suppose that $a_{j}$ is decaying polynomially to $0$ (cf.
[30]), by considering several cases:
1\. Let $a_{j}=\frac{1}{j^{4}}$. Then
$\Phi(x)=\frac{2}{x}\sum_{j\in\mathbb{N}}\left(\frac{1}{j^{4}}-\frac{1}{j^{4}}\cos
xj\right)=\frac{\left(|x|-2\pi\left[\frac{|x|}{2\pi}\right]\right)^{2}}{24x}\left(2\pi-|x|+2\pi\left[\frac{|x|}{2\pi}\right]\right)^{2}.$
Here $[\cdot]$ is the integer part function. By Remark 1.1, $\Phi\in
C^{1}(\mathbb{R},\mathbb{R})$ with the graph on $[-4\pi,4\pi]$:
$\Phi$ has a maximum $\bar{R}=\Phi(x_{0})=\frac{4\pi^{3}}{81}\doteq 1.53117$
at $x_{0}=2\pi/3\doteq 2.0944$. Similar results hold for $a_{j}=j^{-\beta}$
with $\beta>3$ by Remark 1.1.
2\. Let $a_{j}=\frac{1}{j^{3}}$. So we consider the dipole-dipole interaction
(cf. [5, 13, 25, 30]). By Remark 1.1, $\Phi\in C(\mathbb{R},\mathbb{R})$ with
the graph on $[-4\pi,4\pi]$:
$\Phi$ has a maximum $\bar{R}=\Phi(x_{0})\doteq 1.68311$ at $x_{0}\doteq
1.76076$. Next we know that [41]
$\sum_{j\in\mathbb{N}}\frac{1}{j}\cos
xj=-\ln\left|2\sin\frac{x}{2}\right|,\quad 0<x<2\pi.$
Then
$\sum_{j\in\mathbb{N}}\frac{1}{j^{2}}\sin
xj=-\int\limits_{0}^{x}\ln\left|2\sin\frac{s}{2}\right|ds.$
Using $x/2\leq\sin x\leq x$ for $x\geq 0$ small, we derive
$x-x\ln x=-\int\limits_{0}^{x}\ln
s\,ds\leq\sum_{j\in\mathbb{N}}\frac{1}{j^{2}}\sin
xj\leq-\int\limits_{0}^{x}\ln\frac{s}{2}ds=x-x\ln\frac{x}{2}.$
By L’Hopital’s rule, we obtain
$\lim_{x\to 0_{+}}\frac{\Phi(x)}{x}=\lim_{x\to
0_{+}}\frac{4\sum_{j\in\mathbb{N}}\frac{1}{j^{3}}\sin^{2}xj}{x^{2}}=\lim_{x\to
0_{+}}\frac{2\sum_{j\in\mathbb{N}}\frac{1}{j^{2}}\sin 2xj}{x}=+\infty.$
Hence $\Phi$ has no derivative at $x_{0}=0$.
Next, let $a_{j}=j^{-\beta}$ for $2<\beta<3$. By Remark 1.1, $\Phi$ is still
continuous. Since $\Phi(0)=0$ and
$\lim_{x\to 0_{+}}\frac{\Phi(x)}{x}\geq\lim_{x\to
0_{+}}\frac{4\sum_{j\in\mathbb{N}}\frac{1}{j^{3}}\sin^{2}xj}{x^{2}}=+\infty,$
$\Phi(x)$ is continuous but not $C^{1}$-smooth on $\mathbb{R}$.
3\. Let $a_{j}=\frac{1}{j^{2}}$. Then
$\Phi(x)=\frac{2}{x}\sum_{j\in\mathbb{N}}\left(\frac{1}{j^{2}}-\frac{1}{j^{2}}\cos
xj\right)=\frac{\left(|x|-2\pi\left[\frac{|x|}{2\pi}\right]\right)}{2x}\left(2\pi-|x|+2\pi\left[\frac{|x|}{2\pi}\right]\right).$
By Remark 1.1, $\Phi\in C(\mathbb{R}\setminus\\{0\\},\mathbb{R})$ with the
graph on $[-4\pi,4\pi]$:
$\Phi$ is discontinuous at $x_{0}=0$ where it has a supremum $\bar{R}=\pi$.
4\. Let $a_{j}=j^{-\beta}$ for $1<\beta<2$. For $\beta=7/4$, $\Phi$ has the
graph on $[-4\pi,4\pi]$:
Hence $\Phi$ is discontinuous at $x_{0}=0$ with $\lim_{x\to
0_{+}}\Phi(x)=+\infty$. We show that this holds for any $1<\beta<2$. First
suppose $3/2<\beta<2$. Then the series
$\Upsilon(x):=\sum_{j\in\mathbb{N}}\frac{1}{j^{\beta-1}}\sin jx$
converges uniformly on any $[\varepsilon,2\pi-\varepsilon]$ for
$0<\varepsilon<\pi$. But since
$\sum_{j\in\mathbb{N}}\frac{1}{j^{2(\beta-1)}}<\infty$, so $\Upsilon\in
L^{2}\subset L^{1}$. On the other hand, we know [41] that
$\Upsilon(x):=\Gamma(2-\beta)\cos\frac{\pi(\beta-1)}{2}\cdot x^{\beta-2}+O(1)$
on $(0,\pi]$. Hence
$\sum_{j\in\mathbb{N}}\frac{1-\cos
jx}{j^{\beta}}=\int_{0}^{x}\Upsilon(s)ds=\frac{\Gamma(2-\beta)}{\beta-1}\cos\frac{\pi(\beta-1)}{2}\cdot
x^{\beta-1}+O(x)$
on $[0,\pi]$. Consequently, we obtain
$\Phi(x)=\frac{2\Gamma(2-\beta)}{\beta-1}\cos\frac{\pi(\beta-1)}{2}\cdot
x^{\beta-2}+O(1)$
on $(0,\pi]$, which implies $\lim_{x\to 0_{+}}\Phi(x)=+\infty$ for any
$3/2<\beta<2$. Finally, if $1<\beta\leq 3/2$, then
$\Phi(x)\geq\frac{2}{x}\sum_{j\in\mathbb{N}}\frac{1-\cos
jx}{j^{7/4}}=\frac{8}{3}\Gamma\left(\frac{1}{4}\right)\cos\frac{3\pi}{8}\cdot\frac{1}{\sqrt[4]{x}}+O(1)\to+\infty$
as $x\to 0_{+}$. Hence, $\lim_{x\to 0_{+}}\Phi(x)=+\infty$ for any
$1<\beta<2$.
Summarizing, we have the following result.
###### Lemma 2.14.
Let $a_{j}=j^{-\beta}$ for $1<\beta$. Then
* (i)
$\Phi\in C^{1}(\mathbb{R},\mathbb{R})$ for $\beta>3$, and
${\mathcal{R}}\Phi=[-\bar{R},\bar{R}]$ for some $\bar{R}<\infty$.
* (ii)
$\Phi\in C(\mathbb{R},\mathbb{R})$ and $\Phi\notin
C^{1}(\mathbb{R},\mathbb{R})$ for $2<\beta\leq 3$, and
${\mathcal{R}}\Phi=[-\bar{R},\bar{R}]$ for some $\bar{R}<\infty$.
* (iii)
$\Phi\in C(\mathbb{R}\setminus\\{0\\},\mathbb{R})$ and $\Phi\notin
C(\mathbb{R},\mathbb{R})$ for $\beta=2$, and ${\mathcal{R}}\Phi=(-\pi,\pi)$.
* (iv)
$\Phi\in C(\mathbb{R}\setminus\\{0\\},\mathbb{R})$ and $\Phi\notin
C(\mathbb{R},\mathbb{R})$ for $1<\beta<2$, and
${\mathcal{R}}\Phi=(-\infty,+\infty)$.
###### Remark 2.15.
We see that if the interaction is strong, so the case (iv) of Lemma 2.14
holds, then there are continuum many quasi periodic traveling wave solutions
$U(z)$ of Theorem 1.2 for any $\nu\neq 0$, $T>0$ and $r\in(0,1)$ such that
$r\notin\left\\{z-[z]\mid z\in\frac{T}{2\pi}\Phi^{-1}(-\nu)\right\\}$, with
$\|U\|_{Y_{r}}\to\infty$ as $|\nu|\to\infty$ by Remark 2.10. On the other
hand, if the interaction is weak, then we can show in addition quasi periodic
traveling waves with speeds in intervals $(-\infty,-\bar{R})$ and
$(\bar{R},\infty)$ for any $T>0$ and $r\in(0,1)$.
###### Remark 2.16.
For the reader convenience, we present the above graphs of function $\Phi$ to
visualize their quantitative and qualitative changes according to different
choices of values of sequences $\\{a_{j}\\}_{j\in\mathbb{Z}}$ in (3), and
hence with different consequences from Theorems 1.2 and 1.3 for the existence
and bifurcations of quasi periodic traveling wave solutions of (3). Moreover,
these graphs can be compared with similar ones for traveling waves for higher
dimensional DNLS in Section 4 and for traveling waves with frequencies in
Section 5. Finally these examples are motivated by applications mentioned in
the corresponding references.
## 3 Bifurcation of Traveling Wave Solutions
In this section we proceed with the study of (6) when nonresonance of Theorem
1.2 fails, i.e. $r\in\\{\bar{r}_{1},\bar{r}_{2},\cdots,\bar{r}_{m}\\}$. We
scale in (6) the velocity by $\nu\leftrightarrow\nu/(1+\lambda)$ to get
equation
$-\nu\imath
U^{\prime}(z)=(1+\lambda)\left(\sum\limits_{j\in\mathbb{N}}a_{j}\partial_{j}U(z)+f(|U(z)|^{2})U(z)\right)\,,$
(15)
where $\lambda$ is a small parameter, i.e.
$u_{n}(t)=U\left(n-\frac{\nu}{1+\lambda}t\right)$ is a solution of (3). We are
interested in the existence of quasi periodic solutions $U(z)$ of (15) stated
in Theorem 1.3.
### 3.1 Preliminaries
In this subsection we recall some results from critical point theory of [29].
Let $H$ be a Hilbert space with a scalar product $(\cdot,\cdot)$ and the
corresponding norm $\|\cdot\|$. Let $\Theta:S^{1}\to L(H)$ be an isometric
representation of the unit circle $S^{1}$ over $H$, i.e. the following
properties are satisfied
* (R)
$\Theta(0)=\bm{I}$ \- the identity,
$\Theta(\theta_{1}+\theta_{2})=\Theta(\theta_{1})\Theta(\theta_{2})$ for any
$\theta_{1},\theta_{2}\in S^{1}$, $(\theta,h)\to\Theta(\theta)h$ is
continuous, and $\|\Theta(\theta)h\|=\|h\|$ for any $\theta\in S^{1}$ and
$h\in H$.
We set
$\textrm{Fix}(S^{1}):=\left\\{h\in
H\mid\Theta(\theta)h=h\,\forall\theta\in\Theta\right\\}.$
We consider $J_{1},J_{2}\in C^{2}(H,\mathbb{R})$ such that
* (H1)
$J_{2}(0)=0$ and $\nabla J_{1}(0)=\nabla J_{2}(0)=0$.
* (H2)
$\textrm{Hess}\,J_{1}(0)$ is a Fredholm operator, i.e.
$\dim\textrm{Hess}\,J_{1}(0)<\infty$, ${\mathcal{R}}\textrm{Hess}\,J_{1}(0)$
is closed and $\textrm{codim}\,{\mathcal{R}}\textrm{Hess}\,J_{1}(0)<\infty$.
* (H3)
$\dim\ker\textrm{Hess}\,J_{1}(0)\geq 2$ and $\textrm{Hess}\,J_{2}(0)$ is
positive definite on $\ker\textrm{Hess}\,J_{1}(0)$.
* (H4)
$J_{1}$ and $J_{2}$ are $S^{1}$-invariant, i.e.
$J_{1,2}(\Theta(\theta)h)=\Theta(\theta)J_{1,2}(h)$ for any $\theta\in\Theta$
and $h\in H$.
* (H5)
$\ker\textrm{Hess}\,J_{1}(0)\cap\textrm{Fix}(S^{1})=\\{0\\}$.
Now we can state the following [29, Theorem 6.7].
###### Theorem 3.1.
Under the above assumptions (H1)-(H5), for each sufficiently small
$\varepsilon>0$, equation
$\nabla J_{1}(h)+\lambda\nabla J_{2}(h)=0$ (16)
has at leat $\frac{1}{2}\dim\ker\textrm{\rm Hess}\,J_{1}(0)$ of $S^{1}$-orbit
solutions
$\left\\{(\lambda_{k}(\varepsilon),\Theta(\theta))h_{k}(\varepsilon)\mid\theta\in
S^{1}\right\\},\quad k=1,2,\cdots,\frac{1}{2}\dim\ker\textrm{\rm
Hess}\,J_{1}(0)$
such that $J_{2}(h_{k}(\varepsilon))=\varepsilon$ and $h_{k}(\varepsilon)\to
0$, $\lambda_{k}(\varepsilon)\to 0$ as $\varepsilon\to 0$. Clearly
$h_{k}(\varepsilon)\neq 0$.
###### Remark 3.2.
When $\textrm{Hess}\,J_{2}(0)$ is negative definite on
$\ker\textrm{Hess}\,J_{1}(0)$, then Theorem 3.1 holds for $\varepsilon<0$
small.
###### Remark 3.3.
By (H4), $\ker\textrm{Hess}\,J_{1}(0)$ is invariant with respect to $\Theta$.
Using (H5), $\dim\ker\textrm{Hess}\,J_{1}(0)$ is even.
Now assume $H=H_{+}\oplus H_{-}$ be an orthogonal and $\Theta$-invariant
decomposition with the corresponding orthogonal projections $P_{\pm}:H\to
H_{\pm}$. Then $\Theta(\theta)P_{\pm}=P_{\pm}\Theta(\theta)$ for any
$\theta\in\Theta$. Let us consider an equation
$\zeta(\bm{I}_{+}-\bm{I}_{-})h+(1+\lambda)({\mathcal{K}}h+\nabla{\mathcal{F}}(h))=0,$
(17)
where $\zeta\neq 0$ is a constant, $\lambda$ is a small parameter,
$\bm{I}_{\pm}:H_{\pm}\to H_{\pm}$ are the identities. We suppose
* (A)
${\mathcal{K}}:H\to H$ is compact self-adjoint and ${\mathcal{F}}\in
C^{2}(H,\mathbb{R})$ with ${\mathcal{F}}(0)=0$, $\nabla{\mathcal{F}}(0)=0$,
$\textrm{\rm Hess}\,{\mathcal{F}}(0)=0$, and ${\mathcal{K}}$, ${\mathcal{F}}$
are $S^{1}$-invariant. Moreover, ${\mathcal{K}}H_{\pm}\subset H_{\pm}$.
Then
$\begin{gathered}J_{1}(h)=\frac{\zeta}{2}(\|P_{+}h\|^{2}-\|P_{-}h\|^{2})+\frac{1}{2}({\mathcal{K}}h,h)+{\mathcal{F}}(h),\\\
J_{2}(h)=\frac{1}{2}({\mathcal{K}}h,h)+{\mathcal{F}}(h).\end{gathered}$
Hence
$\begin{gathered}J_{1}(0)=J_{2}(0)=0,\quad\nabla J_{1}(0)=\nabla
J_{2}(0)=0,\\\
\textrm{Hess}\,J_{1}(0)=\zeta(\bm{I}_{+}-\bm{I}_{-})+{\mathcal{K}},\quad\textrm{Hess}\,J_{2}(0)={\mathcal{K}}.\end{gathered}$
So assumptions (H1), (H2) and (H4) are satisfied. Since
$P_{\pm}{\mathcal{K}}={\mathcal{K}}P_{\pm}$, equation
$\textrm{Hess}\,J_{1}(0)h=\zeta(\bm{I}_{+}-\bm{I}_{-})h+{\mathcal{K}}h=0$
splits into
${\mathcal{K}}h_{+}=-\zeta h_{+},\quad{\mathcal{K}}h_{-}=\zeta h_{-},\quad
h_{\pm}=P_{\pm}h.$
Consequently, supposing either
* (B+)
$\ker(\zeta\bm{I}+{\mathcal{K}})\cap H_{+}=\\{0\\}$,
$\dim\ker(\zeta\bm{I}-{\mathcal{K}})\cap H_{-}\geq 2$ and
$\ker(\zeta\bm{I}-{\mathcal{K}})\cap H_{-}\cap\textrm{Fix}(S^{1})=\\{0\\}$
or
* (B-)
$\ker(\zeta\bm{I}-{\mathcal{K}})\cap H_{-}=\\{0\\}$,
$\dim\ker(\zeta\bm{I}+{\mathcal{K}})\cap H_{+}\geq 2$ and
$\ker(\zeta\bm{I}+{\mathcal{K}})\cap H_{+}\cap\textrm{Fix}(S^{1})=\\{0\\}$
we get either
$\ker\textrm{Hess}\,J_{1}(0)=\ker(\zeta\bm{I}-{\mathcal{K}})\cap H_{-}$
or
$\ker\textrm{Hess}\,J_{1}(0)=\ker(\zeta\bm{I}+{\mathcal{K}})\cap H_{+}$
and so (H5) holds as well. Finally, we derive
$\textrm{Hess}\,J_{2}(0)|\ker\textrm{Hess}\,J_{1}(0)=\pm\zeta\bm{I}$
and thus (H3) is also verified (cf. Remark 3.2). Summarizing, Theorem 3.1 and
Remark 3.2 is applicable to (17):
###### Corollary 3.4.
Under assumptions (A) and (B±), for each sufficiently small $\varepsilon\neq
0$, $\pm\varepsilon\zeta>0$, equation (17) has at leat
$\frac{1}{2}\dim\ker(\zeta\bm{I}\mp{\mathcal{K}})\cap H_{\mp}$ of
$S^{1}$-orbit solutions
$\left\\{(\lambda_{k}(\varepsilon),\Theta(\theta))h_{k}(\varepsilon)\mid\theta\in
S^{1}\right\\},\quad
k=1,2,\cdots,\frac{1}{2}\dim\ker(\zeta\bm{I}\mp{\mathcal{K}})\cap H_{\mp}$
such that
$\frac{1}{2}({\mathcal{K}}h_{k}(\varepsilon),h_{k}(\varepsilon))+{\mathcal{F}}(h_{k}(\varepsilon))=\varepsilon$
and $h_{k}(\varepsilon)\to 0$, $\lambda_{k}(\varepsilon)\to 0$ as
$\varepsilon\to 0$. Clearly $h_{k}(\varepsilon)\neq 0$.
###### Remark 3.5.
If $\textrm{Fix}(S^{1})=\\{0\\}$ then (B+) holds if
* (i)
$-\zeta\notin\sigma({\mathcal{K}}/H_{+})$,
$\zeta\in\sigma({\mathcal{K}}/H_{-})$ and $\zeta$ has a multiplicity at least
$2$,
while (B-) holds if
* (ii)
$-\zeta\in\sigma({\mathcal{K}}/H_{+})$,
$\zeta\notin\sigma({\mathcal{K}}/H_{-})$ and $-\zeta$ has a multiplicity at
least $2$,
respectively.
### 3.2 Proof of Theorem 1.3
We again assume for simplicity $T=2\pi$. So let $r=\bar{r}_{1}\in(0,1)$ and
the equation
$-\nu=\Phi\left(\bar{r}_{1}+k\right)$
has solutions $k_{1},k_{2},\cdots,k_{m_{1}}\in\mathbb{Z}$ which are either all
nonnegative, or all negative. Next (15) has the form (cf. (9))
$2(\nu\bm{I}_{+}-\nu\bm{I}_{-})-(1+\lambda)\left(\widetilde{K}_{r}{\mathcal{L}}_{r}U+\Psi_{r}(U)\right)=0$
(18)
and
$\begin{gathered}H=X_{r},\quad\zeta=2\nu,\quad H_{\pm}=X_{\pm},\\\
{\mathcal{K}}=-\widetilde{K}_{r}{\mathcal{L}}_{r},\quad{\mathcal{F}}(u)=-\int_{0}^{2\pi}F(|U(z)|^{2})dz.\end{gathered}$
Isometric representation $\Theta$ is naturally given as
$\Theta(\theta)U(z):=U(z+\theta),$
i.e.
$\Theta(\theta)\left(\sum_{k\in\mathbb{Z}}U_{k}{\,\textrm{\rm
e}}^{(\bar{r}_{1}+k)z\imath}\right)=\sum_{k\in\mathbb{Z}}U_{k}{\,\textrm{\rm
e}}^{\theta k\imath}{\,\textrm{\rm e}}^{(\bar{r}_{1}+k)z\imath}.$
Note $\textrm{Fix}(S^{1})=\\{0\\}$. It is easy to verify (R) for $\Theta$. By
results of Section 2, we get both ${\mathcal{K}}H_{\pm}\subset H_{\pm}$ and
assumption (A) holds, and moreover
$\sigma\left({\mathcal{K}}/H_{\pm}\right)=\left\\{\pm 2\Phi(\bar{r}_{1}+k)\mid
k\in\mathbb{Z}_{\pm}\right\\}.$
Note $\mathbb{Z}_{+}=\\{0\\}\cup\mathbb{N}$ and $\mathbb{Z}_{-}=-\mathbb{N}$.
Hence (i) of Remark 3.5 is satisfied if
$-\nu\notin\left\\{\Phi(\bar{r}_{1}+k)\mid
k\in\mathbb{Z}_{+}\right\\},\quad-\nu\in\left\\{\Phi(\bar{r}_{1}+k)\mid
k\in\mathbb{Z}_{-}\right\\},$
while (ii) if
$-\nu\in\left\\{\Phi(\bar{r}_{1}+k)\mid
k\in\mathbb{Z}_{+}\right\\},\quad-\nu\notin\left\\{\Phi(\bar{r}_{1}+k)\mid
k\in\mathbb{Z}_{-}\right\\}.$
But these are precisely assumptions of Theorem 1.3. So its proof is complete
by Corollary 3.4 and Remark 3.5.
## 4 Traveling Waves for Higher Dimensional DNLS
In this section, we first show how to extend previous results for
2-dimensional DNLS (2D DNLS) [10, 11, 22] of forms
$\begin{gathered}\imath\dot{u}_{n,m}=\sum\limits_{(i,j)\in\mathbb{Z}^{2}_{0}}a_{i,j}\Delta_{i,j}u_{n,m}+f(|u_{n,m}|^{2})u_{n,m},\quad(n,m)\in\mathbb{Z}^{2}\\\
=2\sum\limits_{(i,j)\in\mathbb{Z}^{2}_{0}}a_{i,j}\left(u_{n+i,m+j}-u_{n,m}\right)+f(|u_{n,m}|^{2})u_{n,m},\end{gathered}$
(19)
where $u_{n,m}\in\mathbb{C}$,
$\mathbb{Z}_{0}^{2}:=\mathbb{Z}^{2}\setminus\\{(0,0)\\}$,
$\Delta_{i,j}u_{n,m}:=u_{n+i,m+j}+u_{n-i,m-j}-2u_{n,m}$ are $2$-dimensional
discrete Laplacians, $f$ satisfies (H1) and $a_{i,j}=a_{-i,-j}$ along with
$\sum\limits_{(i,j)\in\mathbb{Z}^{2}_{0}}|a_{i,j}|<\infty$ and all $a_{i,j}$
are not zero.
Again, (19) conserves two dynamical invariants
$\begin{gathered}\sum\limits_{(n,m)\in\mathbb{Z}^{2}}|u_{n,m}|^{2}\quad-\textrm{the
norm},\\\
\sum\limits_{(n,m)\in\mathbb{Z}^{2}}\left[-\sum\limits_{(i,j)\in\mathbb{Z}^{2}_{0}}a_{i,j}\left|u_{n+i,m+j}-u_{n,m}\right|^{2}+F(|u_{n,m}|^{2})\right]\quad-\textrm{the
energy}.\end{gathered}$
We look for traveling wave solutions of (19) of the form
$u_{n,m}(t)=U(n\cos\theta+m\sin\theta-\nu t)$ (20)
with a direction $(\cos\theta,\sin\theta)$ [21]. Hence we are interested in
the equation
$-\nu\imath
U^{\prime}(z)=\sum\limits_{(i,j)\in\mathbb{Z}^{2}_{0}}a_{i,j}\partial_{i,j}U(z)+f(|U(z)|^{2})U(z)\,,$
(21)
where $z=n\cos\theta+m\sin\theta-\nu t$, $\nu\neq 0$ and
$\partial_{i,j}U(z):=U(z+i\cos\theta+j\sin\theta)+U(z-i\cos\theta-j\sin\theta)-2U(z).$
We see that (21) has a very similar form like (6). So we can directly repeat
the above arguments, where now instead of $\Phi(x)$ we get
$\Phi_{\theta}(x):=\frac{4}{x}\sum\limits_{(i,j)\in\mathbb{Z}^{2}_{0}}a_{i,j}\sin^{2}\frac{x(i\cos\theta+j\sin\theta)}{2}.$
Set $\bar{R}_{\theta}:=\sup_{\mathbb{R}}\Phi_{\theta}$. Summarizing, Theorems
1.2 and 1.3 have the following analogies:
###### Theorem 4.1.
Let (H1) hold and $T>0$, $\theta\in[0,2\pi)$. Then for almost each
$\nu\in\mathbb{R}\setminus\\{0\\}$ and any rational $r\in\mathbb{Q}\cap(0,1)$,
there is a nonzero periodic traveling wave solution (20) of (19) with $U\in
C^{1}(\mathbb{R},\mathbb{C})$ satisfying (5). Moreover, for any
$\nu\in\mathbb{R}\setminus\\{0\\}$, there is at most a finite number of
$\bar{r}_{1,\theta},\bar{r}_{2,\theta},\cdots,\bar{r}_{m_{\theta},\theta}\in(0,1)$
such that equation
$-\nu=\Phi_{\theta}\left(\frac{2\pi}{T}(\bar{r}_{j,\theta}+k)\right)$
has a solution $k\in\mathbb{Z}$. Then for any
$r\in(0,1)\setminus\\{\bar{r}_{1,\theta},\bar{r}_{2,\theta},\cdots,\bar{r}_{m_{\theta},\theta}\\}$
there is a nonzero quasi periodic traveling wave solution (20) of (19) with
the above properties. In particular, for any $|\nu|>\bar{R}_{\theta}$ and
$r\in(0,1)$, there is such a nonzero quasi periodic traveling wave solution.
###### Theorem 4.2.
Suppose $f\in C^{2}(\mathbb{R}_{+},\mathbb{R})$ with $f(0)=0$. If there are
$\bar{r}_{1,\theta}\in(0,1)$, $T>0$, $\theta\in[0,2\pi)$ and
$\nu\in{\mathcal{R}}\Phi_{\theta}\setminus\\{0\\}$ such that all integer
number solutions $k_{1},k_{2},\cdots,k_{m_{1,\theta}}$ of equation
$-\nu=\Phi_{\theta}\left(\frac{2\pi}{T}(\bar{r}_{1,\theta}+k)\right)$
are either nonnegative or negative, and $m_{1,\theta}>0$. Then for any
$\varepsilon>0$ small there are $m_{1,\theta}$ branches of nonzero quasi
periodic traveling wave solutions (20) of (19) with $U_{j,\varepsilon}\in
C^{1}(\mathbb{R},\mathbb{C})$, $j=1,2,\cdots,m_{1,\theta}$, and nonzero
velocity $\nu_{\varepsilon}$ satisfying $U_{j,\varepsilon}(z+T)={\,\textrm{\rm
e}}^{2\pi\bar{r}_{1}\imath}U(z)_{j,\varepsilon}$, $\forall z\in\mathbb{R}$
along with $\nu_{\varepsilon}\to\nu$ and $U_{j,\varepsilon}\rightrightarrows
0$ uniformly on $\mathbb{R}$ as $\varepsilon\to 0$.
###### Example 4.3.
We consider the discrete 2D Kac-Baker interaction kernel
$a_{i,j}={\,\textrm{\rm e}}^{-|i|-|j|}$ for $(i,j)\in\mathbb{Z}^{2}_{0}$. Then
$\sum\limits_{(i,j)\in\mathbb{Z}^{2}_{0}}{\,\textrm{\rm
e}}^{-|i|-|j|}=\frac{4{\,\textrm{\rm e}}}{(e-1)^{2}}$ and
$\Phi_{\theta}(x)=\left[\frac{({\,\textrm{\rm e}}+1)^{2}}{({\,\textrm{\rm
e}}-1)^{2}}-\frac{({\,\textrm{\rm e}}^{2}-1)^{2}}{(1+{\,\textrm{\rm
e}}^{2}-2{\,\textrm{\rm e}}\cos(x\cos\theta))(1+{\,\textrm{\rm
e}}^{2}-2{\,\textrm{\rm e}}\cos(x\sin\theta))}\right]\frac{4}{x}.$
A numerical evaluation shows that function $(x,\theta)\to\Phi_{\theta}(x)$ has
a maximum $\bar{R}\doteq 9.75047$ at $x_{0}\doteq 1.08205$ and
$\theta_{0}\doteq 0.785398$. To justify this theoretically, we take
$a=x\cos\theta$ and $b=x\sin\theta$ to transform $\Phi_{\theta}(x)$ into
$\Phi(a,b)=\left[\frac{({\,\textrm{\rm e}}+1)^{2}}{({\,\textrm{\rm
e}}-1)^{2}}-\frac{({\,\textrm{\rm e}}^{2}-1)^{2}}{(1+{\,\textrm{\rm
e}}^{2}-2{\,\textrm{\rm e}}\cos a)(1+{\,\textrm{\rm e}}^{2}-2{\,\textrm{\rm
e}}\cos b)}\right]\frac{4}{\sqrt{a^{2}+b^{2}}}\,.$
Note $\Phi(a,b)=\Phi(\pm a,\pm b)=\Phi(b,a)$. A numerical evaluation shows
that function $\Phi_{\theta}(a,b)$ has a maximum $\bar{R}\doteq 9.75047$ at
$a_{0}=b_{0}\doteq 0.765123$ which correspond to $x_{0}$ and $\theta_{0}$. On
the other hand, if $a^{2}+b^{2}\geq 4$ then $\Phi(a,b)\leq
2\frac{({\,\textrm{\rm e}}+1)^{2}}{({\,\textrm{\rm e}}-1)^{2}}\doteq
9.36539<9.75047$, so $\Phi(a,b)$ achieves its maximum in the disc
$D_{2}:=\left\\{a^{2}+b^{2}\leq 4\right\\}$. Next, solving the system
$\frac{\partial}{\partial a}\Phi(a,b)=\frac{\partial}{\partial b}\Phi(a,b)=0$
we derive $\frac{\sin a_{0}}{a_{0}}=\frac{\sin b_{0}}{b_{0}}$ at the maximum
point $(a_{0},b_{0})\in D_{2}$, $a_{0}>0$, $b_{0}>0$. But the function
$\frac{\sin w}{w}$ is decreasing on $[0,2]$, so $a_{0}=b_{0}$, and thus
$\theta_{0}=\pi/4$. An elementary but awkward calculus shows for function
$\Phi_{\pi/4}(x)=\left[\frac{({\,\textrm{\rm e}}+1)^{2}}{({\,\textrm{\rm
e}}-1)^{2}}-\frac{({\,\textrm{\rm e}}^{2}-1)^{2}}{\left(1+{\,\textrm{\rm
e}}^{2}-2{\,\textrm{\rm
e}}\cos\left(x\frac{\sqrt{2}}{2}\right)\right)^{2}}\right]\frac{4}{x}$
with the graph on $[-20,20]$:
that $x_{0}\in(0,2)$ is the only root of $\Phi_{\pi/4}^{\prime}(x_{0})=0$ on
$(0,2)$, and then $\bar{R}=\Phi_{\pi/4}(x_{0})$. So $\bar{R}$ is computed also
analytically in this case.
Summarizing, Theorems 4.1 and 4.2 can be applied in this case for any suitable
nonzero $\nu$, and resonant traveling waves with maximum velocities which are
achieved in the diagonal directions $\pm\theta_{0}=\pm\pi/4$.
Finally, it is now clear how to proceed to 3D DNLS or even to higher
dimensional DNLS, so we omit further details.
## 5 Traveling Waves with Frequencies
We could consider more general traveling wave solutions than above of forms
$\begin{gathered}u_{n}(t)=U(n-\nu t){\,\textrm{\rm e}}^{\imath\omega t},\\\
u_{n,m}(t)=U(n\cos\theta+m\sin\theta-\nu t){\,\textrm{\rm e}}^{\imath\omega
t}\end{gathered}$ (22)
with velocity $\nu\neq 0$ and frequency $\omega\neq 0$ (see [32]). Then, there
is a dispersion relation between the velocity $\nu$ and frequency $\omega$ as
follows. Inserting (22) into (3) and (19), respectively, we are interested in
equations
$\begin{gathered}-\nu\imath
U^{\prime}(z)=\sum\limits_{j\in\mathbb{N}}a_{j}\partial_{j}U(z)+\omega
U(z)+f(|U(z)|^{2})U(z),\\\ -\nu\imath
U^{\prime}(z)=\sum\limits_{(i,j)\in\mathbb{Z}^{2}_{0}}a_{i,j}\partial_{i,j}U(z)+\omega
U(z)+f(|U(z)|^{2})U(z),\end{gathered}$ (23)
respectively. We see that (6), (21) and (23) are very similar, so we can
repeat the above arguments to (23) when instead of $\Phi(x)$ and
$\Phi_{\theta}(x)$ now we have
$\Phi(x,\omega):=\Phi(x)-\frac{\omega}{x}\,,\quad\Phi_{\theta}(x,\omega):=\Phi_{\theta}(x)-\frac{\omega}{x}\,,$
(24)
respectively. Consequently, we have analogies of Theorems 1.2, 1.3, 4.1 and
4.2 to (23) but we do not state them since they are obvious.
###### Example 5.1.
We consider the discrete Kac-Baker interaction kernel from Example 2.12. Then
$\Phi(x,\omega)=\frac{2{\,\textrm{\rm e}}({\,\textrm{\rm e}}+1)(1-\cos
x)}{({\,\textrm{\rm e}}-1)x({\,\textrm{\rm e}}^{2}+1-2{\,\textrm{\rm e}}\cos
x)}-\frac{\omega}{x}\,.$
To be more concrete, we first take $\omega=1$, and then $\Phi(x,1)$ has the
graph on $[-4\pi,4\pi]$:
with $\lim_{x\to 0_{\pm}}\Phi(x,1)=\mp\infty$. A numerical evaluation shows
that function $\Phi(x,1)$ has a maximum $\bar{R}\doteq 0.282071$ on
$(0,\infty)$ at $x_{0}\doteq 1.9905$. Consequently, the analogy of Theorem 1.2
can be applied now to any $\nu\neq 0$ while the analogy of Theorem 1.3 can be
applied for almost any $\nu\in\mathbb{R}\setminus[-0.282071,0.282071]$, while
for nonzero $\nu\in[-0.282071,0.282071]$ could be problematic in general.
On the other hand for $\omega=-1$, $\Phi(x,-1)$ has the graph on
$[-4\pi,4\pi]$:
with $\lim_{x\to 0_{\pm}}\Phi(x,-1)=\pm\infty$. Consequently, the analogy of
Theorem 1.2 can again be applied now to any $\nu\neq 0$ while the analogy of
Theorem 1.3 can now be applied for almost any $\nu\neq 0$. Of course now we
have totally different situations than in Example 2.12 for traveling waves
without frequencies by comparing the above graphs with that one in Example
2.12.
### 5.1 Acknowledgements
Michal Fečkan is partially supported by the Grants VEGA-MS 1/0098/08 and VEGA-
SAV 2/7140/27. Vassilis Rothos is partially supported by Research Grant-
International Relations of AUTH.
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|
arxiv-papers
| 2009-09-09T21:15:18 |
2024-09-04T02:49:05.238074
|
{
"license": "Public Domain",
"authors": "Michal Feckan and Vassilis Rothos",
"submitter": "Vassilis Rothos",
"url": "https://arxiv.org/abs/0909.1833"
}
|
0909.1887
|
# Non-negative Wigner functions for orbital angular momentum states
I. Rigas Departamento de Óptica, Facultad de Física, Universidad Complutense,
28040 Madrid, Spain L. L. Sánchez-Soto Departamento de Óptica, Facultad de
Física, Universidad Complutense, 28040 Madrid, Spain A. B. Klimov
Departamento de Física, Universidad de Guadalajara, 44420 Guadalajara,
Jalisco, Mexico J. Řeháček Department of Optics, Palacky University, 17.
listopadu 50, 772 00 Olomouc, Czech Republic Z. Hradil Department of Optics,
Palacky University, 17. listopadu 50, 772 00 Olomouc, Czech Republic
###### Abstract
The Wigner function of a pure continuous-variable quantum state is non-
negative if and only if the state is Gaussian. Here we show that for the
canonical pair angle and angular momentum, the only pure states with non-
negative Wigner functions are the eigenstates of the angular momentum. Some
implications of this surprising result are discussed.
###### pacs:
03.65.Fd,03.65.Ta,03.65.Sq,03.67.-a
For continuous variables, the Wigner function Wigner (1932) is a very useful
tool that establishes a one-to-one correspondence between quantum states and
joint quasiprobability distributions of canonically conjugate variables in
phase space (position and momentum, in the standard case). However, it can
take on negative values, a property that distinguishes it from a true
probability distribution Hillery et al. (1984); Lee (1995); Zachos et al.
(2005). Indeed, this negative character is associated with the existence of
quantum interference, which itself may be identified as a signal of
nonclassical behavior Kenfack and Życzkowski (2004).
In consequence, the characterization of quantum states that are classical, in
the sense of giving rise to non-negative Wigner functions, is a topic of
undoubted interest. Among pure states, it was proven in a classical paper by
Hudson Hudson (1974) (later generalized by Soto and Claverie Soto and Claverie
(1983) to multipartite systems) that the only states that have non-negative
Wigner functions are Gaussian states Janssen (1984); Lieb (1990). This is one
of the main reasons for the prominent role these states play in modern quantum
information Cerf et al. (2007).
The original definition of the Wigner function has also been extended to
discrete systems (see Ref. Björk et al. (2008) for a comprehensive review).
Again, the classification of states with non-negative Wigner functions is an
amazing problem that has been solved quite recently by Paz and coworkers
Cormick et al. (2006); Cormick and Paz (2006) and Gross Gross (2006, 2007), so
that the role of Gaussian states is now taken on by stabilizer states.
Interestingly, these are the only states that can be simulated efficiently in
classical computers Gottesman (1997).
Between these two cases (whose proofs are otherwise completely different), we
have the interesting situation of canonical pairs, such as the angle and
orbital angular momentum (OAM), for which one variable is continuous while the
other one is discrete Kastrup (2006). The associated phase space is the
discrete cylinder $\mathcal{S}_{1}\times\mathbb{Z}$, where $\mathcal{S}_{1}$
stands for the unit circle (associated to the angle) and the integers
$\mathbb{Z}$ translate the discreteness of the OAM. The physical example we
have in mind is the OAM of photons. This is an emerging field that has given
rise to many developments, ranging from optical tweezers to high-dimensional
quantum entanglement, or fundamental processes in Bose-Einstein condensates,
to cite only a few relevant examples Allen et al. (2003).
The seminal paper of Allen _et al._ Allen et al. (1992) firmly established
that the Laguerre-Gauss modes carry a well-defined OAM. They appear as annular
rings with a zero on-axis intensity and an azimuthal dependence
$\exp(i\ell\phi)$ that gives rise to spiral wave fronts. The index $\ell$
takes only integer values and can be seen as the eigenvalue of the OAM
operator. Since then, several methods have been established to produce light
beams with the required azimuthal phase structure, among these spiral phase
plates, forked holograms, and spatial light modulators are perhaps the most
versatile. In this way, a variety of modes with helical phase fronts but
different transverse patterns (such as Bessel, Mathieu, or hypergeometric
beams) can be routinely generated in the laboratory Franke-Arnold et al.
(2008).
The goal of this work is precisely to determine the pure states of these OAM-
carrying systems for which the Wigner function is non-negative, filling in
this way a long overdue gap.
To be as self-contained as possible, we first introduce some basic notions for
the problem at hand of cylindrical symmetry. We are concerned with the planar
rotations by an angle $\phi$ generated by the angular momentum along the $z$
axis, which for simplicity will be denoted henceforth as $\hat{L}$. We do not
want to enter in a long discussion about the possible existence of an angle
operator Řeháček et al. (2008). For our purposes here, the simplest solution
is to adopt two periodic angular coordinates, e.g., cosine and sine, that we
shall denote by $\hat{C}$ and $\hat{S}$ to make no further assumptions about
the angle itself. One can concisely condense all this information using the
complex exponential of the angle $\hat{E}=\hat{C}+i\hat{S}$, which satisfies
the commutation relation
$[\hat{E},\hat{L}]=\hat{E}\,.$ (1)
In mathematical terms, this defines the Lie algebra of the two-dimensional
Euclidean group E(2), which is precisely the canonical symmetry group for the
cylinder.
The action of $\hat{E}$ on the basis of eigenstates of $\hat{L}$ is
$\hat{E}|\ell\rangle=|\ell-1\rangle$, and it possesses then a simple
implementation by means of a phase mask removing a charge $+1$ from a vortex
state Mair et al. (2001); Hradil et al. (2006). Since the integer $\ell$ runs
from $-\infty$ to $+\infty$, $\hat{E}$ is a unitary operator whose
eigenvectors
$|\phi\rangle=\frac{1}{\sqrt{2\pi}}\sum_{\ell\in\mathbb{Z}}e^{i\ell\phi}|\ell\rangle$
(2)
form a complete basis and describe states with well-defined angle. In the
representation generated by them, $\hat{L}$ acts as $-i\partial_{\phi}$ (in
units of $\hbar=1$).
Given the key role played by the displacement operators in settling the Wigner
function for the harmonic oscillator, we introduce a unitary displacement
operator
$\hat{D}(\ell,\phi)=e^{i\alpha(\ell,\phi)}\,\hat{E}^{-\ell}e^{-i\phi\hat{L}}\,,$
(3)
where $\alpha(\ell,\phi)$ is a phase required to avoid plugging in extra
factors when acting with $\hat{D}$. The conditions of unitarity and
periodicity restrict the possible values of $\alpha$, although a sensible
choice is $\alpha(\ell,\phi)=-\ell\phi/2$. Note that here we cannot rewrite
Eq. (3) as an entangled exponential, since the action of the operator to be
exponentiated would not be well defined.
We use as a guide the analogy with the continuous case and introduce the
mapping Berezin (1975)
$W_{\hat{\varrho}}(\ell,\phi)=\mathop{\mathrm{Tr}}\nolimits[\hat{\varrho}\,\hat{w}(\ell,\phi)]\,,$
(4)
which maps the density operator into a Wigner function via a kernel $\hat{w}$
defined as a double Fourier transform of the displacement operator Plebański
et al. (2000):
$\hat{w}(\ell,\phi)=\frac{1}{(2\pi)^{2}}\sum_{{\ell^{\prime}}\in\mathbb{Z}}\int_{2\pi}\exp[-i(\ell^{\prime}\phi-\ell\phi^{\prime})]\,\hat{D}(\ell^{\prime},\phi^{\prime})\,d\phi^{\prime}\,,$
(5)
where the integral extends to the $2\pi$ interval within which the angle is
defined. This mapping is invertible, so one can reconstruct the density
operator as
$\hat{\varrho}=2\pi\,\sum_{{\ell}\in\mathbb{Z}}\int_{2\pi}\hat{w}(\ell,\phi)\,W_{\hat{\varrho}}(\ell,\phi)\,d\phi\,.$
(6)
The (Hermitian) Wigner kernels $\hat{w}(\ell,\phi)$ are a complete orthonormal
basis (in the trace sense) for the operators acting on the Hilbert space of
the system. In addition, they are explicitly covariant; i.e., they transform
properly under displacements,
$\hat{w}(\ell,\phi)=\hat{D}(\ell,\phi)\,\hat{w}(0,0)\,\hat{D}^{\dagger}(\ell,\phi)$.
In fact, these properties guarantee that the Wigner function defined in Eq.(4)
bears all the good properties required for a probabilistic description. In
particular, it reproduces the proper marginal distributions, that is,
$\sum_{{\ell}\in\mathbb{Z}}W_{\hat{\varrho}}(\ell,\phi)=\langle\phi|\hat{\varrho}|\phi\rangle\,,\quad\int_{2\pi}W_{\hat{\varrho}}(\ell,\phi)\,d\phi=\langle\ell|\hat{\varrho}|\ell\rangle\,.$
(7)
Finally, the overlap of two density operators is proportional to the integral
of the associated Wigner functions:
$\mathop{\mathrm{Tr}}\nolimits(\hat{\varrho}\,\hat{\sigma})\propto\sum_{{\ell}\in\mathbb{Z}}\int_{2\pi}W_{\hat{\varrho}}(\ell,\phi)W_{\hat{\sigma}}(\ell,\phi)\,d\phi\,.$
(8)
This property (often called traciality) offers practical advantages, since it
allows one to predict the statistics of any outcome, once the Wigner function
of the measured state is known.
We remark that this approach to the Wigner function is grounded in the
axiomatic method developed by Stratonovich Stratonovich (1956) and Berezin
Berezin (1975) (see also Ref. Brif and Mann (1998)). It is possible to follow
alternative routes, such as, introducing a Wigner function as the Fourier
transform of some generalized characteristic function Wolf (1996). This has
been pursued also for the group E(2) Nieto et al. (1998). However, these
apparently disjoint formulations turn out to be equivalent for most practical
purposes Chumakov et al. (2000).
To give an explicit form of the Wigner function (4) we need to evaluate it in
a basis. Using the OAM eigenstates, we get
$\displaystyle W_{\hat{\varrho}}(\ell,\phi)$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\sum_{{\ell^{\prime}}\in\mathbb{Z}}e^{-2i\ell^{\prime}\phi}\langle\ell-\ell^{\prime}|\hat{\varrho}|\ell+\ell^{\prime}\rangle$
(9) $\displaystyle+$
$\displaystyle\frac{1}{2\pi^{2}}\sum_{{\ell^{\prime},\ell^{\prime\prime}}\in\mathbb{Z}}\frac{(-1)^{\ell^{\prime\prime}}}{\ell^{\prime\prime}+1/2}e^{-(2\ell^{\prime}+1)i\phi}$
$\displaystyle\times$
$\displaystyle\langle\ell+\ell^{\prime\prime}-\ell^{\prime}|\hat{\varrho}|\ell+\ell^{\prime\prime}+\ell^{\prime}+1\rangle\,.$
This looks rather cumbersome due to the second sum in Eq. (9) and sometimes is
preferable to work in the angle representation, for which one easily finds
$W_{\hat{\varrho}}(\ell,\phi)=\frac{1}{2\pi}\int_{-\pi}^{\pi}\\!\\!\langle\phi-\phi^{\prime}/2|\hat{\varrho}|\phi+\phi^{\prime}/2\rangle\,e^{i\phi^{\prime}\ell}\,d\phi^{\prime}\,.$
(10)
This coincides with the result of Mukunda Mukunda (1979); Mukunda et al.
(2005) (see also Ref. Bizarro (1994)) and bears a resemblance with the
standard Wigner function for position and momentum that is more than evident.
Note that using this latter function in terms of transverse coordinates, as is
often done in classical optics Simon and Agarwal (2000), is not appropriate
for the geometry of the cylinder, which is the natural domain in which the
Wigner function should be defined.
We have now all the ingredients needed to accomplish our program. In what
follows, the Fourier transform of $2\pi$-periodic functions (i.e., with domain
in $\mathcal{S}_{1}$), defined as
$(\mathcal{F}g)(k)=\frac{1}{2\pi}\int_{2\pi}g(\phi)\,e^{i\phi k}\,d\phi\,,$
(11)
with $k\in\mathbb{Z}$, will play a relevant role. We first state our main
result, which can be viewed as analogous to the Hudson theorem for the
canonical pair angle and angular momentum.
###### Theorem (Classical OAM states).
The Wigner function of a pure state $|\psi\rangle$ is non-negative if and only
if $|\psi\rangle$ is an OAM eigenstate $|\ell_{0}\rangle$.
###### Proof.
The sufficiency is obvious since the Wigner function for the state
$|\ell_{0}\rangle$ is
$W_{|\ell_{0}\rangle}(\ell,\phi)=\delta_{\ell\ell_{0}}/(2\pi)$. The delicate
point is to prove the necessity. Before proceeding, we sketch the idea behind
the proof. The first step is to show that the wave function [and thus, the
integrand in Eq. (10)] must be of constant modulus. The second step is then to
corroborate that the Wigner function can only be non-zero for a single value
of $\ell$. Traciality permits us to derive an equation that shows that this
value of $\ell$ cannot vary over $\phi$, and that indeed the only states with
non-negative Wigner functions are the OAM eigenstates. We start with the
following lemma.
###### Lemma 1.
If the Fourier transform of a smooth, complex, $2\pi$-periodic function
$g(\phi)$ is non-negative, then the integration kernel $g(\phi-\phi^{\prime})$
is non-negative.
###### Proof.
By a direct calculation we can check that
$\int_{2\pi}g(\phi-\phi^{\prime})\,e^{-i\phi^{\prime}k}\,d\phi^{\prime}=2\pi\,(\mathcal{F}g)(k)\,e^{-i\phi
k}\,,$ (12)
so, for any smooth test function
$\chi(\phi)=\sum_{{k}\in\mathbb{Z}}\chi(k)\,e^{-i\phi k}$, it holds
$\int_{2\pi}\chi^{\ast}(\phi)\,g(\phi-\phi^{\prime})\,\chi(\phi^{\prime})\,d\phi
d\phi^{\prime}=4\pi^{2}\sum_{{k}\in\mathbb{Z}}|\chi(k)|^{2}\,(\mathcal{F}g)(k)\,.$
(13)
It is clear that the non-negativity of the kernel $g(\phi-\phi^{\prime})$
follows from the non-negativity of the Fourier transform $(\mathcal{F}g)(k)$.
∎
We apply the lemma to
$\chi(\phi)=\frac{1}{2}[\delta_{2\pi}(\phi-c_{1})+\delta_{2\pi}(\phi-
c_{2})]\,,$ (14)
Here, $\delta_{2\pi}$ denotes the periodic delta function (or Dirac comb) of
period $2\pi$ and $c_{1},c_{2}\in\mathcal{S}_{1}$. For this function we have
$|\chi(k)|^{2}=\\{1+\cos[k(c_{1}-c_{2})]\\}/(8\pi^{2})$, so the sum in the
right-hand side of Eq. (13) reduces to
$g(0)/2+[g(c_{1}-c_{2})+g(c_{2}-c_{1})]/4\,.$ (15)
Consequently, for a function $g(\phi)$ whose Fourier transform is non-
negative, the kernel $g(\phi-\phi^{\prime})$ must also be non-negative on the
test functions (14) for all the possible parameters
$c_{1},c_{2}\in\mathcal{S}_{1}$.
For a pure state $|\psi\rangle$, the Wigner function (10) is just the Fourier
transform of $\psi^{\ast}(\phi+\phi^{\prime}/2)\,\psi(\phi-\phi^{\prime}/2)$,
where we have expressed the wave functions in the angle representation. By
Lemma 1, for the test functions (14) the non-negativity of $W_{|\psi\rangle}$
leads to
$|\psi(\phi)|^{2}\geq|\psi(\phi-a/2)|\,|\psi(\phi+a/2)|\,,$ (16)
with $a=c_{1}-c_{2}$. This implies that $|\psi(\phi)|$ cannot have any minima
and the modulus of $\psi$ must thus be flat over $\mathcal{S}_{1}$.
To proceed further we need a technical detail.
###### Lemma 2.
If a function $f(k):\mathbb{Z}\to\mathbb{C}$ has an inverse Fourier transform
of constant modulus over $\phi$, then
$\sum_{{k}\in\mathbb{Z}}f(k)\,f^{\ast}(k+j)=0\qquad\forall j\neq 0\,.$ (17)
###### Proof.
Let us first introduce the operator
$\hat{A}=\sum_{{m,k}\in\mathbb{Z}}f(m-k)\,|m\rangle\langle k|\,.$ (18)
One can check that it can be expressed in a diagonal form in the angle basis,
namely
$\hat{A}=\int_{2\pi}|\phi\rangle\langle\phi|\,(\mathcal{F}^{-1}f)(-\phi)\,d\phi\,.$
(19)
If $(\mathcal{F}^{-1}f)(\phi)$ has constant modulus, it can be written as
$(\mathcal{F}^{-1}f)(\phi)=c\,e^{i\lambda(\phi)}$, where $\lambda$ is a real
function. Therefore, we have
$\hat{A}\,\hat{A}^{\dagger}=|c|^{2}\,\hat{\openone}$. But according to the
definition (18), this is tantamount to the orthogonality relation
$\sum_{{m,k}\in\mathbb{Z}}\sum_{{m^{\prime},k^{\prime}}\in\mathbb{Z}}\langle
n|m\rangle\langle k|f(m-k)|k^{\prime}\rangle\langle
m^{\prime}|f^{\ast}(m^{\prime}-k^{\prime})|n+j\rangle=0\,.$
The Plancherel formula allows one to cancel the diagonal parts, so we are led
to
$\sum_{{k}\in\mathbb{Z}}f(n-k)\,f^{\ast}(n+j-k)=0\,,$ (20)
whence the result follows. ∎
Next, for every $\phi$, we consider the Wigner function of the state as a
function exclusively of the discrete index $\ell$; that is,
$f_{\phi}(\ell)=W_{|\psi\rangle}(\ell,\phi):\mathbb{Z}\to\mathbb{R}$ (in fact,
$W$ is real valued), and make use of the fact that the (inverse) Fourier
transform of $f_{\phi}(\ell)$ has a constant modulus over $\phi$. Then, by
Lemma 2, the orthogonality
$\sum_{{\ell}\in\mathbb{Z}}f_{\phi}(\ell)\,f^{\ast}_{\phi}(\ell+\ell^{\prime})=0\,,\qquad\forall\ell^{\prime}\neq
0\,,$ (21)
must hold for all $\phi\in\mathcal{S}_{1}$. But since $f$ is non-negative on
the whole phase-space, this is only possible if $f$ is equal to zero for all
but one $\ell_{0}$. Note that, in principle, $\ell_{0}$ may depend on $\phi$.
Taking into account the marginal distribution (7), we see that
$W(\ell,\phi)=\delta_{\ell\ell_{0}(\phi)}/(2\pi)$.
We now make use of the fact that the state $|\psi\rangle$ is pure [that is,
$\mathop{\mathrm{Tr}}\nolimits(\hat{\varrho}^{2})=1$]. From the traciality
property, one can show that the Wigner function representing the product of
two density operators $\hat{\varrho}$ and $\hat{\sigma}$ can be expressed as
$\displaystyle\displaystyle
W_{\hat{\varrho}\,\hat{\sigma}}(\ell,\phi)=\frac{1}{2\pi}\sum_{{\ell_{1}\,\ell_{2}}\in\mathbb{Z}}\int_{2\pi}\,W_{\hat{\varrho}}(\ell+\ell_{1},\phi+\psi_{1}/2)$
$\displaystyle\times\,W_{\hat{\sigma}}(\ell+\ell_{2},\phi+\psi_{2}/2)\,e^{i(\ell_{2}\psi_{1}-\ell_{1}\psi_{2})}\,d\psi_{1}d\psi_{2}\,.$
(22)
We apply this to the pure state $|\psi\rangle$ whose Wigner function is of the
form $\delta_{\ell\ell_{0}(\phi)}/(2\pi)$.
Without loss of generality, we can assume that $\ell_{0}(\phi=0)=0$ and may
revert this choice later by a displacement
$|\psi\rangle\to\hat{D}(\ell_{0},0)|\psi\rangle$. Then, Eq. (Proof.) becomes
$\displaystyle\displaystyle W_{|\psi\rangle}(0,0)=\frac{1}{2\pi}$
$\displaystyle\displaystyle=\frac{1}{(2\pi)^{3}}\int_{2\pi}e^{i[\ell_{0}(\psi_{2}/2)\psi_{1}-\ell_{0}(\psi_{1}/2)\psi_{2}]}\,d\psi_{1}d\psi_{2}\,.$
(23)
This means that the integral of the imaginary part must vanish, while the
integral of the real part must be equal $(2\pi)^{2}$. This is only possible if
the exponential is exactly one for all the arguments $(\psi_{1},\psi_{2})$;
i.e.,
$\ell_{0}(\psi_{1}/2)\,\psi_{2}=\ell_{0}(\psi_{2}/2)\,\psi_{1}\,\bmod{2\pi}$.
This is only possible when $\ell_{0}\equiv 0$. ∎
We have shown that if the Wigner function of a pure state is non-negative,
then it is necessarily a Kronecker delta, and thus stems from an OAM
eigenstate, which concludes the long yet instructive proof of our theorem.
It is worth stressing that for the continuous case the notions of coherent
states, Gaussian wave packets, and states with non-negative Wigner functions
(often identified as nonclassical states) are completely equivalent. However,
special care must be paid in extending these ideas to other physical systems
like OAM, since they lose their equivalence.
For example, OAM coherent states $|\ell_{0},\phi_{0}\rangle$ in the cylinder
Kowalski et al. (1996) can be expressed in the angle representation by
$\langle\phi|\ell_{0},\phi_{0}\rangle=\frac{e^{i\ell_{0}(\phi-\phi_{0})}}{\sqrt{\vartheta_{3}\left(0\big{|}\frac{1}{e}\right)}}\vartheta_{3}\left(\frac{\phi-\phi_{0}}{2}\Big{|}\frac{1}{e^{2}}\right)\,,$
where $\vartheta_{3}$ denotes the third Jacobi theta function. However,
despite the key role played by this function in angular problems, a simple
calculation Rigas et al. (2008) immediately reveals that the Wigner function
for them takes negative values.
In the same vein, the states
$\Psi_{\kappa}(\phi)=\frac{1}{\sqrt{2\pi
I_{0}(2\kappa)}}\exp(\kappa\cos\phi)\,,$ (24)
whose associated probability distribution is precisely the von Mises
distribution Řeháček et al. (2008), are usually taken as Gaussians for this
problem. One can easily check that their Wigner function also takes negative
values.
Even with all these cautions, the characterization we have presented of OAM
eigenstates as the only ones with non-negative Wigner function has interest in
its own, although, unfortunately, they cannot be viewed as Gaussian states.
A topic of interest is the characterization of unitaries that preserve the
non-negativity. Obviously, all the displacement operators are of this kind.
But the exponential of an arbitrary real function $f(\hat{L})$ also preserves
non-negativity and this includes quadratic exponentials, which are essential
for a full quantum reconstruction of vortex states Rigas et al. (2008).
Finally, let us mention that a question that naturally arises is whether our
result can be extended to mixed states. Although this question has been
approached by using the notion of the Wigner spectrum Bröcker and Werner
(1995) and explored quite recently for continuous variables Mandilara et al.
(2009), in our case a simple extension seems difficult and will be the object
of our future work.
We acknowledge discussions with Hubert de Guise, José Gracia-Bondía, and Hans
Kastrup. This work was supported by the Spanish Research Directorate, Grants
FIS2005-06714 and FIS-2008-04356, the Mexican Consejo Nacional de Ciencias y
Tecnología (CONACyT), Grant45704, and the Czech Ministry of Education,
Projects MSM6198959213 and LC06007.
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|
arxiv-papers
| 2009-09-10T13:58:49 |
2024-09-04T02:49:05.246609
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "I. Rigas, L. L. Sanchez-Soto, A. B. Klimov, J. Rehacek, and Z. Hradil",
"submitter": "Luis L. Sanchez. Soto",
"url": "https://arxiv.org/abs/0909.1887"
}
|
0909.1907
|
# Time-dependent Ginzburg-Landau theory with floating nucleation kernel;
FIR conductivity in the Abrikosov vortex lattice
Pei-Jen Lin1 P. Lipavský2,3 1NCTS, National Tsing Hua University, Hsinchu 300,
Taiwan
2 Faculty of Mathematics and Physics, Charles University, Ke Karlovu 3, 12116
Prague 2, Czech Republic
3Institute of Physics, Academy of Sciences, Cukrovarnická 10, 16253 Prague 6,
Czech Republic
###### Abstract
We formulate the time-dependent Ginzburg-Landau theory, with the assumption of
local equilibrium made in the reference frame floating with normal electrons.
This theory with floating nucleation kernel is applied to the far infrared
(FIR) conductivity in the Abrikosov vortex lattice. It yields better agreement
with recent experimental data [PRB 79, 174525 (2009)] than the customary time-
dependent Ginzburg-Landau theory.
non-equilibrium superconductivity; time-dependent Ginzburg-Landau theory
###### pacs:
74.40.+k,74.25.Nf,74.25.Qt,74.25.Ha,74.25.Gz
The time-dependent Ginzburg-Landau (TDGL) equation is a useful extension of
the equilibrium Ginzburg-Landau theory. Unfortunately, microscopic derivations
Schmid (1966); Abrahams and Tsuneto (1966); Gor’kov and Eliashberg (1968); Sá
de Melo et al. (1993); Huang et al. (2009) guarantee its validity under such
restrictive conditions that it seems more difficult to find justified
nontrivial applications than to solve it. The TDGL equation is thus most often
applied beyond its nominal range of validity.
As one leaves the familiar vicinity of the superconducting phase transition
and asymptotically slow processes, the intuitive foundation of the theory
becomes shaky. The TDGL theory contains an assumption of local equilibrium,
which is dependent on reference frame; when we adapt the equilibrium-based
equation to non-equilibrium problems, we should at least work in the reference
frame in which electrons are as close to local equilibrium as possible. This
is the frame floating with the normal current in the background of a
superconducting condensate. To this end, in this paper we introduce what we
refer to as a floating nucleation kernel.
The standard TDGL theory is formulated using a kernel static in the laboratory
system. We will show that compared to the TDGL theory in the floating system,
the laboratory formulation lacks a term which is particularly important at
high frequencies of the driving field. We will demonstrate the effects of this
term on the conductivity in the sub-gap far-infrared (FIR) region. Comparing
our results with recent FIR magneto-transmission measurements of Ikebe et al
Ikebe et al. (2009), we will show that use of the floating nucleation kernel
improves agreement between the theory and experimental data.
Let us first describe the magneto-transmission measurement. It is performed on
a thin layer perpendicularly penetrated by the magnetic field in the form of
vortices. The incident FIR light is perpendicular to the surface and its
electric field drives currents which determine the amplitude and phase of the
transmitted light which is measured.
Both the normal and the superconducting electrons are accelerated by the
electric field and experience a friction with the lattice. The friction of the
condensate is much weaker since Joule heat develops only in vortex cores
moving perpendicularly to the electric field. The relative contribution of
these components to the current depends on the frequency of the driving field;
the higher the frequency the higher will be the fraction of the normal
current.
It is useful to inspect characteristic times for NbN, the material used by
Ikebe et al Ikebe et al. (2009). The optical gap $2\Delta=5.3$ meV implies the
maximal sub-gap frequency $\omega<10$ THz. The mean time between two
collisions of the normal electron is $\tau_{n}\sim 5$ fs, therefore during a
single period of the sub-gap FIR field the electron loses momentum more than a
hundred times. At zero magnetic field the condensate suffers no friction. The
field of amplitude $E$ accelerates the condensate to velocity $e^{*}E/\omega
m^{*}$, while a normal electron is accelerated to $eE\tau_{n}/m$. At the
measurement temperature, $T=3$ K and $T_{c}=15$ K, the density of condensed
electrons exceeds the normal density, therefore the condensate clearly
dominates the total current. A different situation obtains, however, for the
Joule heat. The condensate current is out of phase with the driving electric
field and generates no heat. The normal current is in-phase, producing heat.
If the magnetic field penetrates the sample, the condensate generates the
Joule heat due to motion of vortices. We will see that for the sub-gap FIR
frequencies the Joule heat value is much smaller than the amount of heat
generated by normal electrons.
To identify the Joule heat, it is necessary to measure the transmission
coefficient, including its phase. This allows one to determine the complex
conductivity $\sigma$ with ${\rm Im}\,\sigma$ giving the off-phase current and
${\rm Re}\,\sigma$ for the in-phase current. Ikebe et al Ikebe et al. (2009)
achieved this task by splitting short pulses and mixing them again after one
of branches passed through the sample. As mentioned, we will compare their
experimentally established $\sigma$ with theoretical predictions based on the
TDGL theory in the laboratory and the floating coordinate system.
We will use the electric field ${\bf E}(\tau)={\rm Re}\,\left[{\bf E}{\rm
e}^{-i\omega\tau}\right]$ and current ${\bf J}(\tau)={\rm Re}\,\left[{\bf
J}{\rm e}^{-i\omega\tau}\right]$. The complex conductivity is defined via
${\bf J}=\sigma\,{\bf E}$. The current has a small Hall component which we
neglect in our discussion for convenience.
The TDGL equation derived using the static kernel Tinkham (1966),
$\displaystyle{1\over 2m^{*}}\left(\\!-i\hbar\nabla\\!-\\!\frac{e^{*}}{c}{\bf
A}\right)^{2}\psi+\alpha\psi+\beta\left|\psi\right|^{2}\psi=-\Gamma\partial_{\tau}\psi,$
(1)
describes the evolution of the condensate including a relaxation of the GL
function $\psi$ towards its equilibrium value. The vector potential is that of
the internal magnetic field as well as the electric field of the FIR light ;
${\bf B}=\nabla\times{\bf A}$ and ${\bf E}=-(1/c)\partial_{\tau}{\bf A}$. The
electric current
$\displaystyle{\bf j}_{s}={e^{*}\over m^{*}}{\rm
Re}~{}\left[\bar{\psi}\left(-i\hbar\nabla-\frac{e^{*}}{c}{\bf
A}\right)\psi\right]$ (2)
is composed of circulating diamagnetic currents and oscillating response to
the light. We solve Eq. (1) to linear order in $\bf E$ and eliminate the
diamagnetic currents by averaging over the elementary cell of the Abrikosov
vortex lattice; ${\bf J}_{s}=\left\langle{\bf
j}_{s}\right\rangle=(B/\Phi_{0})\int_{\rm cell}{\rm d}x{\rm d}y\,{\bf j}_{s}$.
The supercurrent, ${\bf J}_{s}=\sigma_{s}{\bf E}$, gives the condensate
conductivity
$\sigma_{s}=\frac{3\sigma_{0}}{\beta_{\rm A}}\frac{1-t-b}{b-i\omega\tau_{s}},$
(3)
where $t=T/T_{c}$, $b=B/H_{c2}$ are the dimensionless temperature and magnetic
field, $\sigma_{0}$ is the normal state conductivity, $\beta_{\rm A}=1.16$ is
the Abrikosov constant for hexagonal vortex lattice, and
$\tau_{s}=\Gamma(1-t)/\alpha$. Deriving Eq. (3) we have used the GL parameter
Bel and Rosenstein (2005)
$\Gamma={12\pi\sigma_{0}\alpha\kappa^{2}\xi^{2}\over c^{2}(1-t)^{2}}.$ (4)
The zero-temperature coherence length is determined by the upper critical
field; $\xi^{2}=\Phi_{0}/(2\pi H_{c2}^{0})$. Here $H_{c2}^{0}=15$ T is
obtained via the linear extrapolation $H_{c2}=H_{c2}^{0}(1-t)$ from
experimental data in Fig 3 of Ikebe et al. (2009). The normal-state
conductivity $\sigma_{0}=2\cdot 10^{4}/\Omega$cm, experimentally established
at 20 K Ikebe et al. (2009), has weak temperature dependence and can be used
at 3 K.
Figure 1: Imaginary part of the conductivity giving non-dissipative currents:
Thin lines are the superconducting condensate conductivity ${\rm
Im}\,\sigma_{s}$ (dotted), the TDGL conductivity ${\rm Im}\,\sigma_{\rm GL}$
(full), and the two-fluid modification of the TDGL conductivity ${\rm
Im}\,\sigma_{\rm tf}$ (dashed). The heavy line is the conductivity ${\rm
Im}\,\sigma_{\rm fk}$ evaluated in the floating system. Experimental data of
Ikebe et al Ikebe et al. (2009) at 7 T ($\bullet$) are in the nominal validity
range of the TDGL theory, while the lower magnetic fields 5 T, 3 T, and 1 T
($\circ$) are not.
In Fig. 1 one can see that the imaginary part of $\sigma_{\rm s}$ from formula
(3) reproduces recent experimental data of Ikebe et al Ikebe et al. (2009).
Here we use the GL parameter $\kappa=40$, the only fitting parameter in the
present theory. It is adjusted to fit the imaginary part of the conductivity
at 7 T. Our main interest is in the Joule heat given by the real part of the
conductivity.
Formula (3) was derived for the dense Abrikosov vortex lattice. Theoretically,
the region of nominal validity is $B>4$ T, at the temperature $T=3$ K. It is
therefore somewhat surprising that theoretical curves of ${\rm Im}\,\sigma$
slightly depart from the experimental data only at the lowest magnetic field
$B=1$ T.
Due to the relaxation term $\Gamma\partial_{t}\psi$, the TDGL equation (1)
includes a damping and generates Joule heat Ketterson and Song (1998),
$\dot{Q}=4k_{\rm
B}T\Gamma(\omega/2\pi)\left\langle|\partial_{\tau}\psi|^{2}\right\rangle$,
where the brackets denote the time average:
$\langle\phi\rangle\equiv(\omega/2\pi)\int_{0}^{2\pi/\omega}{\rm d}\tau\phi$.
The left-hand panel of Fig. 2 shows that the supercurrent produces Joule heat
only at vortex cores. The right-hand panel of Fig. 2 presents the spatial
distribution of the power absorbed by the condensate from the electric field
$W=\left\langle{\bf j}_{s}\cdot{\bf E}\right\rangle$. The most intensive
absorption is around vortices in regions elongated in the vertical direction
which is parallel to the electric field. Deep minima of the absorption are
between vortices in horizontal rows. Comparing the two panels shows that the
relation between absorption and heat production is very non-local.
Figure 2: Heat production (left) and the power absorption (right) in the
hexagonal Abrikosov vortex lattice: Crosses denote centers of vortices. The
electric field is polarised vertically so that vortices oscillate horizontally
with amplitude shown by arrows. The Joule heat is produced at vortex cores,
their horizontal motion is responsible for elongation of the heated region.
Absorption of power is rather delocalised. Its maxima are also around vortex
cores but elongated vertically. The rounded minima are between vortices.
Difference of these two maps shows that the ‘rigid’ GL function transfers the
power to be dissipated in cores.
The fraction of Joule heat due to the condensate is small. In Fig. 3 we
compare the real part of the condensate conductivity (3) with experiment.
Indeed, the discrepancy between experimental data and Re $\sigma_{s}$
indicates that the supercurrent produces only a minor part of the Joule heat;
the normal current cannot be neglected .
Figure 3: Real part of the conductivity giving Joule heat: Points are
experimental data of Ikebe et al Ikebe et al. (2009) for 7 T ($\bullet$). The
superconducting condensate contribution (dotted line) given by formula (3) is
by an order of magnitude too small. The time-dependent Ginzburg-Landau theory
(thin line) adds a contribution of normal electrons, see Eq. (5), arriving at
too high values. The two-fluid approach (dashed line) reduces the conductivity
subtracting double-counted condensed electrons from the normal conductivity,
see Eq. (7). The floating kernel approach (heavy line) given by Eq. (11)
removes double-counting from the supercurrents and yields the closest
agreement with experiment.
From microscopic derivations Schmid (1966); Abrahams and Tsuneto (1966);
Gor’kov and Eliashberg (1968); Kopnin (2001) of the GL theory it follows that
the normal current and the supercurrent simply add. Adding the current ${\bf
J}_{n}=\sigma_{0}(1+i\tau_{n}\omega){\bf E}$ which would appear in the normal
state one obtains the TDGL conductivity
$\sigma_{\rm GL}=\sigma_{s}+\sigma_{n},$ (5)
with the normal conductivity $\sigma_{n}=\sigma_{0}(1+i\tau_{n}\omega)$. For
experimentally established values $\sigma_{0}=2\cdot 10^{4}/\Omega$cm and
$\tau_{n}=5$ fs Ikebe et al. (2009), the normal conductivity yields a
negligible contribution to ${\rm Im}\,\sigma_{\rm GL}$, as seen in Fig. 1, but
it provides the dominant contribution to ${\rm Re}\,\sigma_{\rm GL}$. One can
see in Fig. 3 that ${\rm Re}\,\sigma_{\rm GL}$ is much closer to observed
values than ${\rm Re}\,\sigma_{s}$. It is higher than the observed values,
however. This problem becomes more serious at lower magnetic fields, where the
observed real part of total conductivity is further reduced well below the
level of the normal conductivity, see Fig. 4, while the TDGL conductivity is
always larger, ${\rm Im}\,\sigma_{\rm GL}>{\rm Im}\,\sigma_{n}$.
The simple addition of normal current and supercurrent works well close to the
phase transition but it badly overestimates conductivity far from it.
Apparently, it is insufficient simply to add the supercurrent and the normal
current; the electric field accelerates all electrons. Since electrons in the
condensate escape frictional effects, this fraction of electrons must be
removed in order to obtain the normal conductivity. An intuitive way to avoid
double-counting of condensed electrons is to introduce a normal current
reduced in the spirit of the two-fluid model,
$\tilde{\bf j}_{n}=\left(1-{2|\psi|^{2}\over n}\right){\bf J}_{n}.$ (6)
The total current averaged over the elementary vortex lattice cell, ${\bf
J}={\bf J}_{s}+\tilde{\bf J}_{n}$, leads to a conductivity
$\sigma_{\rm tf}=\sigma_{s}+\left(t+b\right)\sigma_{n},$ (7)
where we have evaluated the averaged normal fraction,
$1-2\left\langle|\psi|^{2}\right\rangle/n=t+b$. One can see in Figs. 1 and 3
that the two-fluid conductivity yields the same non-dissipative currents
described by ${\rm Im}\,\sigma_{\rm tf}$ as the TDGL theory, but that it
allows for ${\rm Re}\,\sigma_{\rm tf}$ smaller than the normal conductivity.
In fact ${\rm Re}\,\sigma_{\rm tf}$ is too small, when compared to
experimental data.
The reduced normal current (6) contradicts microscopic studies Schmid (1966);
Abrahams and Tsuneto (1966); Gor’kov and Eliashberg (1968); Sá de Melo et al.
(1993); Huang et al. (2009). Indeed, the total current is derived from the
Nambu-Gor’kov Green function expanded in the gap, $G\approx
G_{0}+G_{0}\Delta^{*}\tilde{G}_{0}\Delta G_{0}$, where $G_{0}$ gives ${\bf
j}_{n}$ and the second term provides the supercurrent. Apparently, the double-
counting has to be remedied within the supercurrent itself.
With this issue in mind we shift to our new formulation of the theory,
expressing the nucleation of superconductivity using the floating nucleation
kernel. The Cooper pairs are created from electrons initially in the normal
state, with mean velocity ${\bf v}={\bf J}_{n}/(en)$. The free energy of
condensation has to supply the kinetic energy which electrons gain going from
the normal component into the condensate, therefore the stability condition
reads
$\displaystyle{1\over
2m^{*}}\\!\left(\\!-i\hbar\nabla\\!-\\!\frac{e^{*}}{c}{\bf A}\\!-\\!m^{*}{\bf
v}\right)^{2}\\!\varphi+\alpha\varphi+\beta\\!\left|\varphi\right|^{2}\\!\varphi=-\\!\Gamma\partial_{\tau}\varphi.$
(8)
We note that quantum kinetic energy is in fact a non-local contribution of the
nucleation kernel. For the floating kernel it depends exclusively on the
velocity differences of the normal and superconducting component Lin and
Lipavský (2008).
The corresponding supercurrent
$\displaystyle\tilde{\bf j}_{s}={e^{*}\over m^{*}}{\rm
Re}~{}{\bar{\varphi}}\left(-i\hbar\nabla-\frac{e^{*}}{c}{\bf A}-m^{*}{\bf
v}\right)\varphi$ (9)
we can write as $\tilde{\bf j}_{s}={\bf j}_{s}-e^{*}{\bf v}|\varphi|^{2}={\bf
j}_{s}-(2|\varphi|^{2}/n){\bf J}_{n}$, therefore this approach is free of
double-counting.
If an effect of velocity $\bf v$ on the GL function is negligible, then
$\varphi=\psi$ and the total current ${\bf j}_{\rm fk}=\tilde{\bf j}_{s}+{\bf
J}_{n}$ obtained with the floating kernel is not different from the current in
the two-fluid approximation ${\bf j}_{\rm tf}={\bf j}_{s}+\tilde{\bf j}_{n}$.
In the presence of vortices, the kinetic energy is non-zero due to diamagnetic
currents and the perturbation enters the TDGL equation in the linear order
leading to changes of the GL function. The averaged total current $\tilde{\bf
J}_{s}+{\bf J}_{n}$ then differs from ${\bf J}_{s}+\tilde{\bf J}_{n}$. The
magneto-transmission thus allows us to test the TDGL theory formulated with
the floating nucleation kernel.
To obtain the conductivity we do not need to evaluate the modified GL
function. The supercurrent modified by the inertial force
$m^{*}\partial_{\tau}{\bf v}$ is readily obtained from the condensate
conductivity (3). The driving force in Eq. (9) is
$\partial_{\tau}\left(-(e^{*}/c){\bf A}-m^{*}{\bf v}\right)=e^{*}{\bf
E}+i(\omega/e^{*}n)\sigma_{n}{\bf E}$, therefore
$\displaystyle\tilde{\bf
J}_{s}=\sigma_{s}\left(1+i\frac{\omega}{e^{*2}n}\sigma_{n}\right){\bf E}.$
(10)
The conductivity corresponding to the current $\tilde{\bf J}_{s}+{\bf J}_{n}$
is given by
$\displaystyle\sigma_{\rm
fk}=\sigma_{s}\left(1+i\frac{\omega}{e^{*2}n}\sigma_{n}\right)+\sigma_{n}.$
(11)
In Fig. 1 we compare ${\rm Im}\,\sigma_{\rm fk}$ with ${\rm Im}\,\sigma_{s}$.
One can see that both values are very close except for at the smallest
magnetic field where ${\rm Im}\,\sigma_{\rm fk}$ is closer to experimental
data.
In contrast, the Joule heat obtained within various approximations is rather
different. In Fig. 4 we compare the standard TDGL theory with the floating
kernel formulation. Although none of the approximations provides satisfactory
values, among the tested approaches our floating kernel prescription leads to
values closest to experiment.
Figure 4: Real part of the conductivity giving the Joule heat: Points are
experimental data of Ikebe et al Ikebe et al. (2009) for 7 T ($\bullet$), 5 T
($\circ$), 3 T ($\Box$), and 1 T ($\triangle$). The time-dependent Ginzburg-
Landau theory (thin line) given by Eq. (5) overestimates the dissipation. The
two-fluid approach given by Eq. (7) reduces the dissipation too much leading
to the underestimate. The floating kernel approach ( heavy line) given by Eq.
(11) yields higher values although still smaller than experimental data.
In summary, we have formulated a version of TDGL theory using a floating
nucleation kernel, meaning that the assumption of local equilibrium is applied
to electrons in the moving reference frame of the normal current.
When compared with standard TDGL theory in the context of far-infrared
spectroscopy, we have found that the floating kernel formulation yields better
agreement with experiment. In particular, recent published measurements of
conductivity were considered; since we have established the GL parameter
$\kappa$ from the non-dissipative response given by the imaginary part of the
conductivity, our theory has no fitting parameters with respect to the Joule
heat given by the real part of the conductivity.
Finally, since use of this new approach does not generally introduce
significant additional complexity, it may be promising in the consideration of
systems farther from equilibrium than is usually amenable to analysis via
standard TDGL theory.
The authors are grateful to Peter Matlock for valuable comments and help in
preparation of the manuscript. This work was supported by research plans MSM
0021620834 and No. AVOZ10100521, by grants GAČR 202/07/0597 and GAAV
100100712.
## References
* Schmid (1966) A. Schmid, Phys. kondens. Materie 5, 302 (1966).
* Abrahams and Tsuneto (1966) E. Abrahams and T. Tsuneto, Phys. Rev. 152, 416 (1966).
* Gor’kov and Eliashberg (1968) L. L. Gor’kov and G. M. Eliashberg, Zh. Eksp. Teor. Fiz. 54, 612 (1968), [JETP Lett. 27, 328 (1968)].
* Sá de Melo et al. (1993) C. A. R. Sá de Melo, M. Randeria, and J. R. Engelbrecht, Phys. Rev. Lett. 71, 3202 (1993).
* Huang et al. (2009) K. Huang, Z.-Q. Yu, and L. Yin, Physical Review A 79, 053602 (2009).
* Ikebe et al. (2009) Y. Ikebe, R. Shimano, M. Ikeda, T. Fukumura, and M. Kawasaki, Physical Review B 79, 174525 (2009).
* Tinkham (1966) M. Tinkham, _Introduction to Superconductivity_ (McGraw Hill, New York, 1966).
* Bel and Rosenstein (2005) G. Bel and B. Rosenstein, arXiv:cond-mat/0509677v2 (2005).
* Ketterson and Song (1998) J. B. Ketterson and S. N. Song, _Superconductivity_ (University Press, Cambridge, 1998).
* Kopnin (2001) N. B. Kopnin, _Theory of Nonequilibrium Superconductivity_ (Claredon Press, Oxford, 2001).
* Lin and Lipavský (2008) P.-J. Lin and P. Lipavský, Physical Review B 77, 144505 (2008).
|
arxiv-papers
| 2009-09-10T09:38:10 |
2024-09-04T02:49:05.251746
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Pei-Jen Lin, P. Lipavsky",
"submitter": "PeiJen Lin",
"url": "https://arxiv.org/abs/0909.1907"
}
|
0909.2080
|
# Subsequence Sums of Zero-sum free Sequences II
Pingzhi Yuan
School of Mathematics, South China Normal University , Guangzhou 510631,
P.R.CHINA
e-mail mcsypz@mail.sysu.edu.cn
###### Abstract
Let $G$ be a finite abelian group, and let $S$ be a sequence of elements in
$G$. Let $f(S)$ denote the number of elements in $G$ which can be expressed as
the sum over a nonempty subsequence of $S$. In this paper, we determine all
the sequences $S$ that contains no zero-sum subsequences and $f(S)\leq
2|S|-1$.
MSC: Primary 11B75; Secondary 11B50.
Key words: Zero-sum problems, Davenport’s constant, zero-sum free sequences.
00footnotetext: Supported by NSF of China (No. 10571180).
## 1 Introduction
Let $G$ be a finite abelian group (written additively)throughout the present
paper. $\mathcal{F}(G)$ denotes the free abelian monoid with basis $G$, the
elements of which are called $sequences$ (over $G$). A sequence of not
necessarily distinct elements from $G$ will be written in the form
$S=g_{1}\cdot\,\cdots\,\cdot g_{k}=\prod_{i=1}^{k}g_{i}=\prod_{g\in
G}g^{\mathsf{v}_{g}(S)}\in\mathcal{F}(G)$, where $\mathsf{v}_{g}(S)\geq 0$ is
called the $multiplicity$ of $g$ in $S$. Denote by $|S|=k$ the number of
elements in $S$ (or the $length$ of $S$) and let ${\rm supp}(S)=\\{g\in
G:\,\mathsf{v}_{g}(S)>0\\}$ be the $support$ of $S$.
We say that $S$ contains some $g\in G$ if $\mathsf{v}_{g}(S)\geq 1$ and a
sequence $T\in\mathcal{F}(G)$ is a $subsequence$ of $S$ if
$\mathsf{v}_{g}(T)\leq\mathsf{v}_{g}(S)$ for every $g\in G$, denoted by $T|S$.
If $T|S$, then let $ST^{-1}$ denote the sequence obtained by deleting the
terms of $T$ from $S$. Furthermore, by $\sigma(S)$ we denote the sum of $S$,
(i.e. $\sigma(S)=\sum_{i=1}^{k}g_{i}=\sum_{g\in G}\mathsf{v}_{g}(S)g\in G$).
By $\sum(S)$ we denote the set consisting of all elements which can be
expressed as a sum over a nonempty subsequence of $S$, i.e.
$\sum(S)=\\{\sigma(T):T\,{\rm is\,a\,nonempty\,subsequence\,of\,}S\\}.$
We write $f(S)=|\sum(S)|$, $<S>$ for the subgroup of $G$ generated by all the
elements of $S$.
Let $S$ be a sequence over $G$. We call $S$ a $zero-sum$ $sequence$ if
$\sigma(S)=0$, a $zero-sum\,free$ $sequence$ if $\sigma(W)\neq 0$ for any
subsequence $W$ of $S$, and $squarefree$ if $\mathsf{v}_{g}(S)\leq 1$ for
every $g\in G$. We denote by $\mathcal{A}^{\star}(G)$ the set of all zero-sum
free sequences in $\mathcal{F}(G)$.
Let $D(G)$ be the Davenport’s constant of $G$, i.e., the smallest integer $d$
such that every sequence $S$ over $G$ with $|S|\geq d$ satisfies
$0\in\sum(S)$. For every positive integer $r$ in the interval
$\\{1,\,\ldots,\,D(G)-1\\}$, let
$f_{G}(r)=\min_{S,\,|S|=r}f(S),$ (1)
where $S$ runs over all zero-sum free sequences of $r$ elements in $G$. How
does the function $f_{G}$ behave?
In 2006, Gao and Leader proved the following result.
Theorem A [5] Let $G$ be a finite abelian group of exponent $m$. Then
(i) If $1\leq r\leq m-1$ then $f_{G}(r)=r$.
(ii) If $\gcd(6,\,m)=1$ and $G$ is not cyclic then $f_{G}(m)=2m-1$.
Recently, Sun[10] showed that $f_{G}(m)=2m-1$ still holds without the
restriction that $\gcd(6,\,m)=1$.
Using some techniques from the author [11], the author [12] proved the
following two theorems.
Theorem B[12, 8] Let $S$ be a zero-sum free sequence over $G$ such that $<S>$
is not a cyclic group, then $f(S)\geq 2|S|-1$.
Theorem C [12] Let $S$ be a zero-sum free sequence over $G$ such that $<S>$ is
not a cyclic group and $f(S)=2|S|-1$. Then $S$ is one of the following forms
(i) $S=a^{x}(a+g)^{y},\,x\geq y\geq 1$, where $g$ is an element of order 2.
(ii) $S=a^{x}(a+g)^{y}g,\,x\geq y\geq 1$, where $g$ is an element of order 2.
(iii) $S=a^{x}b,\,x\geq 1$.
However, Theorem B is an old theorem of Olson and White [8] which has been
overlooked by the author. For more recent progress on this topic, see [4, 9,
13].
The main purpose of the present paper is to determine all the sequences $S$
over a finite abelian group such that $S$ contains no zero-sum subsequences
and $f(S)\leq 2|S|-1$. To begin with, we need the notation of $g$-smooth.
###### Definition 1.1
[7, Definition 5.1.3] A sequence $S\in\mathcal{F}(G)$ is called $smooth$ if
$S=(n_{1}g)(n_{2}g)\cdot\,\cdots\,\cdot(n_{l}g)$, where
$|S|\in\mathbb{N},\,g\in G,\,1=n_{1}\leq\cdots\leq
n_{l},\,n=n_{1}+\cdots+n_{l}<\mbox{ord}(g)$ and $\sum(S)=\\{g,\ldots,\,ng\\}$
( in this case we say more precisely that $S$ is $g$-smooth).
We have
###### Theorem 1.1
Let $G$ be a finite abelian group and let $S$ be a zero-sum free sequence over
$G$ with $f(S)\leq 2|S|-1$. Then $S$ has one of the following forms:
(i) $S$ is $a$-smooth for some $a\in G$.
(ii) $S=a^{k}b$, where $k\in\mathbb{N}$ and $a,b\in G$ are distinct.
(iii) $S=a^{k}b^{l}$, where $k\geq l\geq 2$ and $a,b\in G$ are distinct with
$2a=2b$.
(iv) $S=a^{k}b^{l}(a-b)$, where $k\geq l\geq 2$ and $a,b\in G$ are distinct
with $2a=2b$.
For a sequence $S$ over $G$ we call
$\mathsf{h}(S)=\max\\{\mathsf{v}_{g}(S)|g\in G\\}\in[0,|S|]$
$the\,maximum\,of\,the\,multiplicities\,of\,S.$
Let $S=a^{x}b^{y}T$ with $x\geq y\geq\mathsf{h}(T)$, then Theorem 1.1(i) can
be stated more precisely as that $S$ is $a$-smooth or $b$-smooth.
## 2 Some Lemmas
Let $\emptyset\neq G_{0}\subseteq G$ be a subset of $G$ and $k\in\mathbb{N}$.
Define
$\mathsf{f}(G_{0},\,k)=\min\\{f(S):\,S\in\mathcal{F}(G_{0})\,\,{\rm~{}zero-
sumfree,\,squarefree~{}and~{}}\,|S|=k\\}$
and set $\mathsf{f}(G_{0},\,k)=\infty$, if there are no sequences over $G_{0}$
of the above form.
###### Lemma 2.1
Let $G$ be a finite abelian group.
1. 1.
If $k\in\mathbb{N}$ and $S=S_{1}\cdot\,\cdots\,\cdot
S_{k}\in\mathcal{A}^{\star}(G)$, then
$f(S)\geq f(S_{1})+\cdots+f(S_{k})\,.$
2. 2.
If $G_{0}\subset G$, $k\in\mathbb{N}$ and $\mathsf{f}(G_{0},\,k)>0$, then
$\mathsf{f}(G_{0},\,k)\ \left\\{\begin{array}[]{ll}=1\,,&\mbox{if}\quad
k=1\,,\\\ =3\,,&\mbox{if}\quad k=2\,,\\\ \geq 5\,,&\mbox{if}\quad k=3\,,\\\
\geq 6\,,&\mbox{if}\quad k=3\quad\mbox{and}\quad 2g\neq 0\quad\mbox{for
all}\quad g\in G_{0}\,,\\\ \geq 2k\,,&\mbox{if}\quad k\geq
4\,.\end{array}\right.$
* Proof.
1\. See [6, Theorem 5.3.1].
2\. See [6, Corollary 5.3.4].
$\Box$
###### Lemma 2.2
Let $a,\,b$ be two distinct elements in an abelian group $G$ such that
$a^{2}b^{2}\in\mathcal{A}^{\star}(G),\,2a\neq 2b,a\neq 2b$, and $b\neq 2a$.
Then $f(a^{2}b^{2})=8$.
* Proof.
It is easy to see that $a,\,2a,\,b,\,2b,\,a+b,\,a+2b,\,2a+b,\,2a+2b$ are all
the distinct elements in $\sum(a^{2}b^{2})$. We are done. $\Box$
###### Lemma 2.3
Let $S=a^{k}b$ be a zero-sum free sequence over $G$. If $S=a^{k}b$ is not
$a$-smooth, then $f(S)=2k+1$.
* Proof.
The assertion follows from the fact that
$a,\,\ldots,\,ka,\,b,\,a+b,\,\ldots,\,ka+b$ are all the distinct elements in
$\sum(a^{k}b)$.$\Box$
###### Lemma 2.4
[10, Lemma 4] Let $S$ be a zero-sum free sequence over $G$. If there is some
element $g$ in $S$ with order $2$, then $f(S)\geq 2|S|-1$.
###### Lemma 2.5
Let $k\geq l\geq 2$ be two integers, and let $a$ and $b$ be two distinct
elements of $G$ such that $a^{k}b^{l}\in\mathcal{A}^{\star}(G)$ and
$a^{k}b^{l}$ is not smooth. Then we have
(i) If $2a\neq 2b$, then $f(a^{k}b^{l})\geq 2(k+l)$.
(ii) If $2a=2b$, then $f(a^{k}b^{l})=2(k+l)-1$.
* Proof.
If $nb\neq sa$ for any $n$ and $s$ with $1\leq n\leq l$ and $1\leq s\leq k$,
then $ra+sb,\,r+s\neq 0,\,0\leq r\leq k,\,0\leq s\leq b$ are all the distinct
elements in $\sum(a^{k}b^{l})$, and so
$f(a^{k}b^{l})=kl+k+l\geq 2(k+l).$
Now we assume that $nb=sa$ for some $n$ and $s$ with $1\leq n\leq l$ and
$1\leq s\leq k$. Let $n$ be the least positive integer with $nb=sa,\,1\leq
n\leq l,\,1\leq s\leq k$ . Then $n\geq 2$ and $s\geq 2$ by our assumptions. It
is easy to see that
$a,\,\ldots,\,ka,\,\ldots,\,(k+[\frac{l}{n}]s)a,$
$b,\,a+b,\,\ldots,\,b+ka,\,\ldots,\,b+(k+[\frac{l-1}{n}]s)a,$ $\ldots\ldots$
$(n-1)b,\,\ldots,\,(n-1)b+ka,\,\ldots,\,(n-1)b+(k+[\frac{l-n+1}{n}]s)a$
are all the distinct elements in $\sum(a^{k}b^{l})$, and so
$f(a^{k}b^{l})=k+[\frac{l}{n}]s+1+k+[\frac{l-1}{n}]s+\cdots+1+k+[\frac{l-n+1}{n}]s$
$=n(k-s+1)+ls+s-1.$
Since $n(k-s+1)+ls+s-1-2(k+l)=(n-2)(k-s)+(l-1)(s-2)+n-3$, we have
$f(a^{k}b^{l})\geq 2(k+l)-1$ and the equality holds if and only if $n=s=2$,
that is $2a=2b$. This completes the proof.
$\Box$
Remark: Note that if $a^{k}b^{l}\in\mathcal{A}^{\star}(G),\,k\geq l\geq 2$,
then the conditions that $a^{k}b^{l}$ is smooth and $2a=2b$ cannot hold
simultaneously. Otherwise, we may suppose that $2a=2b$ and $a^{k}b^{l}$ is
$a$-smooth (the case that $a^{k}b^{l}$ is $b$-smooth is similar), then
$b=ta,\,2\leq t\leq(k+1)$. It follows that
$b+(t-2)a=2(t-1)a=2b-2a=0,\,0<t-2\leq k-1$, which contradicts the fact that
$a^{k}b^{l}\in\mathcal{A}^{\star}(G)$.
###### Lemma 2.6
[12, Lemma 2.9]Let $S=a^{k}b^{l}g,\,k\geq l\geq 1$ be a zero-sum free sequence
over $G$ with $b-a=g$ and ${\rm ord}(g)=2$, then $f(S)=2(k+l)+1$.
###### Lemma 2.7
Let $S_{1}\in\mathcal{F}(G)$ and $a,\,g\in G$ such that
$S=S_{1}a\in\mathcal{A}^{\star}(G)$, $S_{1}$ is $g$-smooth and $S$ is not
$g$-smooth. Then $f(S)=2f(S_{1})+1$.
* Proof.
If $a\not\in<g>$, then $\sum(S)=\sum(S_{1})\cup\\{a\\}\cup(\sum(S_{1})+a)$,
and so $f(S)=2f(S_{1})+1$.
If $a\in<g>$, we let $\sum(S_{1})=\\{g,\,\ldots,ng\\}$,
$a=tg,\,t\in\mathbb{N}$, then $t\geq n+2$ by our assumptions. It follows that
$\sum(S)=\\{g,\,\ldots,\,ng,\,tg,\,(t+1)g,\,\ldots,\,(t+n)g\\}$, and so
$f(S)=2f(S_{1})+1$.$\Box$
###### Lemma 2.8
Let $k\geq 2$ be a positive integer and $a,\,b,\,c$ three distinct elements in
$G$ such that $a^{k}bc\in\mathcal{A}^{\star}(G)$ and $a^{k}bc$ is not
$a$-smooth. Then $f(a^{k}bc)\geq 2k+4$.
* Proof.
Observe that $f(a^{k}bc)\geq 2k+4$ when $a^{k}bc$ is $b$ or $c$-smooth. We
consider first the case that $a^{k}b$ is $a$-smooth (the case that $a^{k}c$ is
$a$-smooth is similar). It is easy to see $f(a^{k}b)\geq k+2$, and so
$f(a^{k}bc)=2f(a^{k}b)+1\geq 2k+5$ by Lemma 2.7. Therefore we may assume that
both $a^{k}b$ and $a^{k}c$ are not $a$-smooth in the remaining arguments. We
divide the proof into three cases.
(i) If $a^{k}(b+c)$ is not $a$-smooth, then
$a,\,\ldots,\,ka,\,b,\,b+a,\,\ldots,\,b+c,\,b+c+a,\,\ldots,\,b+c+ka$ are
distinct elements in $\sum(a^{k}bc)$, and so
$f(a^{k}bc)\geq k+k+1+k+1\geq 2k+4.$
(ii) If neither $a^{k}(b-c)$ nor $a^{k}(c-b)$ is $a$-smooth, then
$a,\,\ldots,\,ka,\,b,\,b+a,\,\ldots,\,c,\,c+a,\,\ldots,\,c+ka,\,b+c+ka$ are
distinct elements in $\sum(a^{k}bc)$, and so
$f(a^{k}bc)\geq k+k+1+k+1+1\geq 2k+5.$
(iii) If $a^{k}(b+c)$ is $a$-smooth and $a^{k}(b-c)$ (or $a^{k}(c-b)$ ) is
$a$-smooth, then we have
$b+c=sa,\quad b-c=ta,\quad 1\leq s,\,t\leq k+1,\quad s\neq t.$
It is easy to see that $a,\,\ldots,\,ka,\,(k+1)a,\,\ldots,\,(k+s)a,$
$c,\,c+a,\,\ldots,\,c+(k+t)a$ are all distinct elements in $\sum(a^{k}bc)$,
and so
$f(a^{k}bc)=k+s+k+t+1\geq 2k+4.$
The second equality holds if and only if $(s,\,t)=(1,\,2)$ or $(2,\,1)$. We
are done. $\Box$
The following corollary follows immediately from Lemmas 2.1, and 2.7 and the
proof of Lemma 2.9.
###### Corollary 2.1
Let $k\geq 1$ be a positive integer and $a,\,b,\,c,\,d$ four distinct elements
in $G$ such that $a^{k}bcd\in\mathcal{A}^{\star}(G)$ and $a^{k}bcd$ is not
$a$-smooth. Then $f(a^{k}bcd)\geq 2k+6$.
###### Lemma 2.9
Let $a,b,x$ be three distinct elements in $G$ such that
$a^{k}b^{l}x\in\mathcal{A}^{\star}(G),\,k\geq l\geq 1$, $2a=2b$, and $x\neq
a-b$, then $f(a^{k}b^{l}x)\geq 2(k+l+1)+1$.
* Proof.
If there are no distinct pairs
$(m,\,n)\neq(0,\,0),(m_{1},\,n_{1})\neq(0,\,0),\,0\leq m,\,m_{1}\leq k,\,0\leq
n,\,n_{1}\leq l$ such that $ma+nb=m_{1}a+n_{1}b+x$, then
$\sum(a^{k}b^{l}x)=\sum(a^{k}b^{l})\cup\\{x\\}\cup(\sum(a^{k}b^{l})+x)$, and
so $f(a^{k}b^{l}x)=2f(a^{k}b^{l})+1=4(k+l)-1\geq 2(k+l+1)+1$.
If there are two distinct pairs
$(m,\,n)\neq(0,\,0),(m_{1},\,n_{1})\neq(0,\,0),\,0\leq m,\,m_{1}\leq k,\,0\leq
n,\,n_{1}\leq l$ such that $ma+nb=m_{1}a+n_{1}b+x$, then $x=a-b$ or
$x=ua+b,\,1\leq u\leq(k+l-1)$ or $x=vb,\,v\geq 2$ or $x=ta,\,t\geq 2$.
Let $x=ua+b,1\leq u\leq(k+l-1)$, then
$a,\,\ldots,\,(k+l+u)a,\,b,\,\cdots,\,b+(k+l+u)a$ are all distinct elements in
$\sum(a^{k}b^{l}x)$, and so $f(a^{k}b^{l}x)=2(k+l+u)+1\geq 2(k+l+1)+1$.
Let $x=vb,\,2\leq v\leq(k+l)$ (the case that $x=ta,\,t\geq 2$ is similar). If
$k$ is even, then $b,\,\ldots,\,(k+l+v)b,\,a,\,a+b,\,\ldots,a+(k+l-2+v)b$ are
all distinct elements in $\sum(a^{k}b^{l}x)$, and so
$f(a^{k}b^{l}x)=2(k+l+v-1)+1\geq 2(k+l+1)+1$. If $k$ is odd, then
$b,\,\ldots,\,(k+l+v-1)b,\,a,\,a+b,\,\ldots,a+(k+l-1+v)b$ are all distinct
elements in $\sum(a^{k}b^{l}x)$, and so $f(a^{k}b^{l}x)=2(k+l+v-1)+1\geq
2(k+l+1)+1$. We are done. $\Box$
###### Lemma 2.10
Let $a,b,x$ be three distinct elements in $G$ such that
$a^{k}b^{2}x\in\mathcal{A}^{\star}(G),\,k\geq 2$ and $a^{k}b^{2}x$ is not
$a$-smooth or $b$-smooth, then $f(a^{k}b^{2}x)=2k+5$ if and only if $2a=2b$
and $x=b-a$.
* Proof.
We divide the proof into four cases.
Case 1 $a^{k}b^{2}$ is not smooth and $2b=sa,\,2\leq s\leq k$. If $x=b-a$,
then $a,\,\ldots,\,(k+s)a,\,b-a,\,b,\,\ldots,\,b+(k+s-1)a$ are all the
distinct elements in $\sum(a^{k}b^{2}x)$, and so $f(a^{k}b^{2}x)=2(k+s)+1$. If
$x=ta,\,2\leq t\leq k$, then $a,\,\ldots,\,(k+s+t)a,\,b,\,\ldots,\,b+(k+t)a$
are all the distinct elements in $\sum(a^{k}b^{2}x)$, and so
$f(a^{k}b^{2}x)=2(k+t)+s+1$. If $x=ta+b,\,1\leq t\leq k$, then
$a,\,\ldots,\,(k+s+t)a,\,b,\,\ldots,\,b+(k+t+s)a$ are all the distinct
elements in $\sum(a^{k}b^{2}x)$, and so $f(a^{k}b^{2}x)=2(k+t+s)+1$. Therefore
$f(a^{k}b^{2}x)=2k+5$ if and only if $2a=2b$ and $x=b-a$ in this case.
Case 2 $a^{k}b^{2}$ is not smooth and $2b=sa,\,s>k$ or $2b\not\in<a>$, then
$f(a^{k}b^{2})=3k+2$. If $k\geq 3$, then $f(a^{k}b^{2}x)\geq
f(a^{k}b^{2})+1=3k+3>2k+5$. If $k=2$ and $f(abx)=7$, then $f(a^{2}b^{2}x)\geq
f(abx)+f(ab)=7+3>2k+5$. If $k=2$ and $f(abx)=6$ (i.e., $x=a+b$ or $x=a-b$ or
$x=b-a$), then it is easy to check that $f(a^{2}b^{2}x)>2k+5$.
Case 3 $a^{k}b^{2}$ is smooth and $a^{k}b^{2}x$ is not smooth. If $a^{k}b^{2}$
is $a$-smooth, then $f(a^{k}b^{2}x)=2f(a^{k}b^{2})+1\geq 2(k+2\times
2)+1>2k+5$. If $a^{k}b^{2}$ is $b$-smooth, then
$f(a^{k}b^{2}x)=2f(a^{k}b^{2})+1\geq 2(2+2k)+1>2k+5$.
Case 4 $a^{k}b^{2}x$ is $x$-smooth. We have $f(a^{k}b^{2}x)\geq 1+2k+2\times
3>2k+5$.
This completes the proof of the lemma.
$\Box$
## 3 Proofs of the Main Theorems
To prove the main theorem of the present paper, we still need the following
two obviously facts on smooth sequences.
Fact 1 Let $r$ be a positive integer and $a\in G$. If
$WT_{i}\in\mathcal{A}^{\star}(G)$ is $a$-smooth for all $i=1,\ldots,r$, then
$S=T_{1}\cdot\,\cdots\,\cdot T_{r}W$ is $a$-smooth.
Fact 2 Let $r,\,k,\,l$ be three positive integers and $a,\,b$ two distinct
elements in $G$. If $S\in\mathcal{A}^{\star}(G)$ is $a$-smooth and
$a^{k}b^{l}T_{i}\in\mathcal{A}^{\star}(G)$ is $a$-smooth or $b$-smooth for all
$i=1,\ldots,r$, then the sequence $Sa^{k}b^{l}T_{1}\cdot\,\cdots\,\cdot T_{r}$
is $a$-smooth or $b$-smooth.
Proof of Theorem 1.1:
We start with the trivial case that $S=a^{k}$ with $k\in\mathbb{N}$ and $a\in
G$. Then $\sum(S)=\\{a,\ldots,ka\\}$, and since $S$ is zero-sum free, it
follows that $k<ord(a)$. Thus $S$ is $a$-smooth.
If $S=S_{1}g\in\mathcal{A}^{\star}(G)$, where $g$ is an element of order 2,
then $f(S)\geq 2|S|-1$ by Lemma 2.4, and $f(S)\geq f(S_{1})+2$ since
$\sum(S)\supseteq\sum(S_{1})\cup\\{g,\,g+\sigma(S_{1})\\}$. If
$S=S_{1}g_{1}g_{2}\in\mathcal{A}^{\star}(G)$, where $g_{1}$ and $g_{2}$ are
two elements of order 2, then $f(S)\geq 2|S|$ since
$\sum(S)\supseteq\sum(S_{1}g_{1})\cup\\{g_{2},\,g_{1}+g_{2},\,g_{1}+g_{2}+\sigma(S_{1})\\}$.
Therefore it suffices to determine all $S\in\mathcal{A}^{\star}(G)$ such that
$S$ does not contain any element of order 2 and $f(S)\leq 2|S|-1$, and when
$f(S)\leq 2|S|-1$, determine all $Sg\in\mathcal{A}^{\star}(G)$ such that $g$
is an element of order 2 and $f(Sg)=2|S|+1$.
To begin with, we determine all $S\in\mathcal{A}^{\star}(G)$ such that $S$
does not contain any element of order 2 and $f(S)\leq 2|S|-1$. Let
$S=a^{x}b^{y}c^{z}T$ with $x\geq y\geq z\geq\mathsf{h}(T)$ and
$a,\,b,c\not\in{\rm supp}(T)$. The case that $|{\rm supp}(S)|=2$ follows from
Lemmas 2.3 and 2.5 and the remark after Lemma 2.5. Therefore we may assume
that $|{\rm supp}(S)|\geq 3$ and $S$ does not contain any element of order $2$
in the following arguments.
If $x=y=z$, then $S$ allows the product decomposition
$S=S_{1}\cdot\,\cdots\,\cdot S_{x},$
where $S_{i}=abc\cdot\,\cdots,\,i=1,\,\ldots,\,x$ are squarefree of length
$|S_{i}|\geq 3$. By Lemma 2.1, we obtain
$f(S)\geq\sum_{i=1}^{x}f(S_{i})\geq 2\sum_{i=1}^{x}|S_{i}|=2|S|.$
If $x\geq y>z\geq\mathsf{h}(T)$, or $x>y\geq z\geq\mathsf{h}(T)$, then $S$
allows a product decomposition
$S=T_{1}\cdot\,\cdots\,\cdot T_{r}W$
having the following properties:
* •
$r\geq 1$ and, for every $i\in[2,\,r]$, $S_{i}\in\mathcal{F}(G)$ is squarefree
of length $|S_{i}|=3$.
* •
$W\in\mathcal{F}(G)$ has the form $W=a^{k},\,k\geq 1$ or $W=a^{k}b,\,k\geq 1$
or $W=a^{k}b^{l},\,k\geq l\geq 2$.
We choose a product decomposition such that $k$ is the largest integer in
$W=a^{k}$ (or $a^{k}b$ or $a^{k}b^{l},\,k\geq l\geq 2$) among all such product
decompositions. We divide the remaining proof into three cases.
Case 1 $W=a^{k},\,k\geq 1$. If $T_{i}=xyz$ with $a\not\in\\{x,\,y,\,z\\}$ for
some $i,\,1\leq i\leq r$ such that $a^{k}xyz$ is not $a$-smooth whenever
$k>1$, then $S$ admits the product decomposition
$S=T_{1}\cdot\,\cdots\,\cdot T_{i-1}T_{i}^{\prime}T_{i+1}\cdot\,\cdots\,\cdot
T_{r},$
where $T_{i},\,i=1,\,\ldots,\,r$ have the properties described above and
$T_{i}^{\prime}=a^{k}xyz$. By Lemma 2.1, and Corollary 2.1, we get
$f(S)\geq\sum_{j\neq i}^{r}f(T_{j})+f(T_{i}^{\prime})\geq\sum_{j\neq
i}^{r}2|T_{j}|+2|T_{i}^{\prime}|=2|S|.$
If $T_{i}=axy$ for some $i,\,1\leq i\leq r$ such that $a^{k+1}xy$ is not
$a$-smooth, then $S$ admits the product decomposition
$S=T_{1}\cdot\,\cdots\,\cdot T_{i-1}T_{i}^{\prime}T_{i+1}\cdot\,\cdots\,\cdot
T_{r},$
where $T_{i},\,i=1,\,\ldots,\,r$ have the properties described above and
$T_{i}^{\prime}=a^{k+1}xy$. By Lemmas 2.1 and 2.8, we get
$f(S)\geq\sum_{j\neq i}^{r}f(T_{j})+f(T_{i}^{\prime})\geq\sum_{j\neq
i}^{r}2|T_{j}|+2|T_{i}^{\prime}|=2|S|.$
Therefore we have proved that if $S$ is not $a$-smooth and $W=a^{k}$, then
$f(S)\geq 2|S|$.
Case 2 $W=a^{k}b,\,k\geq 1$.
Let $T_{i}=xyz$ with $a\not\in\\{x,\,y,\,z\\}$ for some $i,\,1\leq i\leq r$.
If $k=1$, then $T_{i}W=abxyz$. If $k=2$, then $T_{i}W=abx\cdot ayz$. If $k\geq
3$ and one sequence among three sequences $a^{k-1}yz,\,a^{k-1}xz$, and
$a^{k-1}xy$, say, $a^{k-1}yz$ is not $a$-smooth, then $T_{i}W=abx\cdot
a^{k-1}yz$. It follows from Lemmas 2.1 and 2.8 that $f(T_{i}W)\geq
2|T_{i}|+2|W|$, and so $f(S)\geq 2|S|$.
Let $T_{i}=bxy$ for some $i,\,1\leq i\leq r$, then $k\geq 2$. If $k=2$, then
$T_{i}W=abx\cdot aby$. If $k>2$ and $a^{k-1}by$ (or $a^{k-1}bx$) is not
$a$-smooth, then $T_{i}W=abx\cdot a^{k-1}by$ (or $T_{i}W=aby\cdot a^{k-1}bx$).
It follows from Lemmas 2.1 and 2.8 that $f(T_{i}W)\geq 2|T_{i}|+2|W|$, and so
$f(S)\geq 2|S|$.
Let $T_{i}=abx$ for some $i,\,1\leq i\leq r$, then $T_{i}W=a^{k+1}b^{2}x$. If
$a^{k+1}b^{2}x$ is not $a$-smooth or $b$-smooth, then by Lemma 2.10 we have
$f(T_{i}W)\geq 2|T_{i}|+2|W|$, and so $f(S)\geq 2|S|$.
Therefore we have proved that if $S$ is not $a$-smooth or $b$-smooth, then
$f(S)\geq 2|S|$ in this case.
Case 3 $W=a^{k}b^{l},\,k\geq l\geq 2$. If $2a\neq 2b$ and $a^{k}b^{l}$ is not
smooth, then by Lemma 2.5 we have $f(W)\geq 2|W|$ and we are done. Note that
the conditions that $2a=2b$ and $a^{k}b^{l}$ is smooth cannot hold
simultaneous. Here we omit the similar arguments as we have done in Case 1.
Subcase 1 $2a=2b$.
Let $T_{i}=xyz$ with $a\not\in\\{x,\,y,\,z\\}$ for some $i,\,1\leq i\leq r$,
then $T_{i}W=abxy\cdot a^{k-1}b^{l-1}z$. It follows from Lemmas 2.1 and 2.9
that $f(T_{i}W)\geq 2|T_{i}|+2|W|$, and so $f(S)\geq 2|S|$.
Let $T_{i}=byz$ for some $i,\,1\leq i\leq r$, then $k\geq l+1$,
$T_{i}W=aby\cdot a^{k-1}b^{l}z$. It follows from Lemmas 2.1 and 2.9 that
$f(T_{i}W)\geq 2|T_{i}|+2|W|$, and so $f(S)\geq 2|S|$.
Let $T_{i}=abx$ for some $i,\,1\leq i\leq r$, then $T_{i}W=a^{k+1}b^{l+1}x$.
If $a^{k+1}b^{2}x$ is not $a$-smooth or $b$-smooth, then by Lemma 2.10 we have
$f(T_{i}W)\geq 2|T_{i}|+2|W|$, and so $f(S)\geq 2|S|$.
Subcase 2 $a^{k}b^{l}$ is smooth, $a\neq 2b$, and $b\neq 2a$. Then
$W=(a^{2}b^{2})^{s}W_{1}$, $W_{1}=a^{k_{1}}$ or $W_{1}=a^{k_{1}}b$. If
$S_{1}=SW^{-1}W_{1}$ is not $a$-smooth or $b$-smooth, then $f(S_{1})\geq
2|S_{1}|$, and so by Lemmas 2.1 and 2.2 $f(S)\geq sf(a^{2}b^{2})+f(S_{1})\geq
8s+2|S_{1}|=2|S|$. If $S_{1}=SW^{-1}W_{1}$ is $a$-smooth or $b$-smooth, then
$S$ is $a$-smooth or $b$-smooth.
Subcase 3 $a=2b$.
Let $T_{i}=xyz$ with $a,\,b\not\in\\{x,\,y,\,z\\}$ for some $i,\,1\leq i\leq
r$, then it is easy to see that $f(T_{i}W)=f(a^{k}b^{l}xyz)=f(b^{2k+l}xyz)$.
It follows from Corollary 2.1 that $b^{2k+l}xyz$ is $b$-smooth or
$f(T_{i}W)\geq 2(|T_{i}|+|W|)$.
Let $T_{i}=bxy$ with $a,\,b\not\in\\{x,\,y\\}$ for some $i,\,1\leq i\leq r$,
then $f(T_{i}W)=f(a^{k}b^{l+1}xy)=f(b^{2k+l+1}xyz)$. It follows from Lemma 2.8
that $b^{2k+l+1}xy$ is $b$-smooth or $f(T_{i}W)\geq 2(|T_{i}|+|W|)$.
Let $T_{i}=abx$ with $a\neq x,\,b\neq x$ for some $i,\,1\leq i\leq r$, then
$f(T_{i}W)=f(a^{k+1}b^{l+1}x)=f(b^{2k+l+3}xyz)$. It follows from Lemma 2.3
that $b^{2k+l+3}x$ is $b$-smooth or $f(T_{i}W)\geq 2(|T_{i}|+|W|)$.
Subcase 4 $b=2a$. Similar to Subcase 3.
Therefore we have proved that if $S$ is not $a$-smooth or $b$-smooth, then
$f(S)\geq 2|S|-1$ and $f(S)=2|S|-1$ if and only if $S=a^{k}b$ or
$S=a^{k}b^{l},\,2a=2b,\,k\geq l\geq 2$.
Finally, when $f(S)\leq 2|S|-1$, we will determine all
$Sg\in\mathcal{A}^{\star}(G)$ such that $g$ is an element of order 2 and
$f(Sg)=2|S|+1$.
(i) If $S$ is $a$-smooth (the case that $S$ is $b$-smooth is similar), we set
$\sum(S)=\\{a,\,\ldots,\,na\\},\,n\leq 2|S|-1$, then $g\not\in\sum(S)$ since
$g$ is an element of order 2 and $Sg\in\mathcal{A}^{\star}(G)$. It follows
that $\sum(Sg)=\sum(S)\cup\\{g\\}\cup\\{g+\sum(S)\\}$, and so $f(Sg)=2n+1$.
Therefore $f(Sg)\leq 2|S|+1$ if and only if $S=a^{k}$.
(ii) $S=a^{k}b$ is not smooth, by Lemma 2.8, $f(a^{k}bg)\leq 2k+1$ only if
$a^{k}bg$ is $a$-smooth, which is impossible since $g$ is an element of order
2 and $a^{k}bg\in\mathcal{A}^{\star}(G)$.
(iii) $S=a^{k}b^{l},\,2a=2b,\,k\geq l\geq 2$. The result follows from Lemmas
2.5 and 2.9.
Therefore we have proved that if
$S=a^{x}b^{y}\cdot\,\cdots\in\mathcal{A}^{\star}(G),\,x\geq y\geq\cdots$,
where $a,\,b,\,\ldots$ are distinct elements of $G$ and $f(S)\leq 2|S|-1$,
then $S$ is $a$-smooth or $b$-smooth or $S=a^{k}b,\,b\not\in\sum(a^{k})$ or
$S=a^{k}b^{l},\,k\geq l\geq 2,2a=2b$ or $S=a^{k}b^{l},\,k\geq l\geq
2,2a=2b,\,g=a-b$. Theorem 1.1 is proved.
$\Box$
Acknowledgement: The author wishes to thank Alfred Geroldinger for sending the
preprint [7] to him. He also thanks the referee for his/her valuable
suggestions.
## References
* [1] J.D. Bovey, P. Erdős, and I. Niven, _Conditions for zero sum modulo $n$_, Canad. Math. Bull. 18 (1975), 27 – 29.
* [2] B. Bollob$\acute{a}$s and I. Leader, _The number of $k$-sums modulo $k$_, J. Number Theory 78(1999), 27-35.
* [3] S.T. Chapman and W.W. Smith, _A characterization of minimal zero-sequences of index one in finite cyclic groups_ , Integers 5(1) (2005), Paper A27, 5pp.
* [4] W. Gao, Y. Li, J. Peng, and F. Sun, _On subsequence sums of a zero-sum free sequence II_ , the Electronic Journal of Combinatorics 15 (2008), $\sharp$R117.
* [5] W.D. Gao and I. Leader, _sums and $k$-sums in an abelian groups of order $k$_, J. Number Theory 120(2006), 26-32.
* [6] A. Geroldinger and F. Halter-Koch, _Non-Unique Factorizations. Algebraic, Combinatorial and Analytic Theory_ , Pure and Applied Mathematics, Vol. 278, Chapman & Hall/CRC, 2006.
* [7] A. Geroldinger, _Additive group theory and non-unique factorizations_ , to appear.
* [8] J. E. Olson and E.T.White, _sums from a sequence of group elements_ , in : Number Theory and Algebra, Academic Press, New York, 1977, pp. 215-222.
* [9] A. Pixton, _Sequences with small subsum sets_ , J. Number Theory 129(2009), 806-817.
* [10] F. Sun, _On subsequence sums of a zero-sum free sequence_ , the Electronic Journal of Combinatorics 14(2007), $\sharp$R52.
* [11] P.Z. Yuan, _On the index of minimal zero-sum sequences over finite cyclic groups_ , J. Combin. Theory Ser. A114(2007), 1545-1551.
* [12] P.Z. Yuan, _Subsequence sums of a zero-sumfree sequence_ , European Journal of Combinatorics, 30(2009), 439-446.
* [13] P.Z. Yuan, _Subsequence Sums of Zero-sum-free Sequences_ , to appear in the Electronic Journal of Combinatorics.
Pingzhi Yuan
School of Mathematics
South China Normal University
Guangdong, Guangzhou 510631
P.R.CHINA
e-mail:mcsypz@mail.sysu.edu.cn
|
arxiv-papers
| 2009-09-11T02:29:48 |
2024-09-04T02:49:05.261218
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Pingzhi Yuan",
"submitter": "Pingzhi Yuan",
"url": "https://arxiv.org/abs/0909.2080"
}
|
0909.2085
|
# $CP$–Violation in $B_{q}$ Decays and Final State Strong Phases
Fayyazuddin
National Centre for Physics &
Department of Physics, Quaid-i-Azam University, Islamabad
fayyazuddins@gmail.com
(PACS: 12.15.Ji, 13.25.Hw, 14.40.Nd)
###### Abstract
Using the unitarity, $SU(2)$ and $C$-invariance of hadronic interactions, the
bounds on final state phases are derived. It is shown that values obtained for
the final state phases relevant for the direct $CP$-asymmetries
$A_{CP}(B^{0}\rightarrow K^{+}\pi^{-},K^{0}\pi^{0})$ are compatiable with
experimental values for these asymmetries. For the decays $B^{0}\rightarrow
D^{(\ast)-}\pi^{+}$ $(D^{(\ast)+}\pi^{-})$ described by two independent single
amplitudes $A_{f}$ and $A_{\bar{f}}^{\prime}$ with differnt weak phases ($0$
and $\gamma$) it is argued that the $C$-invariance of hadronic interactions
implies the equality of the final state phase $\delta_{f}$ and
$\delta_{\bar{f}}^{\prime}$. This in turn implies, the $CP$-asymmetry
$\frac{S_{+}+S_{-}}{2}$ is determined by weak phase ($2\beta+\gamma)$ only
whereas $\frac{S_{+}-S_{-}}{2}=0.$ Assuming factorization for tree graphs, it
is shown that the $B\rightarrow D^{(\ast)}$ form factors are in excellent
agreement with heavy quark effective theory. From the experimental value for
$\left(\frac{S_{+}+S_{-}}{2}\right)_{D^{\ast}\pi},$ the bound
$\sin(2\beta+\gamma)\geq 0.69$ is obtained and
$\left(\frac{S_{+}+S_{-}}{2}\right)_{D_{S}^{\ast-}K^{+}}\approx-(0.41\pm
0.08)\sin\gamma$ is predicted. For the decays described by the amplitudes
$A_{f}\neq A_{\bar{f}}$ such as $B^{0}\longrightarrow\rho^{+}\pi^{-}:$
$A_{\bar{f}}$ and $B^{0}\longrightarrow\rho^{-}\pi^{+}:A_{f}$ where these
amplitudes are given by tree and penguin diagrams with differnt weak phases,
it is shown that in the limit $\delta_{f,\bar{f}}^{T}\rightarrow
0,r_{f,\bar{f}}\cos\delta_{f,\bar{f}}=\cos\alpha$ and
$\frac{S_{\bar{f}}}{S_{f}}=\frac{S+\Delta S}{S-\Delta
S}=-\frac{\sqrt{1-C_{\bar{f}}^{2}}}{\sqrt{1-C_{f}^{2}}}.$
## 1 Introduction
The CP asymmetries in the hadronic decays of B and K mesons involve strong
final state phases. Thus strong interactions in these decays play a crucial
role. The short distance strong interactions effects at quark level are taken
care of by perturbative QCD in terms of Wilson coefficients. The CKM matrix,
which connects the weak eigenstates with mass eigenstates, is another aspect
of strong interactions at quark level. In the case of semi leptonic decays,
the long distance strong interaction effects manifest themselves in the form
factors of final states after hadronization. Likewise the strong interaction
final state phases are long distance effects. These phase shifts essentially
arise in terms of S-matrix which changes an ’in’ state into an ’out’ state
viz.
$|f\rangle_{in}=S|f\rangle_{out}=e^{2i\delta_{f}}|f\rangle_{out}$ (1)
In fact, the CPT invariance of weak interaction Lagrangian gives for the weak
decay $B(\bar{B})\rightarrow f(\bar{f})$
$\bar{A}_{\bar{f}}\equiv_{out}\langle\bar{f}|\mathcal{L}_{w}|\bar{B}\rangle=\eta_{f}e^{2i\delta_{f}}A_{f}{\ast}$
(2)
Taking out the weak phase $\phi$, the amplitude $A_{f}$ can be written as
$A_{f}=e^{i\phi}F_{f}=e^{i\phi}e^{i\delta_{f}}|F_{f}|$ (3)
Then Eq. $\eqref{02}$ implies
$\bar{A}_{\bar{f}}=e^{-i\phi}e^{2i\delta_{f}}F_{f}^{\ast}=e^{-i\phi}F_{f}$
It is difficult to reliably estimate the final state strong phase shifts. It
involves the hadronic dynamics. However, using isospin, C-invariance of
S-matrix and unitarity, we can relate these phases. In this regard, following
cases are of interest:
Case (i): The decays $B^{0}\rightarrow f,\bar{f}$ described by two independent
single amplitudes $A_{f}$ and $A_{\bar{f}}^{\prime}$ with different weak
phases:
$\displaystyle A_{f}$ $\displaystyle=\langle
f\left|\mathcal{L_{W}}\right|B^{0}\rangle=e^{i\phi}F_{f}=e^{i\phi}e^{i\delta_{f}}\bigl{|}F_{f}\bigr{|}$
$\displaystyle A_{\bar{f}}^{\prime}$
$\displaystyle=\langle\bar{f}\left|\mathcal{L_{W}^{\prime}}\right|B^{0}\rangle=e^{i\phi^{\prime}}F_{\bar{f}}^{\prime}=e^{i\phi^{\prime}}e^{i\delta^{\prime}_{\bar{f}}}\bigl{|}F_{\bar{f}}^{\prime}\bigr{|}$
where the states $|\bar{f}\rangle$ and $|f\rangle$ are C conjugate of each
other such as states $D^{(*)-}\pi^{+}(D^{(*)+}\pi^{-})$,
$D_{s}^{(*)-}K^{+}(D_{s}^{(*)+}K^{-})$, $D^{-}\rho^{+}(D^{+}\rho^{-})$.
For case (i), there is an added advantage that the decay amplitudes $A_{f}$
and $A_{\bar{f}}$ are given by tree graphs. Assuming factorization for tree
amplitudes, it is shown that the form factors $f_{0}^{B-D}(m_{\pi}^{2})$,
$A_{0}^{B-D^{\ast}}(m_{\pi}^{2})$, $f_{+}^{B-D}(m_{\rho}^{2})$ obtained from
the experimental branching ratios are in excellent agreement with Heavy Quark
Effective Theory (HQET). Hence factorization assumption is experimentally on
sound footing for these decays.
Case (ii): The weak amplitudes $A_{f}\neq A_{\bar{f}}$,
$\displaystyle A_{f}$ $\displaystyle=\langle
f\left|\mathcal{L_{W}}\right|B^{0}\rangle=\left[e^{i\phi_{1}}F_{1f}+e^{i\phi_{2}}F_{2f}\right]$
$\displaystyle A_{\bar{f}}$
$\displaystyle=\langle\bar{f}\left|\mathcal{L_{W}}\right|B^{0}\rangle=\left[e^{i\phi_{1}}F_{1\bar{f}}+e^{i\phi_{2}}F_{2\bar{f}}\right]$
as is the case for the following decays,
$\displaystyle B^{0}\rightarrow\rho^{-}\pi^{+}(f):A_{f}$ $\displaystyle,\quad
B^{0}\rightarrow\rho^{+}\pi^{-}(\bar{f}):A_{\bar{f}}$ $\displaystyle
B_{s}^{0}\rightarrow K^{\ast-}K^{+}$ $\displaystyle,\quad B_{s}^{0}\rightarrow
K^{\ast+}K^{-}$ $\displaystyle B^{0}\rightarrow D^{\ast-}D^{+}$
$\displaystyle,\quad B^{0}\rightarrow D^{\ast+}D^{-}$ $\displaystyle
B_{s}^{0}\rightarrow D_{s}^{\ast-}D_{s}^{+}$ $\displaystyle,\quad
B_{s}^{0}\rightarrow D_{s}^{\ast+}D_{s}^{-}$
The $C-$ invariance of S-matrix gives $S_{\bar{f}}=S_{f}$ which implies
$\delta_{f}=\delta_{\bar{f}}^{\prime},\qquad\delta_{1f}=\delta_{1\bar{f}},\qquad\delta_{2f}=\delta_{2\bar{f}}$
## 2 Unitarity and Final State Strong Phases
The time reversal invariance gives
$F_{f}=_{out}\langle f|\mathcal{L}_{W}|B\rangle=_{in}\langle
f|\mathcal{L}_{W}|B\rangle^{\ast}$ (4)
where $\mathcal{L}_{W}$ is the weak interaction Lagrangian without the CKM
factor such as $V_{ud}^{\ast}V_{ub}$. From Eq. $\eqref{03}$, we have
$\displaystyle F_{f}^{\ast}=$ ${}_{out}\langle
f|S^{\dagger}\mathcal{L}_{W}|B\rangle$ $\displaystyle=$
$\displaystyle\sum_{n}S_{nf}^{\ast}F_{n}$ (5)
It is understood that the unitarity equation which follows from time reversal
invariance holds for each amplitude with the same weak phase. Above equation
can be written in two equivalent forms:
1. 1.
Exclusive version of Unitarity [1, 2]
Writing
$S_{nf}=\delta_{nf}+iM_{nf}$ (6)
we get from Eq (5) ,
$\text{Im}F_{f}=\frac{1}{2}\sum_{n}M_{nf}^{\ast}F_{n}$ (7)
where $M_{nf}$ is the scattering amplitude for $f\rightarrow n$ and $F_{n}$ is
the decay amplitude for $B\rightarrow n$. In this version, the sum is over all
allowed exclusive channels. This version is more suitable in a situation where
a single exclusive channel is dominant one. To get the final result, one uses
the dispersion relation. In dispersion relation two particle unitarity gives
dominant contribution. From Eq.(7), using two particle unitarity, we get [1],
$Disc\text{ }F(B\rightarrow f^{\prime})\approx\frac{1}{16\pi
s}\int_{-\infty}^{0}M_{f^{\prime}f}^{\ast}F(B\rightarrow f)dt$ (8)
where $t=-2\vec{p}^{2}(1-\cos\theta)$,
$\left|\vec{p}\right|\approx\frac{1}{2}\sqrt{s}.$ Eq.$\left(\ref{6a}\right)$
is especially suitable to calculate rescattering corrections to color
suppressed $T$-amplitude in terms of color favored $T$-amplitude as for
example rescattering correction to color suppressed decay
$B^{0}\rightarrow\pi^{0}\bar{D}^{0}(f)$ in terms of dominant decay mode
$B^{0}\rightarrow\pi^{+}D^{-}(f)$. Before using two particle unitarity in this
form, it is essential to consider two particle scattering processes.
$SU(3)$ or $SU(2)$ and $C$-invariance of $S$-matrix can be used to express
scattering amplitudes in terms of two amplitudes $M^{+}$ and $M^{-}$ which in
terms of Regge trajectories are given by [3, 4, 5]
$\displaystyle M^{(+)}$ $\displaystyle=$ $\displaystyle
P+f+A_{2}=-C_{P}\frac{e^{-i\pi\alpha_{p}(t)/2}}{\sin\pi\alpha_{p}(t)/2}\left(s/s_{0}\right)^{\alpha(t)}$
(9)
$\displaystyle-2C_{\rho}\frac{1+e^{-i\pi\alpha(t)}}{\sin\pi\alpha(t)}\left(s/s_{0}\right)^{\alpha(t)}$
$\displaystyle M^{(-)}$ $\displaystyle=$
$\displaystyle\rho+\omega=2C_{\rho}\frac{1-e^{-i\pi\alpha(t)}}{\sin\pi\alpha(t)}\left(s/s_{0}\right)^{\alpha(t)}$
(10)
For linear Regge trajectories, using exchange degeneracy, we have
$\displaystyle\alpha_{\rho}(t)$ $\displaystyle=$
$\displaystyle\alpha_{A_{2}}(t)=\alpha_{\omega}(t)=\alpha_{f}(t)=\alpha\left(0\right)+\alpha^{\prime}t,$
$\displaystyle\alpha_{p}(t)$ $\displaystyle=$
$\displaystyle\alpha_{p}(0)+\alpha_{p}^{\prime}(t),$ $\displaystyle C_{f}$
$\displaystyle=$ $\displaystyle C_{\omega};\text{ }C_{A_{2}}=C_{\rho};\text{
}C_{\omega}=C_{\rho}$ (11)
We take $\alpha_{0}\approx 1/2$, $\alpha^{\prime}\approx 1$GeV${}^{-2},$
$\alpha_{p}(0)\approx 1,\alpha_{p}^{\prime}\approx 0.25$GeV-2. Using $SU(3)$
and taking $\gamma_{\rho D^{+}D^{-}}=\gamma_{\rho K^{+}K^{-}},$ we get
$C_{\rho}=\gamma_{\rho\pi^{+}\pi^{-}}\gamma_{\rho
K^{+}K^{-}}=\gamma_{\rho\pi^{+}\pi^{-}}\gamma_{\rho
D^{+}D^{-}}=\frac{1}{2}\gamma_{0}^{2},$
$\gamma_{0}=\gamma_{\rho\pi^{+}\pi^{-}};\gamma_{0}^{2}\approx
72\cite[cite]{[\@@bibref{}{3}{}{}]}.$ Hence for $\pi^{+}D^{-}$ or
$\pi^{-}K^{+}$ scattering we get
$\displaystyle M$ $\displaystyle=$ $\displaystyle
M^{(+)}+M^{(-)}=iC_{P}e^{bt}(s/s_{0})$ (12)
$\displaystyle+2\gamma_{0}^{2}ie^{\alpha^{\prime}(\ln(s/s_{0})-i\pi)t}(s/s_{0})^{1/2}$
where $b=\alpha_{P}^{\prime}\ln(s/s_{0})$
For $\pi^{0}\bar{D}^{0}\rightarrow\pi^{+}D^{-}$,
$\pi^{0}K^{0}\rightarrow\pi^{-}K^{+}$
$M=\pm\sqrt{2}M^{(-)}=\pm
i2\sqrt{2}C_{\rho}\frac{e^{-i\pi\alpha(t)/2}}{\cos\alpha(t)/2}(s/s_{0})^{\alpha(t)}$
(13)
From Eq.(8) and (13) with the use of dispersion relation, we obtain
$\displaystyle A(B^{0}$ $\displaystyle\rightarrow$
$\displaystyle\pi^{0}\bar{D}^{0})_{FSI}=\frac{\sqrt{2}\gamma_{0}^{2}(1-i)}{16\pi}\frac{A(B^{0}\rightarrow\pi^{+}D^{-})}{\ln\left(\frac{m_{B}^{2}}{s_{0}}\right)+i\pi/2}\frac{1}{\pi}\int_{(m_{B}+m_{D})^{2}}^{\infty}\frac{ds}{s-m_{B}^{2}}(s/s_{0})^{\alpha(t)}$
(14) $\displaystyle=$ $\displaystyle-\sqrt{2}\epsilon
A(B^{0}\rightarrow\pi^{+}D^{-})e^{i\theta}$
We get $\epsilon\approx 0.06,\theta\approx 33^{\circ}$ by putting $s\approx
m_{B}^{2}$ in $\ln(s/s_{0})$. Now $A(B^{0}\rightarrow\pi^{+}D^{-})=T.$ Hence
with rescattering correction [6]
$\displaystyle A(B^{0}$ $\displaystyle\rightarrow$
$\displaystyle\pi^{0}\bar{D}^{0})=-\frac{1}{\sqrt{2}}C-\sqrt{2}\epsilon
Te^{i\theta}$ (15) $\displaystyle=$
$\displaystyle-\frac{C}{\sqrt{2}}\left[1+\frac{\epsilon}{b}e^{i\theta}\right]$
where $2b=C/T.$ Hence the final state phase shift $\delta_{C}$ for the color
suppressed amplitude induced by the final state interaction is given by
$\tan\delta_{C}=\frac{\epsilon/b\sin\theta}{1+\epsilon/b\cos\theta}\rightarrow\delta_{C}\approx
8^{\circ}$ (16)
with $b\approx 0.174,$ which we get from
$\frac{\Gamma(B^{0}\rightarrow\pi^{+}D^{-})}{\Gamma(B^{+}\rightarrow\pi^{+}\bar{D}^{0})}=\frac{1}{(1+2b)^{2}}\approx
0.55\pm 0.03$ (17)
For $B^{0}\rightarrow\pi^{0}K^{0},$ the color suppressed $T$-amplitude with
rescattering correction is given by
$-\frac{1}{\sqrt{2}}C+\sqrt{2}\epsilon
Te^{i\theta}=-\frac{1}{\sqrt{2}}C\left[1-\frac{\epsilon}{b}e^{i\theta}\right]$
(18)
where $2b=C/T\approx 0.37$ [7]. Hence $\delta_{C}$ generated by the final
state interaction is given by
$\tan\delta_{C}=\frac{-\epsilon/b\sin\theta}{1-\epsilon/b\cos\theta}\rightarrow\delta_{C}\approx-8^{\circ}$
(19)
To conclude: The scattering amplitude $M\left(s,t\right)$ for the two particle
final state obtained in eq.$\left(13\right)$ is used in the unitarity equation
to generate the final state strong phase by rescattering for the color
suppressed tree amplitude.
2. 2.
Inclusive version of Unitarity [2]
This version is more suitable for our analysis. For this case, we write Eq.
(5) in the form
$F_{f}^{\ast}-S_{ff}^{\ast}F_{f}=\sum_{n\neq f}S_{nf}^{\ast}F_{n}$ (20)
Parametrizing S-matrix as $S_{ff}\equiv S=\eta e^{2i\Delta}$[5],
$0\leq\eta\leq 1,$ we get after taking the absolute square of both sides of
Eq.(20)
$\left|F\right|^{2}\left[(1+\eta^{2})-2\eta\cos
2(\delta_{f}-\Delta)\right]=\sum_{n,n^{\prime}}F_{n}S_{nf}^{\ast}F_{n^{\prime}}^{\ast}S_{n^{\prime}f}$
(21)
The above equation is an exact equation. In the random phase approximation
[2], we can put
$\displaystyle\sum_{n^{\prime},n\neq
f}F_{n}S_{nf}^{\ast}F_{n^{\prime}}S_{n^{\prime}f}=$ $\displaystyle\sum_{n\neq
f}|F_{n}|^{2}|S_{nf}|^{2}$ $\displaystyle=$
$\displaystyle\bar{|F_{n}|^{2}}(1-\eta^{2})$ (22)
We note that in a single channel description [5, 8]:
$(Flux)_{in}-(Flux)_{out}=1-|\eta
e^{2i\Delta}|^{2}=1-\eta^{2}=\text{Absorption}$
The absorption takes care of all the inelastic channels.
Similarly for the amplitude $F_{\bar{f}}$, we have
$F_{\bar{f}}^{\ast}-S^{\ast}_{\bar{f}\bar{f}}F_{\bar{f}}=\sum_{\bar{n}\neq\bar{f}}S^{\ast}_{\bar{n}\bar{f}}F_{\bar{n}}$
(23)
The C-invariance of S-matrix gives:
$\displaystyle S_{fn}=$ $\displaystyle\langle f|S|n\rangle=\langle
f|C^{-1}CSC^{-1}C|n\rangle$ $\displaystyle=$
$\displaystyle\langle\bar{f}|S|\bar{n}\rangle=S_{\bar{f}\bar{n}}$ (24)
Thus in particular C-invariance of S-matrix gives
$S_{\bar{f}\bar{f}}=S_{ff}=\eta e^{2i\Delta}$ (25)
Hence from Eq. $\left(\ref{08}\right)$, using Eqs. $(\ref{09}-\ref{12})$, we
get
$\frac{1}{1-\eta^{2}}[(1+\eta^{2})-2\eta\cos
2(\delta_{f,\bar{f}}-\Delta)]=\rho^{2},\bar{\rho}^{2}$ (26)
where
$\rho^{2}=\frac{\overline{\bigl{|}F_{n}\bigr{|}}^{2}}{\bigl{|}F_{f}\bigr{|}^{2}},\qquad\bar{\rho}^{2}=\frac{\overline{\bigl{|}F_{\bar{n}}\bigr{|}}^{2}}{\bigl{|}F_{\bar{f}}\bigr{|}^{2}}$
(27)
From Eq.(26), we get
$\sin(\delta_{f,\bar{f}}-\Delta)=\pm\sqrt{\frac{1-\eta^{2}}{4\eta}}\left[\rho^{2},\bar{\rho}-\frac{1-\eta}{1+\eta}\right]^{1/2}$
(28)
The maximum value for $\rho^{2},\bar{\rho}^{2}$ is 1 and the minimum value for
them is $\frac{1-\eta}{1+\eta}.$ Hence we get the following bounds:
$\displaystyle\frac{1-\eta}{1+\eta}$ $\displaystyle\leq$
$\displaystyle\rho^{2},\bar{\rho}^{2}\leq 1$ $\displaystyle 0$
$\displaystyle\leq$ $\displaystyle\delta_{f,\bar{f}}-\Delta\leq\theta$
$\displaystyle-\theta$ $\displaystyle\leq$ $\displaystyle\delta_{f}-\Delta\leq
0$ (29) $\displaystyle\theta$ $\displaystyle=$
$\displaystyle\sin^{-1}\sqrt{\frac{1-\eta}{2}}$ (30)
From now on, we will confine our self to positve square root in Eq,(28).
The strong interaction parameter $\Delta$ and $\eta$ in the above bounds can
be obtained from the scattering amplitude $M(s,t)$ given in Eq.(12) obtain
from Regge pole analysis. The $s-$wave scattering amplitude $f$ is given by
$f\approx\frac{1}{16\pi s}\int_{-s}^{0}M(s,t)$ (31)
For the scattering amplitude $M=M^{+}+M^{-}$ relevant for
$\pi^{+}D^{-},\pi^{-}K^{+}$ and $\pi^{+}\pi^{-}$, we obtain from Eq.(31) using
Eq.(12)
$\displaystyle f$ $\displaystyle=$ $\displaystyle
f_{P}+f_{\rho}=\frac{1}{16\pi
s}\frac{iC_{P}}{b}\left(\frac{s}{s_{0}}\right)+2\frac{\gamma_{0}^{2}}{16\pi}\frac{1}{\ln(s/s_{0})-i\pi}(s/s_{0})^{-1/2}$
(32) $\displaystyle=$ $\displaystyle\left[\begin{array}[]{c}\text{0.12}i\text{
+ (-0.08+0.08}i\text{)}\\\ \text{0.17}i\text{ +(-0.08+0.08}i\text{)}\\\
\text{0.16}i\text{+(-0.16}\pm\text{0.16}i\text{)}\end{array}\right]$ (36)
where we have used $s\approx m_{B}^{2}\approx(5.27)^{2}$ GeV2. For $C_{P}$ we
have used the values of reference [2] whereas for
$C_{\rho}=\gamma_{\rho\pi^{+}\pi^{-}}\gamma_{\rho
K^{+}K^{-}}=\gamma_{\rho\pi^{+}\pi^{-}}\gamma_{\rho
D^{+}D^{-}}=\frac{1}{2}\gamma_{0}^{2}$ and
$C_{\rho}=\gamma_{\rho\pi^{+}\pi^{-}}\gamma_{\rho\pi^{+}\pi^{-}}=\gamma_{0}^{2}\approx
72$ for $\pi D,\pi K$ and $\pi\pi$ respectively.
Using the relation $S=\eta e^{2i\Delta}=1+2if,$ where $f$ is given by
$Eq.(33)$, the phase shift $\Delta,$ the parameter $\eta$ and the phase angle
$\theta$ can be determined. One gets
$\pi^{+}D^{-}(\pi^{-}D^{+}):\Delta\approx-7^{\circ},\eta\approx
0.62,\theta\approx 26^{\circ}$ $\displaystyle\pi^{-}K^{+}\text{ or
}\pi^{0}K^{0}$ $\displaystyle:$
$\displaystyle\Delta\approx-9^{\circ},\eta\approx 0.52,\theta\approx
29^{\circ}$ $\displaystyle\pi^{+}\pi^{-}$ $\displaystyle:$
$\displaystyle\Delta\approx-21^{\circ},\eta\approx 0.48,\theta\approx
31^{\circ}$ (37)
Hence we get the following bounds
$\displaystyle\pi^{+}D^{-}(\pi^{-}D^{+})$ $\displaystyle:$ $\displaystyle
0\leq\delta_{f,\bar{f}}-\Delta\leq 26^{\circ}$
$\displaystyle\pi^{-}K^{+}\text{ or }\pi^{0}K^{0}$ $\displaystyle:$
$\displaystyle 0\leq\delta_{f}-\Delta\leq 29^{\circ}$ (38)
$\displaystyle\pi^{+}\pi^{-}$ $\displaystyle:$ $\displaystyle
0\leq\delta_{f}-\Delta\leq 31^{\circ}$
Further we note that for these decays, $b$-quark is converted into $c$ or$u$
quark : $b\rightarrow c(u)+\bar{u}+d(s)$. In particular for the tree graph,
the configuration is such that $\bar{u}$ and $d(s)$ essentially go together
into a color singlet state with the third quark $c(u)$ recoiling; there is a
significant probability that the system will hadronize as a two body final
state [9]. This physical picture has been put on the strong theoretical basis
[10, 11], where in these references the QCD factorization have been proved.
For the tree amplitude, factorization implies $\delta_{f}^{T}=0.$ We,
therefore take the point of view that effective final state phase shift is
given by $\delta_{f}-\Delta.$ We take the lower bound for the tree amplitude
so that final state effective phase shift $\delta_{f}^{T}=0.$ Thus for
$\pi^{+}D^{-}(\pi^{-}D^{+}),\delta_{f}^{T}=\delta_{f}^{\prime T}=0.$
The decay $B^{0}\rightarrow\pi^{-}K^{+}$ is described by two amplitudes [7]
$A(B^{0}\rightarrow\pi^{-}K^{+})=-\left[P+e^{i\gamma}T\right]=\left|P\right|\left[1-re^{i(\gamma+\delta_{+-})}\right]$
(39)
where
$P=-\left|P\right|e^{-i\delta_{P}},\text{
}T=\left|T\right|e^{i\delta_{T}}\text{, }\delta_{+-}=\delta_{P}\text{,
}r=\frac{\left|T\right|}{\left|P\right|}$
The decay $B^{0}\rightarrow\pi^{0}K^{0}$ is described by the two amplitudes
[7]
$A(B^{0}\rightarrow\pi^{0}K^{0})=-\frac{1}{\sqrt{2}}\left|P\right|\left[1+r_{0}e^{i\left(\gamma+\delta_{00}\right)}\right]$
(40)
where
$C=\left|C\right|e^{i\delta_{C}},\text{
}\delta_{00}=\delta_{C}+\delta_{P},\text{
}r_{0}=\frac{\left|C\right|}{\left|P\right|}$
For these decays, we use the lower bounds in Eq.(38) for the tree amplitude so
that the effective final state phase $\delta_{T}=0.$ The phase $\delta_{C}$ is
generated by rescattering correction and its value is -8${}^{\circ}.$ For the
direct $CP$ asymmetries, the relevant phases are $\delta_{+-}$ and
$\delta_{00}$. For the penguin amplitude, we assume that the effective final
state phase $\delta_{P}$ has the value near the upper bound. Thus we have
$\delta_{+-}\approx 29^{\circ},$ $\delta_{00}\approx 21^{\circ}.$
Now [7]
$\displaystyle A_{CP}(B^{0}$ $\displaystyle\rightarrow$
$\displaystyle\pi^{-}K^{+})=-\frac{2r\sin\gamma\sin\delta_{+-}}{R}$
$\displaystyle R$ $\displaystyle=$ $\displaystyle
1-2r\cos\gamma\cos\delta_{+-}+r_{+-}^{2}$ (41)
Neglecting the terms of order $r^{2}$, we have
$\tan\gamma\tan\delta_{+-}=\frac{-A_{CP}(B^{0}\rightarrow\pi^{-}K^{+})}{1-R}$
(42)
For $B^{0}\rightarrow\pi^{0}K^{0}$
$\displaystyle A_{CP}(B^{0}$ $\displaystyle\rightarrow$
$\displaystyle\pi^{0}K^{0})=(R_{0}-1)\tan\gamma\tan\delta_{00}$ (43)
$\displaystyle R_{0}$ $\displaystyle=$ $\displaystyle
1+2r_{0}\cos\gamma\cos\delta_{00}+r_{00}^{2}$
Now the experimental values of $A_{CP}$, $R$ and $R_{0}$ are [12]
$\displaystyle A_{CP}(B^{0}$ $\displaystyle\rightarrow$
$\displaystyle\pi^{-}K^{+})=-0.101\pm 0.015\text{ }(-0.097\pm 0.012)$
$\displaystyle A_{CP}(B^{0}$ $\displaystyle\rightarrow$
$\displaystyle\pi^{0}K^{0})=-0.14\pm 0.11\text{ }(-0.00\pm\pm 0.10)$
$\displaystyle R$ $\displaystyle=$ $\displaystyle 0.899\pm 0.048$
$\displaystyle R_{0}$ $\displaystyle=$ $\displaystyle 0.908\pm 0.068$
where the numerical values in the bracket are the latest experimental values
as given in ref [7]. With $\delta_{+-}\approx 29^{\circ},$ we get from
Eq.(42), $\gamma=(60\pm 3)^{\circ}.$ However for $\delta_{+-}\approx
20^{\circ}$ which one gets from Eq.(28) for $\rho^{2}=0.65,\gamma=(69\pm
3)^{\circ}.$We obtain the following values for
$A_{CP}(B^{0}\rightarrow\pi^{0}K^{0})$ from Eqs.(42) and (43)
$\displaystyle A_{CP}(B^{0}$ $\displaystyle\rightarrow$
$\displaystyle\pi^{0}K^{0})=\frac{(1-R_{0})\tan\delta_{00}}{\left(1-R\right)\tan\delta_{+-}}A_{CP}(B^{0}\rightarrow\pi^{-}K^{+})$
$\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{c}\begin{array}[]{c}-0.06\pm 0.01,\text{
\ \ }\delta_{+-}=29^{\circ}\\\ \delta_{00}=21^{\circ}\\\ -0.05\pm 0.01,\text{
\ \ }\delta_{+-}=20^{\circ}\end{array}\\\
\delta_{00}=12^{\circ}\end{array}\right\\}$
We conclude: The phase shift $\delta_{+-}\approx(20-29)^{\circ}$ for
$\pi^{-}K^{+}$ is compatible with experimental value of the direct
$CP-$asymmetry for $\pi^{-}K^{+}$ decay mode. For
$\pi^{+}\pi^{-},\delta_{+-}\sim 31^{\circ}$ is compatible with the value
($33\pm 7_{-10}^{+8}$)∘ obtained by the authors of ref.[7]. Finally we note
that the actual value of the effective phase shift ($\delta_{f}-\Delta)$
depends on one free parameter $\rho$, factorization implies $\delta_{f}^{T}=0$
i.e. $\delta_{f}-\Delta=0$ for the tree amplitude; for the penguin amplitude,
$\delta_{f}^{P}$ depends on $\rho.$ However, from the experimental values of
the direct $CP$-violation for $\pi^{-}K^{+},$ $\pi^{-}\pi^{+}$, it is near the
upper bound.
Finally we note that $\pi^{+}D^{-}(\pi^{-}D^{+}),\pi^{-}K^{+},\pi^{-}\pi^{+}$
decays are $s$-wave decay whereas
$B^{0}\rightarrow\rho^{+}\pi^{-}(\rho^{-}\pi^{+})$ decays are $p-$wave decays.
For $p-$wave, the decay amplitude
$\displaystyle f$ $\displaystyle=$ $\displaystyle\frac{1}{16\pi
s}\int_{-s}^{0}M(s,t)(1+\frac{2t}{s})dt$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi
s}iC_{P}\left[\frac{1}{b}+\frac{2}{b^{2}}\frac{1}{s}\right](s/s_{0})$
$\displaystyle+\frac{2\gamma_{0}^{2}}{16\pi}i\left[\frac{1}{\ln(s/s_{0})-i\pi}-\frac{2}{s}\frac{1}{\left[\ln(s/s_{0})-i\pi\right]^{2}}(s/s_{0})^{-1/2}\right]$
$\displaystyle\approx$ $\displaystyle\frac{1}{16\pi
s}iC_{P}\frac{1}{b}(s/s_{0})+\frac{2\gamma_{0}^{2}}{16\pi}i\frac{1}{\ln(s/s_{0})-i\pi}(s/s_{0})^{-1/2}+O\left(\frac{1}{s}\right)$
to be compared with Eq.(32). Now for the $B\rightarrow\rho\pi$ decay, only
longitudinal polarization of $\rho$ is effectively involved. Since the
longitudinal $\rho$-meson emulates a pseudoscalar meson and if we assume same
couplings as for pions, we conclude that the final state phase for $\rho\pi$
should be of the order $30^{\circ}$; in any case it should not be greater than
$30^{\circ}$. The upper bound $\delta_{f}\leq 30^{0}$ can be used to select
the several possible solutions in Table-2 [Section-4] obtained from the
analysis of weak decays
$B\rightarrow\rho^{+}\pi^{-}\left(\rho^{-}\pi^{+}\right)$.
## 3 CP Asymmetries and Strong Phases
In this section, we discuss the experimental tests to verify the equality
(implied by C-invariance of S-matrix) of phase shifts $\delta_{f}$ and
$\delta_{\bar{f}}$ for the weak decays of B mesons mentioned in section 1.
It is convenient to write the time-dependent decay rates in the form [13, 6]
$\displaystyle\left[\Gamma_{f}(t)-\bar{\Gamma}_{\bar{f}}(t)\right]+\left[\Gamma_{\bar{f}}-\bar{\Gamma}_{f}(t)\right]$
$\displaystyle=$ $\displaystyle e^{-\Gamma t}\left\\{\cos\Delta
mt\left[\left(\left|A_{f}\right|^{2}-\left|\bar{A}_{\bar{f}}\right|^{2}\right)+\left(\left|A_{\bar{f}}\right|^{2}-\left|\bar{A}_{f}\right|^{2}\right)\right]\right.$
$\displaystyle\left.+2\sin\Delta
mt\left[\text{Im}\left(e^{2i\phi_{M}}A_{f}^{\ast}\bar{A}_{f}\right)+\text{Im}\left(e^{2i\phi_{M}}A_{\bar{f}}^{\ast}\bar{A}_{\bar{f}}\right)\right]\right\\}$
$\displaystyle\left[\Gamma_{f}(t)+\bar{\Gamma}_{\bar{f}}(t)\right]-\left[\Gamma_{\bar{f}}(t)+\bar{\Gamma}_{f}(t)\right]$
$\displaystyle=$ $\displaystyle e^{-\Gamma t}\left\\{\cos\Delta
mt\left[\left(\left|A_{f}\right|^{2}+\left|\bar{A}_{\bar{f}}\right|^{2}\right)-\left(\left|A_{\bar{f}}\right|^{2}+\left|\bar{A}_{f}\right|^{2}\right)\right]\right.$
$\displaystyle\left.+2\sin\Delta
mt\left[\text{Im}\left(e^{2i\phi_{M}}A_{f}^{\ast}\bar{A}_{f}\right)-\text{Im}\left(e^{2i\phi_{M}}A_{\bar{f}}^{\ast}\bar{A}_{\bar{f}}\right)\right]\right\\}$
Case (i): Eqs. $\eqref{e1}$ and $\eqref{e2}$ give
$\displaystyle\mathcal{A}\left(t\right)$ $\displaystyle\equiv$
$\displaystyle\frac{[\Gamma_{f}(t)-\bar{\Gamma}_{\bar{f}}(t)]+[\Gamma_{\bar{f}}(t)-\bar{\Gamma}_{f}(t)]}{[\Gamma_{f}(t)+\bar{\Gamma}_{\bar{f}}(t)]+[\Gamma_{\bar{f}}(t)+\bar{\Gamma}_{f}]}$
(48) $\displaystyle=$
$\displaystyle\frac{2\bigl{|}F_{f}\bigr{|}\bigl{|}F_{\bar{f}}^{{}^{\prime}}\bigr{|}}{\bigl{|}F_{f}\bigr{|}^{2}+\bigl{|}F_{\bar{f}}^{{}^{\prime}}\bigr{|}^{2}}\sin\Delta
mt\sin\bigl{(}2\phi_{M}-\phi-\phi^{{}^{\prime}}\bigr{)}\cos\bigl{(}\delta_{f}-\delta_{\bar{f}}^{{}^{\prime}}\bigr{)}$
$\displaystyle\mathcal{F}\left(t\right)$ $\displaystyle\equiv$
$\displaystyle\frac{\left[\Gamma_{f}(t)+\bar{\Gamma}_{\bar{f}}\right]-\left[\Gamma_{\bar{f}}(t)+\bar{\Gamma}_{f}\right]}{\left[\Gamma_{f}(t)+\bar{\Gamma}_{\bar{f}}\right]+\left[\Gamma_{\bar{f}}(t)+\bar{\Gamma}_{{f}}\right]}$
(49) $\displaystyle=$
$\displaystyle\frac{\bigl{|}F_{f}\bigr{|}^{2}-\bigl{|}F_{\bar{f}}^{{}^{\prime}}\bigr{|}^{2}}{\bigl{|}F_{f}\bigr{|}^{2}+\bigl{|}F_{\bar{f}}^{{}^{\prime}}\bigr{|}^{2}}\cos\Delta
mt$ $\displaystyle-$
$\displaystyle\frac{2\bigl{|}F_{f}\bigr{|}\bigl{|}F_{\bar{f}}^{{}^{\prime}}\bigr{|}}{\bigl{|}F_{f}\bigr{|}^{2}+\bigl{|}F_{\bar{f}}^{{}^{\prime}}\bigr{|}^{2}}\sin\Delta
mt\cos\left(2\phi_{M}-\phi-\phi^{{}^{\prime}}\right)\sin\bigl{(}\delta_{f}-\delta_{\bar{f}}^{{}^{\prime}}\bigr{)}$
The effective Lagrangians $\mathcal{L}_{W}$ and
$\mathcal{L}_{W}^{{}^{\prime}}$ are given by $(q=d,s)$
$\displaystyle\mathcal{L}_{W}$
$\displaystyle=V_{cb}V_{uq}^{\ast}[\bar{q}\gamma^{\mu}(1-\gamma^{5})u][\bar{c}\gamma_{\mu}(1-\gamma_{5})b]$
$\displaystyle\mathcal{L}_{W}^{{}^{\prime}}$
$\displaystyle=V_{ub}V_{cq}^{\ast}[\bar{q}\gamma^{\mu}(1-\gamma^{5})c][\bar{u}\gamma_{\mu}(1-\gamma_{5})b]$
(50)
Hence for these decays
$\phi=0,\qquad\phi^{\prime}=\gamma$
and
$\phi_{M}=\begin{cases}-\beta,&\text{for $B^{0}$}\\\ -\beta_{s},&\text{for
$B_{s}^{0}$}\end{cases}$ (52) $\displaystyle A_{f}$ $\displaystyle=\langle
D^{-}\pi^{+}\left|\mathcal{L_{W}}\right|B^{0}\rangle=F_{f}$
$\displaystyle\overset{{}^{\prime}}{A}_{\bar{f}}$ $\displaystyle=\langle
D^{+}\pi^{-}\left|\mathcal{L_{W}}^{\prime}\right|B^{0}\rangle=e^{i\gamma}\overset{{}^{\prime}}{F}_{\bar{f}}$
$\displaystyle A_{f_{s}}$ $\displaystyle=\langle
K^{+}D_{s}^{-}\left|\mathcal{L_{W}}\right|B_{s}^{0}\rangle=F_{f_{s}}$
$\displaystyle\overset{{}^{\prime}}{A}_{\bar{f}_{s}}$ $\displaystyle=\langle
K^{-}D_{s}^{+}\left|\mathcal{L_{W}}^{\prime}\right|B_{s}^{0}\rangle=e^{i\gamma}\overset{{}^{\prime}}{F}_{\bar{f}_{s}}$
(53)
Thus, we get from Eqs. $\eqref{e6}-\eqref{cc3}$ for $B^{0}$ decays,
$\displaystyle\mathcal{A}\left(t\right)$
$\displaystyle=-\frac{2r_{D}}{1+r_{D}^{2}}\sin\Delta
m_{B}t\sin\left(2\beta+\gamma\right)\cos\left(\delta_{f}-\delta_{\bar{f}}^{{}^{\prime}}\right)$
$\displaystyle\mathcal{F}\left(t\right)$
$\displaystyle=\frac{1-r_{D}^{2}}{1+r_{D}^{2}}\cos\Delta
m_{B}t-\frac{2r_{D}}{1+r_{D}^{2}}\sin\Delta
m_{B}t\cos\left(2\beta+\gamma\right)\sin\left(\delta_{f}-\delta_{\bar{f}}^{{}^{\prime}}\right)$
(54)
$\mathcal{A}=\frac{-2r_{D}}{1+r_{D}^{2}}\sin(2\beta+\gamma)\frac{(\Delta
m_{B}/\Gamma)}{1+(\Delta
m_{B}/\Gamma)^{2}}\cos(\delta_{f}-\delta_{\bar{f}}^{{}^{\prime}})$ (55)
where
$r_{D}=\lambda^{2}R_{b}\frac{|F_{\bar{f}}^{{}^{\prime}}|}{|F_{f}|}$ (56)
For the decays,
$\displaystyle\bar{B}_{s}^{0}\left(B_{s}^{0}\right)$
$\displaystyle\rightarrow$ $\displaystyle
D_{s}^{+}K^{-}\left(D_{s}^{-}K^{+}\right)$
$\displaystyle\bar{B}_{s}^{0}\left(B_{s}^{0}\right)$
$\displaystyle\rightarrow$ $\displaystyle
D_{s}^{-}K^{+}\left(D_{s}^{+}K^{-}\right)$
we get,
$\displaystyle\mathcal{A}_{s}\left(t\right)$
$\displaystyle=-\frac{2r_{D_{s}}}{1+r_{D_{s}}^{2}}\sin\Delta
m_{B_{s}}t\sin\left(2\beta_{s}+\gamma\right)\cos\left(\delta_{f_{s}}-\delta_{\bar{f}_{s}}^{{}^{\prime}}\right)$
$\displaystyle\mathcal{F}_{s}(t)$
$\displaystyle=\frac{1-r_{D_{s}}^{2}}{1+r_{D_{s}}^{2}}\cos\Delta
m_{B_{s}}t-\frac{2r_{D_{s}}}{1+r_{D_{s}}^{2}}\sin\Delta
m_{B_{s}}t\cos\left(2\beta_{s}+\gamma\right)\sin\left(\delta_{f_{s}}-\delta_{\bar{f}_{s}}^{{}^{\prime}}\right)$
(57)
where
$r_{D_{s}}=R_{b}\frac{|F_{\bar{f}_{s}}^{{}^{\prime}}|}{|F_{f_{s}}|}$ (58)
We note that for time integrated $CP$-asymmetry,
$\displaystyle\mathcal{A}_{s}\equiv$
$\displaystyle\frac{\int_{0}^{\infty}\left[\Gamma_{fs}\left(t\right)-\bar{\Gamma}_{fs}\left(t\right)\right]dt}{\int_{0}^{\infty}\left[\Gamma_{fs}\left(t\right)+\bar{\Gamma}_{fs}\left(t\right)\right]dt}$
$\displaystyle=$
$\displaystyle-\frac{2r_{D_{s}}r}{1+r_{D_{s}}^{2}}\sin\left(2\beta_{s}+\gamma\right)\frac{\Delta
m_{B_{s}}/\Gamma_{s}}{1+\left(\Delta
m_{B_{s}}/\Gamma_{s}\right)^{2}}\cos(\delta_{f_{s}}-\delta_{\bar{f}_{s}}^{{}^{\prime}})$
(59)
The experimental results for the B decays are as follows [12]
$\begin{array}[]{cccc}&D^{-}\pi^{+}&D^{\ast-}\pi^{+}&D^{-}\rho^{+}\\\
\frac{S_{-}+S_{+}}{2}:&-0.046\pm 0.023&-0.037\pm 0.012&-0.024\pm 0.031\pm
0.009\\\ \frac{S_{-}-S_{+}}{2}:&-0.022\pm 0.021&-0.006\pm 0.016&-0.098\pm
0.055\pm 0.018\end{array}$ (60)
where
$\displaystyle\frac{S_{-}+S_{+}}{2}\equiv$
$\displaystyle-\frac{2r_{D}}{1+r^{2}_{D}}\sin(2\beta+\gamma)\cos(\delta_{f}-\delta^{{}^{\prime}}_{\bar{f}})$
$\displaystyle\frac{S_{-}-S_{+}}{2}\equiv$
$\displaystyle-\frac{2r_{D}}{1+r^{2}_{D}}\cos(2\beta+\gamma)\sin(\delta_{f}-\delta^{{}^{\prime}}_{\bar{f}})$
(61)
For $B_{s}^{0}\rightarrow
D_{s}^{\ast-}K^{+},D_{s}^{-}K^{+},D_{s}^{-}K^{\ast+}$, replace
$r_{D}\rightarrow r_{s}$, $\beta\rightarrow\beta_{s}$,
$\delta_{f}\rightarrow\delta_{f_{s}}$,
$\delta^{{}^{\prime}}_{\bar{f}}\rightarrow\delta^{{}^{\prime}}_{\bar{f}_{s}}$
in Eq. $\eqref{cc8}$.
Since for $B_{s}^{0}$, in the standard model, with three generations, gives
$\beta_{s}=0$, so we have for the CP-asymmetries $\sin\gamma$ or $\cos\gamma$
instead of $\sin(2\beta+\gamma)$, $\cos(2\beta+\gamma)$. Hence
$B_{s}^{0}$-decays are more suitable for testing the equality of phase shifts
$\delta_{f_{s}}$ and $\delta^{{}^{\prime}}_{\bar{f}_{s}}$ as for this case
neither $r_{s}$ nor $\cos\gamma$ is suppressed as compared to the
corresponding quantities for $B^{0}$. To conclude, for $B_{q}^{0}$ decays, the
equality of phases $\delta_{f}$ and $\delta^{{}^{\prime}}_{\bar{f}}$ for
$B_{d}^{0}$ gives
$\displaystyle-\frac{S_{-}+S_{+}}{2}$
$\displaystyle=2r_{D}\sin(2\beta+\gamma)$
$\displaystyle-\frac{S_{-}-S_{+}}{2}$ $\displaystyle=0$ (62)
whereas for $B_{s}^{0}$ decays, we get
$\displaystyle-\frac{S_{-}+S_{+}}{2}$
$\displaystyle=\frac{2r_{D_{s}}}{1+r_{D_{s}}^{2}}\sin(2\beta_{s}+\gamma)$
$\displaystyle-\frac{S_{-}-S_{+}}{2}$ $\displaystyle=0$ (63)
Corresponding to the decays $B_{s}^{0}\rightarrow
D_{s}^{-}K^{+},D_{s}^{+}K^{-}$ described by the tree diagrams, we have the
color suppressed decays $B^{0}\rightarrow\bar{D}^{0}K^{0},D^{0}K^{0}$. For
these decays,
$\displaystyle-\frac{S_{-}+S_{+}}{2}=$
$\displaystyle\frac{2r_{DK}}{1+r_{DK}^{2}}\sin(2\beta+\gamma)\cos(\delta_{\bar{D}^{0}K^{0}_{s}}-\delta^{{}^{\prime}}_{D^{0}\bar{K}^{0}_{s}})$
$\displaystyle-\frac{S_{-}-S_{+}}{2}=$
$\displaystyle\frac{2r_{DK}}{1+r_{DK}^{2}}\cos(2\beta+\gamma)\sin(\delta_{\bar{D}^{0}K^{0}_{s}}-\delta^{{}^{\prime}}_{D^{0}\bar{K}^{0}_{s}})$
$\displaystyle r_{DK}=$ $\displaystyle
R_{b}\frac{\bigl{|}C_{D^{0}K_{s}}^{{}^{\prime}}\bigr{|}}{\bigl{|}C_{\bar{D}^{0}K_{s}}\bigr{|}}$
and the corresponding expression for
$B_{s}^{0}\rightarrow\bar{D}^{0}\phi,D^{0}\phi$. For the color suppressed
decays $B^{0}\rightarrow\bar{D}^{0}\pi^{0},D^{0}\pi^{0}$, we get similar
expression as for $B^{0}\rightarrow D^{-}\pi^{+},D^{+}\pi^{-}$, with
$r_{D}\equiv
r_{D^{-}\pi^{-}},\delta_{D^{-}\pi^{+}},\delta^{{}^{\prime}}_{D^{-}\pi^{+}}\quad\text{replaced
by}\quad
r_{D^{0}\pi^{0}},\delta_{\bar{D}^{0}\pi^{0}},\delta^{{}^{\prime}}_{D^{0}\pi^{0}}$
To determine the parameter $r_{D}$ or $r_{D_{s}}$, we assume factorization for
the tree amplitude [7]. Factorization gives for the decays
$\bar{B}^{0}\rightarrow
D^{+}\pi^{-},D^{\ast+}\pi^{-},D^{+}\rho^{-},D^{+}a_{1}^{-}$:
$\displaystyle|\bar{F}_{\bar{f}}|=|\bar{T}_{\bar{f}}|$
$\displaystyle=G[f_{\pi}(m_{B}^{2}-m_{D}^{2})f_{0}^{B-D}(m_{\pi}^{2}),2f_{\pi}m_{B}|\vec{p}|A_{0}^{B-D^{\ast}}(m_{\pi}^{2}),$
$\displaystyle
2f_{\rho}m_{B}|\vec{p}|f_{+}^{B-D}(m_{\rho}^{2}),2f_{a_{1}}m_{B}|\vec{p}|f_{+}^{B-D}(a_{1}^{2})]$
(64) $\displaystyle|\bar{F}_{f}^{{}^{\prime}}|=|\bar{T}_{f}^{{}^{\prime}}|$
$\displaystyle=G^{{}^{\prime}}[f_{D}(m_{B}^{2}-m_{\pi}^{2})f_{0}^{B-\pi}(m_{D}^{2}),2f_{D^{\ast}}m_{B}|\vec{p}|f^{B-\pi}(m_{D^{\ast}}^{2}),$
$\displaystyle
2f_{D}m_{B}|\vec{p}|A_{0}^{B-\rho}(m_{D}^{2}),2f_{D}m_{B}|\vec{p}|A_{0}^{B-a_{1}}(m_{B}^{2})]$
(65) $\displaystyle G$
$\displaystyle=\frac{G_{F}}{\sqrt{2}}|V_{ud}||V_{cb}|a_{1},\quad
G^{{}^{\prime}}=\frac{G_{F}}{\sqrt{2}}|V_{cd}||V_{ub}|$ (66)
The decay widths for the above channels are given in the table 1
Decay | Decay Width $(10^{-9}$ MeV $\times|V_{cb}|^{2}$) | Form Factor | Form Factors $h(w^{(\ast)})$
---|---|---|---
$\bar{B}^{0}\rightarrow D^{+}\pi^{-}$ | $(2.281)|f_{0}^{B-D}(m_{\pi}^{2})|^{2}$ | $0.58\pm 0.05$ | $0.51\pm 0.03$
$\bar{B}^{0}\rightarrow D^{\ast+}\pi^{-}$ | $(2.129)|A_{0}^{B-D^{*}}(m_{\pi}^{2})|^{2}$ | $0.61\pm 0.04$ | $0.54\pm 0.03$
$\bar{B}^{0}\rightarrow D^{+}\rho^{-}$ | $(5.276)|f_{+}^{B-D}(m_{\rho}^{2})|^{2}$ | $0.65\pm 0.11$ | $0.57\pm 0.10$
$\bar{B}^{0}\rightarrow D^{+}a_{1}^{-}$ | $(5.414)|f_{+}^{B-D}(m_{a_{1}}^{2})|^{2}$ | $0.57\pm 0.31$ | $0.50\pm 0.27$
Table 1: Form Factors
where we have used
$a_{1}^{2}|V_{ud}|^{2}\approx 1,\quad f_{\pi}=131MeV,\quad
f_{\rho}=209MeV,\quad f_{a_{1}}=229MeV$
Using the experimental branching ratios and [12]
$|V_{cb}|=(38.3\pm 1.3)\times 10^{-3}$ (67)
we obtain the corresponding form factors given in Table 1.
In terms of variables [14, 15]:
$\omega=v\cdot v^{{}^{\prime}},\quad v^{2}=v^{{}^{\prime}2}=1,\quad
t=q^{2}=m_{B}^{2}+m_{D^{(\ast)}}^{2}-2m_{B}m_{D^{(\ast)}}\omega$ (68)
the form factors can be put in the following form
$\displaystyle f_{+}^{B-D}(t)$
$\displaystyle=\frac{m_{B}+m_{D}}{2\sqrt{m_{B}m_{D}}}h_{+}(\omega),\quad
f_{0}^{B-D}(t)=\frac{\sqrt{m_{B}m_{D}}}{m_{B}+m_{D}}(1+\omega)h_{0}(\omega)$
$\displaystyle A_{2}^{B-D^{\ast}}(t)$
$\displaystyle=\frac{m_{B}+m_{D^{\ast}}}{2\sqrt{m_{B}m_{D^{\ast}}}}(1+\omega)h_{A_{2}}(\omega),\quad
A_{0}^{B-D^{\ast}}(t)=\frac{m_{B}+m_{D^{\ast}}}{2\sqrt{m_{B}m_{D^{\ast}}}}h_{A_{0}}(\omega)$
$\displaystyle A_{1}^{B-D^{\ast}}(t)$
$\displaystyle=\frac{\sqrt{m_{B}m_{D^{\ast}}}}{m_{B}+m_{D^{\ast}}}(1+\omega)h_{A_{1}}(\omega)$
(69)
Heavy Quark Effective Theory (HQET) gives [14, 15]:
$h_{+}(\omega)=h_{0}(\omega)=h_{A_{0}}(\omega)=h_{A_{1}}(\omega)=h_{A_{2}}(\omega)=\zeta(\omega)$
where $\zeta(\omega)$ is the form factor, with normalization $\zeta(1)=1$. For
$\displaystyle t$ $\displaystyle=m_{\pi}^{2},m_{\rho}^{2},m_{a_{1}}^{2}$
$\displaystyle\omega^{(*)}$ $\displaystyle=1.589(1.504),1.559,1.508$ (70)
In reference [16], the value quoted for $h_{A_{1}}(\omega_{max}^{\ast})$ is
$|h_{A_{1}}(\omega_{max}^{\ast})|=0.52\pm 0.03$ (71)
Since $\omega_{max}^{*}=1.504$, the value for $|h_{A_{0}}(\omega_{\max}^{*})|$
obtained in Table 1 is in remarkable agreement with the value given in Eq.
$\eqref{c10}$ showing that factorization assumption for $B^{0}\rightarrow\pi
D^{(*)}$ decays is experimentally on solid footing and is in agreement with
HQET.
From Eqs. $\eqref{c1}$ and $\eqref{c2}$, we obtain
$\displaystyle r_{D}$
$\displaystyle=\lambda^{2}R_{b}\frac{|\bar{T}_{f}^{{}^{\prime}}|}{|\bar{T}_{\bar{f}}|}$
$\displaystyle=\lambda^{2}R_{b}\left[\frac{f_{D}(m_{B}^{2}-m_{\pi}^{2})f_{0}^{B-\pi}(m_{D}^{2})}{f_{\pi}(m_{B}^{2}-m_{D}^{2})f_{0}^{B-D}(m_{\pi}^{2})},\quad\frac{f_{D^{\ast}}f_{+}^{B-\pi}(m_{D^{\ast}}^{2})}{f_{\pi}A_{0}^{B-D}(m_{\pi}^{2})},\quad\frac{f_{D}A_{0}^{B-\rho}(m_{D}^{2})}{f_{\rho}f_{+}^{B-D}(m_{\rho^{2}})}\right]$
(72)
where
$\frac{|V_{ub}||V_{cd}|}{|V_{cb}||V_{ud}|}=\lambda^{2}R_{b}\approx(0.227)^{2}(0.40)\approx
0.021$ (73)
To determine $r_{D}$, we need information for the form factors
$f_{0}^{B-\pi}(m_{D}^{2}),f_{+}^{B-\pi}(m_{D}^{2}),A_{0}^{B-\rho}(m_{D}^{2})$.
For these form factors, we use the following values [17, 18]:
$\displaystyle A_{0}^{B-\rho}(0)$ $\displaystyle=0.30\pm
0.03,A_{0}^{B-\rho}(m_{D}^{2})=0.38\pm 0.04$ $\displaystyle f_{+}^{B-\pi}(0)$
$\displaystyle=f_{0}^{B-\pi}(0)=0.26\pm 0.04,\quad
f_{+}^{B-\pi}(m_{D^{\ast}}^{2})=0.32\pm 0.05,\quad
f_{0}^{B-D}(m_{D}^{2})=0.28\pm 0.04$
Along with the values of remaining form factors given in Table 1, we obtain
$r_{D^{(\ast)}}=[0.018\pm 0.002,\quad 0.017\pm 0.003,\quad 0.012\pm 0.002]$
(74)
The above value for $r_{D}^{\ast}$ gives
$-\left(\frac{S_{+}+S_{-}}{2}\right)_{D^{\ast}\pi}=2(0.017\pm
0.003)\sin(2\beta+\gamma)$ (75)
The experimental value of the CP asymmetry for $B^{0}\rightarrow D^{\ast}\pi$
decay has the least error. Hence we obtain the following bounds
$\displaystyle\sin(2\beta+\gamma)$ $\displaystyle>0.69$ (76) $\displaystyle
44^{\circ}$ $\displaystyle\leq(2\beta+\gamma)\leq 90^{\circ}$ (77)
$\displaystyle\text{or}\quad 90^{\circ}$ $\displaystyle\leq(2\beta+\gamma)\leq
136^{\circ}$ (78)
Selecting the second solution, and using $2\beta\approx 43^{\circ}$, we get
$\gamma=(70\pm 23)^{\circ}$ (79)
Further, we note that the factorization for the decay $\bar{B}^{0}\rightarrow
D_{s}^{\ast-}\pi^{+}$ gives
$\bar{T}=|V_{ub}||V_{cs}|f_{D_{s}^{\ast}}2m_{B}|\vec{p}|f_{+}^{B-\pi}(m_{D_{s}^{\ast}}^{2})$
(80)
Using the experimental branching ratio for this decay, we get
$\left(\frac{f_{D_{s}^{\ast}}}{f_{\pi}}\right)^{2}\left|\frac{f_{+}^{B-\pi}(m_{D_{s}^{\ast}}^{2})}{f_{+}^{B-\pi}(0)}\right|^{2}=7.7\pm
1.9$ (81)
On using
$\frac{f_{+}^{B-\pi}(0)}{f_{+}^{B-\pi}(m_{D_{s}^{\ast}}^{2})}=0.77\pm 0.09$
(82)
we get
$f_{D_{s}^{\ast}}=279\pm 79MeV$ (83)
Similar analysis for $\bar{B}^{0}\rightarrow D_{s}^{-}\pi^{+}$ gives
$\left(\frac{f_{D_{s}}}{f_{\pi}}\right)^{2}\left|\frac{f_{0}^{B-\pi}(m_{D_{s}}^{2})}{f_{0}^{B-\pi}(0)}\right|^{2}=2.72\pm
0.64$ (84)
On using
$\frac{f_{0}^{B-\pi}(0)}{f_{0}^{B-\pi}(m_{D_{s}^{2}})}=0.93\pm 0.05$ (85)
we get
$f_{D_{s}}=201\pm 47MeV$ (86)
Finally from the experimental branching ratio for the decay
$\bar{B}_{s}^{0}\rightarrow D_{s}^{+}\pi^{-}$, we obtain
$\displaystyle f_{0}^{B_{s}-D_{s}}(0)$ $\displaystyle=0.62\pm 0.18$ (87)
$\displaystyle h_{0}(1.531)$ $\displaystyle=0.55\pm 0.16$ (88)
To end this section, we discuss the decays $\bar{B}_{s}^{0}\rightarrow
D_{s}^{+}K^{-},D_{s}^{\ast+}K^{-}$ for which no experimental data are
available. However, using factorization, we get
$\displaystyle\Gamma(\bar{B}_{s}^{0}\rightarrow D_{s}^{+}K^{-})$
$\displaystyle=(1.75\times
10^{-10})|V_{cb}f_{0}^{B_{s}-D_{s}}(m_{K}^{2})|^{2}MeV$ (89)
$\displaystyle\Gamma(\bar{B}_{s}^{0}\rightarrow D_{s}^{\ast+}K^{-})$
$\displaystyle=(1.57\times
10^{-10})|V_{cb}A_{0}^{B_{s}-D_{s}^{\ast}}(m_{K}^{2})|^{2}MeV$ (90)
SU(3) gives
$\displaystyle|V_{cb}f_{0}^{B_{s}-D_{s}}(m_{K}^{2})|^{2}$
$\displaystyle\approx|V_{cb}||f_{0}^{B-D}(m_{\pi}^{2})|^{2}=(0.50\pm
0.04)\times 10^{-3}$
$\displaystyle|V_{cb}A_{0}^{B_{s}-D_{s}^{\ast}}(m_{K}^{2})|^{2}$
$\displaystyle\approx|V_{cb}||A_{0}^{B-D^{\ast}}(m_{\pi}^{2})|^{2}=(0.56\pm
0.04)\times 10^{-3}$ (91)
From the above equations, we get the following branching ratios
$\frac{\Gamma(\bar{B_{s}}^{0}\rightarrow
D_{s}^{(\ast)+}K^{-})}{\Gamma_{\bar{B}_{s}^{0}}}=(1.94\pm 0.07)\times
10^{-4}[(1.96\pm 0.07)\times 10^{-4}]$ (92)
For $\bar{B}_{s}^{0}\rightarrow D_{s}^{*+}K^{-}$
$r_{D_{s}}=R_{b}\left[\frac{f_{D_{s}^{*}}f_{+}^{B_{s}-K}(m_{D_{s}^{*}}^{2})}{f_{K}A_{0}^{B_{s}-D_{s}^{*}}(m_{K}^{2})}\right]$
(93)
Hence we get
$\displaystyle-(\frac{S_{+}+S_{-}}{2})_{D_{s}^{\ast}K}$
$\displaystyle=(0.41\pm 0.08)\sin(2\beta_{s}+\gamma)$ $\displaystyle=(0.41\pm
0.08)\sin\gamma$ (94)
where we have used
$\displaystyle R_{b}$
$\displaystyle=0.40,\quad\frac{f_{D_{s}}}{f_{K}}=\frac{f_{D_{s}^{\ast}}}{f_{K}}=1.75\pm
0.06,\quad f_{+}^{B_{s}-K}(m_{D_{s}^{\ast}}^{2})=0.34\pm 0.06$ $\displaystyle
A_{0}^{B_{s}-D_{s}^{\ast}}(m_{K}^{2})$
$\displaystyle=A_{0}^{B_{s}-D_{s}^{\ast}}(0)=\frac{m_{B_{s}}+m_{D_{s}^{\ast}}}{2\sqrt{m_{B_{s}m_{D_{s}^{\ast}}}}}\left[h_{0}(\omega_{s}^{\ast}=1.453)=0.52\pm.03\right]$
$\displaystyle=0.58\pm 0.03$ (95)
## 4 CP Asymmetries for $A_{f}\neq A_{\bar{f}}$
We now discuss the decays listed in case (ii) where $A_{f}\neq A_{\bar{f}}$.
Subtracting and adding Eqs. $(\ref{e2})$ and $(\ref{e1})$, we get,
$\displaystyle\frac{\Gamma_{f}(t)-\bar{\Gamma}_{f}(t)}{\Gamma_{f}(t)+\bar{\Gamma}_{f}(t)}=$
$\displaystyle C_{f}\cos\Delta mt+S_{f}\sin\Delta mt$ $\displaystyle=$
$\displaystyle(C-\Delta C)\cos\Delta mt+(S-\Delta S)\sin\Delta mt$ (96)
$\displaystyle\frac{\Gamma_{\bar{f}}(t)-\bar{\Gamma}_{\bar{f}}(t)}{\Gamma_{\bar{f}}(t)+\bar{\Gamma}_{\bar{f}}(t)}=$
$\displaystyle C_{\bar{f}}\cos\Delta mt+S_{\bar{f}}\sin\Delta mt$
$\displaystyle=$ $\displaystyle(C+\Delta C)\cos\Delta mt+(S+\Delta
S)\sin\Delta mt$ (97)
where
$\displaystyle C_{\bar{f},f}$ $\displaystyle=(C\pm\Delta C)$
$\displaystyle=\frac{\bigl{|}A_{\bar{f},f}\bigr{|}^{2}-\bigl{|}\bar{A}_{\bar{f},f}\bigr{|}^{2}}{\bigl{|}A_{\bar{f},f}\bigr{|}^{2}+\bigl{|}\bar{A}_{\bar{f},f}\bigr{|}^{2}}$
$\displaystyle=\frac{\Gamma_{\bar{f},f}-\bar{\Gamma}_{\bar{f},f}}{\Gamma_{\bar{f},f}+\bar{\Gamma}_{\bar{f},f}}$
$\displaystyle=\frac{R_{\bar{f},f}(1-A_{CP}^{\bar{f},f})-R_{\bar{f},f}(1+A_{CP}^{\bar{f},f})}{\Gamma(1\pm
A_{CP})}$ (98) $\displaystyle S_{\bar{f},f}$ $\displaystyle=(S\pm\Delta S)$
(99)
$\displaystyle=\frac{2\text{Im}[e^{2i\phi_{M}}A^{\ast}_{\bar{f},f}\bar{A}_{\bar{f},f}]}{\Gamma_{\bar{f},f}+\bar{\Gamma}_{\bar{f},f}}$
(100) $\displaystyle A_{CP}^{\bar{f}}$
$\displaystyle=\frac{\bar{\Gamma}_{f}-\Gamma_{\bar{f}}}{\Gamma_{\bar{f}}+\bar{\Gamma}_{f}}$
$\displaystyle A_{CP}^{f}$
$\displaystyle=\frac{\bar{\Gamma}_{\bar{f}}-\Gamma_{f}}{\Gamma_{f}+\bar{\Gamma}_{\bar{f}}}$
(101) $\displaystyle A_{CP}$
$\displaystyle=\frac{(\Gamma_{\bar{f}}+\bar{\Gamma}_{\bar{f}})-(\bar{\Gamma_{f}}+\Gamma_{f})}{(\Gamma_{\bar{f}}-\bar{\Gamma}_{\bar{f}})-(\bar{\Gamma_{f}}+\Gamma_{f})}$
(102)
$\displaystyle=\frac{R_{f}A^{f}_{CP}-R_{\bar{f}}A^{\bar{f}}_{CP}}{\Gamma}$
(103)
where
$\displaystyle R_{f}$
$\displaystyle=\frac{1}{2}(\Gamma_{f}+\bar{\Gamma}_{\bar{f}}),\qquad
R_{\bar{f}}=\frac{1}{2}(\Gamma_{\bar{f}}+\bar{\Gamma}_{f})$
$\displaystyle\Gamma$ $\displaystyle=R_{f}+R_{\bar{f}}$ (104)
The following relations are also useful which can be easily derived from above
equations
$\displaystyle\frac{R_{\bar{f},f}}{R_{f}+R_{\bar{f}}}$
$\displaystyle=\frac{1}{2}[(1\pm\Delta C)\pm A_{CP}C]$ (105)
$\displaystyle\frac{R_{\bar{f}}-R_{f}}{R_{f}+R_{\bar{f}}}$
$\displaystyle=[\Delta C+A_{CP}C]$ (106)
$\displaystyle\frac{R_{\bar{f}}A_{CP}^{\bar{f}}+R_{f}A_{CP}^{f}}{R_{f}+R_{\bar{f}}}$
$\displaystyle=[C+A_{CP}\Delta C]$ (107)
For these decays, the decay amplitudes can be written in terms of tree
amplitude $e^{i\phi_{T}}T_{f}$ and the penguin amplitude $e^{i\phi_{P}}P_{f}$:
$\displaystyle A_{f}$
$\displaystyle=e^{i\phi_{T}}e^{i\delta_{f}^{T}}\bigl{|}T_{f}\bigr{|}[1+r_{f}e^{i(\phi_{P}-\phi_{T})}e^{i\delta_{f}}]$
$\displaystyle A_{\bar{f}}$
$\displaystyle=e^{i\phi_{T}}e^{i\delta_{\bar{f}}^{T}}\bigl{|}T_{\bar{f}}\bigr{|}[1+r_{\bar{f}}e^{i(\phi_{P}-\phi_{T})}e^{i\delta_{\bar{f}}}]$
(108)
where
$r_{f,\bar{f}}=\frac{\bigl{|}P_{f,\bar{f}}\bigr{|}}{\bigl{|}T_{f,\bar{f}}\bigr{|}},\quad\delta_{f,\bar{f}}=\delta^{P}_{f,\bar{f}}-\delta^{T}_{f,\bar{f}}$.
$\displaystyle\bar{A}_{\bar{f}}$
$\displaystyle=e^{-i\phi_{T}}e^{i\delta_{f}^{T}}\bigl{|}T_{f}\bigr{|}[1+r_{f}e^{-i(\phi_{P}-\phi_{T})}e^{i\delta_{f}}]$
$\displaystyle\bar{A}_{f}$
$\displaystyle=e^{-i\phi_{T}}e^{i\delta_{\bar{f}}^{T}}\bigl{|}T_{\bar{f}}\bigr{|}[1+r_{\bar{f}}e^{-i(\phi_{P}-\phi_{T})}e^{i\delta_{\bar{f}}}]$
(109)
$\text{For}B^{0}\rightarrow\rho^{-}\pi^{+}:A_{f};\qquad
B^{0}\rightarrow\rho^{+}\pi^{-}:A_{\bar{f}};\quad\phi_{T}=\gamma,\phi_{P}=-\beta$
(110)
$\text{For}B^{0}\rightarrow D^{\ast-}D^{+}:A^{D}_{f};\qquad B^{0}\rightarrow
D^{\ast+}D^{-}:A^{D}_{\bar{f}};\quad\phi_{T}=0,\phi_{P}=-\beta$ (111)
Hence for $B^{0}\rightarrow\rho^{-}\pi^{+},B^{0}\rightarrow\rho^{+}\pi^{-}$,
we have
$\displaystyle A_{f}$
$\displaystyle=\bigl{|}T_{f}\bigr{|}e^{+i\gamma}e^{i\delta_{f}^{T}}[1-r_{f}e^{i(\alpha+\delta_{f})}]$
$\displaystyle A_{\bar{f}}$
$\displaystyle=\bigl{|}T_{\bar{f}}\bigr{|}e^{+i\gamma}e^{i\delta_{\bar{f}}^{T}}[1-r_{\bar{f}}e^{i(\alpha+\delta_{\bar{f}})}]$
(112) $\displaystyle\text{where}\qquad r_{f,\bar{f}}$
$\displaystyle=\frac{|V_{tb}||V_{td}|}{|V_{ub}||V_{ud}|}\frac{\bigl{|}P_{f,\bar{f}}\bigr{|}}{\bigl{|}T_{f,\bar{f}}\bigr{|}}=\frac{R_{t}}{R_{b}}\frac{\bigl{|}P_{f,\bar{f}}\bigr{|}}{\bigl{|}T_{f,\bar{f}}\bigr{|}}$
(113)
and for $\text{B}^{0}\rightarrow D^{*-}D^{+}$, $\text{B}^{0}\rightarrow
D^{*+}D^{-}$, we have
$\displaystyle A_{f}^{D}$
$\displaystyle=\bigl{|}T_{f}^{D}\bigr{|}e^{i\delta_{f}^{TD}}[1-r_{f}^{D}e^{i(-\beta+\delta_{f}^{D})}]$
$\displaystyle A_{\bar{f}}^{D}$
$\displaystyle=\bigl{|}T_{\bar{f}}^{D}\bigr{|}e^{i\delta_{\bar{f}}^{TD}}[1-r_{\bar{f}}^{D}e^{i(-\beta+\delta_{\bar{f}}^{D})}]$
(114) $\displaystyle\text{where}\qquad r_{f,\bar{f}}$
$\displaystyle=R_{t}\frac{\bigl{|}P_{f,\bar{f}}^{D}\bigr{|}}{\bigl{|}T_{f,\bar{f}}^{D}\bigr{|}}$
We now confine ourselves to
$B^{0}(\bar{B}^{0})\rightarrow\rho^{-}\pi^{+},\rho^{+}\pi^{-}(\rho^{+}\pi^{-},\rho^{-},\pi^{+})$
decays only [19, 20]. The experimental results for these decays are [12] as
$\displaystyle\Gamma$ $\displaystyle=R_{f}+R_{\bar{f}}=(22.8\pm 2.5)\times
10^{-6}$ (115) $\displaystyle A_{CP}^{f}$ $\displaystyle=-0.16\pm 0.23,\quad
A_{CP}^{\bar{f}}=0.08\pm 0.12$ (116) $\displaystyle C$ $\displaystyle=0.01\pm
0.14,\quad\Delta C=0.37\pm 0.08$ (117) $\displaystyle S$
$\displaystyle=0.01\pm 0.09,\quad\Delta S=-0.05\pm 0.10$ (118)
With the above values, it is hard to draw any reliable conclusion. Neglecting
the term $A_{CP}C$ in Eqs. $\eqref{ccc9}$ and $\eqref{ccc10}$, we get
$\displaystyle R_{\bar{f},f}$ $\displaystyle=\frac{1}{2}\Gamma(1\pm\Delta C)$
(119) $\displaystyle R_{\bar{f}}-R_{f}$ $\displaystyle=\Delta C$
Using the above value for $\Delta C$, we obtain
$\displaystyle R_{\bar{f}}$ $\displaystyle=(15.6\pm 1.7)\times 10^{-6}$
$\displaystyle R_{f}$ $\displaystyle=(7.2\pm 0.8)\times 10^{-6}$ (120)
We analyze these decays by assuming factorization for the tree graphs [10,
11]. This assumption gives
$\displaystyle T_{\bar{f}}$ $\displaystyle=\bar{T}_{f}\sim
2m_{B}f_{\rho}|\vec{p}|f_{+}(m_{\rho}^{2})$ (121) $\displaystyle T_{f}$
$\displaystyle=\bar{T}_{\bar{f}}\sim 2m_{B}f_{\pi}|\vec{p}|A_{0}(m_{\pi}^{2})$
(122)
Using $f_{+}(m_{\rho}^{2})\approx 0.26\pm 0.04$ and $A_{0}(m_{\pi}^{2})\approx
A_{0}(0)=0.29\pm 0.03$ and $|V_{ub}|=(3.5\pm 0.6)\times 10^{-3}$, we get the
following values for the tree amplitude contribution to the branching ratios
$\displaystyle\Gamma_{\bar{f}}^{\text{tree}}$ $\displaystyle=(15.6\pm
1.1)\times 10^{-6}\equiv|T_{\bar{f}}|^{2}$ (123)
$\displaystyle\Gamma_{f}^{\text{tree}}$ $\displaystyle=(7.6\pm 1.4)\times
10^{-6}\equiv|T_{f}|^{2}$ (124) $\displaystyle t$
$\displaystyle=\frac{T_{f}}{T_{\bar{f}}}=\frac{f_{\pi}A_{0}(m_{\pi}^{2})}{f_{\rho}f_{+}(m_{\rho}^{2})}=0.70\pm
0.12$ (125)
Now
$\displaystyle B_{\bar{f}}$
$\displaystyle=\frac{R_{\bar{f}}}{|T_{\bar{f}}|^{2}}=1-2r_{\bar{f}}\cos\alpha\cos\delta_{\bar{f}}+r_{\bar{f}}^{2}$
(126) $\displaystyle B_{f}$
$\displaystyle=\frac{R_{f}}{|T_{f}|^{2}}=1-2r_{f}\cos\alpha\cos\delta_{f}+r_{f}^{2}$
(127)
Hence from Eqs. $\eqref{ccc25}$ and $\eqref{ccc29}$, we get
$\displaystyle B_{\bar{f}}$ $\displaystyle=1.00\pm 0.12$ $\displaystyle B_{f}$
$\displaystyle=0.95\pm 0.11$ (128)
In order to take into account the contribution of penguin diagram, we
introduce the angles $\alpha_{eff}^{f,\bar{f}}$ [21], defined as follows
$\displaystyle e^{i\beta}A_{f,\bar{f}}$
$\displaystyle=|A_{f,\bar{f}}|e^{-i\alpha_{eff}^{f,\bar{f}}}$ $\displaystyle
e^{-i\beta}\bar{A}_{\bar{f},f}$
$\displaystyle=|\bar{A}_{\bar{f},f}|e^{i\alpha_{eff}^{f,\bar{f}}}$ (129)
With this definition, we separate out tree and penguin contributions:
$\displaystyle e^{i\beta}A_{f,\bar{f}}-e^{-i\beta}\bar{A}_{\bar{f},f}$
$\displaystyle=|A_{f,\bar{f}}|e^{-i\alpha^{f,\bar{f}}}-|\bar{A}_{\bar{f},f}|e^{i\alpha^{f,\bar{f}}}$
$\displaystyle=2iT_{f,\bar{f}}\sin\alpha$ (130) $\displaystyle
e^{i(\alpha+\beta)}A_{f,\bar{f}}-e^{-i(\alpha+\beta)}\bar{A}_{\bar{f},f}$
$\displaystyle=|A_{f,\bar{f}}|e^{-i(\alpha_{eff}^{f,\bar{f}}-\alpha)}-|\bar{A}_{\bar{f},f}|e^{i(\alpha_{eff}^{f,\bar{f}}-\alpha)}$
$\displaystyle=(2iT_{f,\bar{f}}\sin\alpha)r_{f,\bar{f}}e^{i\delta_{f,\bar{f}}}$
$\displaystyle=2iP_{f,\bar{f}}\sin\alpha$ (131)
From Eq. $\eqref{ccc35}$, we get
$\displaystyle 2\frac{|T_{f,\bar{f}}|^{2}}{R_{f,\bar{f}}}\sin^{2}\alpha$
$\displaystyle\equiv\frac{2\sin^{2}\alpha}{B_{f,\bar{f}}}=1-\sqrt{1-A_{CP}^{f,\bar{f}2}}\cos
2\alpha_{eff}^{f,\bar{f}}$ (132) $\displaystyle\sin 2\delta_{f,\bar{f}}^{T}$
$\displaystyle=-A_{CP}^{f,\bar{f}}\frac{\sin
2\alpha_{eff}^{f,\bar{f}}}{1-\sqrt{1-A_{CP}^{f,\bar{f}2}}\cos
2\alpha_{eff}^{f,\bar{f}}}$ (133) $\displaystyle\cos 2\delta_{f,\bar{f}}^{T}$
$\displaystyle=\frac{\sqrt{1-A_{CP}^{f,\bar{f}2}}-\cos
2\alpha_{eff}^{f,\bar{f}}}{1-\sqrt{1-A_{CP}^{f,\bar{f}2}}\cos
2\alpha_{eff}^{f,\bar{f}}}$ (134)
From Eqs. $\eqref{ccc35}$ and $\eqref{ccc36}$, we get
$\displaystyle r_{f,\bar{f}}^{2}$
$\displaystyle=\frac{1-\sqrt{1-A_{CP}^{f,\bar{f}2}}\cos(2\alpha_{eff}^{f,\bar{f}}-2\alpha)}{1-\sqrt{1-A_{CP}^{f,\bar{f}2}}\cos
2\alpha_{eff}^{f,\bar{f}}}$ (135) $\displaystyle
r_{f,\bar{f}}\cos\delta_{f,\bar{f}}$
$\displaystyle=\frac{\cos\alpha-\sqrt{1-A_{CP}^{f,\bar{f}2}}\cos(2\alpha_{eff}^{f,\bar{f}}-\alpha)}{1-\sqrt{1-A_{CP}^{f,\bar{f}2}}\cos
2\alpha_{eff}^{f,\bar{f}}}$ (136) $\displaystyle
r_{f,\bar{f}}\sin\delta_{f,\bar{f}}$
$\displaystyle=\frac{-\frac{A_{CP}^{f,\bar{f}}}{\sin\alpha}}{1-\sqrt{1-A_{CP}^{f,\bar{f}2}}\cos
2\alpha_{eff}^{f,\bar{f}}}$ (137)
Now factorization implies [22]
$\delta_{f}^{T}=0=\delta_{\bar{f}}^{T}$ (138)
Thus in the limit $\delta_{f}^{T}\rightarrow 0$, we get for Eq.
$\eqref{ccc38b}$
$\displaystyle\cos 2\alpha_{eff}^{f,\bar{f}}$
$\displaystyle=-1,\qquad\alpha_{eff}^{f,\bar{f}}=90^{\circ}$ (139)
$\displaystyle r_{f,\bar{f}}\cos\delta_{f,\bar{f}}$ $\displaystyle=\cos\alpha$
(140) $\displaystyle r_{f,\bar{f}}\sin\delta_{f,\bar{f}}$
$\displaystyle=\frac{-A_{CP}^{f,\bar{f}}/\sin\alpha}{1+\sqrt{1-A_{CP}^{f,\bar{f}2}}}$
(141) $\displaystyle r_{f,\bar{f}}^{2}$
$\displaystyle=\frac{1+\sqrt{1-A_{CP}^{f,\bar{f}2}}\cos
2\alpha}{1+\sqrt{1-A_{CP}^{f,\bar{f}2}}}$ (142)
$\displaystyle\approx\cos^{2}\alpha+\frac{1}{4}A_{CP}^{f,\bar{f}2}\sin^{2}\alpha$
(143)
The solution of Eq. $\eqref{ccc44}$ is graphically shown in Fig. 1 for
$\alpha$ in the range $80^{\circ}\leq\alpha<103^{\circ}$ for
$r_{f,\bar{f}}=0.10,015,0.20,0.25,0.30$. From the figure, the final state
phases $\delta_{f,\bar{f}}$ for various values of $r_{f,\bar{f}}$ can be read
for each value of $\alpha$ in the above range. Few examples are given in Table
2
$\alpha$ | $r_{f}$ | $\delta_{f}$ | $A_{CP}^{f}\approx-2r_{f}\sin\delta_{f}\sin\alpha$
---|---|---|---
$80^{\circ}$ | 0.20 | $29^{\circ}$ | -0.19
| 0.25 | $46^{\circ}$ | -0.36
$82^{\circ}$ | 0.15 | $22^{\circ}$ | -0.11
| 0.20 | $46^{\circ}$ | -0.28
$85^{\circ}$ | 0.10 | $29^{\circ}$ | -0.10
| 0.15 | $54^{\circ}$ | -0.24
$86^{\circ}$ | 0.10 | $46^{\circ}$ | -0.14
| 0.15 | $62^{\circ}$ | -0.26
$88^{\circ}$ | 0.10 | $70^{\circ}$ | -0.19
Table 2:
For $\alpha>90^{\circ}$, change $\alpha\rightarrow\pi-\alpha$,
$\delta_{f}\rightarrow\pi-\delta_{f}$. For example, for $\alpha=103^{\circ}$
$\displaystyle r_{f}$ $\displaystyle=0.25,\quad\delta_{f}=154^{\circ},\quad
A_{CP}^{f}\approx-0.22$ $\displaystyle r_{f}$
$\displaystyle=0.30,\quad\delta_{f}=138^{\circ},\quad A_{CP}^{f}\approx-0.40$
These examples have been selected keeping in view that final state phases
$\delta_{f,\bar{f}}$ are not too large. For $A^{f,\bar{f}}_{CP}$, we have used
Eq. $\eqref{ccc45}$ neglecting the second order term. An attractive option is
$A_{CP}^{f}=A_{CP}^{\bar{f}}$ for each value of $\alpha$; although
$A_{CP}^{f}\neq A_{CP}^{\bar{f}}$ is also a possibility.
$A^{f}_{CP}=A_{CP}^{\bar{f}}$ implies
$r_{f}=r_{\bar{f}},\delta_{f}=\delta_{\bar{f}}$.
Neglecting terms of order $r_{f,\bar{f}}^{2}$, we have
$\displaystyle
A_{CP}\approx\frac{2\sin\alpha(r_{\bar{f}}\sin\delta_{\bar{f}}-t^{2}r_{f}\sin\delta_{f})}{1+t^{2}}=-\frac{A_{CP}^{\bar{f}}-t^{2}A_{CP}^{f}}{1+t^{2}}$
(144) $\displaystyle
C\approx-\frac{2t^{2}}{(1+t)^{2}}(A_{CP}^{\bar{f}}+A_{CP}^{f})$ (145)
$\displaystyle\Delta
C\approx\frac{1-t^{2}}{1+t^{2}}-\frac{4t^{2}\cos\alpha}{(1+t^{2})^{2}}(r_{\bar{f}}\cos\delta_{\bar{f}}-r_{f}\cos\delta_{f})$
(146)
Now the second term in Eq. $\eqref{cccc3}$ vanishes and using the value of $t$
given in Eq. $\eqref{ccc30}$, we get
$\Delta C\approx 0.34\pm 0.06$ (147)
Assuming $A_{CP}^{\bar{f}}=A_{CP}^{f}$, we obtain
$\displaystyle A_{CP}$
$\displaystyle=-\frac{1-t^{2}}{1+t^{2}}A_{CP}^{\bar{f}}$
$\displaystyle=(0.34\pm 0.06)(-A_{CP}^{\bar{f}})$ (148) $\displaystyle C$
$\displaystyle\approx-\frac{4t^{2}}{(1+t^{2})^{2}}A_{CP}^{\bar{f}}\approx-(0.88\pm
0.14)A_{CP}^{\bar{f}}$ (149)
Finally the CP asymmetries in the limit $\delta_{f,\bar{f}}^{T}\rightarrow 0$
$\displaystyle S_{\bar{f}}=S+\Delta S$
$\displaystyle=\frac{2\text{Im}[e^{2i\phi_{M}}A_{\bar{f}^{\ast}}\bar{A}_{\bar{f}}]}{\Gamma(1+A_{CP})}$
$\displaystyle=\sqrt{1-C_{\bar{f}}^{2}}\sin(2\alpha_{eff}^{\bar{f}}+\delta)$
$\displaystyle=-\sqrt{1-C_{\bar{f}}^{2}}\cos\delta$ (150) $\displaystyle
S_{f}=S-\Delta S$
$\displaystyle=\frac{2\text{Im}[e^{2i\phi_{M}}A_{f}^{\ast}\bar{A}_{f}]}{\Gamma(1-A_{CP})}$
$\displaystyle=\sqrt{1-C_{f}^{2}}\sin(2\alpha_{eff}^{f}-\delta)$
$\displaystyle=\sqrt{1-C_{f}^{2}}\cos\delta$ (151)
The phase $\delta$ is defined as
$\bar{A}_{\bar{f}}=\frac{|\bar{A}_{\bar{f}|}}{|\bar{A}_{f}|}\bar{A}_{f}e^{i\delta}$
(152)
To conclude:
The final state strong phases essentially arise in terms of $S$-matrix, which
converts an “$in"$ state into an “$out"$ state. The isospin, $C$-invariance of
hadronic dynamics and the unitarity together with two particle scattering
amplitudes in terms of Regge trajectories are used to get information about
these phases. In particular two body unitarity is used to calculate the final
state phase $\delta_{C}$ generated by rescattering for the color suppressed
decays in terms of the color favored decays. In the inclusive version of
unitarity, the information obtained for $s$-wave scattering from Regge
trajectories is used to derive the bounds on the final state phases. In
particular, the value obtained for the final state phases
$\delta_{+-}=\delta^{P}$ $\approx 29^{\circ}-20^{\circ}$ and
$\delta_{00}=\delta^{C}+\delta^{P}\approx 20^{\circ},12^{\circ}$ is found to
be compatible with the experimental values for direct $CP$ asymmetries
$A_{CP}(B^{0}\rightarrow\pi^{-}K^{+},\pi^{0}K^{0})$. For $B^{0}\rightarrow
D^{(\ast)-}\pi^{+}(D^{(\ast)+}\pi^{-})$, $B_{s}^{0}\rightarrow
D_{s}^{(\ast)-}K^{+}(D_{s}^{(\ast)+}K^{-})$ decays described by two
independent single amplitudes $A_{f}$, $A_{\bar{f}}^{{}^{\prime}}$ and
$A_{f_{s}},$ $A_{\bar{f}_{s}}^{{}^{\prime}}$ with different weak phases viz.
$0$ and $\gamma$, equality of phases
$\delta_{f}=\delta_{\bar{f}}^{{}^{\prime}}$ implies, the time dependent CP
asymmetries
$\displaystyle-\left(\frac{S_{+}+S_{-}}{2}\right)$
$\displaystyle=\frac{2r_{D_{\left(s\right)}^{\left(\ast\right)}}}{1+r_{D_{\left(s\right)}^{\left(\ast\right)}}^{2}}\sin(2\beta_{\left(s\right)}+\gamma)$
(153) $\displaystyle\frac{S_{+}-S_{-}}{2}$ $\displaystyle=0$ (154)
An added advantage is that these decays are described by tree graphs. Assuming
factorization, the decay amplitude $A_{f}$ can be determined in term of the
form factors $f_{0}^{B-D}(m_{\pi}^{2})$ and $A_{0}^{B-D^{\ast}}(m_{\pi}^{2})$.
The parameter $r_{D^{\left(\ast\right)}}$ can be expressed in terms of the
ratios of the form factors
$f_{D}f_{0}^{B-\pi}(m_{D}^{2})$/$f_{\pi}f_{0}^{B-D}(m_{\pi}^{2})$ and
$f_{D^{\ast}}f_{+}^{B-\pi}(m_{D^{\ast}}^{2})$/$f_{\pi}A_{0}^{B-D^{\ast}}(m_{\pi}^{2})$.
From the experimental branching ratios, we have obtained the form factors
$f_{0}^{B-D}(m_{\pi}^{2})$ and $A_{0}^{B-D^{\ast}}(m_{\pi}^{2})$ which are in
excellent agreement with the prediction of HQET. We have also determined
$r_{D^{\ast}}$. For $r_{D^{\ast}}$ we get the value $r_{D^{\ast}}=0.017\pm
0.003$. Using this value we get the following bound from the experimental
value of $\frac{S_{+}+S_{-}}{2}$ for $B^{0}\rightarrow D^{\ast-}\pi^{+}$
decay:
$\sin(2\beta+\gamma)>0.69$
Using SU(3), for the form factors for $B_{s}^{0}\rightarrow
D_{s}^{\ast-}K^{+}(D_{s}^{\ast+}K^{-})$ decays, we predict
$\displaystyle-\left(\frac{S_{+}+S_{-}}{2}\right)$ $\displaystyle=(0.41\pm
0.08)\sin(2\beta+\gamma)$ $\displaystyle=(0.41\pm 0.08)\sin\gamma$
in the standard model.
In section-4, the decays
$B\rightarrow\rho^{+}\pi^{-}\left(\rho^{-}\pi^{+}\right)$ for which decay
amplitudes $A_{\bar{f}}$ and $A_{f}$ are given in terms of tree and penguin
diagrams are discussed. We have analyzed these decays assuming factorization
for the tree graph. Factorization implies
$\delta_{f}^{T}=\delta_{\bar{f}}^{T}$. In the limit
$\delta_{f,\bar{f}}^{T}\rightarrow 0$, we have shown that
$\displaystyle r_{f,\bar{f}}\cos\delta_{f,\bar{f}}$ $\displaystyle=\cos\alpha$
$\displaystyle r_{f,\bar{f}}^{2}$
$\displaystyle\approx\cos^{2}\alpha+A_{CP}^{f,\bar{f}2}\sin^{2}\alpha$
The first equation has been solved graphically, from which the final state
phases $\delta_{f,\bar{f}}$ corresponding to various values of $r_{f,\bar{f}}$
can be found for a particular value of $\alpha$. The upper bound
$\delta_{f,\bar{f}}\leq 30^{0}$ obtained in Section-2, using unitarity and
strong interaction dynamics based on Regge pole phenomonalogy can be used to
select the solutions given in Table-2. Neglecting the terms of order
$r_{f,\bar{f}}^{2}$, we get using factorization
$\Delta C=0.34\pm 0.06$
Finally, in the limit $\delta_{f,\bar{f}}^{T}\rightarrow 0$, we get
$\frac{S_{\bar{f}}}{S_{f}}=\frac{S+\Delta S}{S-\Delta
S}=-\frac{\sqrt{1-C_{\bar{f}}^{2}}}{\sqrt{1-C_{f}^{2}}}$
With the present experimental data, it is hard to draw any definite
conclusion.
Acknowledgement The author acknowledges a research grant provided by the
Higher Education Commission of Pakistan as a Distinguished National Professor.
## References
* [1] J. F. Donoghue et.al. Phys. Rev. Lett. 77, 2187 (1996).
* [2] M. Suzuki and L. Wofenstein, Phys. Rev. D 60, 074019 (1999).
* [3] A. Falk et.al. Phys.Rev.D 57, 4290 (1998) hep-ph/9712225.
* [4] I.Caprini, L.Micer and C.Bourrely, Phys.Rev.D 60,074016 (1999) hep-ph/9904214.
* [5] Fayyazuddin, JHEP 09, 055 (2002).
* [6] Fayyazuddin, Phys. Rev. D70, 114018 (2004).
* [7] M.Gronau and J.L. Rosner, hep-ph/0807.3080 v3.
* [8] L. Wolfenstein, hep-ph/0407344 v1 (2004); N. Spokvich, Nuovo Cinento, 26, 186 (1962); K. Gottfried and J. D. Jackson, Nuovo einento 34, 735 (1964); see also, Fayyazuddin and Riazuddin, Quantum Mechanics, Page 140, World Scientific (1990).
* [9] J. D. Bjorken, Topics in B-physics, Nucl. Physics 11 (proc.suppl.) 325 (1989).
* [10] M. Beneke, G. Buchalla, M. Neubart and C. T. Sachrajda, Phys. Rev. Lett, 83, 1914 (1999), Nucl. Phys. B591 313 (2000).
* [11] C. W. Bauer, D. Pirjol and I. W. Stewart, Phys. Rev. Lett. 87, 201806 (2001) hep-ph/0107002.
* [12] Particle Data Group, C. Amsler, et.al, Phys. Lett B667,1 (2008).
* [13] For a review, see for example CP-violation editor: C. Jarlskog, World scientific (1989); H. Quinn, B physics and CP-violation [hep-ph/0111174]. Fayyazuddin and Riazuddin, A Modern Introduction to Particle physics, 2nd edition, world scientific.
* [14] S. Balk, J. G. Korner, G. Thompson, F. Hussain Z. Phys. C 59, 283-293 (1993).
* [15] N. Isgur and M. B. Wise Phys. Lett B 232, 113 (1989) Phys. Lett. B 237, 527 (1990).
* [16] S. Faller et.al. hep-ph/0809.0222 v1.
* [17] P. Ball, R. Zweicky and W. I. Fine hep-ph/0412079 v1.
* [18] G. Duplancic et.al. hep-ph/0801.1796 v2.
* [19] V. Page and D. London, Phys. Rev. D 70, 017501 (2004).
* [20] M. Gronau and J. Zupan: hep-ph/0407002, 2004 Refernces to earlier literature can be found in this ref.
* [21] Y.Grossman and H.R.Quinn, Phys.Rev.D 58, 017504 (1998); J.Charles, Phys.Rev.D 59, 054007 (1999); M.Gronau et.al. Phys.Lett B 514, 315 (2001)
* [22] M. Beneke and M. Neuebert, Nucl. Phys. B675, 333 (2003).
Figure Caption:
Plot of equation $r_{f}\cos\delta_{\left(f\right)}=\cos\alpha$ for different
values of $r.$ For $80^{o}\leq\alpha\leq 103^{o}.$ Where solid curve, dashed
curve, dashed doted curve, dashed bouble doted and double dashed doted curve
are corresponding to $r=0.1,\ r=0.15,\ r=0.2,\ r=0.25$ and $r=0.3$
respectively.
|
arxiv-papers
| 2009-09-11T04:05:53 |
2024-09-04T02:49:05.266564
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Fayyazuddin",
"submitter": "Aqeel Ahmed",
"url": "https://arxiv.org/abs/0909.2085"
}
|
0909.2131
|
# Trilepton production at the CERN LHC: SUSY Signals and Standard Model
Backgrounds
Argonne National Laboratory
E-mail Argonne report number ANL-HEP-CP-09-92. Research supported by the U. S.
Department of Energy under Contract No. DE-AC02-06CH11357. We gratefully
acknowledge the use of JAZZ, a 350-node computer cluster operated by the
Mathematics and Computer Science Division at Argonne as part of the Laboratory
Computing Resource Center. Zack SULLIVAN
Illinois Institute of Technology
E-mail Zack.Sullivan@iit.edu
###### Abstract:
Events with isolated leptons and missing energy in the final state are known
to be signatures of new physics phenomena at high energy collider physics
facilities. Standard model (SM) sources of isolated trilepton final states
include gauge boson pair production such as $WZ$ and $W\gamma^{*}$, and
$t\bar{t}$ production. Symbol $\gamma^{*}$ represents a virtual photon. Our
new contribution is the demonstration that bottom and charm meson decays,
$b\rightarrow lX$ and $c\rightarrow lX$, produce isolated lepton events that
can overwhelm the effects of other processes. We compute contributions from a
wide range of SM heavy flavor processes including $bZ/{\gamma^{*}}$,
$cZ/\gamma^{*}$, $b\bar{b}Z/\gamma^{*}$, $c\bar{c}Z/\gamma^{*}$. We also
include contributions from processes in which a $W$ is produced in association
with one or more heavy flavors such as $tW$, $b\bar{b}W$, $c\bar{c}W$. In all
these cases, one or more of the final observed isolated leptons comes from a
heavy flavor decay. We propose new cuts to control the heavy flavor
backgrounds in the specific case of chargino plus neutralino pair production
in supersymmetric models.
###### pacs:
13.85.Qk, 13.85.Rm, 12.60.Jv
## 1 Introduction
Isolated leptons along with missing transverse energy $/\\!\\!\\!\\!E_{T}$ are
signatures for new physics processes at collider energies. A known example of
charged dilepton production is Higgs boson decay, $H\rightarrow W^{+}W^{-}$
followed by purely leptonic decay of the $W$ intermediate vector bosons.
Charged trilepton production may arise from the associated production of a
chargino $\tilde{\chi}_{1}^{\pm}$ and a neutralino $\tilde{\chi}^{0}_{2}$ in
supersymmetric models, followed by the leptonic decays of the chargino and
neutralino. There are many standard model (SM) sources of isolated leptons,
such as leptonic decays of $W$ and $Z$ bosons produced from standard model
processes. Semi-leptonic decays of heavy flavors (bottom and charm quarks)
also make a very important contribution to the rate of isolated lepton
production. The nature and magnitude of these contributions from heavy flavor
sources are emphasized in our two recent papers [1, 2] and summarized in this
brief report.
## 2 Isolated leptons from heavy flavor decays
Given a lepton track and a cone, in rapidity and azimuthal angle space, of
size $\Delta R$, the lepton is said to be isolated if the sum of the
transverse energy of all other particles within the cone is less than a
predetermined value (either a constant or a value that scales with the
transverse momentum of the lepton). Our simulations based on the known semi-
leptonic decays of bottom and charm mesons show that leptons which satisfy
isolation take a substantial fraction of the momentum of the parent heavy
meson. Moreover, isolation leaves $\sim 7.5\times 10^{-3}$ muons per parent
$b$ quark. The potential magnitude of the background from heavy flavor decays
may be appreciated from the fact that one begins with an inclusive $b\bar{b}$
cross section at LHC energies of about $5\times 10^{8}$ pb. A suppression of
$\sim 10^{-5}$ from isolation still leaves a formidable rate of isolated
dileptons. For the isolated leptons, our simulations show that roughly $1/2$
of the events satisfy isolation because the remnant is just outside whatever
cone is used for the tracking and energy cuts, and another $1/2$ pass because
the lepton took nearly all the energy, meaning there is nothing left to reject
upon. The latter events are not candidates to reject with impact parameter
cuts since they tend to point to the primary vertex. Although the decay
leptons are “relatively” soft, we find that their associated backgrounds
extend well into the region of new physics with relatively large mass scales,
such as a Higgs boson with mass $\sim 160$ GeV.
## 3 SM backgrounds in Higgs boson production and decay
Our analysis of the role of heavy flavor backgrounds in $H\rightarrow
W^{+}W^{-}\rightarrow l^{+}l^{-}+/\\!\\!\\!\\!E_{T}$ at Fermilab Tevatron and
CERN Large Hadron Collider (LHC) energies is presented in Ref. [1]. In
addition to continuum $W^{+}W^{-}$, $Z/\gamma^{*}$, and $t\bar{t}$, we
simulate the contributions from processes with $b$ and $c$ quarks in the final
state, including $b\bar{b}X$, $c\bar{c}X$, $Wc$, $Wb$, $Wb\bar{b}$, as well as
single top quark contributions. Symbol $\gamma^{*}$ represents a virtual
photon (a “Drell-Yan” pair of leptons). We use QCD hard matrix elements fed
through PYTHIA showering. The PYTHIA output is then put through a detector
simulation code. We learn that isolation cuts do not generally remove leptons
from heavy flavor sources as backgrounds to multi-lepton searches. A sequence
of complex physics cuts is needed, conditioned by the new physics one is
searching for. Moreover, the heavy flavor backgrounds cannot be easily
extrapolated from more general samples. The interplay between isolation and
various physics cuts tends to emphasize corners of phase space rather than the
bulk characteristics. Nevertheless, for Higgs boson searches in the mass range
$\sim 160$ GeV, we find that hardening the cut on the momentum of the next-to-
leading lepton serves to suppress heavy flavor backgrounds adequately at LHC
energies [1].
## 4 Trileptons at the LHC
The associated production of a chargino and neutralino, followed by their
leptonic decays, $\tilde{\chi}_{1}^{\pm}\tilde{\chi}^{0}_{2}\rightarrow
l^{+}l^{-}l^{\pm}+/\\!\\!\\!\\!E_{T}$ is a golden signature for supersymmetry.
The LHC collaborations ATLAS and CMS have devised strategies to observe this
signal, as reported in their respective Technical Design Reports (TDRs) [3,
4]. The SM backgrounds examined in detail include continuum $WZ$ and
$W\gamma^{*}$ production and leptonic decay, along with $t\bar{t}$, $tW$, and
$t\bar{b}$ production and decay. In Ref. [2], we repeat the CMS and ATLAS
simulations of the SUSY signals and SM backgrounds, but we include, in
addition, the contributions to the backgrounds from $bZ/\gamma^{*}$,
$b\bar{b}Z/\gamma^{*}$, $cZ/\gamma^{*}$, $c\bar{c}Z/\gamma^{*}$, $b\bar{b}W$,
and $c\bar{c}W$. To touch base with the CMS and ATLAS analyses, we examine the
SUSY trilepton signal and SM backgrounds for four SUSY points labeled LM1,
LM7, LM9, and SU2. Their parameter values may be found in Ref. [2]. These
points may be disfavored by other data, but we adopt them to make contact with
the ATLAS and CMS simulations.
We reproduce the analysis chains described in Refs. [3, 4]. Our hard-
scattering matrix elements are computed with MadEvent [5] at leading-order
(LO) in perturbation theory, so that we retain all spin and angular
correlations. We feed the LO results into PYTHIA in order to include the
effects of showering and hadronization. The LO treatment is perhaps adequate
in view of the rejection for physics reasons of events with hard jets, and
because we want to avoid double-counting of radiation included in PYTHIA. An
alternative approach would begin with next-to-leading (NLO) order matrix
elements and a showering code that deals properly with matching and double
counting aspects of the radiation. Not having this tool available, and
recognizing that any showering code will have its limitations until it has
been tested and tuned against LHC data, we proceed as described. Our MadEvent
results, fed through PYTHIA showering and then through a detector simulation,
reproduce the CMS and ATLAS full detector results to 10%. The important cuts
in the physics analysis are (a) a requirement of 3 isolated leptons with
transverse momenta $p_{T,\mu}>10$ GeV, $p_{T,e}>17$ GeV; (b) a requirement
that there be no jets with $E_{T}>30$ GeV, to reduce effects from $t\bar{t}$
production and from higher mass SUSY sources; and (c) a requirement that the
invariant mass of a pair of opposite-sign, same-flavor (OSSF) leptons
$M_{ll}^{OSSF}<75$ GeV to eliminate backgrounds from real $Z$ bosons. As is
detailed in Ref. [2] the contributions of $Z/\gamma^{*}$ plus heavy flavor
decays produce trileptons 10 times more often than the previously considered
$WZ/\gamma^{*}$ source in the region below the $Z$ peak. The SUSY signals are
overwhelmed.
The number of additional cuts available to reject the background from
$Z/\gamma^{*}+$heavy flavors is limited. In Ref. [1] we recommend raising the
minimum lepton $p_{T}$ threshold since the lepton $p_{T}$ spectrum from $b$
and $c$ decays tends to fall rapidly. In typical trilepton studies, however,
the leptons are soft, and an increase in the cut on the lepton $p_{T}$ tends
to reject too much of the signal. Missing transverse energy
$/\\!\\!\\!\\!E_{T}$ is somewhat discriminatory. The SUSY signals contain
invisible neutralinos which leave a broad range of $/\\!\\!\\!\\!E_{T}$ in the
detector. Trilepton signatures from $t\bar{t}$ production generally have two
neutrinos which lead to large missing energy. The contribution from
$Z/\gamma^{*}+$heavy flavor processes peaks at over 400 times the size of the
LM9 signal at low $/\\!\\!\\!\\!E_{T}$, but it falls rapidly to below the
signal by $/\\!\\!\\!\\!E_{T}>50$ GeV.
We find that the requirement $/\\!\\!\\!\\!E_{T}>30$ GeV removes a reasonable
fraction of the $Z/\gamma^{*}+$heavy flavor backgrounds for a modest loss of
signal. A cut below 20 GeV is not as useful and is likely not achievable at
the LHC. A cut above 40 GeV removes most of the $Z/\gamma^{*}+X$ backgrounds,
but it begins to significantly reduce the signal and is of little additional
help with $WZ/\gamma^{*}$ and $t\bar{t}$ backgrounds. The sharply falling
$/\\!\\!\\!\\!E_{T}$ spectrum in $Z/\gamma^{*}+X$ is sensitive to
uncertainties in the measurement of $/\\!\\!\\!\\!E_{T}$. This uncertainty
makes it difficult to predict absolute cross sections after cuts. On the other
hand, this sensitivity could provide an opportunity to measure the background
in situ and reduce concerns regarding modeling details. The background can be
fit in the data and the $/\\!\\!\\!\\!E_{T}$ cut adjusted to optimize the
purity of the sample.
Since the accuracy of $/\\!\\!\\!\\!E_{T}$ measurements is limited, we examine
also the utility of angular cuts. There are significant angular correlations
in the $Z/\gamma^{*}+$heavy flavor backgrounds that are different from those
in the SUSY trilepton signals or the $WZ/\gamma^{*}$ and $t\bar{t}$
backgrounds. We examine the angular distribution $\theta_{ij}^{\mathrm{CM}}$
between pairs of $p_{T}$-ordered leptons in the trilepton center-of-momentum
(CM) frame. The $Z/\gamma^{*}+$heavy flavor backgrounds have significant peaks
at both small and large angles. The signal and other backgrounds either peak
only at large angles, or are fairly central.
## 5 Summary
We find that the dominant backgrounds to low-momentum trilepton signatures
come from real $b$ and $c$ decays. For the CMS and ATLAS SUSY analyses we
examine, the $Z/\gamma^{*}+$heavy flavor decay backgrounds are a factor of
10–30 larger than $WZ/\gamma^{*}$ or $t\bar{t}$ to trileptons. Large
$/\\!\\!\\!\\!E_{T}$ cuts and angular correlations can be used to
significantly reduce the heavy flavor backgrounds, but we must be mindful of
the modest $/\\!\\!\\!\\!E_{T}$ in the SUSY signal. Along with our results for
dileptons in Ref. [1], we argue that leptons from heavy flavor decays should
be examined for all low-momentum lepton signals. Once normalizations are
measured with LHC data, we may have handles to reduce the effect of these
backgrounds to an acceptable level. The overall message is that precise
understanding of all SM physics processes will enable confident discovery
claims.
## References
* [1] Z. Sullivan and E. L. Berger, Missing heavy flavor backgrounds to Higgs boson production, Phys. Rev. D 74 (2006) 033008 [arXiv:hep-ph/0606271].
* [2] Z. Sullivan and E. L. Berger,Trilepton production at the CERN LHC: Standard model sources and beyond, Phys. Rev. D 78 (2008) 034030 [arXiv:0805.3720 [hep-ph]].
* [3] G. Aad et al. [The ATLAS Collaboration], Expected Performance of the ATLAS Experiment - Detector, Trigger and Physics, arXiv:0901.0512 [hep-ex].
* [4] G. L. Bayatian et al. [CMS Collaboration], CMS Technical Design Report, Vol II: Physics Performance, J. Phys. G 34, 995 (2007).
* [5] F. Maltoni and T. Stelzer, MadEvent: Automatic event generation with MadGraph, JHEP 0302 (2003) 027 [arXiv:hep-ph/0208156].
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arxiv-papers
| 2009-09-11T10:04:08 |
2024-09-04T02:49:05.275200
|
{
"license": "Public Domain",
"authors": "Edmond L Berger (Argonne) and Zack Sullivan (Illinois Institute of\n Technology)",
"submitter": "Edmond Berger",
"url": "https://arxiv.org/abs/0909.2131"
}
|
0909.2244
|
010001 2009 S. A. Cannas 010001
The adsorption of Ar on substrates of Li is investigated within the framework
of a density functional theory which includes an effective pair potential
recently proposed. This approach yields good results for the surface tension
of the liquid-vapor interface over the entire range of temperatures, $T$, from
the triple point, $T_{t}$, to the critical point, $T_{c}$. The behavior of the
adsorbate in the cases of a single planar wall and a slit geometry is analyzed
as a function of temperature. Asymmetric density profiles are found for fluid
confined in a slit built up of two identical planar walls leading to the
spontaneous symmetry breaking (SSB) effect. We found that the asymmetric
solutions occur even above the wetting temperature $T_{w}$ in a range of
average densities $\rho^{*}_{ssb1}\leq\rho^{*}_{av}\leq\rho^{*}_{ssb2}$, which
diminishes with increasing temperatures until its disappearance at the
critical prewetting point $T_{cpw}$. In this way a correlation between the
disappearance of the SSB effect and the end of prewetting lines observed in
the adsorption on a one-wall planar substrate is established. In addition, it
is shown that a value for $T_{cpw}$ can be precisely determined by analyzing
the asymmetry coefficients.
# Correlation between asymmetric profiles in slits and standard prewetting
lines
Salvador A. Sartarelli [inst1] Leszek Szybisz[inst2]-[inst4] E-mail:
asarta@ungs.edu.arE-mail: szybisz@tandar.cnea.gov.ar
(17 June 2009; 28 August 2009)
††volume: 1
99 inst1 Instituto de Desarrollo Humano, Universidad Nacional de General
Sarmiento, Gutierrez 1150, RA–1663 San Miguel, Argentina. inst2 Laboratorio
TANDAR, Departamento de Física, Comisión Nacional de Energía Atómica, Av. del
Libertador 8250, RA–1429 Buenos Aires, Argentina. inst3 Departamento de
Física, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires,
Ciudad Universitaria, RA–1428 Buenos Aires, Argentina. inst4 Consejo Nacional
de Investigaciones Científicas y Técnicas, Av. Rivadavia 1917, RA–1033 Buenos
Aires, Argentina.
## 1 Introduction
The study of physisorption of fluids on solid substrates had led to very
fascinating phenomena mainly determined by the relative strengths of fluid-
fluid ($f$-$f$) and substrate-fluid ($s$-$f$) attractions. In the present work
we shall refer to two of such features. One is the prewetting curve identified
in the study of fluids adsorbed on planar surfaces above the wetting
temperature $T_{w}$ (see, e.g., Pandit, Schick, and Wortis [1]) and the other
is the occurrence of asymmetric profiles of fluids confined in a slit of
identical walls found by van Leeuwen and collaborators in molecular dynamics
calculations [2, 3]. It is known that for a strong substrate (i.e., when the
$s$-$f$ attraction dominates over the $f$-$f$ one) the adsorbed film builds up
continuously showing a complete wetting.In such a case, neither prewetting
transitions nor spontaneous symmetry breaking (SSB) of the profiles are
observed, both these phenomena appear for substrates of moderate strength.
The prewetting has been widely analyzed for adsorption of quantum as well as
classical fluids. A summary of experimental data and theoretical calculations
for 4He may be found in Ref. [4]. Studies of other fluids are mentioned in
Ref. [5]. These investigations indicated that prewetting is present in real
systems such as 4He, H2, and inert gases adsorbed on alkali metals.
On the other hand, after a recent work of Berim and Ruckenstein [6] there is a
renewal of the interest in searching for the SSB effect in real systems. These
authors utilized a density functional (DF) theory to study the confinement of
Ar in a slit composed of two identical walls of CO2 and concluded that SSB
occurs in a certain domain of temperatures. In a revised analysis of this
case, reported in Ref. [7], we found that the conditions for the SSB were
fulfilled because the authors of Ref. [6] had diminished the $s$-$f$
attraction by locating an extra hard-wall repulsion. However, it was found
that inert gases adsorbed on alkali metals exhibit SSB. Results for Ne
confined by such substrates were recently reported [8].
The aim of the present investigation is to study the relation between the
range of temperatures where the SSB occurs and the temperature dependence of
the wetting properties. In this paper we illustrate our findings describing
the results for Ar adsorbed on Li. Previous DF calculations of Ancilotto and
Toigo [9] as well as Grand Canonical Monte Carlo (GCMC) simulations carried
out by Curtarolo et al. [10] suggest that Ar wets Li at a temperature
significantly below $T_{c}$. So, this system should exhibit a large locus of
the prewetting line and this feature makes it very convenient for our study as
it was already communicated during a recent workshop [11].
The paper is organized in the following way. The theoretical background is
summarized in Sec. 2. The results, together with their analysis, are given in
Sec. 3. Sec. 4 is devoted to the conclusions.
## 2 Theoretical background
In a DF theory, the Helmholtz free energy $F_{\rm DF}[\rho({\bf r})]$ of an
inhomogeneous fluid embedded in an external potential $U_{sf}({\bf r})$ is
expressed as a functional of the local density $\rho({\bf r})$ (see, e.g.,
Ref. [12])
$\displaystyle F_{\rm DF}[\rho({\bf r})]$ (1) $\displaystyle=$
$\displaystyle\nu_{\rm id}\,k_{B}\,T\int d{\bf r}\,\rho({\bf
r})\,\\{\ln[\Lambda^{3}\rho({\bf r})]-1\\}$ $\displaystyle+$
$\displaystyle\int d{\bf r}\,\rho({\bf r})\,f_{\rm HS}[\bar{\rho}({\bf
r});d_{\rm HS}]$ $\displaystyle+$ $\displaystyle\frac{1}{2}\int\int d{\bf
r}\,d{\bf r\prime}\,\rho({\bf r})\,\rho({\bf r\prime\prime})\,\Phi_{\rm
attr}(\mid{\bf r}-{\bf r\prime}\mid)$ $\displaystyle+$ $\displaystyle\int
d{\bf r}\,\rho({\bf r})\,U_{sf}({\bf r})\;.$
The first term is the ideal gas free energy, where $k_{B}$ is the Boltzmann
constant and $\Lambda=\sqrt{2\,\pi\,\hbar^{2}/m\,k_{B}\,T}$ the de Broglie
thermal wavelength of the molecule of mass $m$. Quantity $\nu_{\rm id}$ is a
parameter introduced in Eq. (2) of [13] (in the standard theory it is equal
unity). The second term accounts for the repulsive $f$-$f$ interaction
approximated by a hard-sphere (HS) functional with a certain choice for the HS
diameter $d_{\rm HS}$. In the present work we have used for $f_{\rm
HS}[\bar{\rho}({\bf r});d_{\rm HS}]$ the expression provided by the nonlocal
DF (NLDF) formalism developed by Kierlik and Rosinberg [14] (KR), where
$\bar{\rho}({\bf r})$ is a properly averaged density. The third term is the
attractive $f$-$f$ interactions treated in a mean field approximation (MFA).
Finally, the last integral represents the effect of the external potential
$U_{sf}({\bf r})$ exerted on the fluid.
In the present work, for the analysis of physisorption we adopted the ab
initio potential of Chismeshya, Cole, and Zaremba (CCZ) [15] with the
parameters listed in Table 1 therein.
### 2.1 Effective pair attraction
The attractive part of the $f$-$f$ interaction was described by an effective
pair interaction devised in Ref. [5], where the separation of the Lennard-
Jones (LJ) potential introduced by Weeks, Chandler and Andersen (WCA) [16] is
adopted
$\displaystyle\Phi^{\rm WCA}_{\rm attr}(r)$ (4) $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}-\tilde{\varepsilon}_{ff}\;,&r\leq
r_{m}\\\
4\tilde{\varepsilon}_{ff}\biggr{[}\left(\frac{\tilde{\sigma}_{ff}}{r}\right)^{12}-\left(\frac{\tilde{\sigma}_{ff}}{r}\right)^{6}\biggr{]}\;,&r>r_{m}\;.\end{array}\right.$
Here $r_{m}=2^{1/6}\tilde{\sigma}_{ff}$ is the position of the LJ minimum. No
cutoff for the pair potential was introduced. The well depth
$\tilde{\varepsilon}_{ff}$ and the interaction size $\tilde{\sigma}_{ff}$ are
considered as free parameters because the use of the bare values
$\varepsilon_{ff}/k_{B}=119.76$ K and $\sigma_{ff}=3.405$ Å overestimates
$T_{c}$.
So, the complete DF formalism has three adjustable parameters (namely,
$\nu_{id}$, $\tilde{\varepsilon}_{ff}$, and $\tilde{\sigma}_{ff}$), which were
determined by imposing that at $l$-$v$ coexistence, the pressure as well as
the chemical potential of the bulk $l$ and $v$ phases should be equal [i.e.,
$P(\rho_{l})=P(\rho_{v})$ and $\mu(\rho_{l})=\mu(\rho_{v})$]. The procedure is
described in Ref. [5]. In practice, we set $d_{\rm HS}=\tilde{\sigma}_{ff}$
and imposed the coexistence data of $\rho_{l}$, $\rho_{v}$, and
$P(\rho_{l})=P(\rho_{v})=P_{0}$ for Ar quoted in Table X of Ref. [17] to be
reproduced in the entire range of temperatures $T$ between $T_{t}=83.78$ K and
$T_{c}=150.86$ K.
### 2.2 Euler-Lagrange equation
The equilibrium density profile $\rho({\bf r})$ of the adsorbed fluid is
determined by a minimization of the free energy with respect to density
variations with the constraint of a fixed number of particles $N$
$\frac{\delta}{\delta\rho({\bf r})}\biggr{[}F_{\rm DF}[\rho({\bf r})]-\mu\int
d{\bf r}\,\rho({\bf r})\biggr{]}=0\;.$ (6)
Here the Lagrange multiplier $\mu$ is the chemical potential of the system. In
the case of a planar symmetry where the flat walls exhibit an infinite extent
in the $x$ and $y$ directions, the profile depends only on the coordinate $z$
perpendicular to the substrate. For this geometry, the variation of Eq. (6)
yields the following Euler-Lagrange (E-L) equation
$\displaystyle\frac{\delta[(F_{\rm id}+F_{\rm HS})/A]}{\delta\rho(z)}$
$\displaystyle+$
$\displaystyle\int^{L}_{0}dz\prime\rho(z\prime)\bar{\Phi}_{\rm attr}(\mid
z-z\prime\mid)$ (7) $\displaystyle+$ $\displaystyle U_{sf}(z)=\mu\;,$
where
$\frac{\delta(F_{\rm id}/A)}{\delta\rho(z)}=\nu_{\rm
id}\,k_{B}\,T\,\ln{[\Lambda^{3}\,\rho(z)]}\;,$ (8)
and
$\displaystyle\frac{\delta(F_{\rm HS}/A)}{\delta\rho(z)}=f_{\rm
HS}[\bar{\rho}(z);d_{\rm HS}]$ (9) $\displaystyle+$
$\displaystyle\int^{L}_{0}dz\prime\,\rho(z\prime)\,\frac{\delta f_{\rm
HS}[\bar{\rho}(z\prime);d_{\rm
HS}]}{\delta\bar{\rho}(z\prime)}\,\frac{\delta\bar{\rho}(z\prime)}{\delta\rho(z)}\;.$
Here $F_{\rm id}/A$ and $F_{\rm HS}/A$ are free energies per unit of one wall
area $A$. $L$ is the size of the box adopted for solving the E-L equations.
The boundary conditions for the one-wall and slit systems are different and
will be given below. The final E-L equation may cast into the form
$\displaystyle\nu_{\rm id}\,k_{B}\,T\,\ln{[\Lambda^{3}\,\rho(z)]}+Q(z)=\mu\;,$
(10)
where
$\displaystyle Q(z)$ $\displaystyle=$ $\displaystyle f_{\rm
HS}[\bar{\rho}(z);d_{\rm HS}]$ (11) $\displaystyle+$
$\displaystyle\int^{L}_{0}dz\prime\,\rho(z\prime)\,\frac{\delta f_{\rm
HS}[\bar{\rho}(z\prime);d_{\rm
HS}]}{\delta\bar{\rho}(z\prime)}\,\frac{\delta\bar{\rho}(z\prime)}{\delta\rho(z)}$
$\displaystyle+$
$\displaystyle\int^{L}_{0}dz\prime\,\rho(z\prime)\,\bar{\Phi}_{\rm attr}(\mid
z-z\prime\mid)$ $\displaystyle+$ $\displaystyle U_{sf}(z)\;.$
The number of particles $N_{s}$ per unit area, $A$, of the wall is
$N_{s}=\frac{N}{A}=\int^{L}_{0}\rho(z)\,dz\;.$ (12)
In order to get solutions for $\rho(z)$, it is useful to rewrite Eq. (10) as
$\rho(z)=\rho_{0}\,\exp{\left(-\frac{Q(z)}{\nu_{\rm id}\,k_{B}\,T}\right)}\;,$
(13)
with
$\rho_{0}=\frac{1}{\Lambda^{3}}\exp{\left(\frac{\mu}{\nu_{\rm
id}\,k_{B}\,T}\right)}\;.$ (14)
The relation between $\mu$ and $N_{s}$ is obtained by substituting Eq. (13)
into the constraint of Eq. (12)
$\displaystyle\mu$ $\displaystyle=$ $\displaystyle-\nu_{\rm id}\,k_{B}\,T$
(15) $\displaystyle\times$
$\displaystyle\ln{\biggr{[}\frac{1}{N_{s}\Lambda^{3}}\int^{L}_{0}dz\exp{\left(-\frac{Q(z)}{\nu_{\rm
id}k_{B}T}\right)\biggr{]}}}\;.$
When solving this kind of systems, it is usual to define dimensionless
variables $z^{*}=z/{\tilde{\sigma}}_{ff}$ for the distance and
$\rho^{*}=\rho\,{\tilde{\sigma}}^{3}_{ff}$ for the densities. In these units
the box size becomes $L^{*}=L/{\tilde{\sigma}}_{ff}$.
## 3 Results and Analysis
In order to quantitatively study the adsorption of fluids within any
theoretical approach,one must require the experimental surface tension of the
bulk liquid-vapor interface, $\gamma_{lv}$, to be reproduced satisfactorily
over the entire $T_{t}\leq T\leq T_{c}$ temperature range. Therefore, we shall
first examine the prediction for this observable before studying the
adsorption phenomena.
### 3.1 Surface tension of the bulk liquid-vapor interface
Figure 1 shows the experimental data of $\gamma_{lv}$ taken from Table II of
Ref. [18]. In order to theoretically evaluate this quantity the E-L equations
for free slabs of Ar, i.e. setting
$U_{sf}(z)=0\;,$ (16)
were solved imposing periodic boundary conditions $\rho(z=0)=\rho(z=L)$. At a
given temperature $T$, for a sufficiently large system one must obtain a wide
central region with $\rho(z\simeq L/2)=\rho_{l}(T)$ and tails with density
$\rho_{v}(T)$, where the values of $\rho_{l}(T)$ and $\rho_{v}(T)$ should be
those of the liquid-vapor coexistence curve. The surface tension of the
liquid-vapor interface is calculated according to the thermodynamic definition
$\gamma_{lv}=(\Omega+P_{0}\,V)/A=\Omega/A+P_{0}\,L\;,$ (17)
where $\Omega=F_{\rm DF}-\mu\,N$ is the grand potential of the system and
$P_{0}$ the pressure at liquid-vapor coexistence previously introduced. We
solved a box with $L^{*}=40$. The obtained results are plotted in Fig. 1
together with the prediction of the fluctuation theory of critical phenomena
$\gamma_{lv}=\gamma^{0}_{lv}(1-T/T_{c})^{1.26}$ with $\gamma^{0}_{lv}=17.4$
K/Å2 (see, e.g., [19]). One may realize that our values are in satisfactory
agreement with experimental data and the renormalization theory over the
entire range of temperatures $T_{t}\leq T\leq T_{c}$, showing a small
deviation near $T_{t}$.
Figure 1: Surface tension of Ar as a function of temperature. Squares are
experimental data taken from Table II of Ref. [18]. The solid curve
corresponds to the fluctuation theory of critical phenomena and the circles
are present DF results. Figure 2: Adsorption isotherms for the Ar/Li system,
i.e., $\Delta\mu$ as a function of coverage $\Gamma_{\ell}$. Up-triangles
correspond to $T=119$ K; circles to $T=118$ K; diamonds to $T=117$ K; squares
to $T=116$ K; down-triangles to $T=114$ K and stars to $T=112$ K.
### 3.2 Adsorption on one planar wall
It is assumed that the physisorption of Ar on a one wall substrate of Li is
driven by the CCZ potential, i.e.,
$U_{sf}(z)=U_{\rm CCZ}(z)\;.$ (18)
The E-L equations were solved in a box of size $L^{*}=40$ by imposing
$\rho(z>L)=\rho(z=L)$. The solution gives a density profile $\rho(z)$ and the
corresponding chemical potential $\mu$. Adsorption isotherms at a given
temperature were calculated as function of the excess surface density. This
quantity, also termed coverage, is often expressed in nominal layers $\ell$
$\Gamma_{\ell}=(1/\rho^{2/3}_{l})\int_{0}^{\infty}dz[\rho(z)-\rho_{B}]\;,$
(19)
where $\rho_{B}=\rho(z\to\infty)$ is the asymptotic bulk density and
$\rho_{l}$ the liquid density at saturation for a given temperature. By
utilizing the results for $\mu$ obtained from the E-L equation and the value
$\mu_{0}$ corresponding to saturation at a given temperature $T$, the
difference $\Delta\mu=\mu-\mu_{0}$ was evaluated. Figure 2 shows the
adsorption isotherms for temperatures above $T_{w}$, where an equal area
Maxwell construction is feasible. This is just the prewetting region
characterized by a jump in coverage $\Gamma_{\ell}$. The size of this jump
depends on temperate. The largest jump occurs at $T_{w}$ and diminishes for
increasing $T$ until its disappearance at $T_{cpw}$. Density profiles just
below and above the coverage jump for $T=114$ K are displayed in Fig. 3, in
that case $\Gamma_{\ell}$ jumps from 0.5 to 3.6. Therefore, the formation of
the fourth layer may be observed in the plot.
Figure 3: Examples of density profiles of Ar adsorbed on a surface of Li at
$T=114$ K displayed as a function of the distance from the wall located at
$z^{*}=0$. Dashed curves are profiles for $\Gamma_{\ell}$ below the coverage
jump, while solid curves are stable films above this jump. Figure 4:
Prewetting line for Ar adsorbed on Li. The solid curve is the fit to Eq. (20)
and reaches the $\Delta\mu_{pw}/k_{B}=0$ line at $T_{w}=110.1$ K.
The wetting temperature $T_{w}$ can be obtained from the analysis of the
values of $\Delta\mu/k_{B}$ at which the jump in coverage occurs at each
considered temperature. The behavior $\Delta\mu_{pw}/k_{B}\,\rm vs\,T$ is
displayed in Fig. 4. A useful form for determining the temperature $T_{w}$ was
derived from thermodynamic arguments [20]
$\displaystyle\Delta\mu_{pw}(T)$ $\displaystyle=$
$\displaystyle\mu_{pw}(T)-\mu_{0}(T)$ (20) $\displaystyle=$ $\displaystyle
a_{pw}\,(T-T_{w})^{3/2}\;.$
Here $a_{pw}$ is a model parameter and the exponent $3/2$ is fixed by the
power of the van der Walls tail of the adsorption potential $U_{sf}(z)\simeq-
C_{3}/z^{3}$. The fit of the data of $\Delta\mu/k_{B}$ to Eq. (20) yielded
$T_{w}=110.1$ K and $a_{pw}/k_{B}=-0.16$ K-1/2.
On the other hand, according to Fig. 2, the critical prewetting point
$T_{cpw}$ lies between $T=118$ and 119 K. At the latter temperature, the film
already presents a continuous growth.
Our values of $T_{w}$ and $T_{cpw}$ are smaller than those obtained from prior
DF calculations [9] ($T_{w}=123$ K and $T_{cpw}\simeq 130$ K) and GCMC
simulations [10] ($T_{w}=130$ K). The difference with the DF evaluation of
Ref. [9] is due to the use of different effective pair potentials as we
explain in Ref. [5], where the adsorption of Ne is studied. The present
approach gives a reasonable $\gamma_{lv}$, while that of Ref. [9] fails
dramatically close to $T_{t}$. The difference with the GCMC results cannot be
interpreted in a straightforward way.
### 3.3 Confinement in a planar slit
In the slit geometry, where the Ar atoms are confined by two identical walls
of Li the $s$-$f$ potential becomes
$U_{sf}(z)=U_{\rm CCZ}(z)+U_{\rm CCZ}(L-z)\;.$ (21)
The walls were located at a distance $L^{*}=40$, this width guarantees that
the pair interaction between two atoms located at different walls is
negligible. In fact, this width is wider than $L^{*}=29.1$, which was utilized
in the pioneering molecular dynamics calculations [2, 3]. Accordingly, the E-L
equations were solved in a box of size $L^{*}=40$. In this geometry, the
repulsion at the walls causes the profiles $\rho(z=0)$ and $\rho(z=L)$ to be
equal to zero. The solutions were obtained at a fixed dimensionless average
density defined in terms of $N$, $A$, and $L$ as
$\rho^{*}_{av}=N\,{\tilde{\sigma}}^{3}_{ff}/A\,L=N^{*}_{s}/L^{*}$.
Figure 5: Free energy per particle (in units of $k_{B}\,T$) for Ar confined
in a slit of Li with $L^{*}=40$ at $T=115$ K displayed as a function of the
average density. The curve labeled by circles corresponds to symmetric
solutions, while that labeled by triangles corresponds to asymmetric ones. The
SSB occurs in a certain range of average density
$\rho^{*}_{ssb1}\leq\rho^{*}_{av}\leq\rho^{*}_{ssb2}$.
For temperatures below $T_{w}=110.1$ K, we obtained large ranges of
$\rho^{*}_{av}$ where the asymmetric solutions exhibit a lower free energy
than the corresponding symmetric ones. In spite of the fact that there is a
general idea that a connection exists between the SSB effect and nonwetting,
we have found, by contrast, that SSB behavior extends above the wetting
temperature. Furthermore, we have also found a relation between prewetting and
SSB.
Figure 5 shows the free energy per particle, $f_{\rm DF}=F_{\rm DF}/N$, for
both symmetric and asymmetric solutions for the Ar/Li system at $T=115$
K$\,>T_{w}$ as a function of the average density. According to this picture,
the ground state (g-s) exhibits asymmetric profiles between a lower and an
upper limit $\rho^{*}_{ssb1}=0.057\leq\rho^{*}_{av}\leq\rho^{*}_{ssb2}=0.192$.
Out of this range no asymmetric solutions were obtained form the set of Eqs.
(10)-(15). Similar features were obtained for higher temperatures until
$T=118$ K, above this value the profiles corresponding to the g-s are always
symmetric. Figure 6 shows three examples of solutions determined at $T=115$ K.
The result labeled 1 is a small asymmetric profile, that labeled 2 is the
largest asymmetric solution at this temperature. So, by further increasing
$\rho^{*}_{av}$, the SSB effect disappears and the g-s becomes symmetric, as
indicated by the curve labeled 3. When the asymmetric profiles occur, the
situation is denoted as partial (or one wall) wetting. The symmetric solutions
account for a complete (two wall) wetting. These different situations can be
interpreted in terms of the balance of $\gamma_{sl}$, $\gamma_{sv}$ and
$\gamma_{lv}$ surface tensions, carefully discussed in previous works [2, 3,
7]. Here we shall restrict ourselves to briefly outline the main features.
When the liquid is adsorbed symmetrically like in the case of profile 3 in
Fig. 6, there are two $s$-$l$ and two $l$-$v$ interfaces. Hence, the total
surface excess energy may be written as
$\gamma^{sym}_{tot}=2\,\gamma_{sl}+2\,\gamma_{lv}\;.$ (22)
On the other hand, for a asymmetric profile $\gamma^{asy}_{tot}$ becomes
$\gamma^{asy}_{tot}=\gamma_{sl}+\gamma_{lv}+\gamma_{sv}\;.$ (23)
The three quantities of the r.h.s. of this equation are related by Young’s law
(see, e.g., Eq. (2.1) in Ref. [21])
$\gamma_{sv}=\gamma_{sl}+\gamma_{lv}\,\cos{\theta}\>,$ (24)
where $\theta$ is the contact angle defined as the angle between the wall and
the interface between the liquid and the vapor (see Fig. 1 in Ref. [21]). By
using Young’s law, the Eq. (23) may be rewritten as
$\gamma^{asy}_{tot}=2\,\gamma_{sl}+\gamma_{lv}\,(1+\cos{\theta})\;,$ (25)
with $\cos{\theta}=(\gamma_{sv}-\gamma_{sl})/\gamma_{lv}<1$. If one changes
$\gamma_{sl}$ by increasing enough $N_{s}$ (as shown in Fig. 5), and/or $T$,
and/or the strength of $U_{sf}(z)$, eventually the equality
$\gamma_{sv}-\gamma_{sl}=\gamma_{lv}$ may be reached yielding
$\cos{\theta}=1$. Then, the system would undergo a transition to a symmetric
profile where both walls of the slit are wet.
Figure 6: Density profiles of Ar confined in a slit of Li with $L^{*}=40$ at
$T=115$ K. The displayed spectra denoted by 1, 2 and 3 correspond to average
densities $\rho^{*}_{av}=0.074,0.192$ and $0.218$, respectively.
It is important to remark that, indeed, there are two degenerate asymmetric
solutions. Besides that one shown in Fig. 6 where the profiles exhibit the
thicker film adsorbed on the left wall (left asymmetric solutions - LAS),
there is an asymmetric solution with exactly the same free energy but where
the thicker film is located near the right wall (right asymmetric solutions -
RAS).
The asymmetry of density profiles may be measured by the quantity
$\Delta_{N}=\frac{1}{N_{s}}\int^{L/2}_{0}dz\,[\rho(z)-\rho(L-z)]\;.$ (26)
According to this definition, if the profile is completely asymmetrical about
the middle of the slit, i.e. for: (i) $\rho(z<L/2)\neq 0$ and $\rho(z\geq
L/2)=0$; or (ii) $\rho(z<L/2)=0$ and $\rho(z\geq L/2)\neq 0$ this quantity
becomes $+1$ or $-1$, respectively, while for symmetric solutions it vanishes.
Figure 7: Asymmetry parameter for Ar confined by two Li walls separated by a
distance of $L^{*}=40$ as a function of average density. From outside to
inside the curves correspond to temperatures $T=112,114,115,116,117$ and 118
K. The asymmetric solutions occur for different ranges
$\rho^{*}_{ssb1}\leq\rho^{*}_{av}\leq\rho^{*}_{ssb2}$. Figure 8: Circles
stand for both branches of the asymmetry parameter for Ar confined in an
$L^{*}=40$ slit of Li walls for temperatures between $T_{w}$ and $T_{cpw}$.
The solid curve is the fit to Eq. (27) used to determine $T_{cpw}$.
We evaluated the asymmetry coefficients of solutions obtained for increasing
temperatures up to $T=118$ K. The results for LAS profiles at temperatures
larger that $T_{w}$ are displayed in Fig. 7 as a function of the average
density. One may observe how the range
$\rho^{*}_{ssb1}\leq\rho^{*}_{av}\leq\rho^{*}_{ssb2}$ diminishes under
increasing temperatures. The SSB effect persists at most for the critical
$\rho^{*}_{av}(crit)=(17/24)\,{\tilde{\sigma}}^{2}_{ff}\times 10^{-2}\simeq
0.074$ with ${\tilde{\sigma}}_{ff}$ expressed in Å.
We shall demonstrate that by analyzing the data of $\Delta_{N}$ for
$\rho^{*}_{av}(crit)$ it is possible to determine the critical prewetting
point. Figure 8 shows these values for both the LAS and RAS profiles,
calculated at different temperatures, suggesting a rather parabolic shape. So,
we propose a fit to the following quartic polynomial
$T=T_{cpw}+a_{2}\Delta^{2}_{N}+a_{4}\Delta^{4}_{N}\;.$ (27)
This procedure yielded $T_{cpw}=118.4$ K, $a_{2}=-14.14$ K, and $a_{4}=-16.63$
K. The obtained value of $T_{cpw}$ is in agreement with the limits established
when analyzing the adsorption isotherms of the one-wall systems displayed in
Fig. 2. These results indicate that the disappearance of the SSB effect
coincides with the end of the prewetting line.
## 4 Conclusions
We have performed a consistent study within the same DF approach of free slabs
of Ar, the adsorption of these atoms on a single planar wall of Li and its
confinement in slits of this alkali metal. Good results were obtained for the
surface tension of the liquid-vapor interface. The analysis of the
physisorption on a planar surface indicates that Ar wets surfaces of Li in
agreement with previous investigations. The isotherms for the adsorption on
one planar wall exhibit a locus of prewetting in the $\mu-T$ plane. A fit of
such data yielded a wetting temperature $T_{w}=110.1$ K. In addition, these
isotherms also show that the critical prewetting point $T_{cpw}$ lies between
$T=118$ and 119 K. These results for $T_{w}$ and $T_{cpw}$ are slightly below
the values obtained in Refs. [9, 10], the discrepancy is discussed in the
text.
On the other hand, this investigation shows that the profiles of Ar confined
in a slit of Li present SSB. This effect occurs in a certain range of average
densities $\rho^{*}_{ssb1}\leq\rho^{*}_{av}\leq\rho^{*}_{ssb2}$, which
diminishes for increasing temperatures. The main output of this work is the
finding that above the wetting temperature the SSB occurs until $T_{cpw}$ is
reached. To the best of our knowledge this is the first time that such a
correlation is reported. Furthermore, it is shown that by examining the
evolution of the asymmetry coefficient one can precisely determine $T_{cpw}$.
The obtained value $T_{cpw}=118.4$ K lies in the interval established when
analyzing the adsorption on a single wall.
###### Acknowledgements.
This work was supported in part by the Grants PICT 31980/5 from Agencia
Nacional de Promoción Científica y Tecnológica, and X099 from Universidad de
Buenos Aires, Argentina.
## References
* [1] R Pandit, M Schick, M Wortis, Systematics of multilayer adsorption phenomena on attractive substrates Phys. Rev. B 26, 5112 (1982).
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* [3] M J P Nijmeijer, C Bruin, A F Bakker, J M J van Leeuwen, Wetting and drying of an inert wall by a fluid in a molecular-dynamics simulation, Phys. Rev. A 42, 6052 (1990).
* [4] L Szybisz, Adsorption of superfluid 4He films on planar heavy-alkali metals studied with the Orsay-Trento density functional, Phys. Rev. B 67, 132505 (2003).
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|
arxiv-papers
| 2009-09-11T19:08:31 |
2024-09-04T02:49:05.281036
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Salvador A. Sartarelli, Leszek Szybisz",
"submitter": "Luis Ariel Pugnaloni",
"url": "https://arxiv.org/abs/0909.2244"
}
|
0909.2246
|
010002 2009 M. C. Barbosa H. Fort (Universidad de la República, Uruguay)
010002
In this work, it is pointed out that in the mean-field version of majority-
rule opinion dynamics, the dependence of the consensus time on the population
size exhibits two regimes. This is determined by the size distribution of the
groups that, at each evolution step, gather to reach agreement. When the group
size distribution has a finite mean value, the previously known logarithmic
dependence on the population size holds. On the other hand, when the mean
group size diverges, the consensus time and the population size are related
through a power law. Numerical simulations validate this semi-quantitative
analytical prediction.
# A note on the consensus time of mean-field majority-rule dynamics
Damián H. Zanette[inst1] E-mail: zanette@cab.cnea.gov.ar
(6 July 2009; 2 September 2009)
††volume: 1
99 inst1 Consejo Nacional de Investigaciones Científicas y Técnicas, Centro
Atómico Bariloche and Instituto Balseiro, 8400 San Carlos de Bariloche, Río
Negro, Argentina.
Much attention has been recently paid, in the context of statistical physics,
to models of social processes where ordered states emerge spontaneously out of
disordered initial conditions (homogeneity from heterogeneity, dominance from
diversity, consensus from disagreement, etc.) [1]. Not unexpectedly, many of
them are adaptations of well-known models for coarsening in interacting spin
systems, whose dynamical rules are reinterpreted in the framework of social-
like phenomena. The voter model [2, 3] and the majority rule model [4, 5] are
paradigmatic examples. In the latter, consensus in a large population is
reached by accumulative agreement events, each of them involving just a group
of agents. The present note is aimed at briefly revisiting previous results on
the time needed to reach consensus in majority-rule dynamics, stressing the
role of the size distribution of the involved groups. It is found that the
growth of the consensus time with the population size shows distinct behaviors
depending on whether the mean value of the group size distribution is finite
or not.
Consider a population of $N$ agents where, at any given time, each agent has
one of two possible opinions, labeled $+1$ and $-1$. At each evolution step, a
group of $G$ agents ($G$ odd) is selected from the population, and all of them
adopt the opinion of the majority. Namely, if $i$ is one of the agents in the
selected group, its opinion $s_{i}$ changes as
$\displaystyle s_{i}\to{\rm sign}\sum_{j}s_{j},$ (1)
where the sum runs over the agents in the group. Of course, only the agents,
not the majority, effectively change their opinion. In the mean-field version
of this model, the $G$ agents selected at each step are drawn at random from
the entire population.
It is not difficult to realize that the mean-field majority-rule (MFMR)
dynamics is equivalent to a random walk under the action of a force field. For
a finite-size population, this random walk is moreover subject to absorbing
boundary conditions. Think, for instance, of the number $N_{+}$ of agents with
opinion $+1$. As time elapses, $N_{+}$ changes randomly, with transition
probabilities that depend on $N_{+}$ itself, until it reaches one of the
extreme values, $N_{+}=0$ or $N$. At this point, all the agents have the same
opinion, the population has reached full consensus, and the dynamics freezes.
In view of this overall behavior, a relevant quantity to characterize MFMR
dynamics in finite populations is the consensus time, i.e. the time needed to
reach full consensus from a given initial condition. In particular, one is
interested in determining how the consensus time depends on the population
size $N$. The exact solution for three-agent groups ($G=3$) [5] shows that the
average number of steps needed to reach consensus, $S_{c}$, depends on $N$ as
$\displaystyle S_{c}\propto N\log N,$ (2)
for large $N$. The proportionality factor depends in turn on the initial
unbalance between the two opinions all over the population. The analogy of
MFMR dynamics with random walks suggests that this result should also hold for
other values of the group size $G$, as long as $G$ is smaller than $N$. This
can be easily verified by solving a rate equation for the evolution of $N_{+}$
[1]. Numerical results and semi-quantitative arguments [6] show that Eq. (2)
is still valid if, instead of being constant, the value of $G$ is uniformly
distributed over a finite interval.
What would happen, however, if, at each step, $G$ is drawn from a probability
distribution $p_{G}$ that allows for values larger than the population size?
If, at a given step, the chosen group size $G$ is equal to or largen than $N$,
full consensus will be instantly attained and the evolution will cease. In the
random-walk analogy, this step would correspond to a single long jump taking
the walker to one of the boundaries. Is it possible that, for certain forms of
the distribution $p_{G}$, these single large-$G$ events could dominate the
attainment of consensus? If it is so, how is the $N$-dependence of the
consensus time modified?
To give an answer to these questions, assume that $G$ is drawn from a
distribution which, for large $G$, decays as
$\displaystyle p_{G}\sim G^{-\gamma},$ (3)
with $\gamma>1$. Tuning the exponent $\gamma$ of this power-law distribution,
large values of $G$ may become sufficiently frequent as to control consensus
dynamics.
The probability that at the $S$-th step the selected group size is $G\geq N$,
while in all preceding steps $G<N$, reads
$\displaystyle P_{S}=\left(\sum_{G=G_{\rm
min}}^{N-1}p_{G}\right)^{S-1}\sum_{G=N}^{\infty}p_{G},$ (4)
where $G_{\rm min}$ is the minimal value of $G$ allowed for by the
distribution $p_{G}$. The average waiting time (in evolution steps) for an
event with $G\geq N$ is thus
$\displaystyle
S_{w}=\sum_{S=1}^{\infty}SP_{S}=\left(\sum_{G=N}^{\infty}p_{G}\right)^{-1}\propto
N^{\gamma-1},$ (5)
where the last relation holds for large $N$ when $p_{G}$ verifies Eq. (3).
Compare now Eqs. (2) and (5). For $\gamma>2$ (respectively, $\gamma\leq 2$)
and asymptotically large population sizes, one has $S_{w}\gg S_{c}$
(respectively, $S_{w}\ll S_{c}$). This suggests that above the critical
exponent $\gamma_{\rm crit}=2$, the attainment of consensus will be driven by
the asymptotic random-walk features that lead to Eq. (2). For smaller
exponents, on the other hand, consensus will be reached by the occurrence of a
large-$G$ event, in which all the population is entrained at a single
evolution step. Note that $\gamma_{\rm crit}$ stands at the boundary between
the domain for which the mean group size is finite ($\gamma>\gamma_{\rm
crit}$) and the domain where it diverges ($\gamma<\gamma_{\rm crit}$).
In order to validate this analysis, numerical simulations of MFMR dynamics
have been performed for population sizes ranging from $10^{2}$ to $10^{5}$.
The probability distribution for the group size $G$ has been introduced as
follows. First, define $G=2g+1$. Choosing $g=1,2,3,\dots$ ensures that the
group size is odd and $G\geq 3$. Then, take for $g$ the probability
distribution
$\displaystyle p_{g}=\frac{1}{\zeta(\gamma)}g^{-\gamma},$ (6)
where $\zeta(z)$ is the Riemann zeta function. With this choice, $p_{G}$
satisfies Eq. (3). The average waiting time for a large-$G$ event, given by
Eq. (5), can be exactly given as
$\displaystyle S_{w}=\frac{\zeta(\gamma)}{\zeta(\gamma,1+N/2)},$ (7)
where $\zeta(z,a)$ is the generalized Riemann (or Hurwitz [7]) zeta function.
In the numerical simulations, both opinions were equally represented in the
initial condition. The total number of steps needed to reach full consensus,
$S$, was recorded and averaged over series of $10^{2}$ to $10^{6}$
realizations (depending on the population size $N$).
Figure 1: Numerical results for the number of steps needed to reach consensus,
$S$, normalized by the population size $N$, as a function of $N$, for three
values of the exponent $\gamma$. The straight dotted lines emphasize the
validity of Eq. (2) for $\gamma=2.5$ and $3$. For $\gamma=2$ the line is
horizontal, suggesting $S\propto N$.
The two upper data sets in Fig. 1 show the ratio $S/N$ for two values of the
exponent $\gamma>\gamma_{\rm crit}$. Since the horizontal scale is
logarithmic, a linear dependence in this graph corresponds to the
proportionality given by Eq. (2). Dotted straight lines illustrate this
dependence. For these values of $\gamma$, therefore, the relation between the
consensus time and the population size coincides with that of the case of
constant $G$. For the lowest data set, which corresponds to
$\gamma=\gamma_{\rm crit}$, the relation ceases to hold. The horizontal dotted
line suggests that now $S\propto N$, as predicted for $\gamma=2$ by Eq. (5).
Figure 2: Number of steps needed to reach consensus as a function of the
population size, for three values of the exponent $\gamma$. The slope of the
straight dotted line equals one. Full curves correspond to the function
$S_{w}$ given in Eq. (7).
The log-log plot of Fig. 2 shows the number of steps to full consensus as a
function of the population size for three exponents $\gamma\leq\gamma_{\rm
crit}$. The dotted straight line has unitary slope, representing the
proportionality between $S$ and $N$ for $\gamma=2$. For lower exponents, the
full curves are the graphic representation of $S_{w}$ as given by Eq. (5). The
excellent agreement between $S_{w}$ and the numerical results for $S$
demonstrates that, for these values of $\gamma$, the consensus time in actual
realizations of the MFMR process is in fact dominated by large-$G$ events.
Figure 3: Fraction of realizations where consensus is attained through a
large-$G$ event as a function of the population size, for several values of
the exponent $\gamma$.
A further characterization of the two regimes of consensus attainment is given
by the fraction of realizations where consensus is reached through a large-$G$
event. This is shown in Fig. 3 as a function of the population size. For
$\gamma<\gamma_{\rm crit}$, consensus is the result of a step involving the
whole population in practically all realizations. As $N$ grows, the frequency
of such realizations increases as well. The opposite behavior is observed for
$\gamma>\gamma_{\rm crit}$. For the critical exponent, meanwhile, the fraction
of large-$G$ realizations is practically independent of $N$, and fluctuates
slightly around $0.57$.
In summary, it has been shown here that in majority-rule opinion dynamics, the
dependence of the consensus time on the population size exhibits two distinct
regimes. If the size distribution of the groups of agents selected at each
evolution step decays fast enough, one reobtains the logarithmic analytical
result for constant group sizes. If, on the other hand, the distribution of
group sizes decays slowly, as a power law with a sufficiently small exponent,
the dependence of the consensus time on the population size is also given by a
power law. The two regimes are related to two different mechanisms of
consensus attainment: in the second case, in particular, consensus is reached
during events which involve the whole population at a single evolution step.
The logarithmic regime occurs when the mean group size is finite, while in the
power-law regime the mean value of the distribution of group sizes diverges.
In connection with the random-walk analogy of majority-rule dynamics, this is
reminiscent of the contrasting features of standard and anomalous diffusion
[8].
## References
* [1] C Castellano, S Fortunato, V Loreto, Statistical physics of social dynamics, Rev. Mod. Phys. 81, 591 (2009).
* [2] M Scheucher, H Spohn, A soluble kinetic model for spinodal decomposition, J. Stat. Phys. 53, 279 (1988).
* [3] P L Krapivsky, Kinetics of a monomer-monomer model of heterogeneous catalysis, Phys. Rev. A 45, 1067 (1992).
* [4] S Galam, Minority opinion spreading in random geometry, Eur. Phys. J. B 25, 403 (2002).
* [5] P L Krapivsky, S Redner, Dynamics of majority rule in two-state interacting spin systems, Phys. Rev. Lett. 90, 238701 (2003).
* [6] C J Tessone, R Toral, P Amengual, H S Wio, M San Miguel, Neighborhood models of minority opinion spreading, Eur. Phys. J. B 39, 535 (2004).
* [7] J Spanier, K B Oldham, The Hurwitz Function $\zeta(\nu;u)$, In: An Atlas of Functions, pag. 653 Hemisphere, Washington, DC (1987).
* [8] U Frisch, M F Shlesinger, G Zaslavsky, Eds. Lévy Flights and Related Phenomena in Physics, Springer, Berlin (1995).
|
arxiv-papers
| 2009-09-11T19:19:15 |
2024-09-04T02:49:05.285630
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Dami\\'an H. Zanette",
"submitter": "Luis Ariel Pugnaloni",
"url": "https://arxiv.org/abs/0909.2246"
}
|
0909.2276
|
# Nonlinearity in Oscillatory Flows over Sand Ripple
Ruma Dutta111Electronic address: ruma@mps.ohiostate.edu Dept of Civil
engineering & Geodetic Science, The Ohio State Univ Ethan
Kubatko222Electronic Address: kubatko.3@osu.edu Dept. of Civil Engineering &
Geodetic Science, The Ohio State University
###### Abstract
In this report, We investigated the nonlinear phenomena in the study of flow
dynamics of velocity component. In our studies, we observed nonlinear term in
the vertical component of velocity by vittori et al vit The time series
simulation of vertical component which increases with Reynold stress value. We
developed direct numerical simulation under two dimensional grid system to
study the flow dynamics and vorticity parameter. Flow pattern and flow
dynamics near wavy boundary wall in the victinity of ripple bottom was
readdressed under direct numerical simualtion(DNS) framework. Both horizontal
and vertical component of fluid velocity were studied under pulating force of
flow. Vorticity is calculated under complex framework by taking into higher
order interaction term. we tried to carry out similar simulation with same
particle ejection in the viscous bed using DNS simulation for pulsating flow.
Our focus was to observe particle motion using DNS simulation and study the
particle phase under vortex structures formed here.
## I Introduction
Many observations were made cencerning a complex bed form pattern in the
victinity of offshore region. Temporal chaos in fluid turbulence is the
symptotatic of spatial chaos. The chaos in turbulence can be studied can be
studied by many numerical approaches.
As widely discussed, the chaotic pattern is detected in turbulence both for
fixed geometric configuration increasing the flow Reynolds number and for
fixed characteristic of oscillatory flow increasing the amplitude of the wall
waviness. From an analysis of the The tide driven current in the offshore
region is the m major source sediment transport in beach areas. The current
deflects toward the crests due to increase in bottom friction with a
decreasing water depth. Hence the cross ridge velocity increases to satisfy
continuity whereas the along ridge velocity decreases owing to increase of
bottom friction. Huthnance was the first to present a mathematical description
using simple flow model. He used a simplified version of depth average shallow
water equation with a power law relationship for sediment transport corrected
for downhill gravitational transport.
and turbulence pattern under complex bed-fluid intearction mechanism. In fact,
vast sea water(both offshore and near shore) is not clear Underneath sea with
complex bed form having ripple is observed occasionally. sleath
Indeed many observations were made concerning a complex bed form pattern in
the victinity of offshore region. The bottom topography is indeed very complex
in nature. These ripples are formed due to oscillatory nature of turbulence
flow over sand bed form in a complex manner where the bottom topography often
takes form of The chaotic phenomena related to turbulence studies were
observed by blond brick or tile pattern depending on the complexity of the
bed-fluid interaction under turbulence flow. Flow becomes more complex near
the shore region where the flow due to oscillatory nature affects the ripple
formation. It is believed that waves are mainly responsible for sediment
transport where currents carry the entrained sediments away. Field
observations focussing tide driven region indicate the presence of symmetrical
and asymmetrical waves with crest almost perpendicular to the direction of
main current and characterized by wavelengths of few hundered metres. In this
region, full understanding of turbulence with oscillatory flow is needed to
understand the sediment flow. Turbulent fluctuations due to oscillatory nature
of flow are usually confined within a thin oscillating boundary layer. This
situation makes very difficult to take experimental data accurately in this
region which makes sometimes the interpretation of data controversial. There
are different numerical approaches to study these problems such as Reynolds
Averaged Naveir Stokes Equation(RANS), ‘Direct Numerical Simulation(DNS). LES
model is based on solving Navier Stokes equation with large eddy function and
small eddies are neglected in this calculation. It well known for its
simplicity and low computer cost but can not give more insight in the complex
phenomena. On the other hand, Reynolds Averaged Navier Stokes Equation alsoo
known as RANS model is based on On the other hand, DNS simulation does not
involve any turbulence model but uses unsteady flow using grid system that are
sufficiently fine to resolve all scales of motion. A first attempt to explain
the mechanism of this flow made by Hara an Mei hara who developed a three
dimensional model investigating the stability of Stokes layer induced by sea
wave. The associated steady state along the ripple surface showed a tendency
to accumulate sediment particle in various pattern. B Blondeaux & Vittori blon
tried to model sand ripple based on brick pattern form on the basis of three
dimensional simulation model. They studied three dimensional vortex structure
in the oscillatory form of flow under two dimensional ripple bed. In both
cases, a unidirectional oscillatory flow is considered and fluid particles are
considered on top of the bottom boundary to oscillate to and fro. Ripple at
sea bed affects the sediment transport rate and cause additional energy
dissipation enhancing mixing in the vicinity of ripple. However the detail
knowledge of the flow structure and the dynamics of vortex structure generated
by flow seperation is not clear in this region. Recently Scandura & Blondeaux
Scan studied by means of numerical simulations, the flow induced by wavy wall
under uniform oscillatory motion. They observed in the simulation that
velocity is periodic under weak flow and vorticity is shed just above the
crest which has a tendency for pitchfork bifurcation above critical value of
velocity.
In our numerical simulation, we tried to readdress the problem of chaos in
velocity and vorticity field and observed similar bifurcation at critical
value of velocity. Our direct numerical simulation is based on finite
difference scheme for oscillatory flow of fluid and we observe development of
bifurcation in the normal and streamline flow component which increases with
$u_{\rm 0}$. The nonlinearity nature of the vertical component of the velocity
also shows similar pattern.
## II Formulation of the Problem
The problem was formulated in the following way, We consider incompressible
fluid of density $\rho$ and kinematic viscosity $\nu$ induced close to a wavy
wall by a uniform oscillating pressure gradient. We define Cartesian
orthogonal corordinate We start with Navier Stokes equation for an
incompressible fluid flow in rectangular $(x,y,z)$ coordinates. We also
consider wall profile described parametrically by the relationship
$y=-{\frac{h}{2}}[cos({\it k}{\xi})+{\sum_{n=1}}^{N}c_{\rm n}cos({\gamma}_{\rm
n})]$ (1) $x={\xi}+\frac{h}{2}[sin(k{\xi})+{\sum_{n=1}}^{N}c_{\rm
n}sin({\gamma_{n}})]$ (2)
where ${\rm k}=\frac{2{\pi}}{l}$ is the wavenumber of the waviness. $\xi$ is a
dummy variable and $\gamma_{\rm n}=nk{\xi}+{\phi}_{\rm n}$.
$\displaystyle\ \frac{\partial u}{\partial t}+{\bf u}\cdot\bigtriangledown
u=-\frac{1}{\rho}\frac{\partial p}{\partial
x}+{\nu}\bigtriangledown^{2}u+F^{(x)}$ (3) $\displaystyle\ \frac{\partial
w}{\partial t}+{\bf u}\cdot\bigtriangledown w=-\frac{1}{\rho}\frac{\partial
p}{\partial z}+{\nu}\bigtriangledown^{2}w$ (4) $\displaystyle\
\bigtriangledown\cdot{\bf u}=0$ (5)
For Direct numerical simulation algorithm, we choose collocated, nonstaggered
grid system. The algorithm we
$\displaystyle-\frac{1}{\rho}\frac{\partial p}{\partial
y}+{\nu}\bigtriangledown^{2}v$ (6) $\displaystyle\ \frac{\partial w}{\partial
t}+{\bf u}\cdot\bigtriangledown w=-\frac{1}{\rho}\frac{\partial p}{\partial
z}+{\nu}\bigtriangledown^{2}v$ (7) $\displaystyle\ \bigtriangledown\cdot{\bf
u}=0$ (8)
where the field velocities ( u,v,w) are along (x,y,z) directions repectively.
For the sediment particle, the basic equation is controlled by spherical
particle moving under gravity in viscous fluid.
### II.1 Discussion and Conclusion of the Results
We consider the flow of an incompressible viscous fluid of density $\rho$ and
kinematic viscosity $\nu$ induced close to wavy wall by a uniform oscillating
pressure gradient. The nonlinear fluctuation and periodicity was observed here
for velocity component both in streamline and vertical flow field. The
nonlinearity increases with $u_{\rm o}$ above the threshold value. Whwn the
shear stress experienced by the interface betwen the flowing fluid and the
resting particles is low, the flow is unable to entrain the particles lying on
the bed, which then remains immobile. As the shear stress increases,
## III Acknowledgement
This work was supported by Naval Research Lab Grant.
## References
* (1) O.E Landford III, Annual Rev. Fluid Mech., 14, 347 (1982).
* (2) J. Guckenheimer, Annual Rev. FLuid Mech., 18, 15 (1986).
* (3) Hara. T & Mei C.C (1990), Centrifugal Instability of an oscillatory flow over periodic ripples Journal of Fluid Mechanics, 217, 1-32.
* (4) P. Blondeaux & G. Vittori, A route to chaos in an oscillatory flow: Feigenbaum scenario, Phys. Fluids A 3(11), 2492-2495, 1991.
* (5) Blondeaux P. 1990, Sand Ripples under seawaves Part 1, Ripple Formation, J. Fluid Mechanics,218, 1-17.
* (6) Three dimensional oscillatory flow over steep ripples; J. Fluid Mechanics, 412, 355-478.
* (7) P.Scandura, G.Vittori and P. Blondeaux, Bifurcations in the Oscillatory Flow over a Wavy Wall, Mechanics, 37, 305-311, 2002.
|
arxiv-papers
| 2009-09-12T21:38:20 |
2024-09-04T02:49:05.290340
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ruma Dutta and Ethan Kubatko",
"submitter": "Ruma Dutta Dr",
"url": "https://arxiv.org/abs/0909.2276"
}
|
0909.2349
|
Current address:]LPSC-Grenoble, France Current address:]Los Alamos National
Laborotory, New Mexico, NM Current address:]Los Alamos National Laborotory,
New Mexico, NM Current address:]The George Washington University, Washington,
DC 20052 Current address:]Christopher Newport University, Newport News,
Virginia 23606 Current address:]Edinburgh University, Edinburgh EH9 3JZ,
United Kingdom Current address:]College of William and Mary, Williamsburg,
Virginia 23187-8795
The CLAS Collaboration
# Electroexcitation of nucleon resonances from CLAS data on single pion
electroproduction
I.G. Aznauryan Thomas Jefferson National Accelerator Facility, Newport News,
Virginia 23606 Yerevan Physics Institute, 375036 Yerevan, Armenia V.D.
Burkert Thomas Jefferson National Accelerator Facility, Newport News,
Virginia 23606 A.S. Biselli Fairfield University, Fairfield CT 06824 H.
Egiyan Thomas Jefferson National Accelerator Facility, Newport News, Virginia
23606 University of New Hampshire, Durham, New Hampshire 03824-3568 K. Joo
University of Connecticut, Storrs, Connecticut 06269 University of Virginia,
Charlottesville, Virginia 22901 W. Kim Kyungpook National University, Daegu
702-701, Republic of Korea K. Park Thomas Jefferson National Accelerator
Facility, Newport News, Virginia 23606 Kyungpook National University, Daegu
702-701, Republic of Korea L.C. Smith University of Virginia,
Charlottesville, Virginia 22901 M. Ungaro Thomas Jefferson National
Accelerator Facility, Newport News, Virginia 23606 University of Connecticut,
Storrs, Connecticut 06269 Rensselaer Polytechnic Institute, Troy, New York
12180-3590 K. P. Adhikari Old Dominion University, Norfolk, Virginia 23529
M. Anghinolfi INFN, Sezione di Genova, 16146 Genova, Italy H. Avakian
Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606
J. Ball CEA, Centre de Saclay, Irfu/Service de Physique Nucléaire, 91191 Gif-
sur-Yvette, France M. Battaglieri INFN, Sezione di Genova, 16146 Genova,
Italy V. Batourine Thomas Jefferson National Accelerator Facility, Newport
News, Virginia 23606 I. Bedlinskiy Institute of Theoretical and Experimental
Physics, Moscow, 117259, Russia M. Bellis Carnegie Mellon University,
Pittsburgh, Pennsylvania 15213 C. Bookwalter Florida State University,
Tallahassee, Florida 32306 D. Branford Edinburgh University, Edinburgh EH9
3JZ, United Kingdom W.J. Briscoe The George Washington University,
Washington, DC 20052 W.K. Brooks Universidad Técnica Federico Santa María,
Casilla 110-V Valparaíso, Chile Thomas Jefferson National Accelerator
Facility, Newport News, Virginia 23606 S.L. Careccia Old Dominion
University, Norfolk, Virginia 23529 D.S. Carman Thomas Jefferson National
Accelerator Facility, Newport News, Virginia 23606 P.L. Cole Idaho State
University, Pocatello, Idaho 83209 P. Collins Arizona State University,
Tempe, Arizona 85287-1504 V. Crede Florida State University, Tallahassee,
Florida 32306 A. D’Angelo INFN, Sezione di Roma Tor Vergata, 00133 Rome,
Italy Universita’ di Roma Tor Vergata, 00133 Rome Italy A. Daniel Ohio
University, Athens, Ohio 45701 R. De Vita INFN, Sezione di Genova, 16146
Genova, Italy E. De Sanctis INFN, Laboratori Nazionali di Frascati, 00044
Frascati, Italy A. Deur Thomas Jefferson National Accelerator Facility,
Newport News, Virginia 23606 B Dey Carnegie Mellon University, Pittsburgh,
Pennsylvania 15213 S. Dhamija Florida International University, Miami,
Florida 33199 R. Dickson Carnegie Mellon University, Pittsburgh,
Pennsylvania 15213 C. Djalali University of South Carolina, Columbia, South
Carolina 29208 D. Doughty Christopher Newport University, Newport News,
Virginia 23606 Thomas Jefferson National Accelerator Facility, Newport News,
Virginia 23606 R. Dupre Argonne National Laboratory, Argonne, Illinois 60441
A. El Alaoui [ Institut de Physique Nucléaire ORSAY, Orsay, France L.
Elouadrhiri Thomas Jefferson National Accelerator Facility, Newport News,
Virginia 23606 P. Eugenio Florida State University, Tallahassee, Florida
32306 G. Fedotov Skobeltsyn Nuclear Physics Institute, 119899 Moscow, Russia
S. Fegan University of Glasgow, Glasgow G12 8QQ, United Kingdom T.A. Forest
Idaho State University, Pocatello, Idaho 83209 Old Dominion University,
Norfolk, Virginia 23529 M.Y. Gabrielyan Florida International University,
Miami, Florida 33199 G.P. Gilfoyle University of Richmond, Richmond,
Virginia 23173 K.L. Giovanetti James Madison University, Harrisonburg,
Virginia 22807 F.X. Girod Thomas Jefferson National Accelerator Facility,
Newport News, Virginia 23606 CEA, Centre de Saclay, Irfu/Service de Physique
Nucléaire, 91191 Gif-sur-Yvette, France J.T. Goetz University of California
at Los Angeles, Los Angeles, California 90095-1547 W. Gohn University of
Connecticut, Storrs, Connecticut 06269 E. Golovatch Skobeltsyn Nuclear
Physics Institute, 119899 Moscow, Russia R.W. Gothe University of South
Carolina, Columbia, South Carolina 29208 M. Guidal Institut de Physique
Nucléaire ORSAY, Orsay, France L. Guo [ Thomas Jefferson National
Accelerator Facility, Newport News, Virginia 23606 K. Hafidi Argonne
National Laboratory, Argonne, Illinois 60441 H. Hakobyan Universidad Técnica
Federico Santa María, Casilla 110-V Valparaíso, Chile Yerevan Physics
Institute, 375036 Yerevan, Armenia C. Hanretty Florida State University,
Tallahassee, Florida 32306 N. Hassall University of Glasgow, Glasgow G12
8QQ, United Kingdom D. Heddle Christopher Newport University, Newport News,
Virginia 23606 Thomas Jefferson National Accelerator Facility, Newport News,
Virginia 23606 K. Hicks Ohio University, Athens, Ohio 45701 M. Holtrop
University of New Hampshire, Durham, New Hampshire 03824-3568 C.E. Hyde Old
Dominion University, Norfolk, Virginia 23529 Y. Ilieva University of South
Carolina, Columbia, South Carolina 29208 The George Washington University,
Washington, DC 20052 D.G. Ireland University of Glasgow, Glasgow G12 8QQ,
United Kingdom B.S. Ishkhanov Skobeltsyn Nuclear Physics Institute, 119899
Moscow, Russia E.L. Isupov Skobeltsyn Nuclear Physics Institute, 119899
Moscow, Russia S.S. Jawalkar College of William and Mary, Williamsburg,
Virginia 23187-8795 J.R. Johnstone University of Glasgow, Glasgow G12 8QQ,
United Kingdom Thomas Jefferson National Accelerator Facility, Newport News,
Virginia 23606 D. Keller Ohio University, Athens, Ohio 45701 M. Khandaker
Norfolk State University, Norfolk, Virginia 23504 P. Khetarpal Rensselaer
Polytechnic Institute, Troy, New York 12180-3590 A. Klein [ Old Dominion
University, Norfolk, Virginia 23529 F.J. Klein Catholic University of
America, Washington, D.C. 20064 L.H. Kramer Florida International
University, Miami, Florida 33199 Thomas Jefferson National Accelerator
Facility, Newport News, Virginia 23606 V. Kubarovsky Thomas Jefferson
National Accelerator Facility, Newport News, Virginia 23606 S.E. Kuhn Old
Dominion University, Norfolk, Virginia 23529 S.V. Kuleshov Universidad
Técnica Federico Santa María, Casilla 110-V Valparaíso, Chile Institute of
Theoretical and Experimental Physics, Moscow, 117259, Russia V. Kuznetsov
Kyungpook National University, Daegu 702-701, Republic of Korea K. Livingston
University of Glasgow, Glasgow G12 8QQ, United Kingdom H.Y. Lu University of
South Carolina, Columbia, South Carolina 29208 M. Mayer Old Dominion
University, Norfolk, Virginia 23529 J. McAndrew Edinburgh University,
Edinburgh EH9 3JZ, United Kingdom M.E. McCracken Carnegie Mellon University,
Pittsburgh, Pennsylvania 15213 B. McKinnon University of Glasgow, Glasgow
G12 8QQ, United Kingdom C.A. Meyer Carnegie Mellon University, Pittsburgh,
Pennsylvania 15213 T Mineeva University of Connecticut, Storrs, Connecticut
06269 M. Mirazita INFN, Laboratori Nazionali di Frascati, 00044 Frascati,
Italy V. Mokeev Skobeltsyn Nuclear Physics Institute, 119899 Moscow, Russia
Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606
B. Moreno Institut de Physique Nucléaire ORSAY, Orsay, France K. Moriya
Carnegie Mellon University, Pittsburgh, Pennsylvania 15213 B. Morrison
Arizona State University, Tempe, Arizona 85287-1504 H. Moutarde CEA, Centre
de Saclay, Irfu/Service de Physique Nucléaire, 91191 Gif-sur-Yvette, France
E. Munevar The George Washington University, Washington, DC 20052 P. Nadel-
Turonski Catholic University of America, Washington, D.C. 20064 R.
Nasseripour [ University of South Carolina, Columbia, South Carolina 29208
Florida International University, Miami, Florida 33199 C.S. Nepali Old
Dominion University, Norfolk, Virginia 23529 S. Niccolai Institut de
Physique Nucléaire ORSAY, Orsay, France The George Washington University,
Washington, DC 20052 G. Niculescu James Madison University, Harrisonburg,
Virginia 22807 I. Niculescu James Madison University, Harrisonburg, Virginia
22807 M.R. Niroula Old Dominion University, Norfolk, Virginia 23529 M.
Osipenko INFN, Sezione di Genova, 16146 Genova, Italy A.I. Ostrovidov
Florida State University, Tallahassee, Florida 32306 University of South
Carolina, Columbia, South Carolina 29208 S. Park Florida State University,
Tallahassee, Florida 32306 E. Pasyuk Arizona State University, Tempe,
Arizona 85287-1504 S. Anefalos Pereira INFN, Laboratori Nazionali di
Frascati, 00044 Frascati, Italy S. Pisano Institut de Physique Nucléaire
ORSAY, Orsay, France O. Pogorelko Institute of Theoretical and Experimental
Physics, Moscow, 117259, Russia S. Pozdniakov Institute of Theoretical and
Experimental Physics, Moscow, 117259, Russia J.W. Price California State
University, Dominguez Hills, Carson, CA 90747 S. Procureur CEA, Centre de
Saclay, Irfu/Service de Physique Nucléaire, 91191 Gif-sur-Yvette, France Y.
Prok [ University of Virginia, Charlottesville, Virginia 22901 D.
Protopopescu University of Glasgow, Glasgow G12 8QQ, United Kingdom
University of New Hampshire, Durham, New Hampshire 03824-3568 B.A. Raue
Florida International University, Miami, Florida 33199 Thomas Jefferson
National Accelerator Facility, Newport News, Virginia 23606 G. Ricco INFN,
Sezione di Genova, 16146 Genova, Italy M. Ripani INFN, Sezione di Genova,
16146 Genova, Italy B.G. Ritchie Arizona State University, Tempe, Arizona
85287-1504 G. Rosner University of Glasgow, Glasgow G12 8QQ, United Kingdom
P. Rossi INFN, Laboratori Nazionali di Frascati, 00044 Frascati, Italy F.
Sabatié CEA, Centre de Saclay, Irfu/Service de Physique Nucléaire, 91191 Gif-
sur-Yvette, France M.S. Saini Florida State University, Tallahassee, Florida
32306 J. Salamanca Idaho State University, Pocatello, Idaho 83209 R.A.
Schumacher Carnegie Mellon University, Pittsburgh, Pennsylvania 15213 H.
Seraydaryan Old Dominion University, Norfolk, Virginia 23529 N.V. Shvedunov
Skobeltsyn Nuclear Physics Institute, 119899 Moscow, Russia D.I. Sober
Catholic University of America, Washington, D.C. 20064 D. Sokhan Edinburgh
University, Edinburgh EH9 3JZ, United Kingdom S.S. Stepanyan Kyungpook
National University, Daegu 702-701, Republic of Korea P. Stoler Rensselaer
Polytechnic Institute, Troy, New York 12180-3590 I.I. Strakovsky The George
Washington University, Washington, DC 20052 S. Strauch University of South
Carolina, Columbia, South Carolina 29208 The George Washington University,
Washington, DC 20052 R. Suleiman Massachusetts Institute of Technology,
Cambridge, Massachusetts 02139-4307 M. Taiuti INFN, Sezione di Genova, 16146
Genova, Italy D.J. Tedeschi University of South Carolina, Columbia, South
Carolina 29208 S. Tkachenko Old Dominion University, Norfolk, Virginia 23529
M.F. Vineyard Union College, Schenectady, NY 12308 D.P. Watts [ University
of Glasgow, Glasgow G12 8QQ, United Kingdom L.B. Weinstein Old Dominion
University, Norfolk, Virginia 23529 D.P. Weygand Thomas Jefferson National
Accelerator Facility, Newport News, Virginia 23606 M. Williams Carnegie
Mellon University, Pittsburgh, Pennsylvania 15213 M.H. Wood Canisius
College, Buffalo, NY L. Zana University of New Hampshire, Durham, New
Hampshire 03824-3568 J. Zhang Old Dominion University, Norfolk, Virginia
23529 B. Zhao [ University of Connecticut, Storrs, Connecticut 06269
###### Abstract
We present results on the electroexcitation of the low mass resonances
$\Delta(1232)P_{33}$, $N(1440)P_{11}$, $N(1520)D_{13}$, and $N(1535)S_{11}$ in
a wide range of $Q^{2}$. The results were obtained in the comprehensive
analysis of JLab-CLAS data on differential cross sections, longitudinally
polarized beam asymmetries, and longitudinal target and beam-target
asymmetries for $\pi$ electroproduction off the proton. The data were analysed
using two conceptually different approaches, fixed-$t$ dispersion relations
and a unitary isobar model, allowing us to draw conclusions on the model
sensitivity of the obtained electrocoupling amplitudes. The amplitudes for the
$\Delta(1232)P_{33}$ show the importance of a meson-cloud contribution to
quantitatively explain the magnetic dipole strength, as well as the electric
and scalar quadrupole transitions. They do not show any tendency of
approaching the pQCD regime for $Q^{2}\leq 6~{}$GeV2. For the Roper resonance,
$N(1440)P_{11}$, the data provide strong evidence for this state as a
predominantly radial excitation of a 3-quark ground state. Measured in pion
electroproduction, the transverse helicity amplitude for the $N(1535)S_{11}$
allowed us to obtain the branching ratios of this state to the $\pi N$ and
$\eta N$ channels via comparison to the results extracted from $\eta$
electroproduction. The extensive CLAS data also enabled the extraction of the
$\gamma^{*}p\rightarrow N(1520)D_{13}$ and $N(1535)S_{11}$ longitudinal
helicity amplitudes with good precision. For the $N(1535)S_{11}$, these
results became a challenge for quark models, and may be indicative of large
meson-cloud contributions or of representations of this state different from a
3q excitation. The transverse amplitudes for the $N(1520)D_{13}$ clearly show
the rapid changeover from helicity-3/2 dominance at the real photon point to
helicity-1/2 dominance at $Q^{2}>1~{}$GeV2, confirming a long-standing
prediction of the constituent quark model.
###### pacs:
11.55.Fv, 13.40.Gp, 13.60.Le, 14.20.Gk
## I Introduction
The excitation of nucleon resonances in electromagnetic interactions has long
been recognized as an important source of information to understand the strong
interaction in the domain of quark confinement. The CLAS detector at Jefferson
Lab is the first large acceptance instrument designed for the comprehensive
investigation of exclusive electroproduction of mesons with the goal to study
the electroexcitation of nucleon resonances in detail. In recent years, a
variety of measurements of single pion electroproduction on protons, including
polarization measurements, have been performed at CLAS in a wide range of
photon virtuality $Q^{2}$ from 0.16 to 6 GeV2 Joo1 ; Joo2 ; Joo3 ; Egiyan ;
Ungaro ; Smith ; Park ; Biselli . In this work we present the results on the
electroexcitation of the resonances $\Delta(1232)P_{33}$, $N(1440)P_{11}$,
$N(1520)D_{13}$, and $N(1535)S_{11}$, obtained from the comprehensive analysis
of these data.
Theoretical and experimental investigations of the electroexcitation of
nucleon resonances have a long history, and along with the hadron masses and
nucleon electromagnetic characteristics, the information on the
$\gamma^{*}N\rightarrow N^{*}$ transitions played an important role in the
justification of the quark model. However, the picture of the nucleon and its
excited states, which at first seemed quite simple and was identified as a
model of non-relativistic constituent quarks, turned out to be more complex.
One of the reasons for this was the realization that quarks are relativistic
objects. A consistent way to perform the relativistic treatment of the
$\gamma^{*}N\rightarrow N(N^{*})$ transitions is to consider them in the
light-front (LF) dynamics Drell ; Terentev ; Brodsky . The relevant approaches
were developed and used to describe the nucleon and its excited states
Aznquark ; Aznquark1 ; Weber ; Capstick ; Simula ; Simula1 ; Bruno ; AznRoper
. However, much more effort is required to obtain a better understanding of
what are the $N$ and $N^{*}$ LF wave functions and what is their connection to
the inter-quark forces and to the QCD confining mechanism. Another reason is
connected with the realization that the traditional picture of baryons built
from three constituent quarks is an oversimplified approximation. In the case
of the $N(1440)P_{11}$ and $N(1535)S_{11}$, the mass ordering of these states,
the large total width of $N(1440)P_{11}$, and the substantial coupling of
$N(1535)S_{11}$ to the $\eta N$ channel PDG and to strange particles Liu ;
Xie , are indicative of posible additional $q\bar{q}$ components in the wave
functions of these states Riska ; An and (or) of alternative descriptions.
Within dynamical reaction models Yang ; Kamalov ; Sato ; Lee , the meson-cloud
contribution is identified as a source of the long-standing discrepancy
between the data and constituent quark model predictions for the
$\gamma^{*}N\rightarrow\Delta(1232)P_{33}$ magnetic-dipole amplitude. The
importance of pion (cloud) contributions to the transition form factors has
also been confirmed by the lattice calculations Alexandrou . Alternative
descriptions include the representation of $N(1440)P_{11}$ as a gluonic baryon
excitation Li1 ; Li2 and the possibility that nucleon resonances are meson-
baryon molecules generated in chiral coupled-channel dynamics Weise ; Krehl ;
Nieves ; Oset1 ; Lutz . Relations between baryon electromagnetic form factors
and generalized parton distributions (GPDs) have also been formulated that
connect these two different notions to describe the baryon structure GPD1 ;
GPD2 .
The improvement in accuracy and reliability of the information on the
electroexcitation of the nucleon’s excited states over a large range in photon
virtuality $Q^{2}$ is very important for the progress in our understanding of
this complex picture of the strong interaction in the domain of quark
confinement.
Our goal is to determine in detail the $Q^{2}$-behavior of the
electroexcitation of resonances. For this reason, we analyse the data at each
$Q^{2}$ point separately without imposing any constraints on the $Q^{2}$
dependence of the electroexcitation amplitudes. This is in contrast with the
analyses by MAID, for instance MAID2007 MAID , where the electroexcitation
amplitudes are in part constrained by using parameterizations for their
$Q^{2}$ dependence.
The analysis was performed using two approaches, fixed-$t$ dispersion
relations (DR) and the unitary isobar model (UIM). The real parts of the
amplitudes, which contain a significant part of the non-resonant
contributions, are built in these approaches in conceptually different ways.
This allows us to draw conclusions on the model sensitivity of the resulting
electroexcitation amplitudes.
The paper is organized as follows. In Sec. II, we present the data and discuss
the stages of the analysis. The approaches we use to analyse the data, DR and
UIM, were successfully employed in analyses of pion-photoproduction and
low-$Q^{2}$-electroproduction data, see Refs. Azn0 ; Azn04 ; Azn065 . In Sec.
III we therefore discuss only the points that need different treatment when we
move from low $Q^{2}$ to high $Q^{2}$. Uncertainties of the background
contributions related to the pion and nucleon elastic form factors, and to
$\rho,\omega\rightarrow\pi\gamma$ transition form factors are discussed in
Sec. IV. In Sec. V, we present how resonance contributions are taken into
account and explain how the uncertainties associated with higher resonances
and with the uncertainties of masses and widths of the $N(1440)P_{11}$,
$N(1520)D_{13}$, and $N(1535)S_{11}$ are accounted for. All these
uncertainties are included in the total model uncertainty of the final
results. So, in addition to the uncertainties in the data, we have accounted
for, as much as possible, the model uncertainties of the extracted
$\gamma^{*}N\rightarrow~{}\Delta(1232)P_{33}$, $N(1440)P_{11}$,
$N(1520)D_{13}$, and $N(1535)S_{11}$ amplitudes. The results are presented in
Sec. VI, compared with model predictions in Sec. VII, and summarized in Sec.
VIII.
## II Data analysis considerations
The data are presented in Tables 1-4. They cover the first, second, and part
of the third resonance regions. The stages of our analysis are dictated by how
we evaluate the influence of higher resonances on the extracted amplitudes for
the $\Delta(1232)P_{33}$ and for the resonances from the second resonance
region.
In the first stage, we analyse the data reported in Table 1
($Q^{2}=0.3-0.65~{}$GeV2) where the richest set of polarization measurements
is available. The results based on the analysis of the cross sections and
longitudinally polarized beam asymmetries ($A_{LT^{\prime}}$) at $Q^{2}=0.4$
and $0.6-0.65~{}$GeV2 were already presented in Refs. Azn04 ; Azn065 .
However, recently, new data have become available from the JLab-CLAS
measurements of longitudinal target ($A_{t}$) and beam-target ($A_{et}$)
asymmetries for $\vec{e}\vec{p}\rightarrow ep\pi^{0}$ at
$Q^{2}=0.252,~{}0.385,~{}0.611~{}$ GeV2 Biselli . For this reason, we
performed a new analysis on the same data set, including these new
measurements. We also extended our analysis to the available data for the
close values of $Q^{2}=0.3$ and $0.5-0.525~{}$GeV2. As the asymmetries
$A_{LT^{\prime}},~{}A_{t},~{}A_{et}$ have relatively weak $Q^{2}$ dependences,
the data on asymmetries at nearby $Q^{2}$ were also included in the
corresponding sets at $Q^{2}=0.3$ and $0.5-0.525~{}$GeV2. Following our
previous analyses Azn04 ; Azn065 , we have complemented the data set at
$Q^{2}=0.6-0.65~{}$GeV2 with the DESY $\pi^{+}$ cross sections data Alder ,
since the corresponding CLAS data extend over a restricted range in $W$.
In Ref. Azn065 , the analysis of data at $Q^{2}=0.6-0.65~{}$GeV2 was performed
in combination with JLab-CLAS data for double-pion electroproduction off the
proton Fedotov . This allowed us to get information on the electroexcitation
amplitudes for the resonances from the third resonance region. This
information, combined with the $\gamma p\rightarrow N^{*}$ amplitudes known
from photoproduction data PDG , sets the ranges of the higher resonance
contributions when we extract the amplitudes of the $\gamma^{*}p\rightarrow$
$\Delta(1232)P_{33}$, $N(1440)P_{11}$, $N(1520)D_{13}$, and $N(1535)S_{11}$
transitions from the data reported in Table 1.
In the next step, we analyse the data from Table 2 which present a large body
of $\vec{e}p\rightarrow en\pi^{+}$ differential cross sections and
longitudinally polarized electron beam asymmetries at large
$Q^{2}=1.72-4.16~{}$GeV2 Park . As the isospin $\frac{1}{2}$ nucleon
resonances couple more strongly to the $\pi^{+}n$ channel, these data provide
large sensitivity to the electrocouplings of the $N(1440)P_{11}$,
$N(1520)D_{13}$, and $N(1535)S_{11}$ states. Until recently, the information
on the electroexcitation of these resonances at $Q^{2}>1~{}$GeV2 was based
almost exclusively on the (unpublished) DESY data Haidan on $ep\rightarrow
ep\pi^{0}$ ($Q^{2}\approx 2$ and $3~{}$GeV2) which have very limited angular
coverage. Furthermore, the $\pi^{0}p$ final state is coupled more weakly to
the isospin $\frac{1}{2}$ states, and is dominated by the nearby isospin
$\frac{3}{2}$ $\Delta(1232)P_{33}$ resonance. For the $N(1535)S_{11}$, which
has a large branching ratio to the $\eta N$ channel, there is also information
on the $\gamma^{*}N\rightarrow N(1535)S_{11}$ transverse helicity amplitude
found from the data on $\eta$ electroproduction off the proton Armstrong ;
Thompson ; Denizli .
In the range of $Q^{2}$ covered by the data Park (Table 2), there is no
information on the helicity amplitudes for the resonances from the third
resonance region. The data Park cover only part of this region and do not
allow us to extract reliably the corresponding amplitudes (except those for
$N(1680)F_{15}$). For the $\gamma^{*}p\rightarrow N(1440)P_{11}$,
$N(1520)D_{13}$, and $N(1535)S_{11}$ amplitudes extracted from the data Park ,
the evaluation of the uncertainties caused by the lack of information on the
resonances from the third resonance region is described in Sec. V.
Finally, we extract the $\gamma^{*}p\rightarrow\Delta(1232)P_{33}$ amplitudes
from the data reported in Tables 3 and 4. These are low $Q^{2}$ data for
$\pi^{0}$ and $\pi^{+}$ electroproduction differential cross sections Smith
and data for $\pi^{0}$ electroproduction differential cross sections at
$Q^{2}=1.15,1.45~{}$GeV2 Joo1 and $3-6~{}$GeV2 Ungaro . In the analysis of
these data, the influence of higher resonances on the results for the
$\Delta(1232)P_{33}$ was evaluated by employing the spread of the
$\gamma^{*}p\rightarrow N(1440)P_{11}$, $N(1520)D_{13}$, and $N(1535)S_{11}$
amplitudes obtained in the previous stages of our analysis of the data from
Tables 1 and 2.
Although the data for $Q^{2}=0.75-1.45~{}$GeV2 (Table 4) cover a wide range in
$W$, the absence of $\pi^{+}$ electroproduction data for these $Q^{2}$, except
$Q^{2}=0.9~{}$GeV2, does not allow us to extract the amplitudes for the
$N(1440)P_{11}$, $N(1520)D_{13}$, $N(1535)S_{11}$ resonances with model
uncertainties comparable to those for the amplitudes found from the data of
Tables 1 and 2. For $Q^{2}\simeq 0.95~{}$GeV2, there are DESY $\pi^{+}$
electroproduction data Alder , which cover the second and third resonance
regions, allowing us to extract amplitudes for all resonances from the first
and second resonance regions at $Q^{2}=0.9-0.95~{}$GeV2. To evaluate the
uncertainties caused by the higher mass resonances, we have used for
$Q^{2}=0.9-0.95~{}$GeV2 the same procedure as for the data from Table 2.
| | | Number | | | |
---|---|---|---|---|---|---|---
| | | of data | | $\frac{\chi^{2}}{N}$ | |
Obser- | $Q^{2}$ | $W$ | points | | | | Ref.
vable | (GeV2) | (GeV) | ($N$) | DR | | UIM |
$\frac{d\sigma}{d\Omega}(\pi^{+})$ | 0.3 | 1.1-1.55 | 2364 | 2.06 | | 1.93 | Egiyan
$A_{t}(\pi^{0})$ | 0.252 | 1.125-1.55 | 594 | 1.36 | | 1.48 | Biselli
$A_{et}(\pi^{0})$ | 0.252 | 1.125-1.55 | 598 | 1.19 | | 1.23 | Biselli
$\frac{d\sigma}{d\Omega}(\pi^{0})$ | 0.4 | 1.1-1.68 | 3530 | 1.23 | | 1.24 | Joo1
$\frac{d\sigma}{d\Omega}(\pi^{+})$ | 0.4 | 1.1-1.55 | 2308 | 1.92 | | 1.64 | Egiyan
$A_{LT^{\prime}}(\pi^{0})$ | 0.4 | 1.1-1.66 | 956 | 1.24 | | 1.18 | Joo2
$A_{LT^{\prime}}(\pi^{+})$ | 0.4 | 1.1-1.66 | 918 | 1.28 | | 1.19 | Joo3
$A_{t}(\pi^{0})$ | 0.385 | 1.125-1.55 | 696 | 1.40 | | 1.61 | Biselli
$A_{et}(\pi^{0})$ | 0.385 | 1.125-1.55 | 692 | 1.22 | | 1.25 | Biselli
$\frac{d\sigma}{d\Omega}(\pi^{0})$ | 0.525 | 1.1-1.66 | 3377 | 1.33 | | 1.35 | Joo1
$\frac{d\sigma}{d\Omega}(\pi^{+})$ | 0.5 | 1.1-1.51 | 2158 | 1.51 | | 1.48 | Egiyan
$\frac{d\sigma}{d\Omega}(\pi^{0})$ | 0.65 | 1.1-1.68 | 6149 | 1.09 | | 1.14 | Joo1
$\frac{d\sigma}{d\Omega}(\pi^{+})$ | 0.6 | 1.1-1.41 | 1484 | 1.21 | | 1.24 | Egiyan
$\frac{d\sigma}{d\Omega}(\pi^{+})$ | $\simeq 0.6$ | 1.4-1.76 | 477 | 1.72 | | 1.74 | Alder
$A_{LT^{\prime}}(\pi^{0})$ | 0.65 | 1.1-1.66 | 805 | 1.09 | | 1.13 | Joo2
$A_{LT^{\prime}}(\pi^{+})$ | 0.65 | 1.1-1.66 | 812 | 1.09 | | 1.04 | Joo3
$A_{t}(\pi^{0})$ | 0.611 | 1.125-1.55 | 930 | 1.38 | | 1.40 | Biselli
$A_{et}(\pi^{0})$ | 0.611 | 1.125-1.55 | 923 | 1.26 | | 1.28 | Biselli
Table 1: The data sets included in the first stage of the analysis, as discussed in the text. The columns corresponding to DR and UIM show the results for $\chi^{2}$ per data point obtained, respectively, using fixed-$t$ dispersions relations and the unitary isobar model described in Sec. III. | | | Number of | | $\chi^{2}/N$ |
---|---|---|---|---|---|---
Obser- | $Q^{2}$ | $W$ | data points | | |
vable | (GeV2) | (GeV) | ($N$) | DR | | UIM
$\frac{d\sigma}{d\Omega}(\pi^{+})$ | 1.72 | 1.11-1.69 | 3530 | 2.3 | | 2.5
| 2.05 | 1.11-1.69 | 5123 | 2.3 | | 2.2
| 2.44 | 1.11-1.69 | 5452 | 2.0 | | 2.0
| 2.91 | 1.11-1.69 | 5484 | 1.9 | | 2.1
| 3.48 | 1.11-1.69 | 5482 | 1.3 | | 1.4
| 4.16 | 1.11-1.69 | 5778 | 1.1 | | 1.1
$A_{LT^{\prime}}(\pi^{+})$ | 1.72 | 1.12-1.68 | 699 | 2.9 | | 3.0
| 2.05 | 1.12-1.68 | 721 | 3.0 | | 2.9
| 2.44 | 1.12-1.68 | 725 | 3.0 | | 3.0
| 2.91 | 1.12-1.68 | 767 | 2.7 | | 2.7
| 3.48 | 1.12-1.68 | 623 | 2.4 | | 2.3
Table 2: The $\vec{e}p\rightarrow en\pi^{+}$ data from Ref. Park . | | | Number of | | $\chi^{2}/N$ |
---|---|---|---|---|---|---
Obser- | $Q^{2}$ | $W$ | data points | | |
vable | (GeV2) | (GeV) | ($N$) | DR | | UIM
$\frac{d\sigma}{d\Omega}(\pi^{0})$ | 0.16 | 1.1-1.38 | 3301 | 1.96 | | 1.98
$\frac{d\sigma}{d\Omega}(\pi^{+})$ | 0.16 | 1.1-1.38 | 2909 | 1.69 | | 1.67
$\frac{d\sigma}{d\Omega}(\pi^{0})$ | 0.20 | 1.1-1.38 | 3292 | 2.29 | | 2.24
$\frac{d\sigma}{d\Omega}(\pi^{+})$ | 0.20 | 1.1-1.38 | 2939 | 1.76 | | 1.78
$\frac{d\sigma}{d\Omega}(\pi^{0})$ | 0.24 | 1.1-1.38 | 3086 | 1.86 | | 1.82
$\frac{d\sigma}{d\Omega}(\pi^{+})$ | 0.24 | 1.1-1.38 | 2951 | 1.49 | | 1.46
$\frac{d\sigma}{d\Omega}(\pi^{0})$ | 0.28 | 1.1-1.38 | 2876 | 1.56 | | 1.59
$\frac{d\sigma}{d\Omega}(\pi^{+})$ | 0.28 | 1.1-1.38 | 2941 | 1.47 | | 1.44
$\frac{d\sigma}{d\Omega}(\pi^{0})$ | 0.32 | 1.1-1.38 | 2836 | 1.51 | | 1.48
$\frac{d\sigma}{d\Omega}(\pi^{+})$ | 0.32 | 1.1-1.38 | 2922 | 1.39 | | 1.37
$\frac{d\sigma}{d\Omega}(\pi^{0})$ | 0.36 | 1.1-1.38 | 2576 | 1.46 | | 1.42
$\frac{d\sigma}{d\Omega}(\pi^{+})$ | 0.36 | 1.1-1.38 | 2611 | 1.35 | | 1.38
Table 3: The low $Q^{2}$ data from Ref. Smith analysed in the third stage of the analysis. | | | Number of | | $\chi^{2}/N$ |
---|---|---|---|---|---|---
Obser- | $Q^{2}$ | $W$ | data points | | |
vable | (GeV2) | (GeV) | ($N$) | DR | | UIM
$\frac{d\sigma}{d\Omega}(\pi^{0})$ | 0.75 | 1.1-1.68 | 3555 | 1.16 | | 1.18
$\frac{d\sigma}{d\Omega}(\pi^{0})$ | 0.9 | 1.1-1.68 | 3378 | 1.22 | | 1.25
$\frac{d\sigma}{d\Omega}(\pi^{+})$ | $\simeq 0.95$ | 1.36-1.76 | 725 | 1.62 | | 1.66
$\frac{d\sigma}{d\Omega}(\pi^{0})$ | 1.15 | 1.1-1.68 | 1796 | 1.09 | | 1.15
| 1.45 | 1.1-1.62 | 1878 | 1.15 | | 1.18
| 3 | 1.11-1.39 | 1800 | 1.41 | | 1.37
| 3.5 | 1.11-1.39 | 1800 | 1.22 | | 1.24
| 4.2 | 1.11-1.39 | 1800 | 1.16 | | 1.19
| 5 | 1.11-1.39 | 1800 | 0.82 | | 0.88
| 6 | 1.11-1.39 | 1800 | 0.66 | | 0.67
Table 4: The data included in the third stage of the analysis: the data for
$\frac{d\sigma}{d\Omega}(\pi^{0})$ at $Q^{2}=0.75-1.45$ and $3-6~{}$GeV2 are
from Refs. Joo1 and Park , respectively; the data for
$\frac{d\sigma}{d\Omega}(\pi^{+})$ are from Ref. Alder .
## III Analysis approaches
The approaches we use to analyse the data, DR and UIM, are described in detail
in Refs. Azn0 ; Azn04 and were successfully employed in Refs. Azn0 ; Azn04 ;
Azn065 for the analyses of pion-photoproduction and
low-$Q^{2}$-electroproduction data. In this Section we discuss certain aspects
in these approaches that need a different treatment as we move to higher
$Q^{2}$.
### III.1 Dispersion relations
We use fixed-$t$ dispersion relations for invariant amplitudes defined in
accordance with the following definition of the electromagnetic current
$I^{\mu}$ for the $\gamma^{*}N\rightarrow\pi N$ process Devenish :
$\displaystyle
I^{\mu}\equiv\bar{u}(p_{2})\gamma_{5}{\cal{I}}^{\mu}u(p_{1})\phi_{\pi},$ (1)
$\displaystyle{\cal{I}}^{\mu}=\frac{B_{1}}{2}\left[\gamma^{\mu}k/-k/\gamma^{\mu}\right]+2P^{\mu}B_{2}+2q^{\mu}B_{3}$
(2)
$\displaystyle+2k^{\mu}B_{4}-\gamma^{\mu}B_{5}+k/P^{\mu}B_{6}+k/k^{\mu}B_{7}+k/q^{\mu}B_{8},$
where $k,~{}q,~{}p_{1},~{}p_{2}$ are the four-momenta of the virtual photon,
pion, and initial and final nucleons, respectively;
$P=\frac{1}{2}(p_{1}+p_{2}),~{}B_{1}(s,t,Q^{2}),B_{2}(s,t,Q^{2}),...B_{8}(s,t,Q^{2})$
are the invariant amplitudes that are functions of the invariant variables
$s=(k+p_{1})^{2},~{}t=(k-q)^{2},~{}Q^{2}\equiv-k^{2}$; $u(p_{1})$, $u(p_{2})$
are the Dirac spinors of the initial and final state nucleon, and $\phi_{\pi}$
is the pion field.
The conservation of $I^{\mu}$ leads to the relations:
$\displaystyle 4Q^{2}B_{4}=(s-u)B_{2}-2(t+Q^{2}-m_{\pi}^{2})B_{3},$ (3)
$\displaystyle 2Q^{2}B_{7}=-2B^{\prime}_{5}-(t+Q^{2}-m_{\pi}^{2})B_{8},$ (4)
where $B^{\prime}_{5}\equiv B_{5}-\frac{1}{4}(s-u)B_{6}$. Therefore, only six
of the eight invariant amplitudes are independent. In Ref. Azn0 , the
following independent amplitudes were chosen:
$B_{1},B_{2},B_{3},B^{\prime}_{5},B_{6},B_{8}$. Taking into account the
isotopic structure, we have 18 independent invariant amplitudes. For the
amplitudes
$B_{1}^{(\pm,0)},B_{2}^{(\pm,0)},B_{3}^{(+,0)},{B^{\prime}}_{5}^{(\pm,0)},B_{6}^{(\pm,0)},B_{8}^{(\pm,0)}$,
unsubtracted dispersion relations at fixed $t$ can be written. The only
exception is the amplitude $B_{3}^{(-)}$, for which a subtraction is
neccessary:
$\displaystyle
Re~{}B_{3}^{(-)}(s,t,Q^{2})=f_{sub}(t,Q^{2})-ge\frac{F_{\pi}(Q^{2})}{t-m_{\pi}^{2}}$
$\displaystyle-\frac{ge}{4}\left[F_{1}^{p}(Q^{2})-F_{1}^{n}(Q^{2})\right]\left(\frac{1}{s-m^{2}}+\frac{1}{u-m^{2}}\right)$
(5)
$\displaystyle+\frac{P}{\pi}\int\limits_{s_{thr}}^{\infty}Im~{}B_{3}^{(-)}(s^{\prime},t,Q^{2})\left(\frac{1}{s^{\prime}-s}+\frac{1}{s^{\prime}-u}\right)ds^{\prime},$
where $g^{2}/4\pi=13.8$, $e^{2}/4\pi=1/137$, $F_{\pi}(Q^{2})$ is the pion form
factor, $F_{1}^{N}(Q^{2})$ is the nucleon Pauli form factor, and $m$ and
$m_{\pi}$ are the nucleon and pion masses, respectively.
At $Q^{2}=0$, using the relation $B_{3}=B_{2}\frac{s-u}{2(t-m_{\pi}^{2})}$,
which follows from Eq. (3), and DR for the amplitude $B_{2}(s,t,Q^{2}=0)$, one
obtains:
$f_{sub}(t,Q^{2})=4\frac{P}{\pi}\int\limits_{s_{thr}}^{\infty}\frac{Im~{}B_{3}^{(-)}(s^{\prime},t,Q^{2})}{u^{\prime}-s^{\prime}}ds^{\prime},$
(6)
where $u^{\prime}=2m^{2}+m_{\pi}^{2}-Q^{2}-s^{\prime}-t$.
This expression for $f_{sub}(t,Q^{2})$ was successfully used for the analysis
of pion photoproduction and low $Q^{2}=$0.4, 0.65 GeV2 electroproduction data
Azn0 ; Azn04 . However, it turned out that it is not suitable at higher
$Q^{2}$. Using a simple parametrization:
$f_{sub}(t,Q^{2})=f_{1}(Q^{2})+f_{2}(Q^{2})t,$ (7)
a suitable subtraction was found from the fit to the data for
$Q^{2}=1.7-4.5~{}$GeV2 Park . The linear parametrization in $t$ is also
consistent with the subtraction found from Eq. (6) at low $Q^{2}$. Fig. 1
demonstrates smooth transition of the results for the coefficients
$f_{1}(Q^{2}),f_{2}(Q^{2})$ found at low $Q^{2}<0.7~{}$GeV2 using Eq. (6) to
those at large $Q^{2}=1.7-4.5~{}$GeV2 found from the fit to the data Park .
Figure 1: $Q^{2}$ dependence of the coefficients $f_{1}(Q^{2})$ (solid curve)
and $f_{2}(Q^{2})$ (dashed curve) from Eq. (7). The results for
$Q^{2}<0.7~{}$GeV2 were found using Eq. (6), whereas the results for
$Q^{2}=1.7-4.5~{}$GeV2 are from the fit to the data Park .
Fig. 2 shows the relative contribution of $f_{sub}(t,Q^{2})$ compared with the
pion contribution in Eq. (5) at $Q^{2}=0$ and $Q^{2}=2.44~{}$GeV2. It can be
seen that the contribution of $f_{sub}(t,Q^{2})$ is comparable with the pion
contribution only at large $|t|$, where the latter is small. At small $|t|$,
$f_{sub}(t,Q^{2})$ is very small compared to the pion contribution.
Figure 2: The pion contribution in GeV-2 units (solid curves) to the DR for
the amplitude $B_{3}^{(-)}(s,t,Q^{2})$, Eq. (5), compared to
$f_{sub}(t,Q^{2})$ at $Q^{2}=0$ (a) and $Q^{2}=2.44~{}$GeV2 (b). The dashed
curves represent $f_{sub}(t,Q^{2})$ taken in the form of Eq. (6), the dash-
dotted curve corresponds to the results for $f_{sub}(t,Q^{2})$ obtained by
fitting the data Park . At $Q^{2}=2.44~{}$GeV2, the physical region is located
on the right side of the dotted vertical line.
### III.2 Unitary isobar model
The UIM of Ref. Azn0 was developed on the basis of the model of Ref. Drechsel
. One of the modifications made in Ref. Azn0 consisted in the incorporation
of Regge poles with increasing energies. This allowed us to describe pion
photoproduction multipole amplitudes GWU0 ; GWU3 with a unified Breit-Wigner
parametrization of resonance contributions in the form close to that
introduced by Walker Walker . The Regge-pole amplitudes were constructed using
a gauge invariant Regge-trajectory-exchange model developed in Refs. Laget1 ;
Laget2 . This model gives a good description of the pion photoproduction data
above the resonance region and can be extended to finite $Q^{2}$ Laget3 .
The incorporation of Regge poles into the background of UIM, built from the
nucleon exchanges in the $s$\- and $u$-channels and $t$-channel $\pi$, $\rho$
and $\omega$ exchanges, was made in Ref. Azn0 in the following way:
$\displaystyle Background$ (8)
$\displaystyle=[N+\pi+\rho+\omega]_{UIM}~{}at~{}s<s_{0},$
$\displaystyle=[N+\pi+\rho+\omega]_{UIM}\frac{1}{1+(s-s_{0})^{2}}+$
$\displaystyle
Re[\pi+\rho+\omega+b_{1}+a_{2}]_{Regge}\frac{(s-s_{0})^{2}}{1+(s-s_{0})^{2}}~{}at~{}s>s_{0}.$
Here the Regge-pole amplitudes were taken from Refs. Laget1 ; Laget2 and
consisted of reggeized $\pi$, $\rho$, $\omega$, $b_{1}$, and $a_{2}$
$t$-channel exchange contributions. This background was unitarized in the
$K$-matrix approximation. The value of $s_{0}\simeq 1.2~{}$GeV2 was found in
Ref. Azn0 from the description of the pion photoproduction multipole
amplitudes GWU0 ; GWU3 . With this value of $s_{0}$, we obtained a good
description of $\pi$ electroproduction data at $Q^{2}=0.4$ and $0.65~{}$GeV2
in the first, second and third resonance regions Azn04 ; Azn065 . The
modification of Eq. (8) was important to obtain a better description of the
data in the second and third resonance regions, but played an insignificant
role at $\sqrt{s}<1.4~{}$GeV.
When the relation in Eq. (8) was applied for $Q^{2}\geq 0.9~{}$GeV2, the best
description of the data was obtained with $\sqrt{s}_{0}>1.8~{}$GeV.
Consequently, in the analysis of the data Park , the background of UIM was
built just from the nucleon exchanges in the $s$\- and $u$-channels and
$t$-channel $\pi$, $\rho$ and $\omega$ exchanges.
## IV $N,\pi,\rho,\omega$ contributions
In both approaches, DR and UIM, the non-resonant background contains Born
terms corresponding to the $s$\- and $u$-channel nucleon exchanges and
$t$-channel pion contribution, and therefore depends on the proton, neutron,
and pion form factors. The background of the UIM also contains the $\rho$ and
$\omega$ $t$-channel exchanges and, therefore, the contribution of the form
factors $G_{\rho(\omega)\rightarrow\pi\gamma}(Q^{2})$. All these form factors,
except the neutron electric and $G_{\rho(\omega)\rightarrow\pi\gamma}(Q^{2})$
ones, are known in the region of $Q^{2}$ that is the subject of this study.
For the proton form factors we used the parametrizations found for the
existing data in Ref. Melnitchouk . The neutron magnetic form factor and the
pion form factor were taken from Refs. Lung ; Lachniet and Bebek1 ; Bebek2 ;
Horn ; Tadevos , respectively. The neutron electric form factor,
$G_{E_{n}}(Q^{2})$, is measured up to $Q^{2}=1.45~{}$GeV2 Madey , and Ref.
Madey presents a parametrization for all existing data on $G_{E_{n}}(Q^{2})$,
which we used for the extrapolation of $G_{E_{n}}(Q^{2})$ to
$Q^{2}>1.45~{}$GeV2. In our final results at high $Q^{2}$, we allow for up to
a $50\%$ deviation from this parametrization that is accounted for in the
systematic uncertainty. There are no measurements of the form factors
$G_{\rho(\omega)\rightarrow\pi\gamma}(Q^{2})$; however, investigations made
using both QCD sum rules Eletski and a quark model AznOgan predict a $Q^{2}$
dependence of $G_{\rho(\omega)\rightarrow\pi\gamma}(Q^{2})$ close to the
dipole form $G_{D}(Q^{2})=1/(1+\frac{Q^{2}}{0.71GeV^{2}})^{2}$. We used this
dipole form in our analysis and introduced in our final results a systematic
uncertainty that accounts for a $20\%$ deviation from $0.71~{}$GeV2. All
uncertainties, including those arising from the measured proton, neutron and
pion form factors, were added in quadrature and will be, as one part of our
total model uncertainties, referenced as model uncertainties (I) of our
results.
## V Resonance contributions
We have taken into account all well-established resonances from the first,
second, and third resonance regions. These are 4- and 3-star resonances:
$\Delta(1232)P_{33}$, $N(1440)P_{11}$, $N(1520)D_{13}$, $N(1535)S_{11}$,
$\Delta(1600)P_{33}$, $\Delta(1620)S_{31}$, $N(1650)S_{11}$, $N(1675)D_{15}$,
$N(1680)F_{15}$, $N(1700)D_{13}$, $\Delta(1700)D_{33}$, $N(1710)P_{11}$, and
$N(1720)P_{13}$. For the masses, widths, and $\pi N$ branching ratios of these
resonances we used the mean values of the data from the Review of Particle
Physics (RPP) PDG . They are presented in Table 5. Resonances of the fourth
resonance region have no influence in the energy region under investigation
and were not included.
Resonance contributions to the multipole amplitudes were parametrized in the
usual Breit-Wigner form with energy-dependent widths Walker . An exception was
made for the $\Delta(1232)P_{33}$ resonance, which was treated differently.
According to the phase-shift analyses of $\pi N$ scattering, the $\pi N$
amplitude corresponding to the $\Delta(1232)P_{33}$ resonance is elastic up to
$W=1.43~{}$GeV (see, for example, the latest GWU analyses GWU1 ; GWU2 ). In
combination with DR and Watson’s theorem, this provides strict constraints on
the multipole amplitudes $M_{1+}^{3/2}$, $E_{1+}^{3/2}$, $S_{1+}^{3/2}$ that
correspond to the $\Delta(1232)P_{33}$ resonance Azn0 . In particular, it was
shown Azn0 that with increasing $Q^{2}$, the $W$-dependence of $M_{1+}^{3/2}$
remains unchanged and close to that from the GWU analysis GWU3 at $Q^{2}=0$,
if the same normalizations of the amplitudes at the resonance position are
used. This constraint on the large $M_{1+}^{3/2}$ amplitude plays an important
role in the reliable extraction of the amplitudes for the
$\gamma^{*}N\rightarrow\Delta(1232)P_{33}$ transition. It also impacts the
analysis of the second resonance region, because resonances from this region
overlap with the $\Delta(1232)P_{33}$.
$N^{*}$ | | | | $M$(MeV) | | | $\tilde{M}$(MeV) | | | $\Gamma$(MeV) | | | $\tilde{\Gamma}$(MeV) | | | $\beta_{\pi N}(\%)$ | | | $\tilde{\beta}_{\pi N}(\%)$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\Delta(1232)P_{33}$ | | | | $1231-1233$ | | | $1232$ | | | $116-120$ | | | $118$ | | | $100$ | | | $100$ |
$N(1440)P_{11}$ | | | | $1420-1470$ | | | $1440$ | | | $200-450$ | | | $350$ | | | $55-75$ | | | $60$ |
$N(1520)D_{13}$ | | | | $1515-1525$ | | | $1520$ | | | $100-125$ | | | $112$ | | | $55-65$ | | | $60$ |
$N(1535)S_{11}$ | | | | $1525-1545$ | | | $1535$ | | | $125-175$ | | | $150$ | | | $35-55$ | | | $45$ |
$\Delta(1600)P_{33}$ | | | | $1550-1700$ | | | $1600$ | | | $250-450$ | | | $350$ | | | $10-25$ | | | $20$ |
$\Delta(1620)S_{31}$ | | | | $1600-1660$ | | | $1630$ | | | $135-150$ | | | $145$ | | | $20-30$ | | | $25$ |
$N(1650)S_{11}$ | | | | $1645-1670$ | | | $1655$ | | | $145-185$ | | | $165$ | | | $60-95$ | | | $75$ |
$N(1675)D_{15}$ | | | | $1670-1680$ | | | $1675$ | | | $130-165$ | | | $150$ | | | $35-45$ | | | $40$ |
$N(1680)F_{15}$ | | | | $1680-1690$ | | | $1685$ | | | $120-140$ | | | $130$ | | | $65-70$ | | | $65$ |
$N(1700)D_{13}$ | | | | $1650-1750$ | | | $1700$ | | | $50-150$ | | | $100$ | | | $5-15$ | | | $10$ |
$\Delta(1700)D_{33}$ | | | | $1670-1750$ | | | $1700$ | | | $200-400$ | | | $300$ | | | $10-20$ | | | $15$ |
$N(1710)P_{11}$ | | | | $1680-1740$ | | | $1710$ | | | $50-250$ | | | $100$ | | | $10-20$ | | | $15$ |
$N(1720)P_{13}$ | | | | $1700-1750$ | | | $1720$ | | | $150-300$ | | | $200$ | | | $10-20$ | | | $15$ |
Table 5: List of masses, widths, and branching ratios of the resonances
included in our analysis. The quoted ranges are taken from RPP PDG . The
quantities labeled by tildes ($\tilde{M}$, $\tilde{\Gamma}$,
$\tilde{\beta}_{\pi N}$) correspond to the values used in the analysis and in
the extraction of the $\gamma^{*}p\rightarrow N^{*}$ helicity amplitudes.
The fitting parameters in our analyses were the $\gamma^{*}p\rightarrow N^{*}$
helicity amplitudes, $A_{1/2}$, $A_{3/2}$, $S_{1/2}$. They are related to the
resonant portions of the multipole amplitudes at the resonance positions. For
the resonances with $J^{P}=\frac{1}{2}^{-},\frac{3}{2}^{+},...$, these
relations are the following:
$\displaystyle A_{1/2}=-\frac{1}{2}\left[(l+2){\cal E}_{l+}+l{\cal
M}_{l+}\right],$ (9) $\displaystyle
A_{3/2}=\frac{\left[l(l+2)\right]^{1/2}}{2}({\cal E}_{l+}-{\cal M}_{l+}),$
(10) $\displaystyle S_{1/2}=-\frac{1}{\sqrt{2}}(l+1){\cal S}_{l+}.$ (11)
For the resonances with $J^{P}=\frac{1}{2}^{+},\frac{3}{2}^{-},...$:
$\displaystyle A_{1/2}=\frac{1}{2}\left[(l+2){\cal M}_{(l+1)-}-l{\cal
E}_{(l+1)-}\right],$ (12) $\displaystyle
A_{3/2}=-\frac{\left[l(l+2)\right]^{1/2}}{2}({\cal E}_{(l+1)-}+{\cal
M}_{(l+1)-}),$ (13) $\displaystyle S_{1/2}=-\frac{1}{\sqrt{2}}(l+1){\cal
S}_{(l+1)-},$ (14)
where $J$ and $P$ are the spin and parity of the resonance, $l=J-\frac{1}{2}$,
and
$\displaystyle{\cal M}_{l\pm}({\cal E}_{l\pm},{\cal S}_{l\pm})\equiv
aImM^{R}_{l\pm}(E^{R}_{l\pm},S^{R}_{l\pm})(W=M),$ (15) $\displaystyle
a\equiv\frac{1}{C_{I}}\left[(2J+1)\pi\frac{q_{r}}{K}\frac{M}{m}\frac{\Gamma}{\beta_{\pi
N}}\right]^{1/2},$ $\displaystyle
C_{1/2}=-\sqrt{\frac{1}{3}},~{}C_{3/2}=\sqrt{\frac{2}{3}}~{}for~{}\gamma^{*}p\rightarrow\pi^{0}p,$
$\displaystyle
C_{1/2}=-\sqrt{\frac{2}{3}},~{}C_{3/2}=-\sqrt{\frac{1}{3}}~{}for~{}\gamma^{*}p\rightarrow\pi^{+}n.$
Here $C_{I}$ are the isospin Clebsch-Gordon coefficients in the decay
$N^{*}\rightarrow\pi N$; $\Gamma$, $M$, and $I$ are the total width, mass, and
isospin of the resonance, respectively, $\beta_{\pi N}$ is its branching ratio
to the $\pi N$ channel, $K$ and $q_{r}$ are the photon equivalent energy and
the pion momentum at the resonance position in c.m. system. For the transverse
amplitudes $A_{1/2}$ and $A_{3/2}$, these relations were introduced by Walker
Walker ; for the longitudinal amplitudes, they agree with those from Refs.
Arndt ; Capstick ; Kamalov1 .
The masses, widths, and $\pi N$ branching ratios of the resonances are known
in the ranges presented in Table 5. The uncertainties of masses and widths of
the $N(1440)P_{11}$, $N(1520)D_{13}$, and $N(1535)S_{11}$ are quite
significant and can affect the resonant portions of the multipole amplitudes
for these resonances at the resonance positions. These uncertainties were
taken into account by refitting the data multiple times with the width (mass)
of each of the resonances changed within one standard deviation111The standard
deviations were defined as $\sigma_{M}=(M_{max}-M_{min})/\sqrt{12}$ and
$\sigma_{\Gamma}=(\Gamma_{max}-\Gamma_{min})/\sqrt{12}$, with the maximum and
minimum values as shown in Table V. while keeping those for other resonances
fixed. The resulting uncertainties of the $\gamma^{*}p\rightarrow
N(1440)P_{11}$, $N(1520)D_{13}$, $N(1535)S_{11}$ amplitudes were added in
quadrature and considered as model uncertainties (II).
In Sec. II, we discussed that in the analysis of the data reported in Table 2,
there is another uncertainty in the amplitudes for the $N(1440)P_{11}$,
$N(1520)D_{13}$, and $N(1535)S_{11}$, which is caused by the limited
information available on magnitudes of resonant amplitudes in the third
resonance region. To evaluate the influence of these states on the extracted
$\gamma^{*}p\rightarrow N(1440)P_{11}$, $N(1520)D_{13}$, $N(1535)S_{11}$
amplitudes, we used two ways of estimating their strength.
(i) Directly including these states in the fit, taking the corresponding
amplitudes $A_{1/2}$, $A_{3/2}$, $S_{1/2}$ as free parameters.
(ii) Applying some constraints on their amplitudes. Using symmetry relations
within the $[70,1^{-}]$ multiplet given by the single quark transition model
SQTM , we have related the transverse amplitudes for the members of this
multiplet ($\Delta(1620)S_{31}$, $N(1650)S_{11}$, $N(1675)D_{15}$,
$N(1700)D_{13}$, and $\Delta(1700)D_{33}$) to the amplitudes of
$N(1520)D_{13}$ and $N(1535)S_{11}$ that are well determined in the analysis.
The longitudinal amplitudes of these resonances and the amplitudes of the
resonances $\Delta(1600)P_{33}$ and $N(1710)P_{11}$, which have small
photocouplings PDG and are not seen in low $Q^{2}$ $\pi$ and 2$\pi$
electroproduction Azn065 , were assumed to be zero.
The results obtained for $N(1440)P_{11}$, $N(1520)D_{13}$, and $N(1535)S_{11}$
using the two procedures are very close to each other. The amplitudes for
these resonances presented below are the average values of the results
obtained in these fits. The uncertainties arising from this averaging
procedure were added in quadrature to the model uncertainties (II).
## VI Results
Results for the extracted $\gamma^{*}p\rightarrow\Delta(1232)P_{33}$,
$N(1440)P_{11}$, $N(1520)D_{13}$, $N(1535)S_{11}$ amplitudes are presented in
Tables 6-12. Here we show separately the amplitudes obtained in the DR and UIM
approaches. The amplitudes are presented with the fit errors and model
uncertainties caused by the $N,\pi,\rho$, and $\omega$ contributions to the
background, and those caused by the masses and widths of the $N(1440)P_{11}$,
$N(1520)D_{13}$, and $N(1535)S_{11}$, and by the resonances of the third
resonance region. These uncertainties, discussed in Sections IV and V, and
referred to as model uncertainties (I) and (II), were added in quadrature and
represent model uncertainties of the DR and UIM results.
The DR and UIM approaches give comparable descriptions of the data (see
$\chi^{2}$ values in Tables 1-4), and, therefore, the differences in
$A_{1/2},A_{3/2},S_{1/2}$ are related only to the model assumptions. We,
therefore, ascribe the difference in the results obtained in the two
approaches to model uncertainty, and present as our final results in Tables
6-10 and 12 the mean values of the amplitudes extracted using DR and UIM. The
uncertainty that originates from the averaging is considered as an additional
model uncertainty - uncertainty (III). Along with the average values of the
uncertainties (I) and (II) obtained in the DR and UIM approaches, it is
included in quadrature in the total model uncertainties of the average
amplitudes.
In the fit we have included the experimental point-to-point systematics by
adding them in quadrature with the statistical error. We also took into
account the overall normalization error of the CLAS cross sections data which
is about 5%. It was checked that the overall normalization error results in
modifications of all extracted amplitudes, except $M_{1+}^{3/2}$, that are
significantly smaller than the fit errors of these amplitudes. For
$M_{1+}^{3/2}$, this error results in the overall normalization error which is
larger than the fit error. It is about 2.5% for low $Q^{2}$, and increases up
to 3.2-3.3% at $Q^{2}=3-6~{}$GeV2. For $M_{1+}^{3/2}$, the fit error given in
Table 6 includes the overall normalization error added in quadrature to the
fit error.
Examples of the comparison with the experimental data are presented in Figs.
3-12. The obtained values of $\chi^{2}$ in the fit to the data are presented
in Tables 1-4. The relatively large values of $\chi^{2}$ for
$\frac{d\sigma}{d\Omega}(\pi^{0})$ at $Q^{2}=0.16,0.2~{}$GeV2 and for
$\frac{d\sigma}{d\Omega}(\pi^{+})$ at $Q^{2}=0.3,0.4~{}$GeV2 and
$Q^{2}=1.72,2.05~{}$GeV2 are caused by small statistical errors, which for
each data set Smith , Egiyan and Park , increase with increasing $Q^{2}$. The
values of $\chi^{2}$ for $A_{LT^{\prime}}$ at $Q^{2}\geq 1.72~{}$GeV2 are
somewhat large. However, as demonstrated in Figs. 5,6, the description on the
whole is satisfactory.
$~{}~{}Q^{2}$ | $~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}ImM^{3/2}_{1+}$($\sqrt{\mu b}$), W=1.232 GeV |
---|---|---
(GeV2) | |
| DR UIM | Final results
0.3 | $~{}~{}~{}5.173\pm 0.130~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}5.122\pm 0.130\pm 0.004$ | $~{}~{}~{}~{}~{}~{}5.148\pm 0.130\pm 0.026~{}$
0.4 | $~{}~{}~{}4.843\pm 0.122~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}4.803\pm 0.122\pm 0.005$ | $~{}~{}~{}~{}~{}~{}4.823\pm 0.122\pm 0.021~{}$
0.525 | $~{}~{}~{}4.277\pm 0.109~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}4.238\pm 109\pm 0.008$ | $~{}~{}~{}~{}~{}~{}4.257\pm 0.109\pm 0.021~{}$
0.65 | $~{}~{}~{}3.814\pm 0.097~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}3.794\pm 0.097\pm 0.009$ | $~{}~{}~{}~{}~{}~{}3.804\pm 0.097\pm 0.013~{}$
0.75 | $~{}~{}~{}3.395\pm 0.088~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}3.356\pm 0.088\pm 0.011$ | $~{}~{}~{}~{}~{}~{}3.375\pm 0.088\pm 0.022~{}$
0.9 | $~{}~{}~{}3.010\pm 0.078~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}2.962\pm 0.078\pm 0.012$ | $~{}~{}~{}~{}~{}~{}2.986\pm 0.078\pm 0.027~{}$
1.15 | $~{}~{}~{}2.487\pm 0.066~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}2.438\pm 0.066\pm 0.013$ | $~{}~{}~{}~{}~{}~{}2.463\pm 0.066\pm 0.028~{}$
1.45 | $~{}~{}~{}1.948\pm 0.059~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}1.880\pm 0.059\pm 0.014$ | $~{}~{}~{}~{}~{}~{}1.914\pm 0.059\pm 0.037~{}$
3.0 | $~{}~{}~{}0.725\pm 0.022\pm 0.011~{}~{}~{}~{}0.693\pm 0.022\pm 0.016$ | $~{}~{}~{}~{}~{}~{}0.709\pm 0.022\pm 0.023~{}$
3.5 | $~{}~{}~{}0.582\pm 0.018\pm 0.012~{}~{}~{}~{}0.558\pm 0.018\pm 0.017$ | $~{}~{}~{}~{}~{}~{}0.570\pm 0.018\pm 0.021~{}$
4.2 | $~{}~{}~{}0.434\pm 0.014\pm 0.014~{}~{}~{}~{}0.412\pm 0.014\pm 0.018$ | $~{}~{}~{}~{}~{}~{}0.423\pm 0.014\pm 0.021~{}$
5.0 | $~{}~{}~{}0.323\pm 0.012\pm 0.021~{}~{}~{}~{}0.312\pm 0.012\pm 0.023$ | $~{}~{}~{}~{}~{}~{}0.317\pm 0.012\pm 0.024~{}$
6.0 | $~{}~{}~{}0.200\pm 0.012\pm 0.024~{}~{}~{}~{}0.191\pm 0.012\pm 0.027$ | $~{}~{}~{}~{}~{}~{}0.196\pm 0.012\pm 0.027~{}$
Table 6: The results for the imaginary part of $M^{3/2}_{1+}$ at $W=1.232~{}$GeV. For the DR and UIM results, the first and second uncertainties are the statistical uncertainty from the fit and the model uncertainty (I) (see Sec. IV), respectively. For $Q^{2}=0.3-1.45~{}$GeV2, the uncertainty (I) is practically related only to the form factors $G_{\rho,\omega}(Q^{2})$; for this reason it does not affect the amplitudes found using DR. Final results are the average values of the amplitudes found using DR and UIM; here the first uncertainty is statistical, and the second one is the model uncertainty discussed in Sec. VI. $~{}~{}Q^{2}$ | $~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}R_{EM}$($\%$) |
---|---|---
(GeV2) | |
| DR UIM | Final results
0.16 | $~{}~{}-2.0\pm 0.1~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-1.7\pm 0.1\pm 0.04$ | $~{}~{}~{}-1.9\pm 0.1\pm 0.2~{}~{}$
0.2 | $~{}~{}-1.9\pm 0.2~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-1.6\pm 0.2\pm 0.04$ | $~{}~{}~{}-1.8\pm 0.2\pm 0.2~{}~{}$
0.24 | $~{}~{}-2.2\pm 0.2~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-2.1\pm 0.2\pm 0.1$ | $~{}~{}~{}-2.2\pm 0.2\pm 0.1~{}~{}$
0.28 | $~{}~{}-1.9\pm 0.2~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-1.6\pm 0.2\pm 0.1$ | $~{}~{}~{}-1.8\pm 0.2\pm 0.2~{}~{}$
0.3 | $~{}~{}-2.2\pm 0.2~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-2.1\pm 0.2\pm 0.1$ | $~{}~{}~{}-2.1\pm 0.2\pm 0.1~{}~{}$
0.32 | $~{}~{}-1.9\pm 0.2~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-1.6\pm 0.2\pm 0.1$ | $~{}~{}~{}-1.8\pm 0.2\pm 0.2~{}~{}$
0.36 | $~{}~{}-1.8\pm 0.2~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-1.5\pm 0.3\pm 0.1$ | $~{}~{}~{}-1.7\pm 0.3\pm 0.2~{}~{}$
0.4 | $~{}~{}-2.9\pm 0.2~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-2.4\pm 0.2\pm 0.1$ | $~{}~{}~{}-2.7\pm 0.2\pm 0.3~{}~{}$
0.525 | $~{}~{}-2.3\pm 0.3~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-2.0\pm 0.3\pm 0.1$ | $~{}~{}~{}-2.2\pm 0.3\pm 0.2~{}~{}$
0.65 | $~{}~{}-2.0\pm 0.4~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-1.4\pm 0.3\pm 0.1$ | $~{}~{}~{}-1.7\pm 0.4\pm 0.3~{}~{}$
0.75 | $~{}~{}-2.2\pm 0.4~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-1.9\pm 0.4\pm 0.1$ | $~{}~{}~{}-2.1\pm 0.4\pm 0.2~{}~{}$
0.9 | $~{}~{}-2.4\pm 0.5~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-2.1\pm 0.5\pm 0.2$ | $~{}~{}~{}-2.2\pm 0.5\pm 0.3~{}~{}$
1.15 | $~{}~{}-2.0\pm 0.6~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-2.6\pm 0.5\pm 0.2$ | $~{}~{}~{}-2.3\pm 0.6\pm 0.4~{}~{}$
1.45 | $~{}~{}-2.4\pm 0.7~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-2.5\pm 0.7\pm 0.2$ | $~{}~{}~{}-2.5\pm 0.7\pm 0.2~{}~{}$
3.0 | $~{}~{}-1.6\pm 0.4\pm 0.1~{}~{}~{}~{}~{}-2.3\pm 0.4\pm 0.2$ | $~{}~{}~{}-2.0\pm 0.4\pm 0.4~{}~{}$
3.5 | $~{}~{}-1.8\pm 0.5\pm 0.2~{}~{}~{}~{}~{}-1.1\pm 0.5\pm 0.3$ | $~{}~{}~{}-1.5\pm 0.5\pm 0.5~{}~{}$
4.2 | $~{}~{}-2.3\pm 0.8\pm 0.3~{}~{}~{}~{}~{}-2.9\pm 0.7\pm 0.4$ | $~{}~{}~{}-2.6\pm 0.8\pm 0.4~{}~{}$
5.0 | $~{}~{}-2.2\pm 1.4\pm 0.3~{}~{}~{}~{}~{}-3.2\pm 1.5\pm 0.4$ | $~{}~{}~{}-2.7\pm 1.5\pm 0.6~{}~{}$
6.0 | $~{}~{}-2.1\pm 2.5\pm 1.1~{}~{}~{}~{}~{}-3.6\pm 2.6\pm 1.5$ | $~{}~{}~{}-2.8\pm 2.6\pm 1.7~{}~{}$
Table 7: The results for the ratio $R_{EM}\equiv ImE^{3/2}_{1+}/ImM^{3/2}_{1+}$ at $W=1.232~{}$GeV. All other relevant information is as given in the legend of Table 6. $~{}~{}Q^{2}$ | $~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}R_{SM}$($\%$) |
---|---|---
(GeV2) | |
| DR UIM | Final results
0.16 | $~{}~{}-4.8\pm 0.2~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-4.6\pm 0.2\pm 0.04$ | $~{}~{}~{}-4.7\pm 0.2\pm 0.1~{}~{}$
0.2 | $~{}~{}-4.9\pm 0.2~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-4.4\pm 0.2\pm 0.1$ | $~{}~{}~{}-4.7\pm 0.2\pm 0.3~{}~{}$
0.24 | $~{}~{}-4.7\pm 0.3~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-4.5\pm 0.3\pm 0.1$ | $~{}~{}~{}-4.6\pm 0.3\pm 0.1~{}~{}$
0.28 | $~{}~{}-5.6\pm 0.3~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-5.4\pm 0.3\pm 0.1$ | $~{}~{}~{}-5.5\pm 0.3\pm 0.1~{}~{}$
0.3 | $~{}~{}-5.4\pm 0.2~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-5.0\pm 0.2\pm 0.1$ | $~{}~{}~{}-5.2\pm 0.2\pm 0.2~{}~{}$
0.32 | $~{}~{}-5.9\pm 0.3~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-5.5\pm 0.3\pm 0.1$ | $~{}~{}~{}-5.7\pm 0.3\pm 0.2~{}~{}$
0.36 | $~{}~{}-5.5\pm 0.3~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-5.2\pm 0.3\pm 0.1$ | $~{}~{}~{}-5.4\pm 0.3\pm 0.2~{}~{}$
0.4 | $~{}~{}-5.9\pm 0.2~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-5.2\pm 0.2\pm 0.1$ | $~{}~{}~{}-5.5\pm 0.2\pm 0.4~{}~{}$
0.525 | $~{}~{}-6.0\pm 0.3~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-5.5\pm 0.3\pm 0.1$ | $~{}~{}~{}-5.8\pm 0.3\pm 0.3~{}~{}$
0.65 | $~{}~{}-7.0\pm 0.4~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-6.2\pm 0.4\pm 0.2$ | $~{}~{}~{}-6.6\pm 0.4\pm 0.4~{}~{}$
0.75 | $~{}~{}-7.3\pm 0.4~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-6.7\pm 0.4\pm 0.2$ | $~{}~{}~{}-7.0\pm 0.4\pm 0.4~{}~{}$
0.9 | $~{}~{}-8.6\pm 0.4~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-8.1\pm 0.4\pm 0.2$ | $~{}~{}~{}-8.4\pm 0.5\pm 0.3~{}~{}$
1.15 | $~{}~{}-8.8\pm 0.5~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-8.0\pm 0.5\pm 0.2$ | $~{}~{}~{}-8.4\pm 0.5\pm 0.4~{}~{}$
1.45 | $~{}~{}-10.5\pm 0.8~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-9.6\pm 0.8\pm 0.2$ | $~{}~{}~{}-10.1\pm 0.8\pm 0.5~{}~{}$
3.0 | $~{}~{}-12.6\pm 0.6\pm 0.1~{}~{}~{}~{}~{}~{}~{}~{}-11.4\pm 0.6\pm 0.2$ | $~{}~{}~{}-12.0\pm 0.6\pm 0.6~{}~{}$
3.5 | $~{}~{}-12.8\pm 0.8\pm 0.3~{}~{}~{}~{}~{}~{}~{}~{}-12.4\pm 0.8\pm 0.4$ | $~{}~{}~{}-12.6\pm 0.8\pm 0.4~{}~{}$
4.2 | $~{}~{}-17.1\pm 1.2\pm 0.5~{}~{}~{}~{}~{}~{}~{}~{}-15.9\pm 1.3\pm 0.7$ | $~{}~{}~{}-16.5\pm 1.3\pm 0.8~{}~{}$
5.0 | $~{}~{}-26.6\pm 2.7\pm 1.2~{}~{}~{}~{}~{}~{}~{}~{}-25.2\pm 2.7\pm 1.5$ | $~{}~{}~{}-25.9\pm 2.7\pm 1.7~{}~{}$
6.0 | $~{}~{}-26.4\pm 5.2\pm 3.2~{}~{}~{}~{}~{}~{}~{}~{}-25.3\pm 5.3\pm 3.8$ | $~{}~{}~{}-25.9\pm 5.3\pm 3.8~{}~{}$
Table 8: The results for the ratio $R_{SM}\equiv ImS^{3/2}_{1+}/ImM^{3/2}_{1+}$ at $W=1.232~{}$GeV. All other relevant information is as given in the legend of Table 6. $Q^{2}$ | DR | UIM | Final results
---|---|---|---
(GeV2) | | |
| $A_{1/2}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}S_{1/2}$ | $A_{1/2}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}S_{1/2}$ | $A_{1/2}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}S_{1/2}$
0.3 | $-15.5\pm 1.2\pm 1.0~{}~{}~{}~{}31.8\pm 1.8\pm 0.8$ | $~{}~{}-24.0\pm 1.2\pm 2.5~{}~{}~{}~{}37.6\pm 1.9\pm 2.5$ | $~{}~{}~{}~{}~{}-19.8\pm 1.2\pm 4.6~{}~{}~{}~{}34.7\pm 1.8\pm 3.3$
0.4 | $-9.4\pm 1.1\pm 0.9~{}~{}~{}~{}30.1\pm 1.4\pm 0.9$ | $~{}~{}-19.7\pm 1.1\pm 3.1~{}~{}~{}~{}34.8\pm 1.3\pm 3.0$ | $~{}~{}~{}~{}~{}-14.6\pm 1.1\pm 5.5~{}~{}~{}~{}32.5\pm 1.3\pm 3.1$
0.5 | $10.5\pm 1.2\pm 0.9~{}~{}~{}~{}30.6\pm 1.5\pm 0.9$ | $~{}~{}~{}-4.6\pm 1.3\pm 3.4~{}~{}~{}~{}36.9\pm 1.6\pm 3.0$ | $~{}~{}~{}~{}~{}3.0\pm 1.2\pm 7.9~{}~{}~{}~{}33.8\pm 1.5\pm 3.7$
0.65 | $19.5\pm 1.3\pm 1.0~{}~{}~{}~{}27.6\pm 1.3\pm 1.0$ | $~{}~{}~{}~{}5.4\pm 1.2\pm 3.4~{}~{}~{}~{}~{}~{}35.2\pm 1.2\pm 3.4$ | $~{}~{}~{}~{}~{}12.4\pm 1.2\pm 7.4~{}~{}~{}~{}31.4\pm 1.2\pm 4.4$
0.9 | $31.9\pm 2.6\pm 4.3~{}~{}~{}~{}30.6\pm 2.1\pm 4.3$ | $~{}~{}~{}~{}~{}18.7\pm 2.7\pm 4.3~{}~{}~{}~{}36.2\pm 2.1\pm 4.2$ | $~{}~{}~{}~{}~{}25.3\pm 2.7\pm 7.9~{}~{}~{}~{}33.4\pm 2.1\pm 5.1$
1.72 | $72.5\pm 1.0\pm 4.4~{}~{}~{}~{}24.8\pm 1.4\pm 5.4$ | $~{}~{}~{}~{}~{}58.5\pm 1.1\pm 4.3~{}~{}~{}~{}26.9\pm 1.3\pm 5.4$ | $~{}~{}~{}~{}~{}65.5\pm 1.0\pm 8.3~{}~{}~{}~{}25.8\pm 1.3\pm 5.5$
2.05 | $72.0\pm 0.9\pm 4.3~{}~{}~{}~{}21.0\pm 1.7\pm 5.1$ | $~{}~{}~{}~{}~{}62.9\pm 0.9\pm 3.4~{}~{}~{}~{}15.5\pm 1.5\pm 5.0$ | $~{}~{}~{}~{}~{}67.4\pm 0.9\pm 6.0~{}~{}~{}~{}18.2\pm 1.6\pm 5.8$
2.44 | $50.0\pm 1.0\pm 3.4~{}~{}~{}~{}~{}9.3\pm 1.3\pm 4.3$ | $~{}~{}~{}~{}~{}56.2\pm 0.9\pm 3.4~{}~{}~{}~{}11.8\pm 1.4\pm 4.3$ | $~{}~{}~{}~{}~{}53.1\pm 1.0\pm 4.6~{}~{}~{}~{}10.6\pm 1.4\pm 4.5$
2.91 | $37.5\pm 1.1\pm 3.0~{}~{}~{}~{}~{}9.8\pm 2.0\pm 2.6$ | $~{}~{}~{}~{}~{}42.5\pm 1.1\pm 3.0~{}~{}~{}~{}13.8\pm 2.1\pm 2.6$ | $~{}~{}~{}~{}~{}40.0\pm 1.1\pm 3.9~{}~{}~{}~{}11.8\pm 2.1\pm 3.3$
3.48 | $29.6\pm 0.8\pm 2.9~{}~{}~{}~{}~{}4.2\pm 2.5\pm 2.6$ | $~{}~{}~{}~{}~{}32.6\pm 0.9\pm 2.8~{}~{}~{}~{}14.1\pm 2.4\pm 2.4$ | $~{}~{}~{}~{}~{}31.1\pm 0.9\pm 3.2~{}~{}~{}~{}9.1\pm 2.5\pm 5.5$
4.16 | $19.3\pm 2.0\pm 4.0~{}~{}~{}~{}10.8\pm 2.8\pm 4.7$ | $~{}~{}~{}~{}~{}23.1\pm 2.2\pm 4.9~{}~{}~{}~{}17.5\pm 2.6\pm 5.6$ | $~{}~{}~{}~{}~{}21.2\pm 2.1\pm 4.9~{}~{}~{}~{}14.1\pm 2.7\pm 6.1$
Table 9: The results for the $\gamma^{*}p\rightarrow~{}N(1440)P_{11}$ helicity amplitudes in units of $10^{-3}$GeV-1/2. For the DR and UIM results, the first and second uncertainties are, respectively, the statistical uncertainty from the fit and the model uncertainty, which consists of uncertainties (I) (Sec. IV) and (II) (Sec. V) added in quadrature. Final results are the average values of the amplitudes found using DR and UIM; here the first uncertainty is statistical and the second one is the model uncertainty discussed in Sec. VI. $Q^{2}$ | DR | UIM | Final results
---|---|---|---
(GeV2) | | |
| $A_{1/2}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}S_{1/2}$ | $A_{1/2}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}S_{1/2}$ | $A_{1/2}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}S_{1/2}$
0.3 | $89.4\pm 2.1\pm 1.3~{}~{}~{}~{}-11.0\pm 2.1\pm 0.9$ | $~{}~{}90.9\pm 2.3\pm 1.8~{}~{}~{}~{}-13.0\pm 2.2\pm 2.1$ | $~{}~{}~{}~{}~{}90.2\pm 2.2\pm 1.7~{}~{}~{}~{}-12.0\pm 2.2\pm 1.8$
0.4 | $90.6\pm 1.7\pm 1.4~{}~{}~{}~{}-9.5\pm 1.9\pm 0.9$ | $~{}~{}92.9\pm 1.6\pm 2.2~{}~{}~{}~{}-15.9\pm 2.0\pm 2.2$ | $~{}~{}~{}~{}~{}91.8\pm 1.7\pm 2.1~{}~{}~{}~{}-12.7\pm 2.0\pm 3.6$
0.5 | $90.5\pm 1.9\pm 1.6~{}~{}~{}~{}-10.8\pm 2.2\pm 1.2$ | $~{}~{}~{}91.7\pm 2.0\pm 2.7~{}~{}~{}~{}-16.7\pm 2.4\pm 2.4$ | $~{}~{}~{}~{}~{}91.1\pm 2.0\pm 2.2~{}~{}~{}~{}-13.8\pm 2.3\pm 3.5$
0.65 | $90.0\pm 1.7\pm 1.8~{}~{}~{}~{}-12.9\pm 1.8\pm 1.0$ | $~{}~{}~{}~{}91.6\pm 1.8\pm 3.3~{}~{}~{}~{}-14.4\pm 1.9\pm 2.3$ | $~{}~{}~{}~{}~{}90.8\pm 1.8\pm 2.7~{}~{}~{}~{}-13.6\pm 1.9\pm 1.8$
0.9 | $83.3\pm 2.4\pm 4.9~{}~{}~{}~{}-11.2\pm 3.8\pm 4.6$ | $~{}~{}~{}~{}~{}85.5\pm 2.3\pm 5.2~{}~{}~{}~{}-16.4\pm 3.9\pm 4.9$ | $~{}~{}~{}~{}~{}84.4\pm 2.4\pm 5.2~{}~{}~{}~{}-13.8\pm 3.9\pm 5.5$
1.72 | $72.2\pm 1.5\pm 5.0~{}~{}~{}~{}-20.4\pm 1.8\pm 3.5$ | $~{}~{}~{}~{}~{}75.7\pm 1.4\pm 4.9~{}~{}~{}~{}-24.8\pm 1.6\pm 3.3$ | $~{}~{}~{}~{}~{}73.9\pm 1.5\pm 5.2~{}~{}~{}~{}-22.6\pm 1.7\pm 4.0$
2.05 | $59.8\pm 1.6\pm 4.0~{}~{}~{}~{}-14.8\pm 2.0\pm 3.9$ | $~{}~{}~{}~{}~{}65.4\pm 1.7\pm 4.0~{}~{}~{}~{}-19.9\pm 1.9\pm 4.4$ | $~{}~{}~{}~{}~{}62.6\pm 1.7\pm 4.9~{}~{}~{}~{}-17.4\pm 1.9\pm 4.9$
2.44 | $54.5\pm 2.1\pm 3.6~{}~{}~{}~{}-11.3\pm 2.7\pm 4.1$ | $~{}~{}~{}~{}~{}59.8\pm 2.2\pm 3.9~{}~{}~{}~{}-16.7\pm 2.9\pm 4.3$ | $~{}~{}~{}~{}~{}57.2\pm 2.2\pm 4.6~{}~{}~{}~{}-14.0\pm 2.8\pm 5.0$
2.91 | $49.6\pm 2.0\pm 4.0~{}~{}~{}~{}~{}~{}-9.0\pm 2.6\pm 2.9$ | $~{}~{}~{}~{}~{}53.0\pm 1.9\pm 4.5~{}~{}~{}~{}-12.6\pm 2.8\pm 4.2$ | $~{}~{}~{}~{}~{}51.3\pm 2.0\pm 4.6~{}~{}~{}~{}-10.8\pm 2.7\pm 4.0$
3.48 | $44.9\pm 2.2\pm 4.2~{}~{}~{}~{}~{}~{}-6.3\pm 3.2\pm 2.7$ | $~{}~{}~{}~{}~{}41.0\pm 2.4\pm 4.6~{}~{}~{}~{}-11.3\pm 3.4\pm 2.8$ | $~{}~{}~{}~{}~{}43.0\pm 2.3\pm 4.8~{}~{}~{}~{}-8.8\pm 3.3\pm 3.7$
4.16 | $35.5\pm 3.8\pm 4.5~{}~{}~{}~{}~{}~{}-4.5\pm 6.2\pm 3.5$ | $~{}~{}~{}~{}~{}31.8\pm 3.6\pm 4.5~{}~{}~{}~{}-8.9\pm 5.9\pm 3.8$ | $~{}~{}~{}~{}~{}33.7\pm 3.7\pm 4.9~{}~{}~{}~{}-6.7\pm 6.0\pm 4.3$
Table 10: The results for the $\gamma^{*}p\rightarrow~{}N(1535)S_{11}$ helicity amplitudes in units of $10^{-3}$GeV-1/2. The amplitudes are extracted from the data on $\gamma^{*}p\rightarrow\pi N$ using $\beta_{\pi N}(N(1535)S_{11})=0.485$ (see Subsection VII,C). The remaining legend is as for Table 9. $Q^{2}$ | DR | UIM
---|---|---
(GeV2) | |
| $A_{1/2}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}A_{3/2}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}S_{1/2}$ | $A_{1/2}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}A_{3/2}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}S_{1/2}$
0.3 | $-51.8\pm 1.9\pm 0.8~{}~{}77.2\pm 2.2\pm 0.7~{}-43.7\pm 2.4\pm 1.0$ | $~{}~{}~{}~{}-54.1\pm 1.8\pm 1.8~{}~{}75.1\pm 2.2\pm 2.1~{}-48.4\pm 2.4\pm 2.3$
0.4 | $-57.0\pm 1.4\pm 0.9~{}~{}70.5\pm 1.8\pm 0.7~{}-39.7\pm 1.9\pm 1.0$ | $~{}~{}~{}~{}-59.7\pm 2.1\pm 2.4~{}~{}67.6\pm 1.9\pm 2.2~{}-43.6\pm 2.1\pm 2.4$
0.5 | $-60.2\pm 2.0\pm 0.9~{}~{}56.9\pm 1.7\pm 0.8~{}-35.5\pm 2.5\pm 0.8$ | $~{}~{}~{}~{}-60.6\pm 2.2\pm 2.5~{}~{}60.0\pm 1.9\pm 2.4~{}-39.4\pm 2.4\pm 2.8$
0.65 | $-66.0\pm 1.6\pm 1.1~{}~{}52.0\pm 1.4\pm 0.8~{}-32.7\pm 2.1\pm 0.7$ | $~{}~{}~{}~{}-64.5\pm 1.8\pm 2.7~{}~{}54.2\pm 1.6\pm 2.8~{}-37.5\pm 1.9\pm 2.5$
0.9 | $-58.9\pm 2.4\pm 2.7~{}~{}44.8\pm 2.6\pm 2.8~{}-29.0\pm 3.3\pm 2.5$ | $~{}~{}~{}~{}-64.9\pm 2.2\pm 2.9~{}~{}44.1\pm 2.6\pm 3.1~{}-34.3\pm 3.1\pm 3.0$
1.72 | $-42.4\pm 1.2\pm 3.2~{}~{}18.7\pm 1.2\pm 3.2~{}-11.8\pm 1.1\pm 3.1$ | $~{}~{}~{}~{}-38.8\pm 1.3\pm 3.9~{}~{}21.4\pm 1.2\pm 3.5~{}~{}-9.1\pm 1.0\pm 1.8$
2.05 | $-37.3\pm 1.4\pm 2.1~{}~{}15.6\pm 1.5\pm 2.3~{}-9.6\pm 1.6\pm 2.8$ | $~{}~{}~{}~{}-39.7\pm 1.5\pm 3.2~{}~{}18.3\pm 1.6\pm 2.6~{}~{}~{}-6.8\pm 1.5\pm 1.9$
2.44 | $-36.4\pm 1.3\pm 2.4~{}~{}~{}11.2\pm 1.6\pm 2.1~{}~{}-5.5\pm 1.8\pm 1.6$ | $~{}~{}~{}~{}-36.3\pm 1.4\pm 2.6~{}~{}13.4\pm 1.7\pm 1.9~{}~{}~{}-3.6\pm 1.9\pm 1.6$
2.91 | $-32.8\pm 1.8\pm 2.6~{}~{}~{}~{}5.8\pm 2.1\pm 2.9~{}~{}-3.3\pm 2.0\pm 1.5$ | $~{}~{}~{}~{}-31.0\pm 1.9\pm 2.2~{}~{}9.6\pm 2.0\pm 2.7~{}~{}~{}-2.3\pm 2.1\pm 1.6$
3.48 | $-22.4\pm 2.1\pm 2.7~{}~{}~{}~{}5.5\pm 2.0\pm 5.5~{}~{}-5.3\pm 2.5\pm 2.0$ | $~{}~{}~{}~{}-24.9\pm 2.2\pm 2.9~{}~{}~{}8.2\pm 2.2\pm 5.2~{}~{}~{}-2.6\pm 2.6\pm 2.4$
4.16 | $-19.1\pm 3.9\pm 3.0~{}~{}~{}~{}6.4\pm 3.0\pm 7.5~{}~{}-2.6\pm 4.8\pm 3.0$ | $~{}~{}~{}~{}-20.9\pm 4.2\pm 3.2~{}~{}~{}4.6\pm 3.2\pm 6.9~{}~{}~{}-0.7\pm 4.6\pm 3.2$
Table 11: The results for the $\gamma^{*}p\rightarrow~{}N(1520)D_{13}$ helicity amplitudes in units of $10^{-3}$GeV-1/2. The remaining legend is as for Table 9. $~{}~{}Q^{2}$ | $A_{1/2}$ | $A_{3/2}$ | $S_{1/2}$
---|---|---|---
(GeV2) | | |
0.3 | $-52.9\pm 1.8\pm 1.7$ | $76.1\pm 2.2\pm 1.7$ | $-46.1\pm 2.4\pm 2.9$
0.4 | $-58.3\pm 1.8\pm 2.1$ | $69.1\pm 1.8\pm 2.1$ | $-41.7\pm 2.0\pm 2.6$
0.5 | $-60.4\pm 2.1\pm 1.7$ | $58.5\pm 1.8\pm 2.2$ | $-37.5\pm 2.5\pm 2.7$
0.65 | $-65.2\pm 1.7\pm 2.0$ | $53.1\pm 1.5\pm 2.1$ | $-35.1\pm 2.0\pm 2.9$
0.9 | $-61.9\pm 2.3\pm 4.1$ | $44.4\pm 2.6\pm 3.0$ | $-31.6\pm 3.2\pm 3.8$
1.72 | $-40.6\pm 1.2\pm 4.0$ | $20.0\pm 1.2\pm 3.6$ | $-10.5\pm 1.0\pm 2.8$
2.05 | $-38.5\pm 1.5\pm 2.9$ | $17.0\pm 1.5\pm 2.8$ | $-8.2\pm 1.5\pm 2.7$
2.44 | $-36.3\pm 1.3\pm 2.5$ | $12.3\pm 1.7\pm 2.3$ | $-4.6\pm 1.8\pm 1.9$
2.91 | $-31.9\pm 1.8\pm 2.6$ | $7.7\pm 2.0\pm 3.4$ | $-2.8\pm 2.0\pm 1.6$
3.48 | $-23.6\pm 2.2\pm 3.1$ | $6.8\pm 2.1\pm 5.5$ | $-4.0\pm 2.5\pm 2.6$
4.16 | $-20.0\pm 4.1\pm 3.2$ | $5.5\pm 3.1\pm 7.3$ | $-1.6\pm 4.7\pm 3.2$
Table 12: The average values of the $\gamma^{*}p\rightarrow~{}N(1520)D_{13}$
helicity amplitudes found using DR and UIM (in units of $10^{-3}$GeV-1/2). The
first uncertainty is statistical, and the second one is the model uncertainty
discussed in Sec. VI.
The comparison with the data for $\frac{d\sigma}{d\Omega}$ and
$A_{LT^{\prime}}$ is made in terms of the structure functions
$\sigma_{T}+\epsilon\sigma_{L}$, $\sigma_{TT}$, $\sigma_{LT}$,
$\sigma_{LT^{\prime}}$ and their Legendre moments. They are defined in the
following way:
$\displaystyle\frac{d\sigma}{d\Omega}=\sigma_{T}+\epsilon\sigma_{L}+\epsilon\sigma_{TT}\cos{2\phi}$
(16)
$\displaystyle+\sqrt{2\epsilon(1+\epsilon)}\sigma_{LT}\cos{\phi}+h\sqrt{2\epsilon(1-\epsilon)}\sigma_{LT^{\prime}}\sin{\phi},$
where $\frac{d\sigma}{d\Omega}$ is the differential cross section of the
reaction $\gamma^{*}N\rightarrow N\pi$ in its c.m. system, assuming that the
virtual photon flux factor is
$\Gamma=\frac{\alpha}{2\pi^{2}Q^{2}}\frac{(W^{2}-m^{2})E_{f}}{2mE_{i}}\frac{1}{1-\epsilon},$
$E_{i},~{}E_{f}$ are the initial and final electron energies in the laboratory
frame, and $\epsilon$ is the polarization factor of the virtual photon.
$\theta$ and $\phi$ are the polar and azimuthal angles of the pion in the c.m.
system of the reaction $\gamma^{*}N\rightarrow N\pi$, and $h$ is the electron
helicity. The longitudinally polarized beam asymmetry is related to the
structure function $\sigma_{LT^{\prime}}$ by:
$A_{LT^{\prime}}=\frac{\sqrt{2\epsilon(1-\epsilon)}\sigma_{LT^{\prime}}\sin{\phi}}{\frac{d\sigma}{d\Omega}(h=0)}.$
(17)
For the longitudinal target asymmetry $A_{t}$ and beam-target asymmetry
$A_{et}$ we use the relations presented in detail in Ref. Biselli , where the
experimental results on these observables are reported. These relations
express $A_{t}$ and $A_{et}$ through the response functions defined in Ref.
Response .
The Legendre moments of structure functions are defined as the coefficients in
the expansion of these functions over Legendre polynomials
$P_{l}(\cos{\theta})$:
$\displaystyle\sigma_{T}(W,\cos{\theta})+\epsilon\sigma_{L}(W,\cos{\theta})$
$\displaystyle=$ (18) $\displaystyle\sum_{l=0}^{n}$ $\displaystyle
D_{l}^{T+L}(W)P_{l}(\cos{\theta}),$
$\displaystyle\sigma_{LT}(W,\cos{\theta})=\sin{\theta}\sum_{l=0}^{n-1}$
$\displaystyle D_{l}^{LT}(W)P_{l}(\cos{\theta}),$ (19)
$\displaystyle\sigma_{LT^{\prime}}(W,\cos{\theta})=\sin{\theta}\sum_{l=0}^{n-1}$
$\displaystyle D_{l}^{LT^{\prime}}(W)P_{l}(\cos{\theta}),$ (20)
$\displaystyle\sigma_{TT}(W,\cos{\theta})=\sin^{2}{\theta}\sum_{l=0}^{n-2}$
$\displaystyle D_{l}^{TT}(W)P_{l}(\cos{\theta}).$ (21)
The Legendre moments allow us to present a comparison of the results with the
data over all energies and angles in compact form.
The Legendre moment $D_{0}^{T+L}$ represents the $\cos{\theta}$ independent
part of $\sigma_{T}+\epsilon\sigma_{L}$, which is related to the
$\gamma^{*}N\rightarrow\pi N$ total cross section:
$\displaystyle
D_{0}^{T+L}=\frac{1}{4\pi}(\sigma^{T}_{tot}+\epsilon\sigma^{L}_{tot})\equiv\frac{|\bf{q}|}{K}(\tilde{\sigma}_{tot}^{T}+\epsilon\tilde{\sigma}_{tot}^{L}),$
(22)
$\displaystyle\tilde{\sigma}_{tot}^{T}=\tilde{\sigma}_{1/2}+\tilde{\sigma}_{3/2},$
$\displaystyle\tilde{\sigma}_{1/2}=\sum_{l=0}^{\infty}(l+1)(|A_{l+}|^{2}+|A_{(l+1)-}|^{2}),$
$\displaystyle\tilde{\sigma}_{3/2}=\sum_{l=1}^{\infty}\frac{l}{4}(l+1)(l+2)(|B_{l+}|^{2}+|B_{(l+1)-}|^{2}),$
$\displaystyle\tilde{\sigma}_{tot}^{L}=\frac{Q^{2}}{\bf{k}^{2}}\sum_{l=0}^{\infty}(l+1)^{3}(|S_{l+}|^{2}+|S_{(l+1)-}|^{2}).$
Here $\bf{q}$ and $\bf{k}$ are, respectively, the pion and virtual photon
three-momenta in the c.m. system of the reaction $\gamma^{*}N\rightarrow\pi
N$, $K=(W^{2}-m^{2})/2W$, and
$\displaystyle A_{l+}=\frac{1}{2}\left[(l+2){E}_{l+}+l{M}_{l+}\right],$ (23)
$\displaystyle B_{l+}={E}_{l+}-{M}_{l+},$ $\displaystyle
A_{(l+1)-}=\frac{1}{2}\left[(l+2){M}_{(l+1)-}-l{E}_{(l+1)-}\right],$
$\displaystyle B_{(l+1)-}={E}_{(l+1)-}+{M}_{(l+1)-}.$
The resonance structures related to the resonances $\Delta(1232)P_{33}$ and
$N(1520)D_{13}$, $N(1535)S_{11}$ are revealed in $D_{0}^{T+L}$ as
enhancements. It can be seen that with increasing $Q^{2}$, the resonant
structure near $1.5~{}$GeV becomes increasingly dominant in comparison with
the $\Delta(1232)$. At $Q^{2}\geq 1.72~{}$GeV2, there is a shoulder between
the $\Delta$ and $1.5~{}$GeV peaks, which is related to the large contribution
of the broad Roper resonance. As can be seen from Table 9, the transverse
helicity amplitude $A_{1/2}$ for $\gamma^{*}p\rightarrow N(1440)P_{11}$, which
is large and negative at $Q^{2}=0$ PDG , crosses zero between $Q^{2}=0.4$ and
$0.65~{}$GeV2 and becomes large and positive at $Q^{2}=1.72~{}$GeV2. With
increasing $Q^{2}$, this amplitude drops smoothly in magnitude.
There are dips in the Legendre moment $D_{2}^{T+L}$ that are caused by the
$\Delta(1232)P_{33}$ and $N(1520)D_{13}$, $N(1535)S_{11}$ resonances. They are
related to the following contributions to $D_{2}^{T}$:
$D_{2}^{T}=-\frac{|\bf{q}|}{K}\left[4Re(A_{0+}A^{*}_{2-})+|M_{1+}|^{2}\right].$
(24)
When $Q^{2}$ grows the dip related to the $\Delta(1232)P_{33}$ resonance
becomes smaller compared to that near $1.5~{}$GeV.
At $Q^{2}>1.72~{}$GeV2, the relative values of the dip in $D_{2}^{T+L}$ and
the enhancement in $D_{0}^{T+L}$ near $1.5~{}$GeV, and the shoulder between
the $\Delta$ and $1.5~{}$GeV peaks in $D_{0}^{T+L}$, remain approximately the
same with increasing $Q^{2}$. Our analysis shows that this is a manifestation
of the slow falloff of the $A_{1/2}$ helicity amplitudes of the transitions
$\gamma^{*}p\rightarrow~{}$ $N(1440)P_{11}$, $N(1535)S_{11}$, $N(1520)D_{13}$
for these $Q^{2}$.
The enhancement in $D_{0}^{T+L}$ and the dip in $D_{0}^{TT}$ in the $\Delta$
peak are mainly related to the $M_{1+}^{3/2}$ amplitude of the
$\gamma^{*}p\rightarrow~{}\Delta(1232)P_{33}$ transition:
$\displaystyle D_{0}^{T+L}\approx 2\frac{|\bf{q}|}{K}|M_{1+}|^{2},$ (25)
$\displaystyle D_{0}^{TT}\approx-\frac{3}{2}\frac{|\bf{q}|}{K}|M_{1+}|^{2}.$
(26)
In Figs. 7-9, we show the results for the target and double spin asymmetries
for $\vec{e}\vec{p}\rightarrow ep\pi^{0}$ Biselli . The inclusion of these
data into the analysis resulted in a smaller magnitude of the $S_{1/2}$
amplitude for the Roper resonance, and also in the larger $A_{1/2}$ and
smaller $|S_{1/2}|$ amplitudes for the
$\gamma^{*}p\rightarrow~{}N(1535)S_{11}$ transition. These data had minor
impact on the $\gamma^{*}p\rightarrow~{}\Delta(1232)P_{33}$ and
$N(1520)D_{13}$ amplitudes.
Figure 3: Our results for the Legendre moments of the $\vec{e}p\rightarrow
ep\pi^{0}$ structure functions in comparison with experimental data Joo1 for
$Q^{2}=0.4~{}$GeV2. The solid (dashed) curves correspond to the results
obtained using DR (UIM) approach. Figure 4: Our results for the Legendre
moments of the $\vec{e}p\rightarrow en\pi^{+}$ structure functions in
comparison with experimental data Egiyan for $Q^{2}=0.4~{}$GeV2. The solid
(dashed) curves correspond to the results obtained using DR (UIM) approach.
Figure 5: Our results for the Legendre moments of the $\vec{e}p\rightarrow
en\pi^{+}$ structure functions in comparison with experimental data Park for
$Q^{2}=2.44~{}$GeV2. The solid (dashed) curves correspond to the results
obtained using DR (UIM) approach. Figure 6: The same as in Fig. 5 for
$Q^{2}=3.48~{}$GeV2.
Figure 7: $A_{t}$ (left panel) and $A_{et}$ (right panel) as functions of the
invariant mass $W$, integrated over the whole range in $\cos{\theta}$,
$0.252<Q^{2}<0.611~{}$GeV2 and $60^{0}<\phi<156^{0}$. Experimental data are
form Ref. Biselli . Solid and dashed curves correspond to our results obtained
using DR and UIM approaches, respectively. Figure 8: Our results for the
longitudinal target asymmetry $A_{t}$ in comparison with experimental data for
$Q^{2}=0.385~{}$GeV2 Biselli . Solid (dashed) curves correspond to the results
obtained using DR (UIM) approach. Rows correspond to 7 $W$ bins with $W$ mean
values of 1.125, 1.175, 1.225, 1.275, 1.35, 1.45, and 1.55 GeV. Columns
correspond to $\phi$ bins with $\phi=\pm 72^{0},\pm 96^{0},\pm 120^{0},\pm
144^{0},\pm 168^{0}$. The solid circles are the average values of the data for
positive $\phi$’s and those at negative $\phi$’s taken with opposite signs.
Figure 9: Our results for the beam-target asymmetry $A_{et}$ in comparison
with experimental data for $Q^{2}=0.385~{}$GeV2 Biselli . Solid (dashed)
curves correspond to the results obtained using DR (UIM) approach. Rows
correspond to 7 $W$ bins with $W$ mean values of 1.125, 1.175, 1.225, 1.275,
1.35, 1.45, and 1.55 GeV. Columns correspond to $\phi$ bins with $\phi=\pm
72^{0},\pm 96^{0},\pm 120^{0},\pm 144^{0},\pm 168^{0}$. The average values of
the data for positive and negative $\phi$’s are shown by solid circles.
## VII Comparison with theoretical predictions
In Figs. 10 and 13-15, we present our final results from Tables 6-10 and 12;
they are average values of the amplitudes extracted using DR and UIM.
### VII.1 $\Delta(1232)P_{33}$ resonance
The results for the $\gamma^{*}p\rightarrow~{}\Delta(1232)P_{33}$ magnetic
dipole form factor in the Ash convention Ash and for the ratios $R_{EM}\equiv
E_{1+}^{3/2}/M_{1+}^{3/2}$, $R_{SM}\equiv S_{1+}^{3/2}/M_{1+}^{3/2}$ are
presented in Fig. 10. The relationship between $G^{*}_{M,Ash}(Q^{2})$ and the
corresponding multipole amplitude is given by:
$G^{*}_{M,Ash}(Q^{2})=\frac{m}{k_{r}}\sqrt{\frac{8q_{r}\Gamma}{3\alpha}}M_{1+}^{3/2}(Q^{2},W=M),$
(27)
where $M=1232~{}$MeV and $\Gamma=118~{}$MeV are the mean values of the mass
and width of the $\Delta(1232)P_{33}$ (Table 5), $q_{r},k_{r}$ are the pion
and virtual photon three-momenta, respectively, in the c.m. system of the
reaction $\gamma^{*}p\rightarrow p\pi^{0}$ at the $\Delta(1232)P_{33}$
resonance position, and $m$ is the nucleon mass. This definition is related to
the definition of $G^{*}_{M}$ in the Jones-Scadron convention Scadron by:
$G^{*}_{M,J-S}(Q^{2})=G^{*}_{M,Ash}(Q^{2})\sqrt{1+\frac{Q^{2}}{(M+m)^{2}}}.$
(28)
The low $Q^{2}$ data from MAMI MAMI006 ; MAMI02 and MIT/BATES BATES , and
earlier JLab Hall C Frolov and Hall A KELLY1 ; KELLY2 results are also
shown. The form factor $G^{*}_{M}(Q^{2})$ is presented relative to the dipole
form factor, which approximately describes the elastic magnetic form factor of
the proton. The plot shows that new exclusive measurements of
$G^{*}_{M}(Q^{2})$, which now extend over the range $Q^{2}=0.06-6~{}$GeV2,
confirm the rapid falloff of $G^{*}_{M}(Q^{2})$ relative to the proton
magnetic form factor seen previously in inclusive measurements.
Fig. 10 shows the long-standing discrepancy between the measured
$G^{*}_{M}(Q^{2})$ and the constituent quark model predictions; here in
comparison with the LF relativistic quark model of Ref. Bruno . Within
dynamical reaction models Yang ; Kamalov ; Sato ; Lee , the meson-cloud
contribution was identified as the source of this discrepancy. The importance
of the pion (cloud) contribution for the
$\gamma^{*}p\rightarrow~{}\Delta(1232)P_{33}$ transition is confirmed also by
the lattice QCD calculations Alexandrou . In Fig. 10, the results of the
dynamical model of Ref. Sato are plotted. They show the total amplitude
(‘dressed’ form factor) and the amplitude with the subtracted meson-cloud
contribution (‘bare’ form factor). Very close results are obtained within the
dynamical model of Refs. Yang ; Kamalov . The meson-cloud contribution makes
up more than 30% of the total amplitude at the photon point, and remains
sizeable while $Q^{2}$ increases.
Figure 10 also shows the prediction GPD1 obtained in the large-$N_{c}$ limit
of QCD, by relating the $N\rightarrow\Delta$ and $N\rightarrow N$ GPDs. A
quantitative description of $G^{*}_{M}(Q^{2})$ is obtained in the whole
$Q^{2}$ range.
A consistent picture emerges from the data for the ratios $R_{EM}$ and
$R_{SM}$: $R_{EM}$ remains negative, small and nearly constant in the entire
range $0<Q^{2}<6~{}$GeV2; $R_{SM}$ remains negative, but its magnitude
strongly rises at high $Q^{2}$. It should be mentioned that the observed
behavior of $R_{SM}$ at large $Q^{2}$ sharply disagrees with the solution of
MAID2007 MAID based on the same data set. The magnitude of the relevant
amplitude $S_{1+}^{3/2}$ can be directly checked using the data for the
structure function $\sigma_{LT}$, whose $\cos{\theta}$ behavior at
$W=1.23~{}$GeV is dominated by the interference of this amplitude with
$M_{1+}^{3/2}$:
$D_{1}^{LT}(ep\rightarrow
ep\pi^{0})\approx\frac{8}{3}\left(S_{1+}^{3/2}\right)^{*}M_{1+}^{3/2}.$ (29)
The comparison of the experimental data for the $ep\rightarrow ep\pi^{0}$
structure functions with our results and the MAID2007 solution is shown in
Figs. 11 and 12. At $Q^{2}=0.4-1.45~{}$GeV2 (Fig. 11), MAID2007 describes the
angular behavior of $\sigma_{LT}$. However, it increasingly underestimates the
strong $\cos{\theta}$ dependence of this structure function with rising
$Q^{2}$, which is the direct consequence of the small values of $R_{SM}$ in
the MAID2007 solution. At $Q^{2}\geq 3~{}$GeV2 this is demonstrated in Fig.
12. In terms of $\chi^{2}$ per data point for $\sigma_{LT}$ at $W=1.23~{}$GeV,
the situation is presented in Table 13.
$~{}~{}Q^{2}$ | | $\chi^{2}/$d.p. |
---|---|---|---
(GeV2) | | |
| DR | UIM | MAID2007
0.4 | 2.0 | 2.3 | 2.6
0.75 | 1.3 | 1.8 | 1.3
1.45 | 0.9 | 1.1 | 1.0
3 | 1.6 | 1.9 | 4.8
4.2 | 1.5 | 1.8 | 2.9
5 | 1.0 | 1.3 | 2.6
Table 13: Our results obtained within DR and UIM, and the results of the
MAID2007 solution MAID for $\chi^{2}$ per data point for $\sigma_{LT}$ at
$W=1.23~{}$GeV for $ep\rightarrow ep\pi^{0}$ data Joo1 ; Ungaro .
In constituent quark models, the nonzero magnitude of $E_{1+}^{3/2}$ can arise
only due to a deformation of the $SU(6)$ spherical symmetry in the N and (or)
$\Delta(1232)$ wave functions. In this connection it is interesting that both
dynamical models Sato ; Yang give practically zero ‘bare’ values for $R_{EM}$
(as well as for $R_{SM}$). The entire $E_{1+}^{3/2}$ amplitude in these models
is due to the quadrupole deformation that arises through the interaction of
the photon with the meson cloud.
The knowledge of the $Q^{2}$ behavior of the ratios $R_{EM},R_{SM}$ is of
great interest as a measure of the $Q^{2}$ scale where the asymptotic domain
of QCD may set in for this resonance transition. In the pQCD asymptotics
$R_{EM}\rightarrow 100\%$ and $R_{SM}\rightarrow const$. The measured values
of $R_{EM},R_{SM}$ show that in the range $Q^{2}<6~{}$GeV2, there is no sign
of an approach to the asymptotic pQCD regime in either of these ratios.
Figure 10: Left panel: the form factor $G^{*}_{M}$ for the
$\gamma^{*}p\rightarrow~{}\Delta(1232)P_{33}$ transition relative to $3G_{D}$.
Right panel: the ratios $R_{EM},~{}R_{SM}$. The full boxes are the results
from Tables 6-8 obtained in this work from CLAS data (Tables 1, 3, and 4). The
bands show the model uncertainties. Also shown are the results from MAMI
MAMI006 ; MAMI02 \- open triangles, MIT/BATES BATES \- open crosses,
JLab/Hall C Frolov \- open rhombuses, and JLab/Hall A KELLY1 ; KELLY2 \-
open circles. The solid and dashed curves correspond to the ‘dressed’ and
‘bare’ contributions from Ref. Sato ; for $R_{EM},~{}R_{SM}$, only the
‘dressed’ contributions are shown; the ‘bare’ contributions are close to zero.
The dashed-dotted curves are the predictions obtained in the large-$N_{c}$
limit of QCD GPD1 ; Pascalutsa . The dotted curve for $G^{*}_{M}$ is the
prediction of a LF relativistic quark model of Ref. Bruno ; the dotted curves
for $R_{EM},~{}R_{SM}$ are the MAID2007 solutions MAID . Figure 11: Our
results for the $ep\rightarrow ep\pi^{0}$ structure functions (in $\mu$b/sr
units) in comparison with experimental data Joo1 for $W=1.23~{}$GeV. The
columns correspond to $Q^{2}=0.4,~{}0.75,~{}1.45~{}$GeV2. The solid (dashed)
curves correspond to the results obtained using DR (UIM) approach. The dotted
curves are from MAID2007 MAID . Figure 12: Our results for the $ep\rightarrow
ep\pi^{0}$ structure functions (in $\mu$b/sr units) in comparison with
experimental data Ungaro for $W=1.23~{}$GeV. The columns correspond to
$Q^{2}=3,~{}4.2,~{}5~{}$GeV2. The solid (dashed) curves correspond to the
results obtained using DR (UIM) approach. The dotted curves are from MAID2007
MAID .
### VII.2 $N(1440)P_{11}$ resonance
The results for the $\gamma^{*}p\rightarrow~{}N(1440)P_{11}$ helicity
amplitudes are presented in Fig. 13. The high $Q^{2}$ amplitudes
($Q^{2}=1.72-4.16~{}$GeV2) and the results for $Q^{2}=0.4,0.65~{}$GeV2 were
already presented and discussed in Refs. Azn04 ; Roper . In the present paper
the data for $Q^{2}=0.4,0.65~{}$GeV2 were reanalysed taking into account the
recent CLAS polarization measurements on the target and beam-target
asymmetries Biselli . Also included are new results extracted at
$Q^{2}=0.3,0.525,0.9~{}$GeV2.
By quantum numbers, the most natural classification of the Roper resonance in
the constituent quark model is a first radial excitation of the $3q$ ground
state. However, the difficulties of quark models to describe the low mass and
large width of the $N(1440)P_{11}$, and also its photocouplings to the proton
and neutron, gave rise to numerous speculations. Alternative descriptions of
this state as a gluonic baryon excitation Li1 ; Li2 , or a hadronic N$\sigma$
molecule Krehl , were suggested. The CLAS measurements, for the first time,
made possible the determination of the electroexcitation amplitudes of the
Roper resonance on the proton up to $Q^{2}=4.5~{}$GeV2. These results are
crucial for the understanding of the nature of this state. There are several
specific features in the extracted $\gamma^{*}p\rightarrow~{}N(1440)P_{11}$
amplitudes that are very important to test models. First, the specific
behavior of the transverse amplitude $A_{1/2}$, which being large and negative
at $Q^{2}=0$, becomes large and positive at $Q^{2}\simeq 2~{}$GeV2, and then
drops slowly with $Q^{2}$. Second, the relative sign between the longitudinal
$S_{1/2}$ and transverse $A_{1/2}$ amplitudes. And third, the common sign of
the amplitudes $A_{1/2},S_{1/2}$ extracted from the data on
$\gamma^{*}p\rightarrow\pi N$ includes signs from the $\gamma^{*}p\rightarrow
N(1440)P_{11}$ and $N(1440)P_{11}\rightarrow\pi N$ vertices; both signs should
be taken into account while comparing with model predictions. All these
characteristics are described by the light-front relativistic quark models of
Refs. Capstick ; AznRoper assuming that $N(1440)P_{11}$ is the first radial
excitation of the $3q$ ground state. Although the models Capstick ; AznRoper
fail to describe numerically the data at small $Q^{2}$, this can have the
natural explanation in the meson-cloud contributions, which are expected to be
large for low $Q^{2}$ Lee1 .
### VII.3 $N(1535)S_{11}$ resonance
For the first time, the $\gamma^{*}N\rightarrow N(1535)S_{11}$ transverse
helicity amplitude has been extracted from the $\pi$ electroproduction data in
a wide range of $Q^{2}$ (Fig. 14), and the results confirm the
$Q^{2}$-dependence of this amplitude observed in $\eta$ electroproduction.
Numerical comparison of the results extracted from the $\pi$ and $\eta$ photo-
and electroproduction data depends on the relation between the branching
ratios to the $\pi N$ and $\eta N$ channels. Consequently, it contains an
arbitrariness connected with the uncertainties of these branching ratios:
$\beta_{\pi N}=0.35-0.55$, $\beta_{\eta N}=0.45-0.6$ PDG . The amplitudes
extracted from $\eta$ photo- and electroproduction in Refs. Azneta ; Armstrong
; Thompson ; Denizli correspond to $\beta_{\eta N}=0.55$.
The amplitudes found from $\pi$ and $\eta$ data can be used to specify the
relation between $\beta_{\pi N}$ and $\beta_{\eta N}$. From the fit to these
amplitudes at $0\leq Q^{2}<4.5~{}$GeV2, we found
$\frac{\beta_{\eta N}}{\beta_{\pi N}}=0.95\pm 0.03.$ (30)
Further, taking into account the branching ratio to the $\pi\pi N$ channel
$\beta_{\pi\pi N}=0.01-0.1$ PDG , which accounts practically for all channels
different from $\pi N$ and $\eta N$, we find
$\displaystyle\beta_{\pi N}=0.485\pm 0.008\pm 0.023,$ (31)
$\displaystyle\beta_{\eta N}=0.460\pm 0.008\pm 0.022.$ (32)
The first error corresponds to the fit error in Eq. (30) and the second error
is related to the uncertainty of $\beta_{\pi\pi N}$. The results shown in Fig.
14 correspond to $\beta_{\pi N}=0.485,~{}\beta_{\eta N}=0.46$.
The CLAS data on $\pi$ electroproduction allowed the extraction of the
longitudinal helicity amplitude for the $\gamma^{*}N\rightarrow N(1535)S_{11}$
transition with good precision. These results are crucial for testing
theoretical models. It turned out that at $Q^{2}<2~{}$GeV2, the sign of
$S_{1/2}$ is not described by the quark models. Here it should be mentioned
that quark model predictions for the relative signs between the $S_{1/2}$ and
$A_{1/2},A_{3/2}$ amplitudes, are presented for the transitions
$\gamma^{*}N\rightarrow N(1535)S_{11}$ and $N(1520)D_{13}$ (Figs. 14 and 15)
according to the investigation made in Ref. Definitions . Combined with the
difficulties of quark models to describe the substantial coupling of
$N(1535)S_{11}$ to the $\eta N$ channel PDG and to strange particles Liu ;
Xie , the difficulty in the description of the sign of $S_{1/2}$ can be
indicative of a large meson-cloud contribution and (or) of additional
$q\bar{q}$ components in this state An . Alternative representations of the
$N(1535)S_{11}$ as a meson-baryon molecule have been also discussed Weise ;
Nieves ; Oset1 ; Lutz .
### VII.4 $N(1520)D_{13}$ resonance
The results for the $\gamma^{*}p\rightarrow~{}N(1520)D_{13}$ helicity
amplitudes are shown in Fig. 15, where the transverse amplitudes are compared
with those extracted from earlier data. The new data provide much more
accurate results.
Figure 13: Helicity amplitudes for the
$\gamma^{*}p\rightarrow~{}N(1440)P_{11}$ transition. The full circles are the
results from Table 9 obtained in this work from CLAS data (Tables 1-4). The
bands show the model uncertainties. The open boxes are the results of the
combined analysis of CLAS single $\pi$ and 2$\pi$ electroproduction data
Azn065 . The full triangle at $Q^{2}=0$ is the RPP estimate PDG . The thick
curves correspond to the results obtained in the LF relativistic quark models
assuming that $N(1440)P_{11}$ is a first radial excitation of the $3q$ ground
state: Capstick (dashed), AznRoper (solid). The thin dashed curves are
obtained assuming that $N(1440)P_{11}$ is a gluonic baryon excitation (q3G
hybrid state) Li2 . Figure 14: Helicity amplitudes for the
$\gamma^{*}p\rightarrow~{}N(1535)S_{11}$ transition. The legend is partly as
for Fig. 13. The solid boxes are the results extracted from $\eta$ photo- and
electroproduction data in Ref. Azneta , the open boxes show the results from
$\eta$ electroproduction data Armstrong ; Thompson ; Denizli . The data are
presented assuming $\beta_{\pi N}=0.485$, $\beta_{\eta N}=0.46$ (see
Subsection VII,C). The results of the LF relativistic quark models are given
by the dashed Capstick and dashed-dotted Simula1 curves. The solid curves
are the central values of the amplitudes found within light-cone sum rules
using lattice results for light-cone distribution amplitudes of the
$N(1535)S_{11}$ resonance Braun . Figure 15: Helicity amplitudes for the
$\gamma^{*}p\rightarrow~{}N(1520)D_{13}$ transition. The legend is partly as
for Fig. 13. Open circles show the results Foster extracted from earlier DESY
Haidan ; DESY and NINA NINA data. The curves correspond to the predictions
of the quark models: Warns (solid), Santopinto (dashed), and Merten
(dotted). Figure 16: The helicity asymmetry
$A_{hel}\equiv(A^{2}_{1/2}-A^{2}_{3/2})/(A^{2}_{1/2}+A^{2}_{3/2})$ for the
$\gamma^{*}p\rightarrow~{}N(1520)D_{13}$ transition. Triangles show the
results obtained in this work. The solid curve is the prediction of the quark
model with harmonic oscillator potential Isgur . Figure 17: The helicity
amplitudes $A_{1/2}$ for the $\gamma^{*}p\rightarrow~{}N(1440)P_{11}$,
$N(1520)D_{13}$, $N(1535)S_{11}$ transitions, multiplied by $Q^{3}$. The
results obtained in this work from the JLab-CLAS data on pion
electroproduction on the protons are shown by solid circles ($N(1440)P_{11}$),
solid trangles ($N(1520)D_{13}$), and solid boxes ($N(1535)S_{11}$). Open
boxes and crosses are the results for the $N(1535)S_{11}$ obtained in $\eta$
electroproduction, respectively, in HALL B Thompson ; Denizli and HALL C
Armstrong . The solid curve corresponds to the amplitude $A_{1/2}$ for the
$\gamma^{*}p\rightarrow~{}N(1535)S_{11}$ transition found within light-cone
sum rules Braun .
Sensitivity of the earlier data to the
$\gamma^{*}p\rightarrow~{}N(1520)D_{13}$ longitudinal helicity amplitude was
limited. The CLAS data allowed this amplitude to be determined with good
precision and in a wide range of $Q^{2}$.
The obtained results show the rapid helicity switch from the dominance of the
$A_{3/2}$ amplitude at the photon point to the dominance of $A_{1/2}$ at
$Q^{2}>1~{}$GeV2. This is demonstrated in Fig. 16 in terms of the helicity
asymmetry. Such behavior was predicted by a nonrelativistic quark model with
harmonic oscillator potential Close . Quark models also describe the sign and
$Q^{2}$ dependence of the longitudinal amplitude. However, there are some
shortcomings in the quark model description of the details of the $Q^{2}$
dependence of the $\gamma^{*}p\rightarrow~{}N(1520)D_{13}$ amplitudes. The
amplitude $A_{3/2}$ is significantly underestimated in all quark models for
$Q^{2}<2~{}$GeV2. Dynamical models predict large meson-cloud contributions to
this amplitude Lee1 that could explain the discrepancy.
Finally, Fig. 17 shows the helicity amplitudes $A_{1/2}$ for the resonances
$N(1440)P_{11}$, $N(1520)D_{13}$, $N(1535)S_{11}$, multiplied by $Q^{3}$. The
data indicate that starting with $Q^{2}=3~{}$GeV2, these amplitudes have a
$Q^{2}$ dependence close to $1/Q^{3}$. Such behaviour is expected in pQCD in
the limit $Q^{2}\rightarrow\infty$ Carlson . Measurements at higher $Q^{2}$
are needed in order to check a possible $Q^{3}$ scaling of these amplitudes.
## VIII Summary
The electroexcitation amplitudes for the low mass resonances
$\Delta(1232)P_{33}$, $N(1440)P_{11}$, $N(1520)D_{13}$, and $N(1535)S_{11}$
are determined in a wide range of $Q^{2}$ in the comprehensive analysis of
JLab-CLAS data on differential cross sections, longitudinally polarized beam
asymmetries, and longitudinal target and beam-target asymmetries for $\pi$
electroproduction off the proton. A total of about 119,000 data points were
included covering the full azimuthal and polar angle range. With this, we have
complemented the previous analyses Azn04 ; Azn065 ; Roper by including all
JLab-CLAS pion electroproduction data available today. We also have put
significant effort into accounting for model and systematic uncertainties of
the extracted electroexcitation amplitudes, by including the uncertainties of
hadronic parameters, such as masses and widths of the resonances, the
amplitudes of higher lying resonances, the parameters which determine
nonresonant contributions, as well as the point-to-point systematics of the
experimental data and the overall normalization error of the cross sections.
Utilization of two approaches, DR and UIM, allowed us to also estimate the
model dependence of the results, which was taken into account in the total
model uncertainties of the extracted amplitudes.
There are still additional uncertainties in the amplitudes presented in this
paper. These are related to the lack of precise knowledge of the empirical
resonance couplings to the $N\pi$ channel. However, we did not include these
uncertainties in the error budget as this is an overall multiplicative
correction that affects all amplitudes for a given resonance equally, and,
more importantly, the amplitudes can be corrected for these effects once
improved hadronic couplings become available.
The amplitudes for the electroexcitation of the $\Delta(1232)P_{33}$ resonance
are determined in the range $0.16\leq Q^{2}\leq 6~{}$GeV2. The results are in
agreement with the low $Q^{2}$ data from MAMI MAMI006 ; MAMI02 and MIT/BATES
BATES , and the JLab Hall A ($Q^{2}=1~{}$GeV2) KELLY1 ; KELLY2 and Hall C
($Q^{2}=2.8,4.2~{}$GeV2) Frolov data.
The results for the $\Delta(1232)P_{33}$ resonance show the importance of the
meson-cloud contribution to quantitatively explain the magnetic dipole
strength, as well as the electric and scalar quadrupole transitions. They also
do not show any tendency of approaching the asymptotic QCD regime for
$Q^{2}\leq 6~{}$GeV2. This was already mentioned in the original paper Ungaro
, where the analysis was based on the UIM approach only.
The amplitudes for the electroexcitation of the resonances $N(1440)P_{11}$,
$N(1520)D_{13}$, and $N(1535)S_{11}$ are determined in the range $0.3\leq
Q^{2}<4.5~{}$GeV2.
For the Roper resonance, the high $Q^{2}$ amplitudes ($Q^{2}=1.7-4.5~{}$GeV2)
and the results for $Q^{2}=0.4,0.65~{}$GeV2 were already presented and
discussed in Refs. Azn04 ; Roper . In the present paper, the data for
$Q^{2}=0.4,0.65~{}$GeV2 were reanalysed taking into account the recent CLAS
polarization measurements on the target and beam-target asymmetries Biselli .
Also included are the new results at $Q^{2}=0.3,0.525,0.9~{}$GeV2. The main
conclusion for the Roper resonance is, as already reported in Ref. Roper ,
that the data on $\gamma^{*}p\rightarrow~{}N(1440)P_{11}$ available in the
wide range of $Q^{2}$ provide a strong evidence for this state to be
predominantly the first radial excitation of the 3-quark ground state.
For the first time, the $\gamma^{*}p\rightarrow N(1535)S_{11}$ transverse
helicity amplitude has been extracted from the $\pi$ electroproduction data up
to $Q^{2}=4.5~{}$GeV2. The results confirm the $Q^{2}$-dependence of this
amplitude as observed in $\eta$ electroproduction. The transverse amplitude
found from the $\pi$ and $\eta$ data allowed us to specify the branching
ratios to the $\pi N$ and $\eta N$ channels for the $N(1535)S_{11}$.
Due to the CLAS measurements of $\pi$ electroproduction, for the first time
the $\gamma^{*}p\rightarrow N(1520)D_{13}$ and $N(1535)S_{11}$ longitudinal
helicity amplitudes are determined from experimental data. For the
$\gamma^{*}p\rightarrow N(1535)S_{11}$ transition, the sign of $S_{1/2}$ is
not described by quark models at $Q^{2}<2~{}$GeV2. Combined with the
difficulties of quark models to describe the substantial coupling of the
$N(1535)S_{11}$ to the $\eta N$ and strangeness channels, this can be an
indication of a large meson-cloud contribution and/or of additional $q\bar{q}$
components in this state; alternative representations of the $N(1535)S_{11}$
as a meson-baryon molecule are also possible.
The CLAS data provide much more accurate results for the
$\gamma^{*}p\rightarrow~{}N(1520)D_{13}$ transverse helicity amplitudes than
those extracted from earlier DESY and NINA data. The data confirm the
constituent quark model prediction of the rapid helicity switch from the
dominance of the $A_{3/2}$ amplitude at the photon point to the dominance of
$A_{1/2}$ at $Q^{2}>1~{}$GeV2. Quark models also describe the sign and $Q^{2}$
dependence of the longitudinal amplitude.
Starting with $Q^{2}=3~{}$GeV2, the helicity amplitudes $A_{1/2}$ for the
resonances $N(1440)P_{11}$, $N(1520)D_{13}$, and $N(1535)S_{11}$ have a
behaviour close to $1/Q^{3}$. Measurements at higher $Q^{2}$ are needed in
order to check $Q^{3}$ scaling for these amplitudes.
## IX Acknowledgments
This work was supported in part by the U.S. Department of Energy and the
National Science Foundation, the Korea Research Foundation, the French
Commissariat a l’Energie Atomique and CNRS/IN2P3, the Italian Istituto
Nazionale di Fisica Nucleare, the Skobeltsyn Institute of Nuclear Physics and
Physics Department at Moscow State University, and the UK Science and
Technology Facilities Research Council (STFC). Jefferson Science Associates,
LLC, operates Jefferson Lab under U.S. DOE contract DE-AC05-060R23177.
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|
arxiv-papers
| 2009-09-12T16:05:37 |
2024-09-04T02:49:05.296545
|
{
"license": "Public Domain",
"authors": "I. G. Aznauryan, V. D. Burkert, and the CLAS Collaboration",
"submitter": "Inna Aznauryan",
"url": "https://arxiv.org/abs/0909.2349"
}
|
0909.2387
|
On the transcendence of some infinite sums
Pingzhi Yuan∗ Juan Li
00footnotetext: $*$ This author is responsible for communications, and
supported by the Guangdong Provincial Natural Science Foundation (No.
8151027501000114) and NSF of China (No. 10571180).
###### Abstract
In this paper we investigate the infinite convergent sum
$T=\sum_{n=0}^{\infty}\frac{P(n)}{Q(n)}$, where
$P(x)\in\overline{\mathbb{Q}}[x]$, $Q(x)\in\mathbb{Q}[x]$ and $Q(x)$ has only
simple rational zeros. N. Saradha and R. Tijdeman have obtained sufficient and
necessary conditions for the transcendence of $T$ if the degree of $Q(x)$ is
3. In this paper we give sufficient and necessary conditions for the
transcendence of $T$ if the degree of $Q(x)$ is 4 and $Q(x)$ is reduced.
Key words: Transcendental numbers, algebraic numbers, infinite sums
MCS: primary 11J81; secondary 11J86,11J91
## 1 Introduction
In this paper we will investigate the transcendence of the infinite convergent
sum
$T=\sum_{n=0}^{\infty}\frac{P(n)}{Q(n)},$
where $P(x)\in\overline{\mathbb{Q}}[x]$, $Q(x)\in\mathbb{Q}[x]$ and $Q(x)$ has
only simple rational zeros. Owing to the reduction procedure described in
Tijdeman [10, 11], we have
$T=A+S,\quad S=\sum_{n=1}^{\infty}\frac{f(n)}{n},$
where $A\in\overline{\mathbb{Q}}$, we take $q>1$ to be a positive integer and
$f(x)$ is a number theoretic function which is periodic mod $q$ with
$\sum_{i=1}^{q}f(i)=0$, which we will assume throughout the paper.
About forty years ago, Chowla [4] and Erdős (see [7]) formulated some
conjectures related to whether there exists a rational-valued function $f(n)$
periodic with prime period $p$ such that
$\sum_{n=1}^{\infty}\frac{f(n)}{n}=0.$ One of the conjectures was proved by
Baker, Birch and Wirsing [3] in 1973\. They used Baker’s theory on linear
forms in logarithms to establish that $S\neq 0$ if $f(n)$ is a non-vanishing
function defined on the integers with rational values and period $q$ such that
i) $f(r)=0,\ \mathrm{if}\ 1<\mathrm{gcd}(r,q)<q$,
ii) the cyclotomic polynomial $\Phi_{q}$ is irreducible over
$\mathbb{Q}(f(1),\cdots,f(q))$.
They further showed that their result would be false if i) or ii) is omitted
(see [3]).
In 1982, T. Okada [8] established a result which provides a description of all
functions for which ii) holds and $S=0$. Okada’s proof depends on the basic
result on the linear independence of the logarithms of algebraic numbers and
on the non-vanishing of $L(1,\chi)=\sum_{n=1}^{\infty}\frac{\chi(n)}{n}$ if
$\chi$ is a non-principal Dirichlet character. The precise result is stated in
Section 2.
In 2001, S.D. Adhikari, N. Saradha, T.N. Shorey and R. Tijdeman [2] proved
that if $S\neq 0$, then $S$ is transcendental. They used this result to prove
that if $P(x)\in\overline{\mathbb{Q}}[x]$ and $Q(x)\in\mathbb{Q}[x]$, where
$Q(x)$ is a polynomial with simple rational roots which are all in the
interval $[-1,0)$, then the infinite convergent sum
$T=\sum_{n=0}^{\infty}\frac{P(n)}{Q(n)}$ is $0$ or transcendental. Further, if
$Q(x)$ is a polynomial with simple rational roots, then $T$ is a computable
rational number or a transcendental number. For more information on the
developments sketched above we refer to [1] and [10, 11]. In particular, if
the degree of $Q(x)$ is 2, then
$T=\sum_{n=0}^{\infty}\frac{\alpha}{(qn+s_{1})(qn+s_{2})}$
with $q,\ s_{1},\ s_{2}$ integers, $\alpha\in\overline{\mathbb{Q}}$ nonzero,
is transcendental if and only if $s_{1}\not\equiv s_{2}\ (\mathrm{mod}\ q)$.
On the other hand, by above results, it is easy to see that
$\sum_{n=0}^{\infty}\frac{1}{(3n+1)(3n+2)(3n+3)}>0$
and
$\sum_{n=0}^{\infty}\frac{1}{(n+1)(2n+1)(4n+1)}=\frac{\pi}{3}$
are transcendental. The second equality was also proven by Lehmer [6] in 1975.
In 2003, N. Saradha and R. Tijdeman [9] rephrased Okada’s theorem so that it
becomes a decomposition lemma and gave sufficient and necessary conditions for
the transcendence of $T=\sum_{n=0}^{\infty}\frac{P(n)}{Q(n)}$ if the degree of
$Q(x)$ is 3\. They proved that
$T=\sum_{n=0}^{\infty}\frac{\alpha n+\beta}{(qn+s_{1})(qn+s_{2})(qn+s_{3})}$
is transcendental if $s_{1},s_{2},s_{3}$ are not in the same residue class mod
$q$. However, when the degree of $Q(x)$ is 4, the example
$T=\sum_{n=0}^{\infty}\frac{16n^{2}+12n-1}{(4n+1)(4n+2)(4n+3)(4n+4)}=0$
shows that the corresponding result is not valid.
The main purpose of the present paper is to give sufficient and necessary
conditions for the transcendence of $T$ if the degree of $Q(x)$ is 4, that is
$T=\sum_{n=0}^{\infty}\frac{\alpha n^{2}+\beta
n+\gamma}{(qn+s_{1})(qn+s_{2})(qn+s_{3})(qn+s_{4})}$ (1)
where $\alpha,\beta,\gamma\in\overline{\mathbb{Q}}$, $s_{1},s_{2},s_{3},s_{4}$
are distinct integers. By the reduction procedure described in Tijdeman [10,
11], without loss of generality, we may assume that
$0<s_{1},s_{2},s_{3},s_{4}\leq q$, $\mathrm{gcd}(\alpha x^{2}+\beta
x+\gamma,(qx+s_{1})(qx+s_{2})(qx+s_{3})(qx+s_{4}))=1$ and
$\mathrm{gcd}(s_{1},s_{2},s_{3},s_{4},q)=1$ throughout the paper. The
following simple example shows how the reduction procedure works,
$\sum_{n=0}^{\infty}\frac{1}{(2n+1)(2n+2)(2n+3)}=-\frac{1}{2}+\sum_{n=0}^{\infty}\\{\frac{1}{2n+1}-\frac{1}{2n+2}\\}=-\frac{1}{2}+\sum_{n=0}^{\infty}\frac{1}{(2n+1)(2n+2)}.$
In Section 2 we shall give some preliminaries that will be useful for our
further discussions. In Section 3 we prove the following Theorem.
###### Theorem 1.1
Let
$T=\sum_{n=0}^{\infty}\frac{\alpha n^{2}+\beta
n+\gamma}{(qn+s_{1})(qn+s_{2})(qn+s_{3})(qn+s_{4})}$
where $s_{1},s_{2},s_{3},s_{4}$ are distinct positive integers $\leq q$ and
$\alpha,\beta,\gamma\in\overline{\mathbb{Q}}$. Suppose $\mathrm{gcd}(\alpha
x^{2}+\beta x+\gamma,(qx+s_{1})(qx+s_{2})(qx+s_{3})(qx+s_{4}))=1$ ,
$\mathrm{gcd}(s_{1},s_{2},s_{3},s_{4},q)=1$ and $\Phi_{q}$ is irreducible over
$\mathbb{Q}(\alpha,\beta,\gamma)$. Then $T$ is transcendental except when
$T=\sum_{n=0}^{\infty}\frac{16n^{2}+12n-1}{(4n+1)(4n+2)(4n+3)(4n+4)}=0$ (2)
or
$T=\sum_{n=0}^{\infty}\frac{36n^{2}+36n-1}{(6n+1)(6n+2)(6n+4)(6n+5)}=0.$ (3)
## 2 Preliminaries
In this section we shall introduce some notations and state the related
results that will be needed in the sequel. We denote by $\varphi(n)$ the Euler
function and $P$ the set of all primes dividing $q$. We call the polynomial
$Q(x)$ reduced if $Q(x)\in\mathbb{Q}[x]$ and it has only simple rational zeros
which are all in the interval $[-1,0)$. We denote by $v_{p}(n)$ the exponent
to which $p|n$ for any prime $p$ and $n\in\mathbb{Z}$. We write
$J=\\{a\in\mathbb{Z}\ |\ 1\leq a\leq q,\ \mathrm{gcd}(a,q)=1\\},$
$L=\\{r\in\mathbb{Z}\ |\ 1\leq r\leq q,\ 1<\mathrm{gcd}(r,q)<q\\},$
and
$L^{{}^{\prime}}=L\cup\\{q\\}.$
For $p\in P$ and $r\in L^{{}^{\prime}}$, we define
$P(r)=\\{p\in P\ |\ v_{p}(r)\geq v_{p}(q)\\}$
and
$\displaystyle\varepsilon(r,p)=\left\\{\begin{array}[]{ll}v_{p}(q)+\frac{1}{p-1},&p\in
P(r),\\\ v_{p}(r),&\mbox{ otherwise }.\end{array}\right.$
For $r\in L^{{}^{\prime}}$ and $a\in J$, we define
$A(r,a)=\frac{1}{\mathrm{gcd}(r,q)}\prod_{p\in
P(r)}(1-\frac{1}{p^{\varphi(q)}})^{-1}\sum_{n\in
S(r)}\frac{\sigma(r,a,n)}{n},$
where
$S(r)=\\{\prod_{p\in P(r)}p^{\alpha(p)}\ |\ 0\leq\alpha(p)<\varphi(q)\\}$
and
$\displaystyle\sigma(r,a,n)=\left\\{\begin{array}[]{ll}1,&\mbox{ if }\quad
r\equiv an\gcd(r,q)\pmod{q},\\\ 0,&\mbox{ otherwise }.\end{array}\right.$
Theorem A. (Okada [8]). If $\Phi_{q}$ is irreducible over
$\mathbb{Q}(f(1),\cdots,f(q))$, then $S=\sum_{n=1}^{\infty}\frac{f(n)}{n}=0$
if and only if
$f(a)+\sum_{r\in L}f(r)A(r,a)+\frac{f(q)}{\varphi(q)}=0,\qquad\ for\ all\ a\in
J,$ (6)
and
$\sum_{r\in L^{{}^{\prime}}}f(r)\varepsilon(r,p)=0,\qquad\ for\ all\ p\in P.$
(7)
N. Saradha and R. Tijdeman [9] estabished an equivalent version of Theorem A.
###### Lemma 2.1
(Decomposition Lemma [9] ). Let $\Phi_{q}$ be irreducible over
$\mathbb{Q}(f(1),\cdots,f(q))$. Let M be the set of positive integers which
are composed of prime factors of $q$. Then
$S=\sum_{n=1}^{\infty}\frac{f(n)}{n}=0$ if and only if
$\sum_{m\in M}\frac{f(am)}{m}=0,\qquad for\ all\ a\in J,$ (8)
and
$\sum_{r\in L^{{}^{\prime}}}f(r)\varepsilon(r,p)=0,\qquad\ for\ all\ p\in P.$
As a consequence of Lemma 2.1, they derived the following result.
###### Lemma 2.2
([9]) Let $\Phi_{q}$ be irreducible over $\mathbb{Q}(f(1),\cdots,f(q))$.
Suppose $S=\sum_{n=1}^{\infty}\frac{f(n)}{n}=0$. Then
$\sum_{n=1}^{\infty}\frac{f(kn)}{n}=0,\ for\ every\ k\ with\
\mathrm{gcd}(k,q)=1.$
The following result given by S.D. Adhikari, N. Saradha, T.N. Shorey and R.
Tijdeman [2] is essential for the transcendence of
$\sum_{n=0}^{\infty}\frac{P(n)}{Q(n)}$.
Theorem B. ([2]) Let $P(x)\in\overline{\mathbb{Q}}[x]$, and let
$Q(x)\in\mathbb{Q}[x]$ be reduced. If
$T=\sum_{n=0}^{\infty}\frac{P(n)}{Q(n)}$
converges, then $T$ is $0$ or transcendental.
When the degree of $Q(x)$ is 3, N. Saradha and R. Tijdeman [9] obtained
necessary and sufficient conditions for the transcendence of
$T=\sum_{n=0}^{\infty}\frac{P(n)}{Q(n)}$.
Theorem C. ([9]) Let $T=\sum_{n=0}^{\infty}\frac{\alpha
n+\beta}{(qn+s_{1})(qn+s_{2})(qn+s_{3})}$, where $\alpha,\
\beta\in\overline{\mathbb{Q}}$, and $|\alpha|+|\beta|>0$. Let $\Phi_{q}$ be
irreducible over $\mathbb{Q}(\alpha,\beta)$ and $s_{1},s_{2},s_{3}$ be
distinct integers such that $qn+s_{1},\ qn+s_{2},\ qn+s_{3}$ do not vanish
for $n\geq 0$. Assume that $s_{1},s_{2},s_{3}$ are not in the same residue
class $\mathrm{mod}\ q$. Further let $s_{1}\not\equiv s_{2}\ (\mathrm{mod}\
q)$ if $\alpha s_{3}=\beta q$; $s_{1}\not\equiv s_{3}\ (\mathrm{mod}\ q)$ if
$\alpha s_{2}=\beta q$; $s_{2}\not\equiv s_{3}\ (\mathrm{mod}\ q)$ if $\alpha
s_{1}=\beta q$. Then $T$ is transcendental.
The following result in [5] will be useful in Section 3. For the convenience
of the reader, we provide the sketch of a proof suggested by Frazer Jarvis.
###### Lemma 2.3
Let $n,d$, and $r$ be integers such that $n>1$, $d>0$, $d|n$, and
$\gcd(r,d)=1$, then there are precisely
$\varphi(n)/\varphi(d)\geq\varphi(n/d)$ numbers which are coprime to $n$ in
the set $S=\\{r+td,t=1,2,\cdots,\frac{n}{d}\\}$.
* Proof.
For primes $p|d$ there is no condition, but for primes $p|n$ but $p\not|d$,
the congruence classes for $r+td$ are equally distributed mod $p$, so that
$\frac{p-1}{p}$ of the possible numbers are prime to $p$. The Chinese
Remainder Theorem gives an independence result. Since there are $\frac{n}{d}$
numbers considered, the number we seek is
$\frac{n}{d}\cdot\prod_{p|n,p\not|d}(1-\frac{1}{p}),$
and the result easily follows. $\Box$
## 3 Proof of Theorem 1.1
Let
$T=\sum_{n=0}^{\infty}\frac{\alpha_{k}n^{k}+\alpha_{k-1}n^{k-1}+\cdots+\alpha_{0}}{(qn+r_{1})\cdots(qn+r_{m})},$
where $\alpha_{0},\alpha_{1},\cdots,\alpha_{k}\in\overline{\mathbb{Q}}$,
$r_{1},\cdots,r_{m}$ are distinct positive integers and $k\leq m-2$. Our main
purpose is to consider the transcendence of $T$. By the reduction procedure
given in Tijdeman [10, 11], we may restrict ourselves to the case that
i) $r_{1},\cdots,r_{m}\ \mathrm{are}\ \mathrm{distinct}\ \mathrm{positive}\
\mathrm{integers}\ \leq q,\ \mathrm{gcd}(r_{1},\cdots,r_{m},q)=1,$
ii) $\
\mathrm{gcd}(\alpha_{k}x^{k}+\alpha_{k-1}x^{k-1}+\cdots+\alpha_{0},(qx+r_{1})\cdots(qx+r_{m}))=1.$
Therefore we need only consider the case $T=0$ by Theorem B, which we shall
assume from now on. By partial fractions, we get
$T=\sum_{n=0}^{\infty}\\{\frac{A_{1}}{qn+r_{1}}+\frac{A_{2}}{qn+r_{2}}+\cdots+\frac{A_{m}}{qn+r_{m}}\\},$
where
$A_{1},\cdots,A_{m}\in\mathbb{Q}(\alpha_{0},\alpha_{1},\cdots,\alpha_{k})$ are
all nonzero numbers with
$A_{1}+A_{2}+\cdots+A_{m}=0.$
We define $f(n)$ for $n\geq 0$ as follows:
$\displaystyle f(n)=\left\\{\begin{array}[]{ll}A_{1},&n\equiv
r_{1}\pmod{q},\\\ \cdots&\cdots\\\ A_{m},&n\equiv r_{m}\pmod{q},\\\ 0,&\mbox{
otherwise }.\end{array}\right.$
Then $f(n)$ is a periodic function with period $q$ taking only $m$ non-zero
values $f(r_{1}),f(r_{2}),\cdots,f(r_{m})$ with
$f(r_{1})+f(r_{2})+\cdots+f(r_{m})=0$
and
$T=\sum_{n=1}^{\infty}\frac{f(n)}{n}=0.$
It is easy to see that
$\mathbb{Q}(\alpha_{0},\alpha_{1},\cdots,\alpha_{k})=\mathbb{Q}(A_{1},A_{2},\cdots,A_{m})$.
If $\Phi_{q}$ is irreducible over
$\mathbb{Q}(\alpha_{0},\alpha_{1},\cdots,\alpha_{k})$, then $\Phi_{q}$ is
irreducible over $\mathbb{Q}(f(1),\cdots,f(q))$, so (4), (5) and (6) are valid
by Theorem A and Lemma 2.1. We have
###### Proposition 3.1
Suppose
$T=\sum_{n=0}^{\infty}\frac{\alpha_{k}n^{k}+\alpha_{k-1}n^{k-1}+\cdots+\alpha_{0}}{(qn+r_{1})\cdots(qn+r_{m})}=0,$
where $r_{1},\cdots,r_{m}$ are distinct positive integers $\leq q$, $k\leq
m-2$, and $\alpha_{0},\alpha_{1},\cdots,\alpha_{k}\in\overline{\mathbb{Q}}$.
Suppose $\mathrm{gcd}(r_{1},\cdots,r_{m},q)=1,$ and $\
\mathrm{gcd}(\alpha_{k}x^{k}+\alpha_{k-1}x^{k-1}+\cdots+\alpha_{0},(qx+r_{1})\cdots(qx+r_{m}))=1$
and $\Phi_{q}$ is irreducible over $\mathbb{Q}(\alpha_{0},\cdots,\alpha_{k})$.
Then there exists an $r_{i}$ with $1\leq i\leq m$ such that $\gcd(r_{i},q)>1$.
* Proof.
By the above arguments, if all of $\\{r_{1},\cdots,r_{m}\\}$ are coprime to
$q$, then $f(r)=0$ for all $r\in L^{{}^{\prime}}$. Applying (4) with $a\in J$
we have $f(a)=0$ for all $a\in J,$ a contradiction. This completes the
proof.$\Box$
The main purpose of the present paper is to investigate the transcendence of
$T$ in the case that $m=4$, that is
$T=\sum_{n=0}^{\infty}\frac{\alpha n^{2}+\beta
n+\gamma}{(qn+r_{1})(qn+r_{2})(qn+r_{3})(qn+r_{4})}=\sum_{n=0}^{\infty}\\{\frac{A_{1}}{qn+r_{1}}+\frac{A_{2}}{qn+r_{2}}+\frac{A_{3}}{qn+r_{3}}+\frac{A_{4}}{qn+r_{4}}\\},$
and $f(n)$ is a periodic function with period $q$ taking only four non-zero
values $f(r_{1})=A_{1},f(r_{2})=A_{2},f(r_{3})=A_{3},f(r_{4})=A_{4}$
satisfying
$f(r_{1})+f(r_{2})+f(r_{3})+f(r_{4})=0\ \mathrm{and}\
T=\sum_{n=1}^{\infty}\frac{f(n)}{n}=0.$
We divide the proof of Theorem 1.1 into four cases depending on the number
$\rho$ of elements of $\\{r_{1},r_{2},r_{3},r_{4}\\}$ which are coprime to
$q$. By Proposition 3.1, we have $\rho\leq 3$. First suppose that $\rho=3$,
then without loss of generality we may assume that $\mathrm{gcd}(r_{1},q)>1$
and $\mathrm{gcd}(r_{2}r_{3}r_{4},q)=1$. If $p|\mathrm{gcd}(r_{1},q)$ and
$p\nmid r_{i}$, $i=2,3,4$, then by (5) we get
$f(r_{1})\varepsilon(r_{1},p)=0$, and so $f(r_{1})=0$ since
$\varepsilon(r_{1},p)\neq 0$, a contradiction. Consequently if
$T=\sum_{n=0}^{\infty}\frac{\alpha n^{2}+\beta
n+\gamma}{(qn+r_{1})(qn+r_{2})(qn+r_{3})(qn+r_{4})}=0$ and there exists an
integer $r\in\\{r_{1},r_{2},r_{3},r_{4}\\}$ with $\mathrm{gcd}(r,q)>1$, then
there exists at least another integer
$s\in\\{r_{1},r_{2},r_{3},r_{4}\\}\backslash\\{r\\}$ with
$\mathrm{gcd}(r,s,q)>1$.
Now suppose $\rho=2$. We have
###### Proposition 3.2
Suppose
$T=\sum_{n=0}^{\infty}\frac{\alpha n^{2}+\beta
n+\gamma}{(qn+r_{1})(qn+r_{2})(qn+r_{3})(qn+r_{4})}=0,$
where $r_{1},r_{2},r_{3},r_{4}$ are distinct positive integers $\leq q$ and
$\alpha,\beta,\gamma\in\overline{\mathbb{Q}}$. Suppose $\gcd(\alpha
n^{2}+\beta n+\gamma,(qn+r_{1})(qn+r_{2})(qn+r_{3})(qn+r_{4}))=1$,
$\mathrm{gcd}(r_{1},r_{2},r_{3},r_{4},q)=1$ and $\Phi_{q}$ is irreducible over
$\mathbb{Q}(\alpha,\beta,\gamma)$. Suppose $\rho=2$. Then $T$ is
transcendental except when
$T=\sum_{n=0}^{\infty}\frac{16n^{2}+12n-1}{(4n+1)(4n+2)(4n+3)(4n+4)}$
or
$T=\sum_{n=0}^{\infty}\frac{36n^{2}+36n-1}{(6n+1)(6n+2)(6n+4)(6n+5)}.$
* Proof.
Suppose $\rho=2$. Without loss of generality we may assume that
$\mathrm{gcd}(r_{1},q)>1$, $\mathrm{gcd}(r_{2},q)>1$ and
$\mathrm{gcd}(r_{3}r_{4},q)=1$. By the above arguments, we have
$\mathrm{gcd}(r_{1},r_{2},q)=d>1$.
If $\varphi(d)>2$, we let
$a_{i}\equiv r_{3}+i\cdot\frac{q}{d}\ (\mathrm{mod}\ q),\ 0<a_{i}\leq q,\
i=0,1,\cdots,d-1.$
By Lemma 2.3, there are precisely $\varphi(n)/\varphi(n/d)\geq\varphi(d)$
numbers in $\\{a_{0},a_{1},\cdots,a_{d-1}\\}$ which are coprime to $q$. Since
$\varphi(d)>2$, there exist distinct $a_{i_{0}}$, $a_{j_{0}}$ such that
$a_{i_{0}}\neq r_{3}$, $a_{j_{0}}\neq r_{3}$, and
$\mathrm{gcd}(a_{i_{0}},q)=\mathrm{gcd}(a_{j_{0}},q)=1$. Applying (6) with
$a=r_{3}$, $a=a_{i_{0}}$ and $a=a_{j_{0}}$, we get
$\sum_{m\in M}\frac{f(r_{3}m)}{m}=f(r_{3})+\sum_{r_{3}m\equiv r_{1}\
(\mathrm{mod}\ q)\atop m\in M}\frac{f(r_{1})}{m}+\sum_{r_{3}m\equiv r_{2}\
(\mathrm{mod}\ q)\atop m\in M}\frac{f(r_{2})}{m}=0,$ $\sum_{m\in
M}\frac{f(a_{i_{0}}m)}{m}=f(a_{i_{0}})+\sum_{a_{i_{0}}m\equiv r_{1}\
(\mathrm{mod}\ q)\atop m\in M}\frac{f(r_{1})}{m}+\sum_{a_{i_{0}}m\equiv r_{2}\
(\mathrm{mod}\ q)\atop m\in M}\frac{f(r_{2})}{m}=0,$ $\sum_{m\in
M}\frac{f(a_{j_{0}}m)}{m}=f(a_{j_{0}})+\sum_{a_{j_{0}}m\equiv r_{1}\
(\mathrm{mod}\ q)\atop m\in M}\frac{f(r_{1})}{m}+\sum_{a_{j_{0}}m\equiv r_{2}\
(\mathrm{mod}\ q)\atop m\in M}\frac{f(r_{2})}{m}=0.$
Observe that for every $m\in M$, we have
$r_{3}m\equiv r_{i}\ (\mathrm{mod}\ q)\Longleftrightarrow a_{i_{0}}m\equiv
r_{i}\ (\mathrm{mod}\ q)\Longleftrightarrow a_{j_{0}}m\equiv r_{i}\
(\mathrm{mod}\ q),\quad i=1,2.$
It follows that $f(r_{3})=f(a_{i_{0}})=f(a_{j_{0}})\neq 0$, which contradicts
to our assumptions.
Now we consider the case $\varphi(d)\leq 2$, that is $d=2,3,4,6$.
Case 1. $d=2$. First we consider the subcase of $2\|q$. If $2\|q$, we choose
$u_{0}$ to be the smallest positive integer such that $2^{u_{0}}\equiv 1\
(\mathrm{mod}\ \frac{q}{2})$. It is easy to see that
$\varepsilon(r_{1},2)=\varepsilon(r_{2},2)=2$, applying (5) with $p=2$, we get
$f(r_{1})+f(r_{2})=0.$ (10)
Now we prove the following Claim:
Claim: If there are positive integers $k$ and $c\in J$ such that $r_{1}\equiv
2^{k}c\ (\mathrm{mod}\ q)$, then $f(c)\neq 0$.
Otherwise, if $f(c)=0$, applying (6) with $a=c$ we have
$\sum_{m\in M}\frac{f(cm)}{m}=\sum_{cm\equiv r_{1}\ (\mathrm{mod}\ q)\atop
m\in M}\frac{f(r_{1})}{m}+\sum_{cm\equiv r_{2}\ (\mathrm{mod}\ q)\atop m\in
M}\frac{f(r_{2})}{m}=0.$ (11)
Since $\mathrm{gcd}(r_{1},r_{2},q)=2$, then $cm\equiv r_{i}\ (\mathrm{mod}\
q)$, $i=1,2$ can occur only when $m=2^{x}$ for some positive integer $x$. If
the congruence $r_{2}\equiv 2^{x}c\ (\mathrm{mod}\ q)$ has no solution $x$,
then by (8), we have
$f(r_{1})\sum_{cm\equiv r_{1}\ (\mathrm{mod}\ q)\atop m\in M}\frac{1}{m}=0,$
and so $f(r_{1})=0$, a contradiction. If the congruence $r_{2}\equiv 2^{x}c\
(\mathrm{mod}\ q)$ has solutions, we take $l$ to be the smallest positive
integer solution, then all positive solutions can be expressed as $l+tu_{0}$,
$t=0,1,2,\cdots$. Let $k_{0}$ be the smallest positive integer solution of the
congruence $r_{1}\equiv 2^{k}c\ (\mathrm{mod}\ q)$. Then (8) becomes
$\frac{f(r_{1})}{2^{k_{0}}}\frac{1}{1-2^{-u_{0}}}+\frac{f(r_{2})}{2^{l}}\frac{1}{1-2^{-u_{0}}}=0.$
(12)
Combining (7) and (9), we get $k_{0}=l$, which implies that $r_{1}\equiv
r_{2}\ (\mathrm{mod}\ q)$, a contradiction. We have proved the Claim.
For given positive integers $n,a,$ and $i$ with $\mathrm{gcd}(a,q)=1$, since
$2\|q$, then the congruence $2^{n}a\equiv 2^{i}x_{i}\ (\mathrm{mod}\ q)$ has
precisely one solution $x_{i}$ such that $0<x_{i}<q$ and
$\mathrm{gcd}(x_{i},q)=1$. On the other hand, if $1\leq i<j\leq u_{0}$, then
$x_{i}\neq x_{j}$. Indeed, if $2^{i}x_{i}\equiv 2^{j}x_{i}\equiv 2^{n}a\
(\mathrm{mod}\ q)$, it follows that $2^{j-i}\equiv 1\ (\mathrm{mod}\
\frac{q}{2})$, $u_{0}|j-i$, a contradiction. Let $r_{1}=2^{k}R_{1},\
r_{2}=2^{l}R_{2}$, where $k,l,R_{1},R_{2}$ are positive integers and
$\mathrm{gcd}(R_{1}R_{2},q)=1$. Let $x_{i}$ be the unique solution of
congruence
$2^{k}R_{1}\equiv 2^{i}x_{i}\ (\mathrm{mod}\ q),\ 0<x_{i}<q,\
\mathrm{gcd}(x_{i},q)=1,\ i=1,2,\cdots,u_{0}.$
By the Claim and the above arguments we have $f(x_{i})\neq 0$,
$i=1,2,\cdots,u_{0}$, $\mathrm{gcd}(x_{i},q)=1$ and $x_{i}\neq x_{j}\ (i\neq
j)$, and so $u_{0}\leq 2$ since we have $f(x)=0$ for $x\in
J\backslash\\{r_{3},r_{4}\\}$. If $u_{0}=1$, then $q=2$, a contradiction. If
$u_{0}=2$, then $q=6$. Without loss of generality we may assume that
$r_{1}=2,r_{2}=4,r_{3}=1,r_{4}=5$. Applying (5) with $p=2$ and (6) with
$a=r_{3}$ and $a=r_{4}$, we have
$\displaystyle\left\\{\begin{array}[]{l}f(r_{1})+f(r_{2})=0,\\\
f(r_{3})+\frac{f(r_{1})}{2}\frac{1}{1-2^{-2}}+\frac{f(r_{2})}{4}\frac{1}{1-2^{-2}}=0,\\\
f(r_{4})+\frac{f(r_{1})}{4}\frac{1}{1-2^{-2}}+\frac{f(r_{2})}{2}\frac{1}{1-2^{-2}}=0.\end{array}\right.$
Hence
$f(r_{2})=-f(r_{1}),f(r_{3})=-\frac{1}{3}f(r_{1}),f(r_{4})=\frac{1}{3}f(r_{1}).$
By Lemma 2.1 we get
$T=\frac{1}{3}f(r_{1})\sum_{n=0}^{\infty}\\{\frac{3}{6n+2}-\frac{3}{6n+4}-\frac{1}{6n+1}+\frac{1}{6n+5}\\}$
$=\frac{2}{3}f(r_{1})\sum_{n=0}^{\infty}\frac{36n^{2}+36n-1}{(6n+1)(6n+2)(6n+4)(6n+5)}=0.$
Next we consider the case that $q=4$, without loss of generality we may assume
that $r_{1}=2,r_{2}=4,r_{3}=1,r_{4}=3$. Applying (5) with $p=2$ and (6) with
$a=r_{3}$ and $a=r_{4}$, we have
$\displaystyle\left\\{\begin{array}[]{l}f(r_{1})+3f(r_{2})=0,\\\
f(r_{3})+\frac{f(r_{1})}{2}+\frac{f(r_{2})}{4}\frac{1}{1-\frac{1}{2}}=0,\\\
f(r_{4})+\frac{f(r_{1})}{2}+\frac{f(r_{2})}{4}\frac{1}{1-\frac{1}{2}}=0.\end{array}\right.$
Hence
$f(r_{1})=-3f(r_{2}),f(r_{3})=f(r_{2}),f(r_{4})=f(r_{2}).$
By Lemma 2.1 we have
$T=f(r_{2})\sum_{n=0}^{\infty}\\{\frac{-3}{4n+2}+\frac{1}{4n+4}+\frac{1}{4n+1}+\frac{1}{4n+3}\\}$
$=f(r_{2})\sum_{n=0}^{\infty}\frac{16n^{2}+12n-1}{(4n+1)(4n+2)(4n+3)(4n+4)}=0.$
Now we deal with the case $4|q$ and $q>4$. Since
$d=\mathrm{gcd}(r_{1},r_{2},q)=2$, $4|q$, without loss of generality we may
assume that $q=2^{\alpha_{0}}Q$, $r_{1}=2R_{1}$, and $r_{2}=2^{l}R_{2}$, where
$Q,l,R_{1},R_{2},\alpha_{0}$ are positive integers, $\alpha_{0}\geq 2$,
$2\nmid Q$, $l\geq 1$ and $\mathrm{gcd}(R_{1}R_{2},q)=1$. Let
$a_{i}\equiv r_{3}+i\cdot\frac{q}{2}\ (\mathrm{mod}\ q),\ 0<a_{i}\leq q,\
i=0,1.$
Since $4|q$, then $\mathrm{gcd}(a_{0}a_{1},q)=1$. Note that
$\\{m\in M|\ ma_{0}\equiv r_{i}\ (\mathrm{mod}\ q)\\}=\\{m\in M|\ ma_{1}\equiv
r_{i}\ (\mathrm{mod}\ q)\\},\ i=1,2.$
Applying (6) with $a=a_{0}$ and $a=a_{1}$ we get
$f(a_{0})=f(a_{1}).$
Since $a_{0}=r_{3}$ and $f(x)=0$ for $x\in J\backslash\\{r_{3},r_{4}\\}$, we
have $a_{1}=r_{4}$ and $f(r_{3})=f(r_{4})$. Note that
$M_{1}=\\{m\in M|\ R_{1}m\equiv r_{1}=2R_{1}\ (\mathrm{mod}\ q)\\}=\\{2\\}$
and
$M_{2}=\\{m\in M|\ R_{1}m\equiv r_{2}\ (\mathrm{mod}\ q)\\}=\\{2^{n}\in M|\
R_{1}2^{n}\equiv r_{2}\ (\mathrm{mod}\ q)\\}.$
If the congruence $r_{2}\equiv 2^{x}R_{1}\ (\mathrm{mod}\ q)$ has no solution,
then by applying (6) with $a=R_{1}$ we get $f(R_{1})+\frac{f(r_{1})}{2}=0$,
and so $f(R_{1})=-\frac{f(r_{1})}{2}\neq 0$, it follows that
$f(R_{1})=f(r_{4})=f(r_{3})=-\frac{f(r_{1})}{2}$. Since
$f(r_{1})+f(r_{2})+f(r_{3})+f(r_{4})=0$, so $f(r_{2})=0$, a contradiction. Now
we assume that $l^{{}^{\prime}}$ is the smallest positive solution of the
congruence $r_{2}\equiv 2^{x}R_{1}\ (\mathrm{mod}\ q)$. Let $u_{0}$ be the
smallest positive integer such that $2^{u_{0}}\equiv 1\ (\mathrm{mod}\ Q)$. We
consider the following four subcases.
(i) If $f(R_{1})=0$ and $l\geq\alpha_{0}$. Applying (5) with $p=2$ and (6)
with $a=R_{1}$, we get
$\displaystyle\left\\{\begin{array}[]{l}f(r_{1})+(\alpha_{0}+1)f(r_{2})=0,\\\
\frac{f(r_{1})}{2}+\frac{f(r_{2})}{2^{l^{{}^{\prime}}}}\frac{1}{1-2^{-u_{0}}}=0,\end{array}\right.$
then $\alpha_{0}+1=\frac{2^{u_{0}-l^{{}^{\prime}}+1}}{2^{u_{0}}-1}$, and so
$\alpha_{0}=l^{{}^{\prime}}=u_{0}=1$, a contradiction.
(ii) If $f(R_{1})=0$ and $l<\alpha_{0}$, then $l=l^{{}^{\prime}}$ and
$M_{2}=\\{2^{l^{{}^{\prime}}}\\}$. Applying (5) with $p=2$ and (6) with
$a=R_{1}$, we get
$\displaystyle\left\\{\begin{array}[]{l}f(r_{1})+lf(r_{2})=0,\\\
\frac{f(r_{1})}{2}+\frac{f(r_{2})}{2^{l^{{}^{\prime}}}}=0,\end{array}\right.$
then $l=\frac{1}{2^{l^{{}^{\prime}}-1}}$, and so $l=l^{{}^{\prime}}=1$ and
$r_{2}\equiv 2R_{1}\equiv r_{1}\ (\mathrm{mod}\ q)$, a contradiction.
(iii) If $f(R_{1})\neq 0$ and $l\geq\alpha_{0}$. Similarly, we have
$\displaystyle\left\\{\begin{array}[]{l}f(r_{3})=f(r_{4}),\\\
f(r_{1})+(\alpha_{0}+1)f(r_{2})=0,\\\
f(r_{1})+f(r_{2})+f(r_{3})+f(r_{4})=0,\\\
f(r_{3})+\frac{f(r_{1})}{2}+\frac{f(r_{2})}{2^{l^{{}^{\prime}}}}\frac{1}{1-2^{-u_{0}}}=0,\end{array}\right.$
then $2^{l^{{}^{\prime}}-1}(1-2^{-u_{0}})=1$, and so
$u_{0}=1,l^{{}^{\prime}}=2,q=4$, a contradiction.
(iv) If $f(R_{1})\neq 0$ and $l<\alpha_{0}$, then $M_{1}=\\{2\\}$ and
$M_{2}=\\{2^{l^{{}^{\prime}}}\\}$. Similarly, we have
$\displaystyle\left\\{\begin{array}[]{l}f(r_{3})=f(r_{4}),\\\
f(r_{1})+lf(r_{2})=0,\\\ f(r_{1})+f(r_{2})+f(r_{3})+f(r_{4})=0,\\\
f(r_{3})+\frac{f(r_{1})}{2}+\frac{f(r_{2})}{2^{l^{{}^{\prime}}}}=0,\end{array}\right.$
then $2^{l{{}^{\prime}}}=2$, $l^{{}^{\prime}}=1$, $l=l^{{}^{\prime}}=1$ by the
definition of $l$ and $l^{{}^{\prime}}$, and so $r_{2}\equiv 2R_{1}=r_{1}\
(\mathrm{mod}\ q)$, again a contradiction.
Case 2. $d=3$. Let
$a_{j}\equiv r_{3}+j\frac{q}{3}\ (\mathrm{mod}\ q),\ 0<a_{j}\leq q,\ j=0,1,2,$
and let
$M_{ij}=\\{m\in M\ |\ ma_{j}\equiv r_{i}\ (\mathrm{mod}\ q)\\},\ i=1,2,\
j=0,1,2.$
If $9|q$, then
$M_{10}=M_{11}=M_{12},\ M_{20}=M_{21}=M_{22}.$
Applying (6) with $a=a_{0},a_{1}$ and $a_{2}$, we have
$f(r_{3})=f(a_{0})=f(a_{1})=f(a_{2}),$
and $a_{0},a_{1},a_{2}$ are distinct, which contradicts to the fact that
$f(x)=0$ for $x\in J\backslash\\{r_{3},r_{4}\\}$.
If $3\|q$, then by Lemma 2.3 we can choose $a_{j_{0}}\in\\{a_{1},a_{2}\\}$
such that $\mathrm{gcd}(a_{j_{0}},q)=1$. Similarly, we have
$f(a_{0})=f(a_{j_{0}}),$
so $a_{j_{0}}=r_{4}$ and $f(r_{3})=f(r_{4})$. Applying (5) with $p=3$, we get
$f(r_{1})+f(r_{2})=0$. Combining with $f(r_{3})=f(r_{4})$,
$f(r_{1})+f(r_{2})+f(r_{3})+f(r_{4})=0$, we have $f(r_{3})=0$, a
contradiction.
The cases $d=4$ and $d=6$ are similar to $d=3$, and we omit the details. This
completes the proof.$\Box$
###### Proposition 3.3
Suppose that
$T=\sum_{n=0}^{\infty}\frac{\alpha n^{2}+\beta
n+\gamma}{(qn+r_{1})(qn+r_{2})(qn+r_{3})(qn+r_{4})}=0,$
where $r_{1},r_{2},r_{3},r_{4}$ are distinct positive integers $\leq q$ and
$\alpha,\beta,\gamma\in\overline{\mathbb{Q}}$. Suppose $\gcd(\alpha
n^{2}+\beta n+\gamma,(qn+r_{1})(qn+r_{2})(qn+r_{3})(qn+r_{4}))=1$ ,
$\mathrm{gcd}(r_{1},r_{2},r_{3},r_{4},q)=1$ and $\Phi_{q}$ is irreducible over
$\mathbb{Q}(\alpha,\beta,\gamma)$. Then $\rho\neq 1$.
* Proof.
Suppose $\rho=1$. Without loss of generality we may assume that
$\mathrm{gcd}(r_{i},q)>1$, $i=1,2,3$, and $\mathrm{gcd}(r_{4},q)=1$.
First we consider the case that there exist distinct integers
$r_{i},r_{j}\in\\{r_{1},r_{2},r_{3}\\}$, such that
$\varphi(\mathrm{gcd}(r_{i},r_{j},q))>1$, say
$\varphi(\mathrm{gcd}(r_{2},r_{3},q))>1$. Let
$a_{i}=1+i\cdot\frac{q}{\mathrm{gcd}(r_{2},r_{3},q)},\
i=0,1,\cdots,\mathrm{gcd}(r_{2},r_{3},q)-1.$
By Lemma 2.3, we may choose $a_{i_{0}}$ such that $a_{i_{0}}\neq 1$ and
$\mathrm{gcd}(a_{i_{0}},q)=1$. Applying Lemma 2.2 with $k=a_{i_{0}}$, we have
$\sum_{n=1}^{\infty}\frac{f(a_{i_{0}}n)}{n}=\sum_{n=1}^{\infty}{\\{\frac{f(r_{1})}{nq+r_{1}^{{}^{\prime}}}+\frac{f(r_{2})}{nq+r_{2}}+\frac{f(r_{3})}{nq+r_{3}}+\frac{f(r_{4})}{nq+r_{4}^{{}^{\prime}}}\\}}=0,$
(19)
where $r_{1}^{{}^{\prime}}\equiv a_{i_{0}}^{-1}r_{1}$,
$r_{4}^{{}^{\prime}}\equiv a_{i_{0}}^{-1}r_{4}\ (\mathrm{mod}\ q)$ and
$0<r_{1}^{{}^{\prime}},r_{4}^{{}^{\prime}}<q$. Obviously $r_{4}\neq
r_{4}^{{}^{\prime}}$ since $a_{i_{0}}\not\equiv 1\ (\mathrm{mod}\ q)$ and
$\mathrm{gcd}(r_{4},q)=1$. Subtracting $T$ from (10), we obtain
$T^{{}^{\prime}}=\sum_{n=1}^{\infty}{\\{\frac{f(r_{1})}{nq+r_{1}^{{}^{\prime}}}-\frac{f(r_{1})}{nq+r_{1}}+\frac{f(r_{4})}{nq+r_{4}^{{}^{\prime}}}}-\frac{f(r_{4})}{nq+r_{4}}\\}=0.$
If $r_{1}=r_{1}^{{}^{\prime}}$, then
$T^{{}^{\prime}}=f(r_{4})\sum_{n=1}^{\infty}\\{\frac{1}{r_{4}^{{}^{\prime}}}-\frac{1}{r_{4}}\\}\neq
0$, a contradiction. If $r_{1}\neq r_{1}^{{}^{\prime}}$, then there are
precisely two integers $r_{4},r_{4}^{{}^{\prime}}$ in
$\\{r_{1},r_{1}^{{}^{\prime}},r_{4},r_{4}^{{}^{\prime}}\\}$ which are coprime
to $q$. By Proposition 3.2 we have
$T^{{}^{\prime}}=\sum_{n=0}^{\infty}\\{\frac{-3}{4n+2}+\frac{1}{4n+4}+\frac{1}{4n+1}+\frac{1}{4n+3}\\},\
\ q=4,\ $
or
$T^{{}^{\prime}}=\sum_{n=0}^{\infty}\\{\frac{3}{6n+2}+\frac{-3}{6n+4}+\frac{-1}{6n+1}+\frac{1}{6n+5}\\},\
q=6.$
The first equality is impossible since $q>4$. If the second equality holds,
then $q=6$, $\\{r_{2},r_{3}\\}=\\{3,6\\}$ and $3\not|r_{1}$. Applying (5) with
$p=3$, we get $f(r_{2})+f(r_{3})=0$, which implies that $f(r_{1})+f(r_{4})=0$.
But in the second equality we have $f(r_{1})=3f(r_{4})$ or
$f(r_{1})=-3f(r_{4})$, a contradiction.
Now we assume that $\varphi{(\mathrm{gcd}(r_{i},r_{j},q))}\leq 1$ for all
distinct integers $r_{i},r_{j}\in{\\{r_{1},r_{2},r_{3}\\}}$, then
$\mathrm{gcd}(r_{1},r_{2},q)=\mathrm{gcd}(r_{1},r_{3},q)=\mathrm{gcd}(r_{2},r_{3},q)=2$.
(i) If $2\|q$, then applying (5) with $p=2$, we have
$f(r_{1})+f(r_{2})+f(r_{3})=0$, and so $f(r_{4})=0$, a contradiction.
(ii) If $4|q$, let $a_{1}=1+\frac{q}{2}$, then $a_{1}\neq 1$ ,
$\mathrm{gcd}(a_{1},q)=1$, and $a_{1}r_{i}\equiv r_{i}\ (\mathrm{mod}\ q)$,
$i=1,2,3$. Applying Lemma 2.2 with $k=a_{1}$, we have
$\sum_{n=1}^{\infty}\frac{f(a_{1}n)}{n}=\sum_{n=0}^{\infty}{\\{\frac{f(r_{1})}{nq+r_{1}}+\frac{f(r_{2})}{nq+r_{2}}+\frac{f(r_{3})}{nq+r_{3}}+\frac{f(r_{4})}{nq+r_{4}^{{}^{\prime}}}\\}}=0,$
(20)
where $r_{4}^{{}^{\prime}}\equiv r_{4}+\frac{q}{2}\ (\mathrm{mod}\ q)$ and
$0<r_{4}^{{}^{\prime}}<q$. Subtracting $T$ from (11), we obtain
$\sum_{n=0}^{\infty}{\\{\frac{f(r_{4})}{nq+r^{\prime}_{4}}-\frac{f(r_{4})}{nq+r_{4}}}\\}=0,$
which contradicts to $r_{4}^{{}^{\prime}}\neq r_{4}$. The proof is complete.
$\Box$
###### Proposition 3.4
Suppose that
$T=\sum_{n=0}^{\infty}\frac{\alpha n^{2}+\beta
n+\gamma}{(qn+r_{1})(qn+r_{2})(qn+r_{3})(qn+r_{4})}=0,$
where $r_{1},r_{2},r_{3},r_{4}$ are distinct positive integers $\leq q$ , and
$\alpha,\beta,\gamma\in\overline{\mathbb{Q}}$. Suppose $\gcd(\alpha
n^{2}+\beta n+\gamma,(qn+r_{1})(qn+r_{2})(qn+r_{3})(qn+r_{4}))=1$ ,
$\mathrm{gcd}(r_{1},r_{2},r_{3},r_{4},q)=1$ and $\Phi_{q}$ is irreducible over
$\mathbb{Q}(\alpha,\beta,\gamma)$. Then $\rho\neq 0$.
* Proof.
Suppose $\rho=0$. We divide the proof into two cases.
Case 1. There exist distinct integers
$r_{i},r_{j},r_{k}\in\\{r_{1},r_{2},r_{3},r_{4}\\}$ such that
$\mathrm{gcd}(r_{i},r_{j},r_{k},q)>1$, say,
$d=\mathrm{gcd}(r_{1},r_{2},r_{3},q)>1$. Let
$a_{i}=1+i\cdot\frac{q}{d},\ i=0,1,\cdots,d-1.$
If $\varphi(d)>1$, we may choose
$a_{i_{0}}\in\\{a_{0},a_{1},\cdots,a_{d-1}\\}$ such that $a_{i_{0}}\neq 1$ and
$a_{i_{0}}\in J$ by Lemma 2.3. Note that
$a_{i_{0}}r_{j}\equiv r_{j}\ (\mathrm{mod}\ q),\ j=1,2,3.$
Applying Lemma 2.2 with $k=a_{i_{0}}$, we obtain
$\sum_{n=1}^{\infty}\frac{f(a_{i_{0}}n)}{n}=\sum_{n=0}^{\infty}{\\{\frac{f(r_{1})}{nq+r_{1}}+\frac{f(r_{2})}{nq+r_{2}}+\frac{f(r_{3})}{nq+r_{3}}+\frac{f(r_{4})}{nq+r_{4}^{{}^{\prime}}}}\\}=0,$
(21)
where $r_{4}^{{}^{\prime}}\equiv a_{i_{0}}^{-1}r_{4}\ (\mathrm{mod}\ q)$ and
$0<r_{4}^{{}^{\prime}}<q$. It is easy to check that $r_{4}\neq
r_{4}^{{}^{\prime}}$ since $\mathrm{gcd}(r_{4},d,q)=1$. Subtracting the second
equality of (12) from $T$, we have
$\sum_{n=0}^{\infty}{\\{\frac{f(r_{4})}{nq+r_{4}}-\frac{f(r_{4})}{nq+r_{4}^{{}^{\prime}}}}\\}=0$,
which is impossible since $r_{4}\neq r_{4}^{{}^{\prime}}$.
If $\varphi(d)=1$, then $d=2=\mathrm{gcd}(r_{1},r_{2},r_{3},q)$.
(i) If $2\|q$, applying (5) with $p=2$ we get $f(r_{1})+f(r_{2})+f(r_{3})=0$,
and so $f(r_{4})=0$, a contradiction.
(ii) If $4|q$, we take $b_{1}=1+\frac{q}{2}$, then $\mathrm{gcd}(b_{1},q)=1$,
and $b_{1}r_{i}\equiv r_{i}\ (\mathrm{mod}\ q)$, $i=1,2,3$. Applying (6) with
$a=b_{1}$ we have
$\displaystyle\sum_{n=1}^{\infty}\frac{f(b_{1}n)}{n}=\sum_{n=0}^{\infty}{\\{\frac{f(r_{1})}{nq+r_{1}}+\frac{f(r_{2})}{nq+r_{2}}+\frac{f(r_{3})}{nq+r_{3}}+\frac{f(r_{4})}{nq+r_{4}^{{}^{\prime}}}\\}}=0,$
where $r_{4}^{{}^{\prime}}\equiv b_{1}^{-1}r_{4}\ (\mathrm{mod}\ q)$ and
$0<r_{4}^{{}^{\prime}}<q$. Similarly, $r_{4}^{{}^{\prime}}\neq r_{4}$.
Subtracting the above second equality from $T$, we obtain
$\sum_{n=0}^{\infty}{\\{\frac{f(r_{4})}{nq+r_{4}}-\frac{f(r_{4})}{nq+r_{4}^{{}^{\prime}}}}\\}=0$,
which is also impossible since $r_{4}\neq r_{4}^{{}^{\prime}}$.
Case 2. If $\mathrm{gcd}(r_{i},r_{j},r_{k},q)=1$ for all distinct
$r_{i},r_{j},r_{k}\in{\\{r_{1},r_{2},r_{3},r_{4}\\}}$, then there exist
distinct $r_{i},r_{j}\in{\\{r_{1},r_{2},r_{3},r_{4}\\}}$ such that
$\varphi(\mathrm{gcd}(r_{i},r_{j},q))>1$, say,
$\varphi(\mathrm{gcd}(r_{1},r_{2},q))>1$. Otherwise, we would have
$gcd(r_{i},\,r_{j},q)=1$ or 2 for all distinct
$r_{i},\,r_{j}\in\\{r_{1},\,r_{2},\,r_{3},\,r_{4}\\}$, and it would follow
that $\gcd(r_{1},q)\gcd(r_{2},q)\gcd(r_{3},q)\gcd(r_{4},q)$ has only one prime
divisor 2 by the argument of the paragraph above Proposition 3.2, and this
would mean that $\gcd(r_{1},\,r_{2},\,r_{3},\,r_{4},\,q)=2$ since $\rho=0$,
contradicting our assumptions. Let
$c_{i}=1+i\cdot\frac{q}{\mathrm{gcd}(r_{1},r_{2},q)},\
i=0,1,\cdots,\mathrm{gcd}(r_{1},r_{2},q)-1.$
By Lemma 2.3 we can choose $c_{i_{0}}$ such that $c_{i_{0}}\neq 1$ and
$\mathrm{gcd}(c_{i_{0}},q)=1$. Note that $c_{i_{0}}r_{j}\equiv r_{j}\
(\mathrm{mod}\ q)$, $j=1,2$. Similarly, applying Lemma 2.2 with $k=c_{i_{0}}$
and subtracting, we obtain
$T^{{}^{\prime}}=\sum_{n=0}^{\infty}{\\{\frac{f(r_{3})}{nq+r_{3}}-\frac{f(r_{3})}{nq+r_{3}^{{}^{\prime}}}}+\frac{f(r_{4})}{nq+r_{4}}-\frac{f(r_{4})}{nq+r_{4}^{{}^{\prime}}}\\}=0,$
where $r_{3}^{{}^{\prime}}\equiv c_{i_{0}}^{-1}r_{3}\pmod{q}$,
$r_{4}^{{}^{\prime}}\equiv c_{i_{0}}^{-1}r_{4}\ (\mathrm{mod}\ q)$,
$0<r_{3}^{{}^{\prime}},r_{4}^{{}^{\prime}}<q$. Note that $r_{3}\neq
r_{3}^{{}^{\prime}}$, $r_{4}\neq r_{4}^{{}^{\prime}}$,
$\mathrm{gcd}(r_{3},q)=\mathrm{gcd}(r_{3}^{{}^{\prime}},q)$ and
$\mathrm{gcd}(r_{4},q)=\mathrm{gcd}(r_{4}^{{}^{\prime}},q)$.
(i) If $\mathrm{gcd}(r_{3},r_{4},q)=1$. Since $\mathrm{gcd}(r_{3},q)>1$ and
$\mathrm{gcd}(r_{4},q)>1$, without loss of generality, we may assume that
$\mathrm{gcd}(r_{3},q)>2$, that is $\varphi(\mathrm{gcd}(r_{3},q))>1$. Let
$d_{j}=1+j\cdot\frac{q}{\mathrm{gcd}(r_{3},q)},\
j=0,1,\cdots,\mathrm{gcd}(r_{3},q)-1.$
Similarly, we can choose $d_{j_{0}}\neq 1$ and $\mathrm{gcd}(d_{j_{0}},q)=1$
such that
$\sum_{n=0}^{\infty}{\\{\frac{f(r_{3})}{nq+r_{3}}-\frac{f(r_{3})}{nq+r_{3}^{{}^{\prime}}}+\frac{f(r_{4})}{nq+r_{4}^{{}^{\prime\prime}}}-\frac{f(r_{4})}{nq+r_{4}^{{}^{\prime\prime\prime}}}\\}}=0,$
where $r_{4}^{{}^{\prime\prime}}\equiv d_{j_{0}}^{-1}r_{4}\pmod{q}$ ,
$r_{4}^{{}^{\prime\prime\prime}}\equiv d_{j_{0}}^{-1}r_{4}^{{}^{\prime}}$
$(\mathrm{mod}\ q)$,
$0<r_{4}^{{}^{\prime}},r_{4}^{{}^{\prime\prime}},r_{4}^{{}^{\prime\prime\prime}}<q$.
It follows that
$T^{{}^{\prime\prime}}=\sum_{n=0}^{\infty}{\\{\frac{f(r_{4})}{nq+r_{4}}-\frac{f(r_{4})}{nq+r_{4}^{{}^{\prime}}}-\frac{f(r_{4})}{nq+r_{4}^{{}^{\prime\prime}}}+\frac{f(r_{4})}{nq+r_{4}^{{}^{\prime\prime\prime}}}\\}}=0.$
Note that $r_{4}^{{}^{\prime}}\neq r_{4}^{{}^{\prime\prime\prime}}$,
$r_{4}\neq r_{4}^{{}^{\prime\prime}}$, $r_{4}\neq r_{4}^{\prime}$,
$r_{4}^{{}^{\prime\prime}}\neq r_{4}^{{}^{\prime\prime\prime}}$ and
$\mathrm{gcd}(r_{4},q)=\mathrm{gcd}(r_{4}^{{}^{\prime}},q)=\mathrm{gcd}(r_{4}^{{}^{\prime\prime}},q)=\mathrm{gcd}(r_{4}^{{}^{\prime\prime\prime}},q)$
since $\gcd(c_{i_{0}}d_{j_{0}},q)=1$. Now we have
$T^{{}^{\prime\prime}}=\frac{f(r_{4})}{\mathrm{gcd}(r_{4},q)}\sum_{n=0}^{\infty}\\{\frac{1}{nq^{{}^{\prime}}+a}+\frac{-1}{nq^{{}^{\prime}}+b}+\frac{-1}{nq^{{}^{\prime}}+c}+\frac{1}{nq^{{}^{\prime}}+e}\\}=0,$
where
$q^{{}^{\prime}}=\frac{q}{\mathrm{gcd}(r_{4},q)},a=\frac{r_{4}}{\mathrm{gcd}(r_{4},q)},b=\frac{r_{4}^{{}^{\prime}}}{\mathrm{gcd}(r_{4},q)},c=\frac{r_{4}^{{}^{\prime\prime}}}{\mathrm{gcd}(r_{4},q)},e=\frac{r_{4}^{{}^{\prime\prime\prime}}}{\mathrm{gcd}(r_{4},q)}$.
Obviously all of $a,b,c,e$ are coprime to $q^{\prime}$ and
$\Phi_{q^{{}^{\prime}}}$ is irreducible over $\mathbb{Q}$ .
It is easy to check that $a,b,c,e$ are distinct. Otherwise, since $a\neq b$,
$a\neq c$, $c\neq e$, $b\neq e$, we have $a=e$ or $b=c$. If $a=e$ and $b=c$
both hold, then
$T^{{}^{\prime\prime}}=\frac{f(r_{4})}{\mathrm{gcd}(r_{4},q)}\sum_{n=0}^{\infty}\\{\frac{2}{nq^{{}^{\prime}}+a}+\frac{-2}{nq^{{}^{\prime}}+b}\\}\neq
0,$
which is a contradiction. If $a=e$ and $b\neq c$, then by Theorem C we have
$T^{{}^{\prime\prime}}=\frac{f(r_{4})}{\mathrm{gcd}(r_{4},q)}\sum_{n=0}^{\infty}\\{\frac{2}{nq^{{}^{\prime}}+a}+\frac{-1}{nq^{{}^{\prime}}+b}+\frac{-1}{nq^{{}^{\prime}}+c}\\}\neq
0,$
which is also a contradiction. Similarly, the case that $a\neq e$ and $b=c$ is
impossible. Therefore $T^{{}^{\prime\prime}}=0$ is impossible by Proposition
3.1.
(ii) If $\mathrm{gcd}(r_{3},r_{4},q)>1$, applying (5) with some prime $p$
satisfying $p|\mathrm{gcd}(r_{3},r_{4},q)$, we get
$vf(r_{3})+uf(r_{4})=0,$
where $u,v$ are positive rational numbers, then we may re-write
$T^{{}^{\prime}}$ as
$T^{{}^{\prime}}=\frac{f(r_{4})}{\mathrm{gcd}(r_{3},r_{4},q)}\sum_{n=0}^{\infty}{\\{\frac{-\frac{u}{v}}{nq^{{}^{\prime}}+a}+\frac{\frac{u}{v}}{nq^{{}^{\prime}}+b}+\frac{1}{nq^{{}^{\prime}}+c}+\frac{-1}{nq^{{}^{\prime}}+e}\\}},$
where $q^{{}^{\prime}}=\frac{q}{\mathrm{gcd}(r_{3},r_{4},q)},\
a=\frac{r_{3}}{\mathrm{gcd}(r_{3},r_{4},q)},\
b=\frac{r_{3}^{{}^{\prime}}}{\mathrm{gcd}(r_{3},r_{4},q)},\
c=\frac{r_{4}}{\mathrm{gcd}(r_{3},r_{4},q)},\
e=\frac{r_{4}^{{}^{\prime}}}{\mathrm{gcd}(r_{3},r_{4},q)}$. It is easy to see
that $\mathrm{gcd}(a,q^{{}^{\prime}})=\mathrm{gcd}(b,q^{{}^{\prime}}),\
\mathrm{gcd}(c,q^{{}^{\prime}})=\mathrm{gcd}(e,q^{{}^{\prime}})$, and
$\mathrm{gcd}(a,c,q^{{}^{\prime}})=1$. Note that $\Phi_{q^{{}^{\prime}}}$ is
irreducible over $\mathbb{Q}$ and $a,b,c,e$ are distinct integers by the same
arguments as above.
By Proposition 3.1, we have either $\mathrm{gcd}(a,q^{{}^{\prime}})>1$ or
$\mathrm{gcd}(c,q^{{}^{\prime}})>1$.
If precisely one of
$\mathrm{gcd}(a,q^{{}^{\prime}}),\mathrm{gcd}(c,q^{{}^{\prime}})$ is 1, then
without loss of generality we may assume that
$\mathrm{gcd}(c,q^{{}^{\prime}})=1$ and $\mathrm{gcd}(a,q^{{}^{\prime}})>1$.
Then by Proposition 3.2 we have that
$T^{{}^{\prime}}=\sum_{n=0}^{\infty}\\{\frac{-3}{4n+2}+\frac{1}{4n+4}+\frac{1}{4n+1}+\frac{1}{4n+3}\\}=0,\
q^{\prime}=4,$ (22)
or
$T^{{}^{\prime}}=\sum_{n=0}^{\infty}\\{\frac{3}{6n+2}+\frac{-3}{6n+4}+\frac{-1}{6n+1}+\frac{1}{6n+5}\\}=0,\
q^{\prime}=6.$ (23)
(13) is impossible since
$\\{-3,1,1,1\\}\neq\\{-\frac{hu}{v},\frac{hu}{v},h,-h\\}$ for all
$h\in\mathbb{Q}$. If (14) holds, then $q^{{}^{\prime}}=6$,
$\\{a,b\\}=\\{2,4\\}$ and $\frac{u}{v}=3$, that is
$q=6\mathrm{gcd}(r_{3},r_{4},q)$. Note that
$\gcd(r_{3},q)=\gcd(r_{3},r_{4},q)\gcd(a,q^{\prime})$ and
$\gcd(\gcd(r_{1},r_{2},q),\gcd(r_{3},q))=1$. It follows that
$\mathrm{gcd}(r_{1},r_{2},q)\mathrm{gcd}(a,q^{{}^{\prime}})|6.$
Since $\mathrm{gcd}(a,q^{{}^{\prime}})>1$ and
$\varphi(\mathrm{gcd}(r_{1},r_{2},q))>1$, then
$\mathrm{gcd}(a,q^{{}^{\prime}})=2$ and $\mathrm{gcd}(r_{1},r_{2},q)=3$. If
$\mathrm{gcd}(r_{3},r_{4},q)\neq 2$, then
$\varphi(\mathrm{gcd}(r_{3},r_{4},q))>1$. Similarly, using the same argument
as above we obtain that $\mathrm{gcd}(r_{3},r_{4},q)=3$, a contradiction. If
$\mathrm{gcd}(r_{3},r_{4},q)=2$, then $q=12$ and
$\\{r_{1},r_{2}\\}=\\{3,9\\}$. Applying (5) with $p=3$ we have
$f(r_{1})+f(r_{2})=0$. It follows that $f(r_{3})+f(r_{4})=0$ which contradicts
with $f(r_{3})=-3f(r_{4})$ since $\frac{u}{v}=3$.
If $\mathrm{gcd}(a,q^{{}^{\prime}})>1,\mathrm{gcd}(c,q^{{}^{\prime}})>1$.
Since $\mathrm{gcd}(a,c,q^{{}^{\prime}})=1$,
$\mathrm{gcd}(a,q^{{}^{\prime}})=\mathrm{gcd}(b,q^{{}^{\prime}})$ and
$\mathrm{gcd}(c,q^{{}^{\prime}})=\mathrm{gcd}(e,q^{{}^{\prime}})$, then one of
$\gcd(a,b,q^{\prime})$ and $\gcd(c,e,q^{\prime})$ is larger than 2, say,
$\gcd(a,b,q^{\prime})>2$. Let
$l_{j}=1+j\cdot\frac{q^{\prime}}{\gcd(a,b,q^{\prime})},j=1,\cdots,\frac{q^{\prime}}{\gcd(a,b,q^{\prime})}-1.$
Similarly, we can choose $l_{j_{0}}\neq 1$ and $\gcd(l_{j_{0}},q^{\prime})=1$
such that
$\sum_{n=0}^{\infty}{\\{\frac{-\frac{u}{v}}{nq^{{}^{\prime}}+a}+\frac{\frac{u}{v}}{nq^{{}^{\prime}}+b}+\frac{1}{nq^{{}^{\prime}}+c^{\prime}}+\frac{-1}{nq^{{}^{\prime}}+e^{\prime}}\\}}=0,$
where $c^{\prime}\equiv l_{j_{0}}^{-1}c,e^{\prime}\equiv l_{j_{0}}^{-1}e$,
$0<c^{\prime},e^{\prime}<q^{\prime}$. It follows that
$T_{1}=\sum_{n=0}^{\infty}{\\{\frac{1}{nq^{{}^{\prime}}+c}+\frac{1}{nq^{{}^{\prime}}+e^{\prime}}-\frac{1}{nq^{{}^{\prime}}+c^{\prime}}-\frac{1}{nq^{{}^{\prime}}+e}\\}}=0,$
$c\neq c^{\prime},e\neq
e^{\prime},\gcd(c,q^{\prime})=\gcd(e,q^{\prime})=\gcd(c^{\prime}q^{\prime})=\gcd(e^{\prime},q^{\prime})>1$.
The remaining argument is the same line as in Case 2 (i). This completes the
proof.$\Box$
Proof of Theorem 1.1: By the above propositions 3.1-3.4, we have proven
Theorem 1.1.
Acknowledgement: The authors wish to thank the referee for helpful comments on
this paper.
## References
* [1] S.D. Adhikari, Transcendental Infinite sums and related questions,Number theory and discrete mathematics, proceedings of conference, Chandigarh, 2000 (Hindustan Book Agency, 2002), 169-178.
* [2] S.D. Adhikari, N. Saradha, T.N. Shorey, R. Tijdeman, Transcendental Infinite Sums, Indag. Math. (N.S) 12 (2001), 1-14.
* [3] A. Baker, B.J. Birch, E.A. Wirsing, On a problem of Chowla, J. Number Theory 5 (1973), 224-236.
* [4] S. Chowla, The Riemann zeta and allied functions, Bull. Amer. Math. Soc. 58 (1952),287-305.
* [5] Chao Ko and Qi Sun, Introduction to Number Theory(I) (in Chinese), Higher Education Press, 2001.1, 53-54.
* [6] D. H. Lehmer, Euler constants for arithmetical progressions, Acta Arith. 27 (1975), 125-142.
* [7] A.E. Livingston, The series $\sum_{n=1}^{\infty}\frac{f(n)}{n}$ for periodic $f$, Canad. Math. Bull. 8 (1965), 413-432.
* [8] T. Okada, On a certain infinite sums for a periodic arithmetical functions, Acta Arith.40(1982), 143-153.
* [9] N. Saradha, R. Tijdeman,On the transcendence of infinite sums of values of rational functions, J. London. Math. Soc.(3) 67 (2003), 580-592.
* [10] R. Tijdeman, Some applications of diophantine approximation, Number Theory for the Millennium II,Proceedings of conference, Urbana, IL, 2000-Vol. III( A.K. Peters, Natick MA, 2002), 261-284.
* [11] R. Tijdeman, On irrationality and transcendency of infinite sums of rational numbers, Shorey Proc., to appear. Pingzhi Yuan Juan Li School of Mathematics Department of Mathematics South China Normal University Sun Yat-sen University Guangzhou $510631$ Guangzhou $510275$ P.R.CHINA P.R.CHINA email:mcsypz@mail.sysu.edu.cn email:lijuan6@mail2.sysu.edu.cn
|
arxiv-papers
| 2009-09-13T02:33:27 |
2024-09-04T02:49:05.308500
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Pingzhi Yuan, Juan Li",
"submitter": "Pingzhi Yuan",
"url": "https://arxiv.org/abs/0909.2387"
}
|
0909.2388
|
# Davenport constant with weights
Pingzhi Yuan
School of Mathematics, South China Normal University , Guangzhou 510631,
P.R.CHINA
e-mail mcsypz@mail.sysu.edu.cn
Xiangneng Zeng
Department of Mathematics, Sun Yat-Sen University, Guangzhou 510275, P.R.CHINA
###### Abstract
For the cyclic group $G=\mathbb{Z}/n\mathbb{Z}$ and any non-empty
$A\in\mathbb{Z}$. We define the Davenport constant of $G$ with weight $A$,
denoted by $D_{A}(n)$, to be the least natural number $k$ such that for any
sequence $(x_{1},\cdots,x_{k})$ with $x_{i}\in G$, there exists a non-empty
subsequence $(x_{j_{1}},\,\cdots,\,x_{j_{l}})$ and $a_{1},\,\cdots,\,a_{l}\in
A$ such that $\sum_{i=1}^{l}a_{i}x_{j_{i}}=0$. Similarly, we define the
constant $E_{A}(n)$ to be the least $t\in\mathbb{N}$ such that for all
sequences $(x_{1},\,\cdots,\,x_{t})$ with $x_{i}\in G$, there exist indices
$j_{1},\,\cdots,\,j_{n}\in\mathbb{N},1\leq j_{1}<\cdots<j_{n}\leq t$, and
$\vartheta_{1},\,\cdots,\,\vartheta_{n}\in A$ with
$\sum^{n}_{i=1}\vartheta_{i}x_{j_{i}}=0$. In the present paper, we show that
$E_{A}(n)=D_{A}(n)+n-1$. This solve the problem raised by Adhikari and Rath
[3], Adhikari and Chen [2], Thangadurai [12] and Griffiths [10].
MSC: 11B50
Key words: Zero-sum problems, weighted EGZ, zero-sum free sequences.
00footnotetext: Supported by the Guangdong Provincial Natural Science
Foundation (No. 8151027501000114) and NSF of China (No. 10571180).
## 1 Introduction
For an abelian group $G$, the Davenport constant $D(G)$ is defined to be the
smallest natural number $k$ such that any sequence of $k$ elements in $G$ has
a non-empty subsequence whose sum is zero (the identity element). Another
interesting constant $E(G)$ is defined to be the smallest natural number $k$
such that any sequence of $k$ elements in $G$ has a subsequence of length
$|G|$ whose sum is zero.
The following result due to Gao [7] connects these two invariants.
###### Theorem 1.1
If $G$ is a finite abelian group of order $n$, then $E(G)=D(G)+n-1$.
For a finite abelian group $G$ and any non-empty $A\in\mathbb{Z}$, Adhikari
and Chen [2] defined the Davenport constant of $G$ with weight $A$, denoted by
$D_{A}(G)$, to be the least natural number $k$ such that for any sequence
$(x_{1},\cdots,x_{k})$ with $x_{i}\in G$, there exists a non-empty subsequence
$(x_{j_{1}},\,\cdots,\,x_{j_{l}})$ and $a_{1},\,\cdots,\,a_{l}\in A$ such that
$\sum_{i=1}^{l}a_{i}x_{j_{i}}=0$. Clearly, if $G$ is of order $n$, it is
equivalent to consider $A$ to be a non-empty subset of $\\{0,1,\cdots,n-1\\}$
and cases with $0\in A$ are trivial.
Similarly, for any such set $A$, for a finite abelian group $G$ of order $n$,
the constant $E_{A}(G)$ is defined to be the least $t\in\mathbb{N}$ such that
for all sequences $(x_{1},\,\cdots,\,x_{t})$ with $x_{i}\in G$, there exist
indices $j_{1},\,\cdots,\,j_{n}\in\mathbb{N},1\leq j_{1}<\cdots<j_{n}\leq t$,
and $\vartheta_{1},\,\cdots,\,\vartheta_{n}\in A$ with
$\sum^{n}_{i=1}\vartheta_{i}x_{j_{i}}=0$.
For the group $G=\mathbb{Z}/n\mathbb{Z}$, we write $E_{A}(n)$ and $D_{A}(n)$
respectively for $E_{A}(G)$ and $D_{A}(G)$. In the cases
$A=\\{1\\},\,\\{-1,\,1\\},\mathbb{Z}_{n}^{\star}$ or $A=(a_{1},\cdots,a_{r})$
with $\gcd(a_{2}-a_{1},\cdots,a_{r}-a_{1},n)=1$ or $n=p$ is a prime, it is
proved that $E_{A}(n)=D_{A}(n)+n-1$. The following conjecture has been raised
by Adhikari and Rath [3], Adhikari and Chen [2], Thangadurai [12] and
Griffiths [10], they seems to believe that Conjecture 1.1 is true and have
proved it in some special cases.
###### Conjecture 1.1
For any non-empty set $A\in\mathbb{Z}$, $E_{A}(n)=D_{A}(n)+n-1$.
The main purpose of the present paper is to prove Conjecture 1.1. By using the
main theorem of Devos, Goddyn and Mohar [5] and a recently proved theorem of
the authors [13], we shall prove the following theorem
###### Theorem 1.2
For any non-empty set $A\in\mathbb{Z}$, $E_{A}(n)=D_{A}(n)+n-1$.
Throughout this paper, let $G$ be an additive finite abelian group.
$\mathcal{F}(G)$ denotes the free abelian monoid with basis $G$, the elements
of which are called $sequences$ (in $G$). A sequence of not necessarily
distinct elements from $G$ will be written in the form
$S=g_{1}\,\cdots\,g_{k}=\prod_{i=1}^{k}g_{i}=\prod_{g\in
G}g^{\mathsf{v}_{g}(S)}\in\mathcal{F}(G)$, where $\mathsf{v}_{g}(S)\geq 0$ is
called the $multiplicity$ of $g$ in $S$. We call $|S|=k$ the $length$ of $S$,
$\mathsf{h}(S)=\max\\{\mathsf{v}_{g}(S)|g\in G\\}\in[0,|S|]$ the maximum of
the multiplicities of $S$, ${\rm supp}(S)=\\{g\in G:\,\mathsf{v}_{g}(S)>0\\}$
the $support$ of $S$. For every $g\in G$ we set
$g+S=(g+g_{1})\cdots(g+g_{k})$.
We say that $S$ contains some $g\in G$ if $\mathsf{v}_{g}(S)\geq 1$ and a
sequence $T\in\mathcal{F}(G)$ is a $subsequence$ of $S$ if
$\mathsf{v}_{g}(T)\leq\mathsf{v}_{g}(S)$ for every $g\in G$, denoted by $T|S$.
Furthermore, by $\sigma(S)$ we denote the sum of $S$, (i.e.
$\sigma(S)=\sum_{i=1}^{k}g_{i}=\sum_{g\in G}\mathsf{v}_{g}(S)g\in G$). For
every $k\in\\{1,2,\cdots,\,|S|\\}$, let
$\sum_{k}(S)=\\{g_{i_{1}}+\cdots+g_{i_{k}}|1\leq
i_{1}<\cdots<i_{k}\leq|S|\\},\,\sum_{\leq k}(S)=\cup_{i=1}^{k}\sum_{i}(S)$,
and let $\sum(S)=\sum_{\leq|S|}(S)$.
Let $S$ be a sequence in $G$. We call $S$ a $zero-sum$ $sequence$ if
$\sigma(S)=0$.
Also, we follow the same terminologies and notations as in the survey article
[8] or in the book [9].
## 2 Lemmas
First, we need a result on the sum of $l$ finite subsets of $G$. If ${\bf
A}=(A_{1},A_{2},\cdots,A_{m})$ is a sequence of finite subsets of $G$, and
$l\leq m$, we define
$\sum_{l}({\bf A})=\\{a_{i_{1}}+\cdots+a_{i_{l}}:1\leq i_{1}<\cdots<i_{l}\leq
m\,\mbox{ and}\,a_{i_{j}}\in A_{i_{j}}\,\mbox{ for every}\,1\leq j\leq l\\}.$
So $\sum_{l}({\bf A})$ is the set of all elements which can be represented as
a sum of $l$ terms from distinct members of ${\bf A}$. The following is the
main result of Devos, Goddyn and Mohar [5].
Theorem DGM Let ${\bf A}=(A_{1},A_{2},\cdots,A_{m})$ be a sequence of finite
subsets of $G$, let $l\leq m$, and let $H=stab(\sum_{l}({\bf A}))$. If
$\sum_{l}({\bf A})$ is nonempty, then
$|\sum_{l}({\bf A})|\geq|H|(1-l+\sum_{Q\in
G/H}\min\\{l,|\\{i\in\\{1,\cdots,m\\}:A_{i}\cap Q\neq\emptyset\\}|\\}).$
We still need the following new result on Davenport’s constant [13].
Theorem YZ Let $G$ be a finite abelian group of order $n$ and Davenport
constant $D(G)$. Let $S=0^{\mathsf{h}(S)}\prod_{g\in
G}g^{\mathsf{v}_{g}(S)}\in\mathcal{F}(G)$ be a sequence with a maximal
multiplicity $\mathsf{h}(S)$ attained by $0$ and $|S|=t\geq n+D(G)-1$. Then
there exists a subsequence $S_{1}$ of $S$ with length $|S_{1}|\geq t+1-D(G)$
and $0\in\sum_{k}(S_{1})$ for every $1\leq k\leq|S_{1}|$. In particular, for
every sequence $S$ in $G$ with length $|S|\geq n+D(G)-1$, we have
$0\in\sum_{km}(S),\,\mbox{ for every }\quad 1\leq k\leq(|S|+1-D(G))/m,$
where $m$ is the exponent of $G$.
## 3 Proof of Theorem 1.2
* Proof.
The proof of $E_{A}(n)\geq D_{A}(n)+n-1$ is easy, so it is sufficient to prove
the reverse inequality.
For any non-empty set $A=\\{a_{1},\cdots a_{r}\\}\subset\mathbb{Z}$ and a
cyclic group $G=\mathbb{Z}/n\mathbb{Z}$, let $t=D_{A}(n)+n-1$ and
$S=x_{1}\cdots x_{t}$ is any sequence in $G$ with length $|S|=t=D_{A}(n)+n-1$.
Put
$A_{i}=Ax_{i}=\\{a_{1}x_{i},\cdots,a_{r}x_{i}\\}\mbox{ for}\,i=1,\cdots,t$
and ${\bf A}=(A_{1},\,\cdots,\,A_{t})$. It suffices to prove that
$0\in\sum_{n}({\bf A})$.
We shall assume (for a contradiction) that the theorem is false and choose a
counterexample $(A,G,\,S)$ so that $n=|G|$ is minimum, where $G$ is a cyclic
group of order $n$, $A$ is a finite subset of $\mathbb{Z}$ and $S=x_{1}\cdots
x_{t}$ is a sequence in $G$ such that
$0\not\in\sum_{n}({\bf A}).$
Next we will show that our assumptions imply $H=stab(\sum_{n}({\bf
A}))=\\{0\\}$. Suppose (for a contradiction) that $H=stab(\sum_{n}({\bf
A}))\neq\\{0\\}$ and let $\varphi:\,G\longrightarrow G/H$ denote the canonical
homomorphism and $\varphi(x_{i})$ the image of $x_{i}$ for $1\leq i\leq t$.
Let ${\bf A_{\varphi}}=(\varphi(A_{1}),\cdots,\varphi(A_{t}))$. By our
assumption for the minimal of $|G|$, the theorem holds for
$(A,\varphi(G),\,\varphi(S))$. Since $n>|\varphi(G)|+D_{A}(\varphi(G))-1$,
$|\varphi(G)||n$ and $D_{A}(G)\geq D_{A}(\varphi(G))$, repeated applying the
theorem to the sequence $\varphi(S)=\varphi(x_{1}),\cdots,\varphi(x_{t})$ we
have
$\varphi(0)=\varphi(H)\in\sum_{n}({\bf A_{\varphi}}),$
thus $0\in H\subset\sum_{n}({\bf A})$. This contradiction implies that
$H=stab(\sum_{n}({\bf A}))=\\{0\\}$.
If there is an element $a\in G$ such that $|\\{j\in\\{1,\cdots,t\\}:a\in
A_{j}\\}|\geq n$, then $0\in\sum_{n}({\bf A})$, a contradiction. Therefore we
may assume that for every $a\in G$, $|\\{j\in\\{1,\cdots,t\\}:a\in
A_{j}\\}|\leq n$. Let $r$ be the number of $i\in\\{1,\cdots,t\\}$ with
$|A_{i}|=1$, by Theorem DGM and the assumptions, we have
$n-1\geq\sum_{n}({\bf A})\geq 1-n+\sum_{a\in
G}\min\\{n,|\\{j\in\\{1,\cdots,t\\}:a\in A_{j}\\}|\\}$
$=1-n+\sum_{i=1}^{t}|A_{i}|\geq 1-n+2(n+D_{A}(G)-1-r)+r.$
It follows that
$r\geq 2D_{A}(G).$ (1)
Without loss of generality, we may assume that $x_{1},\cdots,x_{r}$ are all
the elements in $\\{x_{1},\,\cdots,\,x_{t}\\}$ such that $|A_{i}|=1$, and
$x_{1}$ is the element in $\\{x_{1},\,\cdots,\,x_{r}\\}$ such that
$a_{1}x_{1}$ attains the maximal multiplicity in the sequence
$S_{1}=(a_{1}x_{1})\cdots(a_{1}x_{r})$. Observe that $\sum_{n}({\bf
A})=\sum_{n}(A(x_{1}-x_{u}),\,\cdots,\,A(x_{t}-x_{u}))$ for every $1\leq u\leq
r$. Therefore without loss of generality we may assume that $a_{1}x_{1}=0$ and
$\mathsf{v}_{0}(S_{1})=\mathsf{h}(S_{1})$ for the sequence
$S_{1}=(a_{1}x_{1})\cdots(a_{1}x_{r})=0^{\mathsf{h}(S_{1})}(a_{1}x_{\mathsf{h}(S_{1})+1})\cdots(a_{1}x_{r}).$
(2)
Let $H_{1}=<x_{1},\cdots,x_{r}>$ be the group generated by
$x_{1},\cdots,x_{r}$, $H=a_{1}H_{1}$. We have the following claim.
Claim: $D_{A}(G)\geq D_{A}(H_{1})\geq D(H)=|H|$.
The last equality of the Claim follows from the fact that $H$ is a subgroup of
the cyclic group $G$. The first inequality in the Claim is obvious, so we only
need to prove that $D_{A}(H_{1})\geq D(H)$. Suppose that $W=y_{1}\cdots
y_{D(H)-1}$ is a zero-sum free sequence in $H$. Since $H=a_{1}H_{1}$, we have
$y_{i}=a_{1}w_{i},w_{i}\in H_{1},i=1,\cdots r$. Further, it is easy to see
that $Aw_{i}=a_{1}w_{i},\,i=1,\cdots,r$ by the definition of $H_{1}$, so
$w_{1}\cdots w_{D(H)-1}$ is a zero-sum free sequence in $H_{1}$ with respect
to the weight $A$, thus $D_{A}(H_{1})\geq D(H)$ and the Claim follows.
By the Claim, (1), (2) and Theorem YZ, $S_{1}$ has a subsequence $S_{2}$ of
length $|S_{2}|=s\geq r+1-|H|$ such that $0\in\sum_{l}(S_{2})$ for every
$1\leq l\leq s$. Without loss of generality, we may assume that
$S_{2}=(a_{1}x_{1})\cdots(a_{1}x_{s})$.
If $s\geq n$ then $0\in\sum_{n}(S_{2})\subset\sum_{n}({\bf A})$, we are done.
If $s<n$, then $|x_{s+1}\cdots x_{t}|=t-s=n-1+D_{A}(G)-s\geq D_{A}(G)$.
Repeated using the definition of $D_{A}(G)$, there exists an integer $v$ such
that $v\leq n,\,t-s-v\leq D_{A}(G)-1$ and
$0\in\sum_{v}((A_{s+1},\cdots A_{t})).$
Since $\sum_{l}(S_{2})=\sum_{l}((A_{1},\cdots,A_{s}))$ and
$0\in\sum_{l}((A_{1},\cdots,A_{s}))$ for every $1\leq l\leq s$, we have
$0\in\sum_{v+k}({\bf A})\quad\mbox{ for every }0\leq k\leq s.$
Therefore $0\in\sum_{n}({\bf A})$ since $v+s\geq t+1-G_{A}(G)\geq n$. This
completes the proof of the theorem.
$\Box$
## References
* [1] S.D. Adhikari, Y.G. Chen, J.B. Friedlander, S.V. Konyagin, F. Pappalardi, Contributions to zero-sum problems, Discrete Math. 306 (2006) 1-10.
* [2] S.D. Adhikari, Y.G. Chen, Davenport constant with weights and some related questions, II, J. Combin. Theory Theory Ser.A 115(2008), 178-184.
* [3] S.D. Adhikari, P. Rath, Davenport constant with weights and some related questions, Integers, Paper A 6 (2006) 30.
* [4] S.D. Adhikari, P. Rath, Zero-sum problems in Combinatorial Number Theory, in: R. Balasubramanian, K. Srinivas (Eds.), The Riemann Zeta Function and Related Themes: Papers in Honour of Professor K. Ramachandra, Proceedings of International Conf. Held at National Institute of Advanced Studies, Bangalore, 13-15 December, 2003, in: Ramanujan Math. Soc. Lect. Notes Ser., vol. 2, 2006.
* [5] M. DeVos, L. Goddyn and B. Mohar, A generalization of Kneser’s addition theorem, Advance in Mathematics (to appear).
* [6] P. Erd$\ddot{o}$s, A. Ginzburg, A. Ziv, Theorem in the additive number theory, Bull. Res. Council Israel 10F (1961) 41-43.
* [7] W.D. Gao, A combinatorial problem on finite abelian groups, J. Number Theory 58 (1996) 100-103.
* [8] W.D. Gao, A. Geroldinger, Zero-sum problems in finite abelian groups: A survey, Expo. Math. 24 (2006) 337-369.
* [9] A. Geroldinger, F. Halter-Koch, Non-Unique Factorizations, Chapman and Hall/CRC, 2006.
* [10] The Erd$\ddot{o}$s-Ginzberg-Ziv theorem with units, Discrete Mathematics 308(2008), 5473-5484.
* [11] Florian Luca, A generalization of a classical zero-sum problem, Discrete Math. 307(2007) 1672-1678.
* [12] Thangadurai R, A variant of Davenport’s constant, Proc. Indian Acad. Sci. (Math. Sci.) 117(2007), 147-158.
* [13] P. Yuan and X. Zeng, A new result on Davenport’s constant, submitted.
|
arxiv-papers
| 2009-09-13T02:42:25 |
2024-09-04T02:49:05.314975
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Pingzhi Yuan, Xiangneng Zeng",
"submitter": "Pingzhi Yuan",
"url": "https://arxiv.org/abs/0909.2388"
}
|
0909.2404
|
Effect of localizing groups on electron transport through single conjugated
molecules
Santanu K. Maiti1,2,∗
1Theoretical Condensed Matter Physics Division, Saha Institute of Nuclear
Physics,
1/AF, Bidhannagar, Kolkata-700 064, India
2Department of Physics, Narasinha Dutt College, 129, Belilious Road,
Howrah-711 101, India
Abstract
Electron transport properties through single conjugated molecules sandwiched
between two non-superconducting electrodes are studied by the use of Green’s
function technique. Based on the tight-binding model, we do parametric
calculations to characterize the electron transport through such molecular
bridges. The electron transport properties are significantly influenced by (a)
the existence of localizing groups in these conjugated molecules and (b) the
molecule to electrode coupling strength, and, here we focus our results in
these two aspects.
PACS No.: 73.23.-b; 81.07.Nb; 85.65.+h
Keywords: Conjugated molecules; Localizing groups; Conductance; $I$-$V$
characteristic.
∗Corresponding Author: Santanu K. Maiti
Electronic mail: santanu.maiti@saha.ac.in
## 1 Introduction
Molecular electronics and transport have attracted much more attention since
molecules constitute promising building blocks for future generation of
nanoelectronic devices. Following experimental developments, theory can play a
major role in understanding the new mechanisms of conductance. The single-
molecule electronics plays a significant role in designing and developing the
future nanoelectronic circuits, but, the goal of developing a reliable
molecular-electronics technology is still over the horizon and many key
problems, such as device stability, reproducibility and the control of single-
molecule transport need to be solved. Electronic transport through molecules
was first studied theoretically in $1974$ [1]. Then lot of experiments [2, 3,
4, 5, 6] have been performed through molecules placed between two metallic
electrodes with few nanometer separation. The operation of such two-terminal
devices is due to an applied bias. Current passing across the junction is
strongly nonlinear function of applied bias voltage and its detailed
description is a very complex problem. The complete knowledge of the
conduction mechanism in this scale is not well understood even today. In many
molecular devices, electronic transport is dominated by conduction through
broadened HOMO or LUMO states. In contrast here we find that the transport
through single conjugated molecules can be controlled very sensitively by
introducing the localizing groups in these molecules. This sensitivity opens
up new possibilities for novel single-molecule sensors. Electron conduction
through molecules strongly depends on (a) the delocalization of the molecular
electronic orbitals and (b) their coupling strength to the two electrodes. In
a very recent experiment, Tali Dadosh et al. [2] have measured conductance of
single conjugated molecules and predicted that the existence of localizing
groups in a conjugated molecule suppresses the electrical conduction through
the molecule. These results motivate us to study the electron transport
through such conjugated molecules.
The aim of the present paper is to reproduce an analytic approach based on the
tight-binding model to investigate the electron transport properties for the
model of single conjugated molecules taken in their experiment [2]. Several ab
initio methods are used for the calculation of conductance [7, 8, 9, 10, 11,
12], yet it is needed the simple parametric approaches [13, 14, 15, 16, 17,
18, 19, 20, 21] for this calculation. The parametric study is motivated by the
fact that it is much more flexible than that of the ab initio theories since
the later theories are computationally very expensive and here we focus our
attention on the qualitative effects rather than the quantitative ones. This
is why we restrict our calculations on the simple analytical formulation of
the transport problem.
The scheme of the paper is as follow. In Section $2$, we give a very brief
description for the calculation of transmission probability and current
through a finite size conductor sandwiched between two one-dimensional ($1$D)
metallic electrodes. Section $3$ focuses the results of conductance-energy
($g$-$E$) and current-voltage ($I$-$V$) characteristics for the single
conjugated molecules and study the effects of localizing groups in the above
mentioned quantities. Finally, we summarize our results in Section $4$.
## 2 A glimpse onto the theoretical formulation
Here we describe very briefly about the methodology for the calculation of
transmission probability ($T$), conductance ($g$) and current ($I$) through a
finite size conducting system attached to two semi-infinite metallic
electrodes by using the Green’s function technique.
Let us first consider a $1$D conductor with $N$ number of atomic sites (array
of filled circles) connected to two semi-infinite electrodes, namely, source
and drain, as presented in Fig. 1. The conducting system in between
Figure 1: Schematic view of a $1$D conductor with $N$ number of atomic sites
(filled circles) attached to two electrodes through the sites $1$ and $N$,
respectively.
the two electrodes can be an array of few quantum dots, or a single molecule,
or an array of few molecules, etc. At low voltages and temperatures, the
conductance of the conductor can be written by using the Landauer conductance
formula,
$g=\frac{2e^{2}}{h}T$ (1)
where $g$ is the conductance and $T$ is the transmission probability of an
electron through the conductor. The transmission probability can be expressed
in terms of the Green’s function of the conductor and the coupling of the
conductor to the two electrodes by the expression,
$T={\mbox{Tr}}\left[\Gamma_{S}G_{C}^{r}\Gamma_{D}G_{C}^{a}\right]$ (2)
where $G_{C}^{r}$ and $G_{C}^{a}$ are respectively the retarded and advanced
Green’s function of the conductor. $\Gamma_{S}$ and $\Gamma_{D}$ are the
coupling terms of the conductor due to the coupling to the source and drain,
respectively. For the complete system, i.e., the conductor and the two
electrodes, the Green’s function is defined as,
$G=\left(\epsilon-H\right)^{-1}$ (3)
where $\epsilon=E+i\eta$. $E$ is the injecting energy of the source electron
and $\eta$ is a very small number which can be put as zero in the limiting
approximation. The above Green’s function corresponds to the inversion of an
infinite matrix which consists of the finite conductor and two semi-infinite
electrodes. It can be partitioned into different sub-matrices those correspond
to the individual sub-systems.
The effective Green’s function for the conductor can be written as,
$G_{C}=\left(\epsilon-H_{C}-\Sigma_{S}-\Sigma_{D}\right)^{-1}$ (4)
where $H_{C}$ is the Hamiltonian for the conductor sandwiched between the two
electrodes. The single band tight-binding Hamiltonian for the conductor within
the non-interacting picture can be written in the following form,
$H_{C}=\sum_{i}\epsilon_{i}c_{i}^{\dagger}c_{i}+\sum_{<ij>}t\left(c_{i}^{\dagger}c_{j}+c_{j}^{\dagger}c_{i}\right)$
(5)
where $c_{i}^{\dagger}$ ($c_{i}$) is the creation (annihilation) operator of
an electron at site $i$, $\epsilon_{i}$’s are the site energies and $t$ is the
nearest-neighbor hopping integral. Here
$\Sigma_{S}=h_{SC}^{\dagger}g_{S}h_{SC}$ and
$\Sigma_{D}=h_{DC}g_{D}h_{DC}^{\dagger}$ are the self-energy terms due to the
two electrodes. $g_{S}$ and $g_{D}$ are respectively the Green’s function for
the source and drain. $h_{SC}$ and $h_{DC}$ are the coupling matrices and they
will be non-zero only for the adjacent points in the conductor, $1$ and $N$ as
shown in Fig. 1, and the electrodes respectively. The coupling terms
$\Gamma_{S}$ and $\Gamma_{D}$ for the conductor can be calculated through the
expression,
$\Gamma_{\\{S,D\\}}=i\left[\Sigma_{\\{S,D\\}}^{r}-\Sigma_{\\{S,D\\}}^{a}\right]$
(6)
where $\Sigma_{\\{S,D\\}}^{r}$ and $\Sigma_{\\{S,D\\}}^{a}$ are the retarded
and advanced self-energies, respectively, and they are conjugate to each
other. Datta et al. [22] have shown that the self-energies can be expressed
like,
$\Sigma_{\\{S,D\\}}^{r}=\Lambda_{\\{S,D\\}}-i\Delta_{\\{S,D\\}}$ (7)
where $\Lambda_{\\{S,D\\}}$ are the real parts of the self-energies which
correspond to the shift of the energy eigenstates of the conductor and the
imaginary parts $\Delta_{\\{S,D\\}}$ of the self-energies represent the
broadening of these energy levels. This broadening is much larger than the
thermal broadening and this is why we restrict our all calculations only at
absolute zero temperature. By doing some simple algebra these real and
imaginary parts of self-energies can also be determined in terms of coupling
strength ($\tau_{\\{S,D\\}}$) between the conductor and two electrodes,
injection energy ($E$) of the transmitting electron, site energy
($\epsilon_{0}$) of the electrodes and hopping strength ($v$) between nearest-
neighbor sites in the electrodes. Thus the coupling terms $\Gamma_{S}$ and
$\Gamma_{D}$ can be written in terms of the retarded self-energy as,
$\Gamma_{\\{S,D\\}}=-2{\mbox{Im}}\left[\Sigma_{\\{S,D\\}}^{r}\right]$ (8)
Now all the information regarding the conductor to electrode coupling is
included into the two self energies as stated above and is analyzed through
the use of Newns-Anderson chemisorption theory [13, 14]. The detailed
description of this theory is obtained in these two references.
By calculating the self-energies, the coupling terms $\Gamma_{S}$ and
$\Gamma_{D}$ can be easily obtained and then the transmission probability $T$
can be computed from the expression as mentioned in Eq. 2.
Since the coupling matrices $h_{SC}$ and $h_{DC}$ are non-zero only for the
adjacent points in the conductor, $1$ and $N$ as shown in Fig. 1, the
transmission probability becomes,
$T(E,V)=4\Delta_{11}^{S}(E,V)\Delta_{NN}^{D}(E,V)|G_{1N}(E,V)|^{2}$ (9)
The current passing through the conductor is depicted as a single-electron
scattering process between the two reservoirs of charge carriers. The current-
voltage relation is evaluated from the following expression [23],
$I(V)=\frac{e}{\pi\hbar}\int\limits_{E_{F}-eV/2}^{E_{F}+eV/2}T(E,V)dE$ (10)
where $E_{F}$ is the equilibrium Fermi energy. For the sake of simplicity,
here we assume that the entire voltage is dropped across the conductor-
electrode interfaces and this assumption does not significantly change the
qualitative behaviors of the $I$-$V$ characteristics. Using the expression of
$T(E,V)$ as in Eq. 9 the final form of $I(V)$ becomes,
$\displaystyle I(V)$ $\displaystyle=$
$\displaystyle\frac{4e}{\pi\hbar}\int\limits_{E_{F}-eV/2}^{E_{F}+eV/2}\Delta_{11}^{S}(E,V)\Delta_{NN}^{D}(E,V)$
(11) $\displaystyle\times~{}|G_{1N}(E,V)|^{2}dE$
Eq. 1, Eq. 9 and Eq. 11 are the final working formulae for the calculation of
conductance $g$ and current-voltage characteristics, respectively, for any
finite size conductor sandwiched between two electrodes.
With the help of the above formulation, we shall describe the electron
transport properties through some conjugated molecules (Fig. 2). For the sake
of simplicity throughout this article we use the unit $c=e=h=1$.
## 3 Results and discussion
This section focuses the conductance-energy ($g$-$E$) and current-voltage
($I$-$V$) characteristics of three short single conjugated molecules. These
molecules are specified as:
Figure 2: Structures of the three molecules: $1$,$4$-benzenedimethanethiol
(BDMT), $4$,$4^{\prime}$-biphenyldithiol (BPD) and
bis-($4$-mercaptophenyl)-ether (BPE) those are attached to two electrodes by
thiol (S-H) groups.
$1$,$4$-benzenedimethanethiol (BDMT), in which the molecular conjugation is
broken near the contacts by a methylene group;
$4$,$4^{\prime}$-biphenyldithiol (BPD), a fully conjugated molecule; and
bis-($4$-mercaptophenyl)-ether (BPE), where the molecular conjugation is
broken by an oxygen atom at the center. The schematic representations of these
three molecules, with thiol groups at the two extreme ends of each molecules,
are shown in Fig. 2. These molecules are contacted to the two semi-infinite
$1$D electrodes by thiol (S-H) groups via single channels (same as shown
schematically in Fig. 1). In actual experimental arrangement, two electrodes
are constructed by using gold (Au) substance and molecule attached to the
electrodes by thiol (S-H)
Figure 3: Conductance $g$ as a function of the injecting electron energy $E$
in the weak-coupling limit, where (a), (b) and (c) are respectively for the
BDMT, BPD and BPE molecules.
groups in the chemisorption technique where hydrogen (H) atoms remove and
sulfur (S) atoms reside. The electron transport through such conjugated
molecules significantly influenced by the presence of localizing groups in the
molecules and the molecule-to-electrode coupling strength. Here, we shall
investigate our results in these aspects. Throughout the article, we discuss
the results in two limiting regimes depending on the coupling strength of the
molecule to the electrodes. One is defined as $\tau_{\\{S,D\\}}<<t$, the so-
called weak-coupling limit. The other one is $\tau_{\\{S,D\\}}\sim t$, the so-
called strong-coupling limit. The parameters $\tau_{S}$ and $\tau_{D}$
correspond to the coupling of the molecules to the source and drain,
respectively. The values of the different parameters used in our calculations
in these two limiting regimes are assigned as: $\tau_{S}=\tau_{D}=0.5$;
$t=2.5$ (weak coupling) and $\tau_{S}=\tau_{D}=2$; $t=2.5$ (strong-coupling).
For the side attached electrodes the on-site energy ($\epsilon_{0}$) and the
nearest-neighbor hopping strength ($v$) are fixed to $0$ and $4$,
respectively. The Fermi energy $E_{F}$ is set at $0$.
In Fig. 3 we display conductance ($g$) as function of injecting electron
energy ($E$) for the three molecular bridge systems in the limit of weak
molecular coupling. Figures 3(a), (b) and (c) correspond to
Figure 4: Conductance $g$ as a function of the injecting electron energy $E$
in the strong molecule-to-electrode coupling limit, where (a), (b) and (c) are
respectively for the BDMT, BPD and BPE molecules.
the results for the bridges with BDMT, BPD and BPE molecules, respectively.
Conductance shows very sharp resonant peaks for some particular energy values,
while in almost all other cases it drops to zero. These resonant peaks are
associated with the energy eigenstates of the individual molecules that
bridges the two reservoirs. Therefore, the conductance spectrum manifests
itself the electronic structure of the molecule. At resonances, the
conductance ($g$) achieves the value $2$, and accordingly, the transmission
probability ($T$) goes to unity since we have the relation $g=2T$ from the
Landauer conductance formula with $e=h=1$ in our present formulation. For the
bridges with BDMT and BPD molecules, we see that the resonant peaks have very
narrow widths, while for the bridge with BPE molecule the width of the peaks
is almost zero. Thus, fine tuning in the energy scale is necessary to get the
electron conduction through these bridges, specially in the case of BPE
molecule for the weak-coupling limit. The most significant result is that, the
BPD molecule conducts
Figure 5: Current $I$ as a function of the applied bias voltage $V$ in the
weak molecule-to-electrode coupling, where the solid, dotted and dashed lines
are respectively for the BPD, BDMT and BPE molecules.
electron across the zero energy value, while the other two conduct beyond some
critical energy values (see Fig. 3). Therefore, we can tune the electron
conduction through the molecular bridge in a very controllable way.
In the strong molecular coupling limit, the resonant peaks in the conductance
spectra get substantial widths as shown in Fig. 4. This enhancement of the
resonant widths is due to the broadening of the molecular energy eigenstates
caused by the coupling of the molecules to the side attached electrodes in the
strong-coupling limit, where the contribution comes from the imaginary parts
of the self-energies [23]. Though for the molecular bridges with BDMT and BPD
molecules the resonant peaks get substantial widths, but, for the bridge with
BPE molecule, the increment of the widths is very small. For this BPE molecule
since the increment of the width of the resonant peak across the energy $E=2$
is comparatively higher than for the other energy values, we observe only one
peak across this energy value ($E=2$, Fig. 4(c)). Thus, for this molecular
bridge electron conduction takes place across a particular energy value, while
in all other energies no electron conduction takes place. This aspect may be
used to describe switching action in an electronic circuit.
Thus we see that the electron conduction strongly depends on the molecule
itself and also on the strength of the molecular coupling to the side attached
electrodes. The behavior of electron transfer through the molecular junction
becomes much more clearly observed from the current-voltage characteristics.
Current passing through the molecular system is computed from the integration
procedure of the transmission function $T$. The nature of the transmission
function is exactly similar to that of the conductance spectrum since $g=2T$
(from the Landauer formula), differ only in magnitude by the factor $2$.
Figure 6: Current $I$ as a function of the applied bias voltage $V$ in the
strong molecule-to-electrode coupling, where the solid, dotted and dashed
curves are respectively for the BPD, BDMT and BPE molecules.
In Fig. 5, we plot the current-voltage characteristics of these three
molecular bridges in the weak-coupling limit. The solid, dotted and dashed
curves correspond to the results for the molecular bridges with BPD, BDMT and
BPE molecules, respectively. The current shows staircase-like structure with
fine steps as a function of the applied bias voltage. This is due to the
discreteness of molecular resonances as shown in Fig. 3. With the increase of
the bias voltage, the electrochemical potentials on the electrodes are shifted
and eventually cross one of the molecular energy level. Accordingly, a current
channel is opened and a jump in the $I$-$V$ curve appears. The significant
observation is that, for the molecular bridge with BPD molecule (free from
localizing group), the current amplitude is much higher (see solid curve of
Fig. 5) compared two the other two bridges. This is due to the fact that the
localizing groups (both in BDMT and BPE molecules) interfere with the
conjugated aromatic systems and suppress the overall conductance through the
molecules. On the other hand, the another important feature is that, in purely
conjugate molecule (BPD) the electron conduction takes place as long as the
bias voltage is applied, while for the other two molecules it appears beyond
some finite values of $V$. This behavior gives a key idea in the fabrication
of molecular devices.
The shape and height of these current steps depend on the width of the
molecular resonances. With the increase of molecule-to-electrode coupling
strength, current gets a continuous variation with the applied bias voltage
and achieves much higher values (compared to the current amplitude in the weak
coupling case), as plotted in Fig. 6, where the solid, dotted and dashed
curves correspond to the same meaning as in Fig. 5. In this strong molecular
coupling limit, the current amplitude for the molecular bridge with BPE
molecule is negligibly small compared to the other two bridges and the other
features are also similar to the case of weak molecular coupling limit.
## 4 Concluding remarks
In summary, we have studied electron transport, at absolute zero temperature,
through three short single conjugated molecules based on the tight-binding
framework. We have used parametric approach, since we are interested only on
the qualitative behaviors instead of the quantitative ones, rather than the ab
initio theories since the later theories are computationally too expensive.
This technique can be used to study the electronic transport in any
complicated molecular system.
Electronic transport is significantly affected by (a) the molecule itself and
(b) molecule-to-electrode coupling strength and in this article we have
studied our results in these aspects. In the weak-coupling limit conductance
shows sharp resonant peaks, while these peaks get broadened in the limit of
strong molecular coupling. These results predict that by tuning the molecular
coupling strength one can control the electron conduction very sensitively
through the molecular bridges. In the study of current we have seen that the
current shows step-like behavior with sharp steps in the weak molecular
coupling, while it becomes continuous in the strong-coupling limit as a
function of applied bias voltage. Both for the two limiting cases our results
have clearly described that the localizing groups suppress the current
amplitude in large amount compared to the current amplitude in case of purely
conjugate molecule. Another significant observation is that the threshold bias
voltage of electron conduction across a molecular bridge strongly depends on
the molecule itself. These results provide key ideas for fabrication of
different molecular devices, especially in the fabrication of molecular
switches.
Some assumptions have been taken into account for this present study. More
studies are expected to take the Schottky effect which comes from the charge
transfer across the molecule-electrode interfaces, the static Stark effect,
which is taken into account for the modification of the electronic structure
of the bridge system due to the applied bias voltage (essential especially for
higher voltages). However, all these effects can be included into our
framework by a simple generalization of the presented formalism. In this
article we have also neglected the effects of inelastic scattering processes
and electron-electron correlation to characterize the electron transport
through such bridges.
## References
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* [3] R. M. Metzger et al., J. Am. Chem. Soc. 119, 10455 (1997).
* [4] C. M. Fischer, M. Burghard, S. Roth and K. V. Klitzing, Appl. Phys. Lett. 66, 3331 (1995).
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* [9] Y. Xue, S. Datta and M. A. Ratner, J. Chem. Phys. 115, 4292 (2001).
* [10] J. Taylor, H. Gou and J. Wang, Phys. Rev. B 63, 245407 (2001).
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* [13] V. Mujica, M. Kemp and M. A. Ratner, J. Chem. Phys. 101, 6849 (1994).
* [14] V. Mujica, M. Kemp, A. E. Roitberg and M. A. Ratner, J. Chem. Phys. 104, 7296 (1996).
* [15] M. P. Samanta, W. Tian, S. Datta, J. I. Henderson and C. P. Kubiak, Phys. Rev. B 53, R7626 (1996).
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* [17] R. Baer and D. Neuhauser, Chem. Phys. 281, 353 (2002).
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* [20] K. Walczak, Cent. Eur. J. Chem. 2, 524 (2004).
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* [23] S. Datta, Electronic transport in mesoscopic systems, Cambridge University Press, Cambridge (1997).
|
arxiv-papers
| 2009-09-13T08:25:39 |
2024-09-04T02:49:05.319242
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Santanu K. Maiti",
"submitter": "Santanu Maiti Kumar",
"url": "https://arxiv.org/abs/0909.2404"
}
|
0909.2450
|
Fast and Flexible Selection with a Single Switch
Tamara Broderick1,∗, David J. C. MacKay2
1 Department of Statistics, University of California, Berkeley, California,
USA
2 Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
$\ast$ E-mail: tab@stat.berkeley.edu
## Abstract
Selection methods that require only a single-switch input, such as a button
click or blink, are potentially useful for individuals with motor impairments,
mobile technology users, and individuals wishing to transmit information
securely. We present a single-switch selection method, “Nomon,” that is
general and efficient. Existing single-switch selection methods require
selectable options to be arranged in ways that limit potential applications.
By contrast, traditional operating systems, web browsers, and free-form
applications (such as drawing) place options at arbitrary points on the
screen. Nomon, however, has the flexibility to select any point on a screen.
Nomon adapts automatically to an individual’s clicking ability; it allows a
person who clicks precisely to make a selection quickly and allows a person
who clicks imprecisely more time to make a selection without error. Nomon
reaps gains in information rate by allowing the specification of beliefs
(priors) about option selection probabilities and by avoiding tree-based
selection schemes in favor of direct (posterior) inference. We have developed
both a Nomon-based writing application and a drawing application. To evaluate
Nomon’s performance, we compared the writing application with a popular
existing method for single-switch writing (row-column scanning). Novice users
wrote 35% faster with the Nomon interface than with the scanning interface. An
experienced user (author TB, with $>$ 10 hours practice) wrote at speeds of
9.3 words per minute with Nomon, using 1.2 clicks per character and making no
errors in the final text.
## Introduction
In single-switch communication, user input consists of repeated clicks,
distinguished only by timing information; these clicks might be generated by
pressing a button or blinking. For instance, the range of movement of
individuals with severe motor impairments may be limited to a single muscle.
Alternatively, a crowded or jostled mobile technology user may be able to
click precisely while other actions are difficult or sloppy. A single switch
may also be useful when information conveyed, such as a PIN, is sensitive and
hand location on a normal keyboard might betray this content. Our method,
Nomon (Figures 1, 2), expands the application scope of existing methods and
facilitates faster writing than the most common single-switch writing
interface.
Existing single-switch communication methods include scanning [1, 2, 3, 4, 5,
6, 7, 8, 9] and One-Button Dasher [10, 11, 12, 13, 14, 15]. (Morse Code does
not fall under the strict definition of a single switch interface since it
requires either click duration information or multiple switches.) Scanning is
the most popular single-switch selection method. In a scanning interface,
options such as letters are arranged in a grid (Figure 3). For standard row-
column scanning, each row of the grid is highlighted in turn, with the
highlight moving to the next row at fixed time intervals, a.k.a. scanning
delays. When a click is made, the columns of the selected row are then
highlighted in turn, typically iterating at the same fixed time intervals. To
select a column, and thereby make a final selection, the user clicks when the
highlight is on that column. A variety of customizable commercial scanning
software exists for writing and computer navigation [16, 17, 18, 19] although
customization is often not single-switch accessible. The Gnome Onscreen
Keyboard [19], by contrast, can generate a grid for new applications “on the
fly.”
While the scanning method can be used to select anything that can be arranged
in a grid, One-Button Dasher is limited to writing with alphabetic character
sets. Dasher works by arranging all possible character strings in alphabetic
order and having the user zoom in on the desired string. More likely strings,
according to the language model, are given relatively more space and are thus
easier to select.
Scanning and One-Button Dasher require options to be arranged in a particular
configuration. By contrast, traditional operating systems, web browsers, and
free-form applications such as drawing place options at arbitrary points on
the screen. Scanning, the most popular single-switch communication method, is
limited in further ways by its grid structure. For instance, the grid options
may theoretically be reordered after any selection to allow the most likely
options to be selected the most quickly. However, in practice this reordering
requires that users either learn many grid arrangements or search the grid for
their desired option upon each reordering.
Even scanning a grid that maintains a fixed layout at all times has drawbacks.
Previous studies suggest that, at least among children, scanning a fixed grid
demands a higher cognitive load than direct selection [20, 21, 22, 23]—though
an earlier study found no difference [24]. One implicated factor is the need
for a user to divide her attention between the scanning highlight and the
desired option [20, 23]. Another issue in scanning is the possibility of
distraction, and loss of the target from working memory, while highlighting
progresses [25, 23].
Therefore, we seek a single-switch selection method that is not limited to
certain forms of option placement. We want our method to work for any number
of options; to be able to effectively reorder the set of selections without
imposing additional cognitive load; and to allow the user to attend only to
the desired target.
Below, we begin by describing such a method, which we call “Nomon.” We also
describe how our method can adapt to individuals’ clicking abilities and how
it can incorporate prior beliefs about option selection frequency. In order to
evaluate our method’s performance, we note that much single-switch research
has focused on optimizing writing speed [1, 2, 3, 4, 5, 6] and the number of
clicks per output symbol [7, 8, 9] in scanning interfaces. In light of these
studies, we developed a writing application, the Nomon Keyboard (Figure 2),
using our method and compared its performance with a popular commercial
scanning interface, The Grid 2 [16] (Figure 3). We examined the study
participants’ writing speeds, error rates, and number of clicks made per
character as well as the subjective ratings of their experiences.
The full technical report describing Nomon is available online at
http://www.inference.phy.cam.ac.uk/nomon/files/nomon_tech_report.pdf. The
Nomon Keyboard, as well as a drawing application (Nomon Draw) and instructions
for the use of both applications, is available for download at
http://www.inference.phy.cam.ac.uk/nomon/ under the GNU General Public License
3.0.
### A New Method
Figure 1: An example Nomon application for selecting between 16 points on
screen (screenshot). The horizontal and vertical positions of the option
points were chosen uniformly at random in the box shown to illustrate the
flexibility of the method.
Nomon, a new single-switch communication method, does not limit the user to
selecting options that can be arranged in a grid or alphabetically. Rather, it
can be used to select among any points of interest on a screen. The trademark
of a Nomon application is a set of small clocks, one clock associated with
each selectable option. Each clock appears alongside its corresponding option
on the screen. For instance, Figure 1 illustrates clocks corresponding to 16
arbitrary option locations. Another example might be a drawing application
where a clock appears at every “pixel” on the canvas and also next to each
menu option. In a writing application (the Nomon Keyboard), a clock appears
next to each character, word completion, or text editing function (Figure 2).
Figure 2: The Nomon Keyboard, a writing application (screenshot). Words that
are prefixed by the concatenation of the current context and the letter X
appear next to the letter X. Underscore represents a space. Options for
period, a character-deletion function, and an undo function are also
available.
Just as menu options and drawing tools in a point-and-click interface are
accessed in the same way by the mouse, all Nomon clocks are selected in the
same way by a single switch. Each Nomon clock features a moving hand and a
fixed line at noon. All moving hands rotate at the same, fixed speed but, at
any time, are located at a variety of angles relative to noon. The user tries
to click precisely when the moving hand on her desired clock is at noon. She
repeats this action until the clock is selected. Selection is signalled by the
desired clock being highlighted with a darker color and the entire application
flashing a lighter color; there may also be audio feedback. Between clicks (if
more than one click is required to select a clock), the clock angular offsets
are adjusted by a heuristic to maximize the expected information content of
the user’s next click.
Row-column scanning can be viewed as a special case of the Nomon selection
method where clocks are arranged in a grid, moving-hand angular offsets are
aligned alternately across rows and columns of clocks, and each selection is
based only on the times of the last two clicks. But this synchrony does not
take full advantage of the continuous, periodic representation of the clock
and imposes an order on the set of options relative to their positions
onscreen. Rather, by allowing more general clock hand positions, we can,
effectively, completely reorder the set of selections after each click without
demanding any extra cognitive load from the user.
Similarly, the independent movement of the clock hands frees the user to
attend only to the desired target, in contrast to the need, in scanning, for
the user to attend both to the desired target and the moving highlight.
Further, the scanning user may forget her target as highlighting progresses.
But in Nomon, once the target is located visually, the user is free (without
suffering a performance penalty) to focus on selecting a single, fixed clock.
Since the clock periods are usually much shorter than a full scanning
rotation, there is also no significant penalty for missing a potential click
time.
In Nomon, by contrast with scanning, we assume that the user will not always
click perfectly at the desired time. The details of Nomon operation are
described more fully in the Nomon Operation section below and outlined here.
Nomon can learn a user’s probability of clicking at different (typically
small) offsets relative to noon. This learning is accomplished via an
approximate Parzen window estimator, with contributions from more recent
clicks weighted more strongly to allow adaptation to a user whose skill
changes over time. We can also specify a prior probability distribution over
clocks according to a predictive model of user choices. For instance, in the
writing application tested below, our language model assigned prior
probabilities to letters and word completions based on the British National
Corpus word-frequency list [26]. These prior probabilities could also be
adaptive and context dependent.
During a particular selection process, the posterior probability of any clock
given the clicks thus far can be calculated from Bayes’ theorem. When the
probability of a single clock is sufficiently high, we declare it the winner.
The probability threshold for winning is an adjustable parameter of the model;
it can vary according to context or from clock to clock. A higher threshold
can ensure greater safety for critical actions.
## Results
Figure 3: The scanning grid from The Grid 2 used in this comparison study
(screenshot). The six long rectangles on the left hold word completions. The
remaining options are fixed and include letters, an underscore for space, a
period, a character-deletion function, and a word-deletion function.
We developed a writing program using the Nomon method, the Nomon Keyboard
(Figure 2), and conducted a study to compare writing with Nomon to writing
with a popular commercial scanning interface, The Grid 2 [16] (Figure 3). To
that end, sixteen study participants with no previous experience of either
interface wrote with Nomon and The Grid 2. In each of two sessions, a
participant used one of the interfaces to write short phrases appearing on
screen. A session was divided into four blocks, each lasting approximately
$14$ minutes. During the first three (of four) writing blocks, each
participant was allowed to adjust the rotation-period or scanning-delay
parameter, as appropriate to the current interface, at the end of each written
phrase. No changes were allowed during the final block. For each interface,
cash prizes were won by the faster half of participants in the final block.
In total, we collected 34 hours of data from 16 novice participants and one
experienced single-switch user (TB, with $>10$ hours experience in each
interface). We compared three objective measures of the novice participants’
performance between the two interfaces: text-entry rate, error rate, and click
load (clicks per character). We also examined subjective ratings of the two
interfaces given by the novice participants.
### Text-entry Rate
Figure 4: Mean entry rate (left) and click load (right) across interface
blocks. Mean entry rate is measured in words per minute, and click load is
measured in clicks per output character. In both panels, error bars represent
95% confidence intervals for the novice user means, and the average
experienced user (TB) performance is illustrated by horizontal lines for
comparison.
We calculated text-entry rate in words per minute, where a word is defined as
five consecutive characters in the output text. At the beginning of each
fourteen-minute block, the participants were asked to write two periods “..”
using the interface for that session. This action signalled that they were
ready to begin and initiated the display of the first target phrase. Timing
started once the two periods were written. After every phrase, participants
wrote two periods to signal that they were ready for a new phrase. Timing
stopped after the final two periods following the last phrase were written.
All periods except the first two in a block were counted as characters in what
follows, and the time spent writing them was counted as well.
The left panel of Figure 4 shows the novice participants’ mean entry rates
across the four blocks for each interface. Also shown, for comparison, is the
performance of the experienced user. Participants wrote faster with Nomon than
with The Grid 2 during the first block ($F_{1,15}=129$, $p=9.3\cdot 10^{-9}$).
The total session time was short for both interfaces, but participants’
writing speed with each interface improved with practice. Participants became
faster at writing using the Nomon Keyboard during the Nomon session
($F_{3,45}=59$, $p=1.4\cdot 10^{-15}$) and became faster at writing using The
Grid 2 during the scanning session ($F_{3,45}=122$, $p<10^{-15}$). In the
final block we see that participants remained faster at writing with Nomon
than with The Grid 2 ($F_{1,15}=135$, $p=6.8\cdot 10^{-9}$). In this fourth
block, participants wrote $35\%$ faster with Nomon than with the scanning
interface; participants were writing at $4.3$ words per minute on average with
The Grid 2 and $5.8$ words per minute with the Nomon Keyboard. The experienced
user wrote, on average, at $9.3$ words per minute with the Nomon Keyboard and
$5.9$ words per minute with The Grid 2.
While the alphabetic layout was easy for novices to use, a computer simulating
writing from a conversational corpus with no errors has been shown to achieve
a $19\%$ faster writing speed with a frequency-ordered layout than with an
alphabetic layout [6]. Even if we artificially inflate the novice writing
speeds using The Grid 2 by $19\%$, novices remain faster at writing with Nomon
($F_{1,15}=19.14$, $p=5.4\cdot 10^{-4}$).
### Error Rate
To find the error rate during a block, we begin by computing the character-
level Levenshtein distance [27] $d_{i}$ between the $i^{\rm th}$ target phrase
in the block and the text written by the participant; $d_{i}$ is also known as
the edit distance. We define the error rate for the block to be
$\sum_{i}d_{i}/\sum_{i}n_{i}$, where $n_{i}$ is the number of characters in
the $i^{\rm th}$ target phrase.
The average novice character-level error rate (over all blocks) for the Nomon
Keyboard was $0.43\%$, and the average novice error rate for The Grid 2 was
$0.34\%$. There was no significant difference in novice error rate between the
two interfaces ($F_{1,15}=0.71$, $p=0.41$). The experienced user made no
errors while using Nomon ($\sum_{i}d_{i}=0$) and made one error while using
The Grid 2, for a mean scanning block error rate of $0.06\%$.
We believe that the participants’ output errors were mostly caused by poor
recall of the target sentence. For instance, one participant pluralized “head”
in “head_shoulders_knees_and_toes” and wrote “reading_week_is_almost_here”
instead of “reading_week_is_just_about_here”.
### Click Load
The click load is the number of clicks per output-text character. Other names
for this measure include “keystrokes per character” [28] and “gestures per
character” [14]. The click load is calculated as the number of button presses
in a block divided by the number of characters in the output. Clicking often
can be tiring for any user and especially so for some users with specific
motor impairments.
While the inclusion of word-completion options in a scanning grid has been
shown to have no positive effect on writing speed with a scanning interface
[7], other studies confirm that word completion options yield substantial
click-rate savings over the baseline (mistake-free) row-column click load of
two clicks per character [8, 9]. Therefore, we included six word-completion
options in the leftmost row of our scanning grid (consistent with the default
layouts in The Grid 2 [16]). These were ordered from top to bottom and filled
in automatically by the software.
Click loads are illustrated in the right panel of Figure 4. The average novice
rate (over all blocks) for the Nomon Keyboard was $1.58$ clicks per character,
and the average novice rate for The Grid 2 was $1.55$ clicks per character.
There was no significant difference in novice click load between the two
interfaces ($F_{1,15}=0.49$, $p=0.49$).
While the experienced user required, on average, $1.51$ clicks per character
in The Grid 2, she required only $1.18$ clicks per character using the Nomon
Keyboard. For comparison, writing with the same character set on a normal
keyboard requires at least one key press for each character and thus at least
1 click per character (possibly more due to error correction). To compare to
Morse code, we find letter, space, and period frequencies directly from our
phrase set. We assume the Morse encoding of [17, 29]. In this case, an error-
free Morse code click load estimate is $3.0$ clicks per character. This load
is over twice as high as the click load of the experienced user on the Nomon
Keyboard.
### Subjective Ratings
We assessed novice participants’ opinions with a questionnaire immediately
after writing with an interface was completed. The questionnaires for each
interface were identical (except for the name of the interface). Participants
were asked to rate how much they agreed with a series of statements on a scale
from 1 (strongly disagree) to 7 (strongly agree). These statements were
largely the same as those in [30]. Participants were encouraged to write any
thoughts about the interfaces in an “Open Comments” box.
Participants’ responses to selected statements are summarized in Table 1. Not
only did participants like using the Nomon Keyboard in aggregate, but every
participant individually liked using Nomon at least as much as The Grid 2.
Contributing factors for why the Nomon Keyboard was preferred became apparent
in the remaining responses. Participants found it easier to select word
completions and easier to correct errors with the Nomon Keyboard. These
responses corroborate our objective findings above.
While many written comments agreed with participants’ numerical ratings,
unique to the open comments section was the sentiment that Nomon looks unusual
at first but is worth getting to know. One participant remarked,
“Surprisingly, I found this more user-friendly.” Another noted, “The writing
system looks intimidating when it first comes up on screen but is actually
very easy to use.”
Table 1: Subjective ratings of the two interfaces by novice participants.
Statement | Nomon | The Grid 2
---|---|---
| mean (sd) | mean (sd)
I liked writing using X. | 5.6 (1.4) | 3.9 (1.5)
It was easy to select word completions (the, and, cat, …). | 6.1 (0.7) | 4.8 (1.3)
It was easy to correct errors. | 4.5 (1.8) | 3.9 (1.7)
Each response to the lefthand statements was on a scale from 1 (strongly
disagree) to 7 (strongly agree). In the questionnaires, the interface name was
substituted for X. Mean responses are shown with standard deviations in
parentheses. Boldface is used to highlight the means corresponding to a more
positive user experience.
## Discussion
Nomon benefits in this comparison from its nice scaling properties and clock-
position flexibility. Our posterior-based selection method implies that the
time taken to make a selection in Nomon scales logarithmically with the number
of clocks if the prior over clocks is uniform. The entropy of the discrete
uniform distribution, which happens to be the highest-entropy (finite)
discrete distribution, scales logarithmically with respect to the number of
points in the support. Figure 5 shows that, generally, $2$ clicks are required
by an experienced Nomon user (TB, with $>10$ hours experience) to make a
selection in a 30-clock application. In a Nomon application with uniform prior
and $401$ clocks, $3$ clicks are generally required for this user to make a
selection. The difference in entropy between the prior for the $401$-clock
application and the highest-entropy prior for the $30$-clock application is
about $3.5$ bits, in agreement with $\log_{2}(401/30)=3.7$.
Not only does the number of clicks to selection in Nomon scale well, but
including additional options with small prior probabilities has little effect
on clicks-to-selection for more-likely clocks. Therefore, we could place many
more word completions on screen than would be feasible for a scanning
interface. We limited ourselves to three per character so as to allow fast
reading of the three relevant options. Placing word completions next to
letters in Nomon was feasible since clock position onscreen does not affect
Nomon operation. Interspersing word completions with letters in row-column
scanning would increase the number of scanning steps required to reach many
options.
While a Nomon writing application allows a straightforward comparison of Nomon
with existing single-switch communication methods, the Nomon selection method
is not limited to writing. For example, Nomon can be used for internet
browsing by placing a Nomon clock next to each link. Or Nomon can be used for
drawing by placing a dense grid of, say, hundreds of clocks on a canvas. (The
Nomon Draw application works in this way.) A user can draw a line by selecting
points directly from the canvas. Options for colors, shape drawing, saving,
and printing can likewise be accessed with clocks. A general graphical user
interface can be navigated with Nomon by placing clocks at the points where a
user might traditionally point and click.
It is worth pointing out that the flexibility of Nomon is not specific to our
clock display choice. Other local periodic representations of the global set
of options would also allow the arbitrary placement of options onscreen. For
instance, the clocks could be replaced by bouncing balls at different points
in their trajectories; instead of clicking at noon, the user would click when
the desired ball hits the ground. It remains to be studied whether such
alternative display choices might facilitate even faster or easier use of this
system.
Figure 5: Entropy of the estimated probability distribution over clocks for
two Nomon applications. Entropy is shown as a function of clicks remaining to
selection. Each solid line represents a single selection process. Dotted lines
decreasing to zero at respective rates of $1$ (lower) and $3$ (upper) bits per
click are illustrated for reference. Left: 25 selections on the Nomon
Keyboard: 30 clocks, non-trivial prior $p(c)$, clock period $2.0$ seconds,
switch input from joystick button. Right: 25 selections on another Nomon
application: 401 clocks, uniform prior $p(c)$, clock period $2.0$ seconds,
switch input from space bar. Data was generated by the experienced user (TB).
## Materials and Methods
We begin by detailing the experimental method used in the study above and
follow with a description of how Nomon functions.
### Experiment
#### Participants
We recruited sixteen participants from the university community across a wide
range of academic disciplines. All participants gave written informed consent.
In accordance with the University of Cambridge ethical review procedure as
defined in the Cambridge Psychology Research Ethics Committee Handbook
(http://www.bio.cam.ac.uk/sbs/psyres/), the experimental design received an
internal peer review within the department, where it was decided that ethics
approval from the committee was not necessary.
The participants’ ages ranged from 22 to 39 (mean = 26, sd = 4). Eight were
women, and eight were men. Participants were screened for motor or cognitive
difficulties; in particular, no participant had dyslexia or RSI. None of the
participants had used a scanning or Nomon interface before. No participant had
regularly used any single-switch interface before. Twelve of the participants
had used word completion (e.g. on cell phones).
In addition to the sixteen novice participants, an experienced user of Nomon
and The Grid 2 ($>10$ hours writing with each interface) was run through the
same experimental procedure for comparison.
#### Apparatus and Software
All sessions were run on a Dell Latitude XT Tablet PC with a partitioned hard
drive. The 12.1 inch color screen had a physical screen size of 261 $\times$
163 mm. The single-switch hardware device in all cases was the trigger button
of a Logic3 Tornado USB joystick. Participants operated the trigger button
with the first finger of their left or right hand. None of the other joystick
inputs was used. For both writing interfaces, automated spoken feedback was
provided as the user wrote.
##### Nomon Keyboard
We ran the Nomon Keyboard (Figure 2) on an Ubuntu 8.10 operating system
running the Linux kernel. The screen resolution was 1280 $\times$ 800 pixels,
and the physical size of the keyboard display was 224 $\times$ 85 mm (1125
$\times$ 416 pixels). The interface was docked in the upper part of the
screen. A text box and phrase box were located below the keyboard in the same
window. The keys of the keyboard were arranged in six rows and five columns.
Each key contained a principal character, with letters in alphabetical order
(across and then down) first, followed by four special characters: an
underscore (representing space), a period, a character-deletion function, and
an undo function. Each letter key also contained up to three word completions.
The undo function undid the previous selection if it was a character
selection, word-completion selection, or deletion.
The clock rotation period $T$ could be set to $2.0\cdot 0.9^{j}$ seconds for
$j\in\\{-4,-3,\ldots,18\\}$. Higher $j$ corresponded to faster rotation. The
initial setting of the period for novices was $T=2.0$ ($j=0$). The experienced
user initially chose $T=1.06$ ($j=6$).
##### The Grid 2
We ran The Grid 2 (Figure 3) on a Windows Vista Service Pack 1 operating
system. The screen resolution was again 1280 $\times$ 800 pixels. The physical
size of The Grid 2 display, using the scanning grid we designed for this
experiment, was 261 $\times$ 102 mm (1280 $\times$ 500 pixels). The interface
was docked in the upper part of the screen, and the text box and phrase box
were docked immediately below. Six word-completion boxes appeared on the left
side of the main interface. The remaining space was divided into six rows and
five columns of keys. Each key contained a single character. First were
letters arranged in alphabetical order (across and then down), followed by an
underscore, a period, a character-deletion function, and a word-deletion
function.
The Grid 2 allowed scanning delay values $d$ at $0.1(10-j)$ for
$j\in\\{\ldots,-1,0,1,\ldots,9\\}$. Higher $j$ corresponded to faster
scanning. The initial setting of the delay for novices was $d=1.0$ ($j=0$).
The experienced user initially chose $d=0.5$ ($j=5$).
#### Procedure
The experiment consisted of two identical sessions, one for each interface.
The starting interface was balanced across participants, and sessions were
spaced at least four hours apart.
Each session proceeded according to the same schedule. The first ten minutes
were introductory. First, the supervisor either explained or reviewed the
experimental procedure according to the session number. Then the participant
was shown how to use one of the interfaces. The demonstration included basic
writing, word completion, and error correction.
The next hour was divided into four 14-minute blocks, separated by short
breaks. During the blocks, participants were asked to write phrases drawn from
a modified version of the phrase set provided by [31], with British spellings
and words substituted for their American counterparts. For each participant, a
different random ordering of the initial phrase set was generated. Phrases
appeared one at a time in the phrase box at the bottom of the screen. Once a
participant finished a phrase, writing the period character twice would cause
a new target phrase to appear and the text box to empty. Participants were
instructed that no changes relevant to a particular phrase could be made after
the two periods were written.
During the first three (of four) writing blocks, each participant was allowed
to adjust the rotation-period or scanning-delay parameter at the end of each
written phrase. In particular, immediately after writing two periods and
receiving the new target phrase, the participant could increment or decrement
$j$ (defined above) by one. The experienced user incremented to $j=7$
($T=0.96$) after two blocks using the Nomon Keyboard and incremented to $j=6$
($d=0.4$) after two blocks using The Grid 2. No other changes were made by
this user.
Novice participants were paid £10 for each of the two sessions; the
experienced participant was not paid. Novice participants were informed at the
beginning of the study that they could receive a £5 bonus for achieving a
writing speed among the top half of novice participants for each interface.
They were further informed that, for the purposes of the bonus, writing speed
would be measured only during the final writing block. They were told that
they would not be allowed to change the rotation-period or scanning-delay
parameter during this block and thus would have to calibrate it as they saw
fit during the previous blocks. Information about their own writing speeds
across full blocks and also phrase-by-phrase was made available to
participants during the break after each block.
We performed seven significance tests with a family-level significance of
$0.05$. Observing the Bonferroni correction, we performed each individual test
at a significance level of $\alpha=0.007$. Wherever $F$ values are quoted, an
analysis of variance (ANOVA) test for repeated measures was performed.
### Nomon Operation
We here describe the prior over clocks, click likelihood (given a clock), and
the resulting posterior over clocks in turn. While we focus on a prior for a
specific application (the Nomon Keyboard), the likelihood and posterior
discussions are germane to a general Nomon application.
#### Prior
In the absence of information about clock probabilities, we use a uniform
prior $p(c)$ over clocks $c:1\leq c\leq C$. We can choose a more informative
prior for our Nomon writing application, the Nomon Keyboard (Figure 2). This
interface features four special characters (underscore representing space;
period; Delete; and Undo), 26 letters, and up to three word completions per
letter. We assign fixed prior probabilities to the special characters and
assign the remaining priors according to Laplace smoothing out of the leftover
probability mass $p_{\rm alpha}$. Let $l_{1}\cdots l_{N}$ ($N\geq 0$) be the
context (all letters from the end of the current output text) before the user
begins to make another selection. Let $\mathcal{W}_{\rm on}$ be the set of
word completions appearing on screen, and set $C_{\rm on}=|\mathcal{W}_{\rm
on}|+26$. To form our corpus, we begin with the British National Corpus word
list [26], then we remove single-letter words besides “I” and “a” and keep
only words appearing with some small minimum frequency ($>5$ appearances in
the corpus).
When an appropriate word completion is offered, the user may nevertheless
choose the next single letter; the following model assumes that the user is
equally likely to choose either of these options. If $f_{w}$ is the number of
occurrences of word $w$ in our corpus, we define a context frequency
$f(l_{1}\cdots l_{N})=\sum_{w}f_{w}\mathbbm{1}\\{l_{1}\cdots l_{n}\textrm{
prefixes }w\\}$ and a screen word-completion summed frequency
$f(\mathcal{W}_{\rm on})=\sum_{w^{\prime}\in\mathcal{W}_{\rm
on}}f(w^{\prime})$. If $c(l^{\prime})$ is the clock corresponding to letter
$l^{\prime}$ and $c(w)$ the clock corresponding to word $w$,
$\displaystyle p(c(l^{\prime}))$ $\displaystyle=$ $\displaystyle p_{\rm
alpha}\times\frac{f(l_{1}\cdots l_{N}l^{\prime})+1}{f(l_{1}\cdots
l_{N})+f(\mathcal{W}_{\rm on})+C_{\rm on}}$ (1) $\displaystyle p(c(w))$
$\displaystyle=$ $\displaystyle p_{\rm alpha}\times\frac{f(w)+1}{f(l_{1}\cdots
l_{N})+f(\mathcal{W}_{\rm on})+C_{\rm on}}$ (2)
To model an ideal user, we would subtract the count of words onscreen prefixed
by $l_{1}\cdots l_{N}l^{\prime}$ from the numerator of $p(c(l^{\prime}))$, and
both denominators would equal $f(l_{1}\cdots l_{N})+C_{\rm on}$. Finally,
while the number of letters is fixed at 26, $C_{\rm on}$ is variable since,
for any letter, we include only those word completions among the three most
probable above a certain threshold. It was judged that requiring
$f_{w}/f(l_{1}\cdots l_{N})>0.001$ yielded a reasonable balance between
displaying common words and not cluttering the screen.
#### Click Distribution
Any particular clock $c$ defines a desired click time at noon. We wish to
estimate a user’s click time distribution relative to noon $p(t|c)$, where we
distinguish $t$ only up to the clock period $T$ and set $t|c$ to zero at noon.
To that end, we begin with a broad, and slightly offset, initial setting of
our estimate for $p(t|c)$: $\hat{g}_{0}(t)=\mathcal{N}(t;0.05T,(0.14T)^{2})$.
The $T$-dependence ensures the estimate will be nontrivial at any user-chosen
period. We update the $\hat{g}_{0}$ distribution with a (modified) Parzen
window estimator—with width given below—and a damping factor $\lambda$ that
allows learning to continue over time. After any selection is made, we modify
the distribution estimate with the data from the $n_{\rm delay}^{\rm th}$
selection before the latest one (here $n_{\rm delay}=2$). This delay allows
the user to choose Undo after a selection, in which case we do not use the
clicks toward that selection for learning. Once a selection occurred $n_{\rm
delay}$ rounds in the past, we assume that it was correctly chosen. With the
clock choice $c$ known for the $s^{\rm th}$ selection, we are able to
calculate click times around noon $t_{s,r}$ for each click that was made
toward this selection. We treat these as data from the distribution $g$ we are
estimating. To calculate our estimate $\hat{g}_{s}$ for $g$ after the $s^{\rm
th}$ selection, we make use of the unnormalized distributions $\tilde{g}_{s}$.
$\tilde{g}_{s}(t)=\lambda\tilde{g}_{s-1}(t)+\sum_{r=1}^{R_{s}}\mathcal{N}\left(t;t_{s,r},\hat{\sigma}^{2}_{\rm{NS},s}\right)\textrm{
with }\tilde{g}_{0}(t)=n_{\lambda}\hat{g}_{0}(t)$ (3)
The update equation specifies that, after each selection, $\tilde{g}$ is
damped by the factor $\lambda$. The next term is a sum over clicks $r$ leading
to the $s^{\rm th}$ selection. Within the summation is a normal density
centered at the click time $t_{s,r}$, as in Parzen window estimation. The
width for this Parzen-window term is given by $\hat{\sigma}_{\textrm{NS},s}$,
which is derived from the normal scale rule estimate [32, 33] for the Parzen
window. That is,
$\hat{\sigma}_{\textrm{NS},s}=1.06n_{\lambda}^{-0.2}\hat{\sigma}_{s},$ (4)
where $\hat{\sigma}^{2}_{s}$ is the standard (Gaussian maximum likelihood)
variance estimator obtained from the last $n_{\lambda}$ clicks before the
$s^{\rm th}$ round. The factor $n_{\lambda}=(1-\lambda)^{-1}$ in the initial
$\tilde{g}_{0}$ definition is an effective number of samples derived from the
damping factor. Using this factor and the unnormalized update, we ensure that
the initial estimate $\hat{g}_{0}$ dominates $\hat{g}_{s}$ even after the
first few selections. Without the $n_{\lambda}$ factor, the Parzen window term
for the first click,
$\mathcal{N}\left(t;t_{1,1},\hat{\sigma}^{2}_{\rm{NS},1}\right)$, would have
nearly equal weight with the initial estimate.
This estimate for $p(t|c)$ allows us to save the estimated distribution and
update it quickly and easily during operation of the application. As a result,
users can start the Nomon application immediately, without a waiting or
calibration period, but they can also enjoy an experience tailored to their
abilities. For instance, a user need not click at noon (or any offset)
exactly. Their personal offset, reflecting reaction time, is learned by this
method rather than hard-coded and, as long as it is not too close to 6
o’clock, will make no difference to program operation. The precision around
this personal offset determines the number of clicks necessary to make a
selection.
#### Posterior
With a prior and likelihood, we may calculate the posterior probability of
each clock $c$ given the $R$ clicks thus far using Bayes’ theorem:
$p_{c,R}=p(c|t_{1:R})\propto p(c)\prod_{r=1}^{R}p(t_{r}|c)$. In practice, we
store the unnormalized log probabilities for each $p_{c,R}$. Checking that the
highest clock probability $p_{(C),R}$ exceeds some threshold would require
exponentiating every stored value and summing over the results. Noting that
$p_{(C),R}>1-p_{\rm error}$ is equivalent to
$p_{(C),R}>\alpha\sum_{c\neq(C)}p_{c,R}$ for some $\alpha$, we instead declare
a winner when $p_{(C),R}>\alpha p_{(C-1),R}$. The choice of $\alpha=99$
represents a desired upper bound on error fraction, per selection, of $0.01$.
In a sample of 1,714 consecutive selections made by an experienced Nomon user
(TB) on the Nomon Keyboard under this setting, the average value of
$p_{(C),R}/p_{(C-1),R}$ over all selections after the deciding click was
$0.001$, and the average value of $p_{(C),R}/\sum_{c\neq(C)}p_{c,R}$ was
$0.002$, suggesting our heuristic stopping criterion is a reasonable
approximation to the desired one. In the 1,714 selections, 3 (non-consecutive)
selections were Undo, indicating mistakes and giving an empirical error rate
of about $0.002$, in line with the calculated rate.
## Acknowledgments
This research was supported by donations from the Nine Tuna Foundation and
Nokia. We thank Sensory Software for generously providing the 60-day free
trial of The Grid 2 used in our performance comparison. We are grateful to Per
Ola Kristensson, Geoffrey Hinton, Keith Vertanen, Philipp Hennig, Carl
Scheffler, and Philip Sterne for helpful discussions. TB’s research is
supported by a Marshall Scholarship.
## References
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* 3. Evreinov G, Raisamo R (2004) Optimizing menu selection process for single-switch manipulation. In: Proceedings of the 9th International Conference on Computers Helping People with Special Needs (ICCHP 2004). Paris, France: Springer-Verlag Berlin/Heidelberg, pp. 836–844.
* 4. Szeto AYJ, Allen EJ, Littrell MC (1993) Comparison of speed and accuracy for selected electronic communication devices and input methods. Augmentative and Alternative Communication 9: 229–242.
* 5. Baljko M, Tam A (2006) Indirect text entry using one or two keys. In: Assets ‘06: Proceedings of the 8th International ACM SIGACCESS Conference on Computers and Accessibility. New York, NY, USA: ACM, pp. 18–25. doi:http://doi.acm.org/10.1145/1168987.1168992.
* 6. Venkatagiri HS (1999) Efficient keyboard layouts for sequential access in augmentative and alternative communication. Augmentative and Alternative Communication 15: 126–134.
* 7. Koester H, Levine S (1994) Learning and performance of able-bodied individuals using scanning systems with and without word prediction. Assistive Technology 6: 42–53.
* 8. Koester H, Levine S (1996) Effect of a word prediction feature on user performance. Augmentative and Alternative Communication 12: 155–168.
* 9. Lesher G, Moulton B, Higginbotham D (1998) Optimal character arrangements for ambiguous keyboards. IEEE Transactions on Rehabilitation Engineering 6: 415–423.
* 10. MacKay DJC, Ball CJ, Donegan M (2004) Efficient communication with one or two buttons. In: Fischer R, Preuss R, von Toussaint U, editors, Maximum Entropy and Bayesian Methods. Melville, NY, USA: American Institute of Physics, volume 735 of _AIP Conference Proceedings_ , pp. 207–218.
* 11. MacKay DJC, Ball CJ (2006) Dasher’s one-button dynamic mode—theory and preliminary results. Technical report, Cavendish Laboratory, University of Cambridge. Available at http://www.inference.phy.cam.ac.uk/mackay/abstracts/OneButton.html.
* 12. MacKay DJC (2007) Another one-button dynamic mode for Dasher: ‘two-click mode’. Technical report, Cavendish Laboratory, University of Cambridge. Available at http://www.inference.phy.cam.ac.uk/mackay/abstracts/OneButton2.html.
* 13. Mead JMG, Cowans PJ, MacKay DJC (2009) Efficient communication through the timings of one or two buttons. Technical report, Cavendish Laboratory, University of Cambridge. Available at http://www.inference.phy.cam.ac.uk/mackay/buttondasher2b.pdf.
* 14. Ward DJ, Blackwell AF, MacKay DJC (2000) Dasher—a data entry interface using continuous gestures and language models. In: UIST ’00: Proceedings of the 13th Annual ACM Symposium on User Interface Software and Technology. New York, NY, USA: ACM, pp. 129–137. doi:http://doi.acm.org/10.1145/354401.354427.
* 15. Ward DJ, MacKay DJC (2002) Fast hands-free writing by gaze direction. Nature 418: 838.
* 16. Sensory Software International Ltd (2008) The Grid 2: Version 2.4. Sensory Software International Ltd. URL http://www.sensorysoftware.com/.
* 17. Words+, Inc (2000) EZ Keys™User Manual. Words+, Inc. URL http://www.words-plus.com/website/products/manuals/manual.htm.
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* 25. Petersen K, Reichle J, Johnston SS (2000) Examining preschoolers’ performance in linear and row-column scanning techniques. Augmentative and Alternative Communication 16: 27–36.
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* 29. ITU Radiocommunication Assembly (2004). International Morse code, Recommendation ITU-R M.1677.
* 30. Kristensson PO, Denby LC (2009) Text entry performance of state of the art unconstrained handwriting recognition: a longitudinal user study. In: CHI ’09: Proceedings of the 27th International Conference on Human Factors in Computing Systems. New York, NY, USA: ACM, pp. 567–570. doi:http://doi.acm.org/10.1145/1518701.1518788.
* 31. MacKenzie IS, Soukoreff RW (2003) Phrase sets for evaluating text entry techniques. In: CHI ’03 Extended Abstracts on Human Factors in Computing Systems. New York, NY, USA: ACM, pp. 754–755. doi:http://doi.acm.org/10.1145/765891.765971.
* 32. Wand MP, Jones MC (1995) Kernel smoothing. London, UK: Chapman & Hall/CRC.
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|
arxiv-papers
| 2009-09-13T21:39:55 |
2024-09-04T02:49:05.327264
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Tamara Broderick and David John Cameron MacKay",
"submitter": "Tamara Broderick",
"url": "https://arxiv.org/abs/0909.2450"
}
|
0909.2509
|
Electron transport through a quantum wire coupled with a mesoscopic ring
Santanu K. Maiti1,2,∗
1Theoretical Condensed Matter Physics Division, Saha Institute of Nuclear
Physics,
1/AF, Bidhannagar, Kolkata-700 064, India
2Department of Physics, Narasinha Dutt College, 129, Belilious Road,
Howrah-711 101, India
Abstract
Electronic transport through a quantum wire sandwiched between two metallic
electrodes and coupled to a quantum ring, threaded by a magnetic flux $\phi$,
is studied. An analytic approach for the electron transport through the bridge
system is presented based on the tight-binding model. The transport properties
are discussed in three aspects: (a) presence of an external magnetic filed,
(b) strength of the wire to electrode coupling, and (c) presence of in-plane
electric field.
PACS No.: 73.23.-b; 73.63.-b; 73.21.Hb
Keywords: Green’s function; Conductance; $I$-$V$ characteristic; Electric
field.
∗Corresponding Author: Santanu K. Maiti
Electronic mail: santanu.maiti@saha.ac.in
## 1 Introduction
With the advancement in nanoscience and nanotechnology, the fabrication of
sub-micron devices has become possible and has allowed one to study the
electron transport through quantum systems in a very controllable way. These
quantum systems have attracted much more attention since they constitute
promising building blocks for future generation of electronic devices and
directed attention on the study of discrete structures, such as a single
molecule, arrays of molecules, quantum dots, quantum wires and mesoscopic
rings. The electron transport through a bridge system was first studied
theoretically in $1974$ [1]. Later, several numerous experiments [2, 3, 4, 5,
6] have been performed through quantum systems placed between two metallic
electrodes with few nanometer separation. The operation of such two-terminal
devices is due to an applied bias. Current passing across the junction is
strongly nonlinear function of applied bias voltage and its detailed
description is a very complex problem. Though lot of theoretical as well as
experimental papers have been available in the literature, yet the complete
knowledge of the conduction mechanism in this scale is not well understood
even today. The transport properties of these systems are associated with some
quantum effects like, quantization of energy levels, quantum interference of
electron waves, etc. A quantitative understanding of the physical mechanisms
underlying the operation of nanoscale devices remains a major challenge in the
present nanoelectronics research.
The aim of the present article is to reproduce an analytic approach based on
the tight-binding model to investigate the electronic transport properties
through a quantum wire coupled to a mesoscopic ring. There exist some ab
initio methods for the calculation of conductance [7, 8, 9, 10, 11, 12], yet
it is needed the simple parametric approaches [13, 14, 17, 15, 16, 18, 19, 20,
21, 22, 23] for this calculation, especially for the case of larger molecular
bridge systems. The parametric study is motivated by the fact that the ab
initio theories are computationally too expensive and here we focus our
attention on the qualitative effects rather than the quantitative ones. This
is why we restrict our calculations on the simple analytical formulation of
the transport problem.
We organize the paper as follow. Following the introduction (Section $1$), in
Section $2$, we present the model system under consideration and give a very
brief description for the calculation of conductance and current-voltage
characteristics through the bridge system. Section $3$ presents the results of
the system taken into account. Finally, we summarize our results in Section
$4$.
## 2 The model and a brief description onto the theoretical formulation
We begin by referring to Fig. 1. The system considered here is a quantum wire
coupled to a mesoscopic ring with $N$ atomic sites and the wire is attached to
two semi-infinite one-dimensional ($1$D) metallic electrodes, namely, source
and drain.
Figure 1: Schematic view of a quantum wire coupled to a mescopic ring,
threaded by a magnetic flux $\phi$, and the wire is attached to two $1$D
metallic electrodes.
The full system (quantum wire with ring) is described by a single-band tight-
binding Hamiltonian within a non-interacting electron picture, and it can be
written in the form,
$H_{C}=H_{W}+H_{R}+H_{WR}$ (1)
where, $H_{W}$, $H_{R}$ and $H_{WR}$ correspond to the Hamiltonians for the
wire, ring and wire-to-ring coupling, respectively, and they can be expressed
as,
$H_{W}=\sum_{i}\epsilon_{i}d_{i}^{\dagger}d_{i}+\sum_{<ij>}t_{w}\left(d_{i}^{\dagger}d_{j}+d_{j}^{\dagger}d_{i}\right)$
(2)
$H_{R}=\sum_{k}\epsilon_{k}c_{k}^{\dagger}c_{k}+\sum_{<kl>}t_{r}\left(c_{k}^{\dagger}c_{l}e^{i\theta}+c_{l}^{\dagger}c_{k}e^{-i\theta}\right)$
(3) $H_{WR}=t_{0}\left(c_{1}^{\dagger}d_{0}+d_{0}^{\dagger}c_{1}\right)$ (4)
Here, $\epsilon_{i}$’s ($\epsilon_{k}$’s) are the on-site energies of the ring
(wire), $d_{i}^{\dagger}$ and $c_{k}^{\dagger}$ are the creation operators of
an electron at site $i$ and $k$ in the wire and ring. $\theta=2\pi\phi/N$ is
the phase factor due to the flux $\phi$ threaded by the ring. $t_{w}$
($t_{r}$) is the hopping integral between two nearest-neighbor sites in the
ring (wire) and $t_{0}$ is the wire-to-ring tunneling coupling.
At much low temperatures and bias voltage, the linear conductance of the wire-
ring system can be calculated by using one-channel Landauer conductance
formula,
$g=\frac{2e^{2}}{h}T$ (5)
where $T$ is the transmission probability of an electron from the source to
drain through the wire including the ring, and it is defined as [24],
$T(E,V)={\mbox{Tr}}\left[\left(\Sigma_{S}^{r}-\Sigma_{S}^{a}\right)G^{r}\left(\Sigma_{D}^{a}-\Sigma_{D}^{r}\right)G^{a}\right]$
(6)
Now the Green’s function $G$ of the full system (wire with ring) is given by
the relation,
$G=\left[E-H_{C}-\Sigma_{S}-\Sigma_{D}\right]^{-1}$ (7)
where $E$ is the energy of injecting electrons from the source and $H$ is the
Hamiltonian of the full system described above (Eq. 1). In Eq. 7,
$\Sigma_{S}=h_{SC}^{\dagger}g_{S}h_{SC}$ and
$\Sigma_{D}=h_{DC}g_{D}h_{DC}^{\dagger}$, are the self-energy terms due to the
two electrodes. $g_{S}$ and $g_{D}$ correspond to the Green’s functions for
the source and drain, respectively. $h_{SC}$ and $h_{DC}$ are the coupling
matrices and they are non-zero only for the adjacent points of the quantum
wire and the electrodes. The coupling terms $\Gamma_{S}$ and $\Gamma_{D}$ for
the full system can be calculated through the expression [24],
$\Gamma_{\\{S,D\\}}=i\left[\Sigma_{\\{S,D\\}}^{r}-\Sigma_{\\{S,D\\}}^{a}\right]$
(8)
where $\Sigma_{\\{S,D\\}}^{r}$ and $\Sigma_{\\{S,D\\}}^{a}$ are the retarded
and advanced self-energies respectively and they are conjugate to each other.
Datta et al. [25] have shown that the self-energies can be expressed like,
$\Sigma_{\\{S,D\\}}^{r}=\Lambda_{\\{S,D\\}}-i\Delta_{\\{S,D\\}}$ (9)
where $\Lambda_{\\{S,D\\}}$ are the real parts of the self-energies which
correspond to the shift of the energy eigenvalues of the full system (quantum
wire with ring) and the imaginary parts $\Delta_{\\{S,D\\}}$ of the self-
energies represent the broadening of the energy levels. Since this broadening
is much larger than the thermal broadening, we restrict our all calculations
only at absolute zero temperature. By doing some simple calculations, these
real and imaginary parts of the self-energies can be determined in terms of
the coupling strength ($\tau_{\\{S,D\\}}$) between the wire and two
electrodes, injecting electron energy ($E$) and hopping strength ($v$) between
nearest-neighbor sites in the electrodes. Using Eq. 9, the coupling terms
$\Gamma_{S}$ and $\Gamma_{D}$ can be written in terms of the retarded self-
energy as,
$\Gamma_{\\{S,D\\}}=-2{\mbox{Im}}\left[\Sigma_{\\{S,D\\}}^{r}\right]$ (10)
All the information regarding the wire to electrode coupling are included into
the two self energies stated above and is analyzed through the use of Newns-
Anderson chemisorption theory [13, 14]. The detailed description of this
theory is obtained in these two references.
Thus, by calculating the self-energies, the coupling terms $\Gamma_{S}$ and
$\Gamma_{D}$ can be easily obtained and then the transmission probability $T$
will be calculated from the expression given in Eq. 6.
The current passing through the bridge is depicted as a single-electron
scattering process between the two reservoirs of charge carriers. The current-
voltage relation is evaluated from the following expression [24],
$I(V)=\frac{e}{\pi\hbar}\int\limits_{E_{F}-eV/2}^{E_{F}+eV/2}T(E,V)~{}dE$ (11)
where $E_{F}$ is the equilibrium Fermi energy. For the sake of simplicity,
here we assume that the entire voltage is dropped across the wire-electrode
interfaces and this assumption does not greatly affect the qualitative aspects
of the $I$-$V$ characteristics. Throughout the article we set $E_{F}$ to $0$
and use the units $c=e=h=1$.
## 3 Results and discussion
Here we describe conductance-energy and current-voltage characteristics
through the quantum wire coupled to a mesoscopic ring at absolute zero
temperature. Electron transport properties through the system are strongly
affected by the magnetic flux $\phi$, wire-to-electrode coupling strength and
the in-plane electric field. In the presence of in-plane electric filed and
assuming it along the perpendicular direction of the wire, the dependence of
the site energies on the electric field $\mathcal{E}$ is written within the
tight-binding approximation as [15],
$\displaystyle\epsilon_{i}$ $\displaystyle=$
$\displaystyle\left(e\mathcal{E}aN/2\pi\right)\cos\left[2\pi(i-1)/N\right]$
(12) $\displaystyle=$
$\displaystyle\left(et_{r}\right)\left(\mathcal{E}^{\star}N/2\pi\right)\cos\left[2\pi(i-1)/N\right]$
where, $a$ is lattice spacing in the mesoscopic ring and $\mathcal{E}^{\star}$
is the dimensionless electric field strength defined by $\mathcal{E}a/t_{r}$.
For simplicity, here we assume $t_{w}$, $t_{r}$ and $t_{0}$ are identical to
each other in magnitude and specify them by the symbol $t$. We investigate all
the essential features of electron transport for the two limiting cases. One
is the weak-coupling limit, defined as $\tau_{\\{S,D\\}}<<t$ and the other one
is the
Figure 2: Conductance $g$ as a function of energy $E$ in the weak-coupling
limit for the system with ring size $N=10$, where (a) in the absence of any
electric filed with $\phi=0$ (solid line) and $0.4$ (dotted line) and (b) in
the presence of $\phi=0.4$ with $\mathcal{E}=2$ (solid line) and $4$ (dotted
line).
strong-coupling limit and defined it as $\tau_{\\{S,D\\}}\sim t$. The
parameters $\tau_{S}$ and $\tau_{D}$ correspond to the couplings of the wire
to the source and drain, respectively. The common set of values of these
parameters in the two limiting cases are as follow: $\tau_{S}=\tau_{D}=0.5$,
$t=3$ (weak-coupling) and $\tau_{S}=\tau_{D}=2$, $t=3$ (strong-coupling).
In Fig. 2, we plot the conductance ($g$) as a function of the injecting
electron energy ($E$) for the bridge system in the limit of weak-coupling.
Figure 2(a) corresponds to the spectrum in the absence of any electric filed
where, the solid and dotted curves are respectively for $\phi=0$ and $0.4$. In
Fig. 2(b), the spectrum is shown for the non-zero value of the electric field
with $\phi=0.4$ where, the solid and dotted curves represent the results for
the electric filed strengths $\mathcal{E}=2$ and $4$, respectively.
Conductance vanishes almost for all energies except at resonances where it
approaches to $2$. At these resonances, the transmission probability $T$
becomes unity, since $g=2T$ (from the Landauer formula with $e=h=1$). The
resonant peaks in the conductance spectrum coincide with eigenenergies of the
system (wire including the ring), and thus the spectrum manifests itself the
energy levels of the system. For zero electric field strength and in the
absence of magnetic flux $\phi$, the conductance exhibits a single resonant
peak across $E=0$ (see solid curve of Fig. 2(a)), while, in the presence of
$\phi$ more resonant peaks appear in the spectrum (see dotted curve of Fig.
2(a)). It reveals that for non-zero value of $\phi$ more resonating states
appear in the system. This is due to the removal of all the degeneracies in
the energy eigenstates for any non-zero value of $\phi$. In the presence of
in-plane electric field, these resonant peaks are shifted and the conductance
spectrum becomes asymmetric with respect to the energy $E$ (see Fig. 2(b)).
For the strong wire-to-electrode coupling, resonant peaks get substantial
widths as presented in Fig. 3 where, the solid and dotted curves
Figure 3: Conductance $g$ as a function of energy $E$ in the strong-coupling
limit for the system with ring size $N=10$, where (a) in the absence of any
electric filed with $\phi=0$ (solid line) and $0.4$ (dotted line) and (b) in
the presence of $\phi=0.4$ with $\mathcal{E}=2$ (solid line) and $4$ (dotted
line).
correspond to the identical meaning as earlier. The increment of the resonant
widths is due to the broadening of the energy levels of the wire including the
ring, where the contribution comes from the imaginary parts of the two self-
energies [24].
The scenario of electron transfer through the bridge becomes much more clearly
visible by studying the current $I$ as a function of the applied bias voltage
$V$.
Figure 4: Current $I$ as a function of bias voltage $V$ in the limit of weak
wire-to-electrode coupling for the system with ring size $N=10$ and
$\phi=0.4$. The solid and dotted lines correspond to the currents for
$\mathcal{E}=0$ and $3$, respectively.
The Current is computed from the integration procedure of the transmission
function $T$ which shows the same variation, differ only in magnitude by the
factor $2$,
Figure 5: Current $I$ as a function of bias voltage $V$ in the limit of strong
wire-to-electrode coupling for the system with ring size $N=10$ and
$\phi=0.4$. The solid and dotted curves correspond to the currents for
$\mathcal{E}=0$ and $3$, respectively.
like as the conductance spectra (Figs. 2 and 3). The current-voltage
characteristic in the weak-coupling limit for the bridge system is shown in
Fig. 4 where, the solid curve corresponds to the current in the absence of any
electric field and the dotted curve denotes the same for $\mathcal{E}=3$. Here
we take $\phi=0.4$. The current shows staircase-like behavior with sharp
steps, which is associated with the discrete nature of the resonant spectrum
(Fig. 2). The shape and width of the current steps depend on the width of the
resonant spectrum since the hight of a step in $I$-$V$ curve is directly
proportional to the area of the corresponding peak in the conductance
spectrum. On the other hand, the current varies continuously with the applied
bias voltage and achieves much bigger values in the strong-coupling limit, as
shown in Fig. 5 where, the solid and dotted curves correspond to the same
meaning as earlier. From both Figs. 4 and 5 it is clearly observed that the
in-plane electric field suppresses the current amplitude (see the dotted
curves). This feature may be utilized to control externally the amplitude of
the current through the bridge system.
## 4 Concluding remarks
To summarize, we have introduced parametric approach based on the tight-
binding model to investigate the electron transport properties at absolute
zero temperature through a quantum wire coupled to a mesoscopic ring threaded
by a magnetic flux $\phi$. A simple parametric approach is given to study
electron transport properties through the system, and it can be used to study
the transport behavior in any complicated molecular bridge system. Electronic
conduction through the quantum wire is strongly influenced by the flux $\phi$
threaded by the ring and the wire-to-electrode coupling strength. The effects
of in-plane electric field have also been studied in this context and it has
been predicted that the current amplitude can be controlled externally through
the bridge system by means of this electric field.
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|
arxiv-papers
| 2009-09-14T10:10:52 |
2024-09-04T02:49:05.335091
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Santanu K. Maiti",
"submitter": "Santanu Maiti Kumar",
"url": "https://arxiv.org/abs/0909.2509"
}
|
0909.2522
|
paperfile
# (non)commutative F-un geometry
Lieven Le Bruyn Department of Mathematics, University of Antwerp
Middelheimlaan 1, B-2020 Antwerp (Belgium)
lieven.lebruyn@ua.ac.be
###### Abstract.
Stressing the role of dual coalgebras, we modify the definition of affine
schemes over the ’field with one element’. This clarifies the appearance of
Habiro-type rings in the commutative case, and, allows a natural
noncommutative generalization, the study of representations of discrete groups
and their profinite completions being our main motivation.
## 1\. Commutative F-un geometry
In this section we will recall the definition of affine schemes over the
mythical field $\mathbb{F}_{1}$ with one element, originally due to Christophe
Soulé [14] and refined later by Alain Connes and Katia Consani [4]. This
approach is based on functors from abelian groups to sets satisfying a
universal property with respect to an integral- and a complex affine scheme.
We will modify this definition slightly by replacing these affine schemes by
integral- resp. complex dual coalgebras. This amounts to restricting to étale
local data of the affine schemes and has the additional advantage that the
definition can be extended verbatim to the noncommutative world as we will
outline in the next section. Another advantage of the coalgebra approach is
that it inevitably leads to the introduction of the Habiro ring [7] in the
easiest example, that of the multiplicative group. This might be compared to
recent work by Yuri I. Manin [12] and Matilde Marcolli [13].
### 1.1.
For a commutative ring k we will denote with k-calg, resp. k-alg, the
category of all commutative k-algebras, resp. the category of all
k-algebras. and with morphisms all k-algebra morphisms. For two objects $A,B$
in k-alg we will denote the set of all k-algebra morphisms from $A$ to $B$
by $(A,B)_{{\text{\em k}}}$.
### 1.2.
Grothendieck introduced the category k-caff of all affine schemes living over
a commutative ring k to be the category dual to the category k-calg of all
commutative k-algebras, that is, ${\text{\em k-caff}}=({\text{\em
k-calg}})^{o}$. One way to realize this duality is to associate to a
commutative k-algebra $A$ a covariant functor, the functor of points
${\text{\em h}}_{A}$,
${\text{\em h}}_{A}~{}:~{}{\text{\em k-calg}}\rTo{\text{\em sets}}\qquad
B\mapsto(A,B)_{{\text{\em k}}}$
Alternatively, one can associate to $A$ a more classical geometric object, the
affine scheme ${\text{\em spec}}(A)$. This consists of a topological space
$spec(A)$, the set of all prime ideals of $A$ equipped with the Zariski
topology, together with a sheaf of rings $\mathcal{O}_{A}$ on it, called the
structure sheaf of $A$. The ring $A$ is recovered as the ring of global
sections. Whereas both approaches are equivalent, it should be clear that the
functorial point of view lends itself more easily to generalizations.
### 1.3.
F-un or $\mathbb{F}_{1}$, the field with one element, is a virtual object
which might be thought of as a ’ring’ living under $\mathbb{Z}$.
$\mathbb{F}_{1}$-believers base their f-unny intuition on the following two
mantras :
* •
$\mathbb{F}_{1}$ forgets about additive data and retains only multiplicative
data.
* •
$\mathbb{F}_{1}$-objects only acquire flesh when extended to $\mathbb{Z}$ (or
$\mathbb{C}$).
As an example, an $\mathbb{F}_{1}$-vectorspace is merely a set $V$ as there is
no addition of vectors and just one element to use for scalar multiplication.
Hence, the dimension of $V$ equals the cardinality of $V$ as a set. Next one
should specify the classical objects one obtains after ’extending’ $V$ to the
integers or to the complex numbers. The correct integral version of a
vectorspace is a lattice, so one defines $V\otimes_{\mathbb{F}_{1}}\mathbb{Z}$
to be the free $\mathbb{Z}$-lattice $\mathbb{Z}V$ on $V$. Analogously, one
defines the extension of $V$ to the complex numbers,
$V\otimes_{\mathbb{F}_{1}}\mathbb{C}$ to be the complex vectorspace
$\mathbb{C}V$ with basis the set $V$.
But then, linear maps between $\mathbb{F}_{1}$-vectorspaces will be just set-
maps and invertible maps are bijections, whence the group
$GL_{n}(\mathbb{F}_{1})$ is the symmetric group $S_{n}$. For a group $G$, an
$n$-dimensional representation over $\mathbb{F}_{1}$ will then be a
groupmorphism $\rho:G\rTo S_{n}$, that is, a permutation representation of
$G$. Irreducible $G$-representations over $\mathbb{F}_{1}$ are then transitive
permutation representations, and so on.
### 1.4.
In analogy with the finite field case, one expects there to be a unique
$n$-dimensional field extension of $\mathbb{F}_{1}$ which we will denote by
$\mathbb{F}_{1^{n}}$. This has to be a set with $n$ elements allowing a
multiplication, whence the proposal to take $\mathbb{F}_{1^{n}}=C_{n}$ the
cyclic group of order $n$. Extending $\mathbb{F}_{1^{n}}$ to the integers or
complex numbers we should obtain a commutative algebra of rank resp. dimension
$n$. Christophe Soulé [14] proposed to take the integral- and complex group-
algebras
$\mathbb{F}_{1^{n}}\otimes_{\mathbb{F}_{1}}\mathbb{Z}\simeq\mathbb{Z}C_{n}\quad\text{and}\quad\mathbb{F}_{1^{n}}\otimes_{\mathbb{F}_{1}}\mathbb{C}\simeq\mathbb{C}G$
More generally, he proposed to take as the category of all commutative
$\mathbb{F}_{1}$-algebras the category of all finite (!) abelian groups, that
is, $\mathbb{F}_{1}-{\text{\em calg}}={\text{\em abelian}}$. For any abelian
group $G$ we then have to make sense of the extended algebras which we take
again to be the group-algebras
$G\otimes_{\mathbb{F}_{1}}\mathbb{Z}\simeq\mathbb{Z}G\quad\text{and}\quad
G\otimes_{\mathbb{F}_{1}}\mathbb{C}\simeq\mathbb{C}G$
Having a notion for commutative $\mathbb{F}_{1}$-algebras, Soulé takes
Grothendieck functor of points approach to define affine
$\mathbb{F}_{1}$-schemes. This should be a covariant functor
$X~{}:~{}{\text{\em abelian}}\rTo{\text{\em sets}}$
connecting nicely to the functor of points of an affine integral- and complex-
scheme. More precisely, Soulé [14] and later Connes and Consani [4] require
the following data
* •
a complex affine commutative algebra $A\in\mathbb{C}-{\text{\em calg}}$
* •
an integral algebra $B\in\mathbb{Z}-{\text{\em calg}}$ such that
$B\otimes_{\mathbb{Z}}\mathbb{C}\rInto A$
* •
a natural transformation $ev:X\rTo h_{A}$, called the ’evaluation’ map
* •
an inclusion of functors $i:X\rInto h_{B}$
satisfying the following universal property : given any integral algebra
$C\in\mathbb{Z}-{\text{\em calg}}$, any natural transformation $f:X\rTo h_{C}$
and any natural transformation $g:h_{A}\rTo h_{C\otimes_{Z}\mathbb{C}}$ making
the upper square commute
$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{ev}$$\scriptstyle{f}$$\scriptstyle{i}$$\textstyle{h_{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{h_{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{-\otimes\mathbb{C}}$$\textstyle{h_{C\otimes\mathbb{C}}}$$\textstyle{h_{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\exists}$$\scriptstyle{-\otimes\mathbb{C}}$$\textstyle{h_{B\otimes\mathbb{C}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
there ought to be a natural transformation $h_{B}\rTo h_{C}$ making the entire
diagram commute. This means that ${\text{\em spec}}(B)$ is the best affine
integral scheme approximating the functor $X$. Note that by Yoneda’s lemma
this means that one can reconstruct from the $\mathbb{C}$-algebra morphism
$\psi:C\otimes\mathbb{C}\rTo A$ determining the natural transformation
$g=-\circ\psi$ a $\mathbb{Z}$-algebra morphism $\phi:C\rTo B$ compatible with
the inclusion $B\otimes\mathbb{C}\rInto A$. This means that for every abelian
group $G$ we have a commuting diagram
$\textstyle{X(G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{ev}$$\scriptstyle{f}$$\scriptstyle{i}$$\textstyle{(A,\mathbb{C}G)_{\mathbb{C}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{-\circ\psi}$$\textstyle{(C,\mathbb{Z}G)_{\mathbb{Z}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{-\otimes\mathbb{C}}$$\textstyle{(C\otimes\mathbb{C},\mathbb{C}G)_{\mathbb{C}}}$$\textstyle{(B,\mathbb{Z}G)_{\mathbb{Z}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{-\circ\phi}$$\scriptstyle{-\otimes\mathbb{C}}$$\textstyle{(B\otimes\mathbb{C},\mathbb{C}G)_{\mathbb{C}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
### 1.5.
The archetypical example being the multiplicative group. Consider the
forgetful functor
$\mathbb{G}_{m}~{}:~{}{\text{\em abelian}}\rTo{\text{\em sets}}\qquad G\mapsto
G$
Take $A=\mathbb{C}[q^{\pm}]$ and $B=\mathbb{Z}[q^{\pm}]$, then their functors
of points are exactly the multiplicative group scheme, that is give the groups
of units
$h_{A}(D)=D^{*}\quad\text{and}\quad h_{B}(C)=C^{*}$
for all $D\in\mathbb{C}-{\text{\em calg}}$ and $C\in\mathbb{Z}-{\text{\em
calg}}$. We can then take both $i$ and $ev$ the natural transformation taking
$F(G)=G$ to the subgroup of units
$G\subset(\mathbb{Z}G)^{*}\subset(\mathbb{C}G)^{*}$.
Remains only to prove the universal property. Let the natural transformation
$g:h_{\mathbb{C}[q^{\pm}]}\rTo h_{C\otimes\mathbb{C}}$ be determined by the
$\mathbb{C}$-algebra morphism $\psi:C\otimes\mathbb{C}\rTo\mathbb{C}[q^{\pm}]$
and let $N$ be a natural number larger than the degree of all $\psi(c)$ where
$c$ is one of the $\mathbb{Z}$-algebra generators of $C$. Consider the finite
cyclic group $C_{N}=\langle g\rangle$, then tracing the element $g$ around the
above diagram gives the commutative diagram
$\begin{diagram}$
where $\phi=f(g)$. Repeating this argument, $\pi(\psi(c))=\psi(c)=\phi(c)$ for
all $\mathbb{Z}$-generators of $C$, whence we have that
$\psi(C)\subset\mathbb{Z}[q^{\pm}]$ giving the required natural transformation
$h_{\mathbb{Z}[q^{\pm}]}\rTo h_{C}$.
### 1.6.
Observe that Soulé uses only finite abelian groups and hence we do not require
the full functor of points, but rather the restricted functors
${\text{\em h}}^{\prime}_{A}~{}:~{}{\text{\em k-fd.calg}}\rTo{\text{\em
sets}}\qquad B\mapsto(A,B)_{{\text{\em k}}}$
where k-fd.calg is the category of all finite dimensional commutative
k-algebras. On the ’geometric’ level we might still use the affine scheme
${\text{\em spec}}(A)$ as this object contains more information than
${\text{\em h}}^{\prime}_{A}$, but we’d rather use a slimmer geometric object
having the same amount of information as the restricted functor of points. It
will turn out that the object we propose can be extended verbatim to the
noncommutative world, whereas trying to extend affine schemes is known to lead
to major difficulties.
### 1.7.
Let us consider the complex case first. For $A\in\mathbb{C}-{\text{\em
calg}}$, we define the (finite) dual coalgebra $A^{o}$ to be the collection of
all $\mathbb{C}$-linear maps $\lambda:A\rTo\mathbb{C}$ whose kernel contains a
cofinite ideal $I\triangleleft A$. The dual maps to the multiplication and
unit map of $A$ then define a coalgebra structure on $A^{o}$, see for example
Sweedler’s monograph [15]. For $B$ a finite dimensional $\mathbb{C}$-algebra,
any $\mathbb{C}$-algebra morphism $A\rTo B$ dualizes to a
$\mathbb{C}$-coalgebra map $B^{*}\rTo A^{o}$ and as a coalgebra is the limit
of its finite dimensional sub-coalgebras we see that the dual coalgebra
$A^{o}$ contains the same information as the restricted functor of points
${\text{\em h}}^{\prime}_{A}$. We will now turn $A^{o}$ into our desired
’geometric’ object.
As $A$ is commutative, any finite dimensional quotient $A/I\simeq
L_{\mathfrak{m}_{1}}\oplus\ldots\oplus L_{\mathfrak{m}_{k}}$ splits into a
direct sum of locals and hence the dual subcoalgebra $(A/I)^{*}$ is the direct
sum of pointed coalgebras $(L_{\mathfrak{m}})^{*}$ which are subcoalgebras of
the enveloping algebra of the abelian Lie-algebra of tangent-vectors
$(\mathfrak{m}/\mathfrak{m}^{2})^{*}$. Taking limits we have that
$A^{o}=\bigoplus_{\mathfrak{m}\in{\text{\em max}}(A)}P_{\mathfrak{m}}$
with $P_{\mathfrak{m}}\subset U((\mathfrak{m}/\mathfrak{m}^{2})^{*})$. In
particular, we obtain the maximal ideals ${\text{\em max}}(A)$ as the group-
like elements of $A^{o}$, or equivalently, as the direct factors of the
coradical $corad(A^{o})$. Elements of $A$ naturally evaluate on $A^{o}$ (and
hence on the coradical) and induce the usual Zariski topology on ${\text{\em
max}}(A)$.
We thus recover from the dual coalgebra $A^{o}$ the maximal ideal spectrum of
$A$. But, $A^{o}$ contains a lot more local information. This is best seen by
taking the full dual algebra $A^{o*}$ of $A^{o}$ giving rise to a Taylor-
embedding (sending a function to its Taylor series expansions in all points)
$A\rInto A^{o*}=\prod_{\mathfrak{m}\in{\text{\em
max}}(A)}\hat{\mathcal{O}}_{A,\mathfrak{m}}$
where $\hat{\mathcal{O}}_{A,\mathfrak{m}}$ is the $\mathfrak{m}$-adic
completion of $A$ (that is the stalk of the structure sheaf in the étale
topology).
Concluding, the restricted functor of points ${\text{\em h}}^{\prime}_{A}$, or
equivalently the dual coalgebra $A^{o}$, contains enough information to
recover the analytic (or étale) local information in all the closed points of
${\text{\em spec}}(A)$.
### 1.8.
An affine F-un scheme $X:{\text{\em abelian}}\rTo{\text{\em sets}}$ connects
to the complex picture via the evaluation natural transformation
$ev:X\rTo{\text{\em h}}^{\prime}_{A}$. The discussion above leads to the
introduction of an analytic ring of functions $\mathbb{F}_{1}[X]^{an}$ of
which we now have a complex interpretation
$\mathbb{F}_{1}[X]^{an}\otimes_{\mathbb{F}_{1}}\mathbb{C}=\bigcap_{\mathfrak{m}\in
Im(ev)}\hat{\mathcal{O}}_{A,\mathfrak{m}}$
With $Im(ev)$ we denote the images of all maps ${\text{\em
max}}(\mathbb{C}G)\rTo{\text{\em max}}(A)$ coming from the algebra maps
$A\rTo\mathbb{C}G$ contained in $ev(F(G))\subset{\text{\em
h}}^{\prime}_{A}(\mathbb{C}G)$.
For the example 1.5 of the forgetful functor, we have $A=\mathbb{C}[q^{\pm}]$
and hence ${\text{\em max}}(A)=\mathbb{C}^{*}$ and
$\mathbb{C}[q^{\pm}]^{o*}=\prod_{\alpha\in\mathbb{C}^{*}}\mathbb{C}[[q-\alpha]]$
For any finite abelian group $G$, ${\text{\em max}}(\mathbb{C}G)$ is the set
of characters of $G$ and under the evaluation map an element $g\in F(G)=G$
maps a character $\chi$ to its value $\chi(g)$, which are of course all roots
of unity. Hence, if we vary over all finite abelian groups we obtain
$\mathbb{F}_{1}[q^{\pm}]^{an}\otimes_{\mathbb{F}_{1}}\mathbb{C}=\bigcap_{\lambda\in\mu_{\infty}}\mathbb{C}[[q-\lambda]]$
Observe that $\mu_{\infty}$, the set of all roots of unity, is a Zariski dense
set in ${\text{\em max}}(\mathbb{C}[q^{\pm}])=\mathbb{C}^{*}$.
### 1.9.
Whereas the new complex picture based on the dual coalgebra is still pretty
close to the usual affine scheme, this changes drastically in the integral
picture. For a $\mathbb{Z}$-algebra $B$ we have to consider the restricted
functor of points
${\text{\em h}}^{\prime}_{B}~{}:~{}\mathbb{Z}-{\text{\em
fp.calg}}\rTo{\text{\em sets}}\qquad C\mapsto(B,C)_{\mathbb{Z}}$
where $\mathbb{Z}-{\text{\em fp.calg}}$ is the category of all commutative
$\mathbb{Z}$-algebras which are finite projective $\mathbb{Z}$-modules. Again,
this restricted functor contains the same information as the dual
$\mathbb{Z}$-coalgebra
$B^{o}=\underset{\rightarrow}{lim}~{}Hom_{\mathbb{Z}}(B/I,\mathbb{Z})$
where the limit is taken over all ideals $I\triangleleft B$ such that $B/I$ is
a projective $\mathbb{Z}$-module of finite rank. If we try to mimic the
complex description of the dual coalgebra we are led to consider a certain
subset of all coheight one prime ideals of $B$
${\text{\em submax}}(B)=\\{P\in spec(B)~{}|~{}\text{$B/P$ is a free
$\mathbb{Z}$-module of finite rank}\\}$
Note that closed points in ${\text{\em spec}}(B)$ are not contained in
${\text{\em submax}}(B)$. Therefore we face the problem that different
elements $P,P^{\prime}\in{\text{\em submax}}(B)$ are usually not comaximal and
hence that we no longer have a direct sum decomposition of $B^{o}$ over this
set (as was the case for the complex dual coalgebra).
As we will recall in the next section, we are familiar with such situations in
noncommutative algebra, where even maximal ideals can belong to the same
’clique’, that is, that the corresponding simple representations have
nontrivial extensions. Using this noncommutative intuition, we therefore
impose a clique-relation on the elements of ${\text{\em submax}}(B)$
$P\leftrightarrow P^{\prime}\qquad\text{iff}\qquad P+P^{\prime}\not=B$
This relation should be thought of as a ’nearness’ condition. Observe that any
$P\in{\text{\em submax}}(B)$ determines a finite collection of points in
${\text{\em max}}(B\otimes_{\mathbb{Z}}\mathbb{C})$ and hence we can extend
this nearness relation on the points of ${\text{\em max}}(B)$. Observe that
this relation is clearly invariant under the action of the absolute Galois
group $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$.
The different cliques determine the direct sum decomposition of the
$\mathbb{Z}$-coalgebra $B^{o}$ and hence also of the Taylor-like ring of
functions $B^{o*}$. Fully describing the dual $\mathbb{Z}$-coalgebra $B^{o}$
usually is a very difficult task and therefore, as in the complex case, when
we are studying F-un geometry we restrict to that part determined by the
elements in $Im(i)$ where $i~{}:~{}F\rTo{\text{\em h}}^{\prime}_{B}$ is the
inclusion of functors determined by the affine F-un scheme $F~{}:~{}{\text{\em
abelian}}\rTo{\text{\em sets}}$.
### 1.10.
Let us consider again the example of the multiplicative group and indicate how
the $\mathbb{Z}$-coalgebra approach leads to the introduction of the Habiro
ring.
The ideals $I\triangleleft B=\mathbb{Z}[q^{\pm}]$ such that $B/I$ is a free
$\mathbb{Z}$-module of finite rank are precisely the principal ideals
$I=(f(q))$ where $f(q)$ is a monic polynomial. Hence,
${\text{\em submax}}(\mathbb{Z}[q^{\pm}])=\\{(p(q))~{}:~{}\text{$p(q)$ is
monic and irreducible}\\}$
Because $\mathbb{Z}[q^{\pm}]$ is a unique factorization domain we can
decompose any monic polynomial uniquely into irreducible factors
$f(q)=p_{1}(q)^{n_{1}}\ldots p_{k}(q)^{n_{k}}$
and we would like to use this fact, as in the complex case, to decompose the
(linear duals) finite rank $\mathbb{Z}$-algebra quotients over ${\text{\em
submax}}(\mathbb{Z}[q^{\pm}])$. However,
$\frac{\mathbb{Z}[q^{\pm}]}{(f(q))}\not=\frac{\mathbb{Z}[q^{\pm}]}{(p_{1}(q))^{n_{1}}}\oplus\ldots\oplus\frac{\mathbb{Z}[q^{\pm}]}{(p_{k}(q))^{n_{k}}}$
as the different primes $(p_{i}(q))$ and $(p_{j}(q))$ do not have to be
comaximal. This problem makes it impossible to split the description of the
dual coalgebra over the ’points’ as in the complex case. Hence, we have no
other option but to describe it as a direct limit
$\mathbb{Z}[q^{\pm}]^{o}=\underset{\rightarrow}{lim}~{}(\frac{\mathbb{Z}[q^{\pm}]}{(f(q))})^{*}$
where the limit is considered with respect to divisibility of polynomials as
there are natural inclusions of $\mathbb{Z}$-coalgebras
$(\frac{\mathbb{Z}[q^{\pm}]}{(f(q))})^{*}\rInto(\frac{\mathbb{Z}[q^{\pm}]}{(g(q))})^{*}\qquad\text{whenever}\qquad
f(q)|g(q)$
As in the complex case we are then interested in the dual algebra of
$\mathbb{Z}[q^{\pm}]^{o}$ and the natural algebra map
$\mathbb{Z}[q^{\pm}]^{o}\rInto(\mathbb{Z}[q^{\pm}]^{o})^{*}=\underset{\leftarrow}{lim}~{}\frac{\mathbb{Z}[q^{\pm}]}{(f(q))}$
and it is clear that in the description of the algebra on the right-hand side
completions at principal ideals will constitute a main ingredient.
While we can do all these calculations to some extend, we are primarily
interested in that part of ${\text{\em submax}}(\mathbb{Z}[q^{\pm}])$ in the
image of the inclusion functor, that is
$Im(i)=\mathbb{N}=\\{(\Phi_{1}(q)),(\Phi_{2}(q)),\ldots,(\Phi_{n}(q)),\ldots\\}\subset{\text{\em
submax}}(\mathbb{Z}[q^{\pm}])$
We will confuse the natural number $n$ with the corresponding cyclotomic
polynomial $\Phi_{n}(q)$ or with the height one prime generated by it. With
this identification $\mathbb{N}$ is the integral analog of the set of all
roots of unity $\boldsymbol{\mu}_{\infty}$ in the complex case.
In the case of cyclotomic polynomials we have complete information about
possible co-maximality
* •
If $\frac{m}{n}\not=p^{k}$ for some prime number $p$, then
$(\Phi_{m}(q),\Phi_{n}(q))=1$ that is the cyclotomic prime ideals are
comaximal.
* •
If $\frac{m}{n}=p^{k}$ for some prime number $p$, then
$\Phi_{m}(q)\equiv\Phi_{n}(q)^{d}~{}{\text{\em mod}}~{}(p)$ for some integer
$d$, hence the cyclotomic primes are not comaximal.
Therefore, the relevant clique-relation is
$n\leftrightarrow m\qquad\text{if and only if}\qquad\frac{m}{n}=p^{\pm k}$
inducing on the complex level the
$Gal(\overline{\mathbb{Q}}/\mathbb{Q})$-invariant nearness condition on roots
of unity $\lambda,\mu\in\mu_{\infty}$
$\lambda\leftrightarrow\mu\qquad\text{iff}\qquad\frac{\lambda}{\mu}~{}\text{is
of order $p^{k}$}$
for some prime number $p$.
Yuri I. Manin argues in [12] that we should take the analogy between the
integral affine scheme ${\text{\em spec}}(\mathbb{Z}[q^{\pm}])$ and the
(complex) affine plane more seriously and that, besides the arithmetic axis,
one should also consider a projection to the ’geometric axis’ (which should
then be viewed as the affine $\mathbb{F}_{1}$-scheme corresponding to
$\mathbb{F}_{1}[q^{\pm}]$. He proposed that the zero sets of the cyclotomic
polynomials $\Phi_{n}(q)$ for all integers $n$ should be considered as the
union of the fibers in this second projection. That is, we should have the
following picture :
[1]00100100 (10,30),(200,30) (10,30),(10,200) (10,200),(200,200)
(200,30),(200,200) 2pt green (10,0),(200,0) (20,30),(20,200) (30,30),(30,200)
(50,30),(50,200) (70,30),(70,200) (130,30),(130,200) (10,0),(200,0) 6pt
(20,170),(20,200) (30,170),(30,200) (50,170),(50,200) (70,170),(70,200)
(130,170),(130,200) 2pt red (10,50),(200,50) blue (10,70),(200,70)
(10,60),(200,60) (10,80),(200,80) (10,140),(200,140) (-20,30),(-20,200) 6pt
(170,70),(200,70) (170,60),(200,60) (170,80),(200,80) (170,140),(200,140) 2pt
=5pt (20,0),(30,0),(50,0),(70,0),(130,0),(-20,60),(-20,70),(-20,80),(-20,140)
[cc](20,-10)(2) [cc](30,-10)(3) [cc](50,-10)(5) [cc](70,-10)(7)
[cc](130,-10)(p) [cc](130,210)${\text{\em spec}}(\mathbb{F}_{p}[q^{\pm}])$
[cl](210,37)${\text{\em spec}}(\mathbb{Z}[q^{\pm}])$
[cc](220,20)$\begin{diagram}$ [cl](210,0)${\text{\em spec}}(\mathbb{Z})$
[cc](110,-30)ARITHMETIC AXIS [cc](-50,115) GEOMETRIC AXIS [cc](-30,60) $1$
[cc](-30,70) $2$ [cc](-30,80) $3$ [cc](-30,140) $n$ [cc](210,140)
$(\Phi_{n}(q))$ [cc](230,140) ${\text{\em spec}}(\mathbb{Z}[\zeta_{n}])$
[cc](10,230) $\begin{diagram}$
Note that this is an over-simplification. Whereas the different green fibers
for the projection to the arithmetic axis are clearly comaximal, the blue
fibers are not. For example, the zero sets $\mathbb{V}(\Phi_{2}(q))$ and
$\mathbb{V}(\Phi_{1}(q))$ share the maximal ideal $(2,q-1)$. The clique-
relation encodes how the blue fibers intersect each other.
The clique-relation is important to relate different completions occurring in
the F-un determined part of the algebra $(\mathbb{Z}[q^{\pm}]^{o})^{*}$ as was
proved by Kazuo Habiro [7]. Let us define for any subset $S\subset\mathbb{N}$
the completion
$\mathbb{Z}[q^{\pm}]^{S}=\underset{\underset{p\in\Phi_{S}^{*}}{\leftarrow}}{lim}~{}\frac{\mathbb{Z}[q^{\pm}]}{(p)}$
where $\Phi_{S}^{*}$ is the set of monic polynomials generated by all
$\Phi_{n}(q)$ for $n\in S$. Among the many precise results proved in [7] we
mention these two
1. (1)
If $S^{\prime}\subset S$ and if every clique-component of $S$ contains an
element from $S^{\prime}$, then the natural map is an inclusion
$\rho^{S}_{S^{\prime}}~{}:~{}\mathbb{Z}[q^{\pm}]^{S}\rInto\mathbb{Z}[q^{\pm}]^{S^{\prime}}$
2. (2)
If $S$ is a saturated subset of $\mathbb{N}$ meaning that for every $n\in S$
also its divisor-set $\langle n\rangle=\\{m|n\\}$ is contained in $S$, then
$\mathbb{Z}[q^{\pm}]^{S}=\bigcap_{n\in S}\mathbb{Z}[q^{\pm}]^{\langle
n\rangle}=\bigcap_{n\in S}\widehat{\mathbb{Z}[q^{\pm}]}_{(q^{n}-1)}$
where the terms on the right-hand side are the $I$-adic completions where
$I=(q^{n}-1)$.
Using these properties it is then natural to define the integral version of
the ring of analytic functions on the multiplicative group scheme over
$\mathbb{F}_{1}$ to be
$\mathbb{F}_{1}[q^{\pm}]^{an}\otimes_{\mathbb{F}_{1}}\mathbb{Z}\simeq\bigcap_{n\in\mathbb{N}}\widehat{\mathbb{Z}[q^{\pm}]}_{(q^{n}-1)}=\mathbb{Z}[q^{\pm}]^{\mathbb{N}}$
This ring has a description very similar to that of the profinite integers
replacing factorials by q-factorials
$\mathbb{Z}[q^{\pm}]^{\mathbb{N}}=\underset{\underset{n}{\leftarrow}}{lim}~{}\frac{\mathbb{Z}[q^{\pm}]}{((q^{n}-1)(q^{n-1}-1)\ldots(q-1))}$
and as such its elements have a unique description as formal Laurent
polynomials over $\mathbb{Z}$ of the form
$\sum_{n=0}^{\infty}a_{n}(q)(q^{n}-1)(q^{n-1}-1)\ldots(q-1)\in\mathbb{Z}[[q^{\pm}]]\qquad\text{with}\qquad
deg(a_{n}(q))<n$
We observe that any such formal power series can be evaluated at a root of
unity. Some elements of $\mathbb{Z}[q^{\pm}]^{\mathbb{N}}$ have been
discovered before. For example, Maxim Kontsevich observed in his
investigations on Feynman integrals that the formal power series
$\sum_{n=0}^{\infty}(1-q)(1-q^{2})\ldots(1-q^{n})$
has a properly defined value in every root of unity. Subsequently, Don Zagier
[17] proved the strange equality
$\sum_{n=0}^{\infty}(1-q)(1-q^{2})\ldots(1-q^{n})=-\frac{1}{2}\sum_{n=1}^{\infty}n\chi(n)q^{(n^{2}-1)/24}$
where $\chi$ is the quadratic character of conductor $12$. The strange fact
about this equality is that the two sides never make sense simultaneously. The
left hand side diverges for all points within the unit circle and outside the
unit circle and can be evaluated at roots of unity whereas the right hand side
converges only within the unit circle and diverges everywhere else. What
Zagier meant by this equality is that for all
$\alpha\in\boldsymbol{\mu}_{\infty}$ the evaluation of the left hand side
coincides with the radial limit of the function on the right hand side. Don
Zagier says that the function on the right ’leak through roots of unity’.
## 2\. Noncommutative F-un geometry
In this section we will extend Soulé’s definition of an affine
$\mathbb{F}_{1}$-scheme to the noncommutative case. Our main motivation is the
study of finite dimensional representations of discrete groups, such as the
braid groups or the modular group. We have seen that irreducible finite
dimensional $\mathbb{F}_{1}$-representations of a group $\Gamma$ are exactly
the finite transitive permutation representations $\Gamma/\Lambda$ where
$\Lambda$ is of finite index in $\Gamma$. That is, all finite dimensional
$\mathbb{F}_{1}$-representation theory of $\Gamma$ comes from its profinite
completion $\hat{\Gamma}=\underset{\leftarrow}{lim}~{}\Gamma/\Lambda$, the
limit taken over all finite index normal subgroups.
In the previous section we have worked out the special case when
$\Gamma=\mathbb{Z}$. Here, the simple representations of $\hat{\mathbb{Z}}$
are the roots of unity $\mu_{\infty}$ and they are Zariski closed in all
simples $\mathbb{C}^{*}={\text{\em simp}}(\mathbb{Z})$. The clique-relation on
$\mu_{\infty}$ was compatible with the action of the absolute Galois group and
the Habiro ring ’feels’ the inclusion $\mu_{\infty}\subset\mathbb{C}^{*}$,
that is it contains the tangent information in a Galois-compatible way.
Here we extend some of these results to the case of a non-Abelian discrete
group $\Gamma$ satisfying the property $\bullet$ : for every finite collection
of elements $\\{g_{1},\ldots,g_{k}\\}\subset\Gamma$ there is a finite index
subgroup $\Lambda\subset\Gamma$ such that the natural projection map gives an
embedding $\\{g_{1},\ldots,g_{k}\\}\rInto\Gamma/\Lambda$. We will prove that
such groups determine a noncommutative affine $\mathbb{F}_{1}$-scheme, the
F-un information being given by the finite dimensional permutation
representations, or equivalently, the representation theory of the profinite
completion $\hat{\Gamma}$. We will show that ${\text{\em simp}}(\hat{\Gamma})$
is Zariski dense in ${\text{\em simp}}(\Gamma)$ and compute the tangent
information of this embedding. That is, to a finite dimensional permutation
representation $P=\Gamma/\Lambda$ we will associate a noncommutative gadget (a
quiver, relations and a dimension vector) encoding all possible deformations
of $P$ which are still $\Gamma$-representations. In relevant situations,
including the case when $\Gamma$ is the modular group
$\operatorname{PSL}_{2}(\mathbb{Z})$ (in which case the permutation
representations are Grothendieck’s ’dessins d’enfants’) some subsidiary
noncommutative gadgets can be derived from this tangent information, such as
the necklace Lie algebra [2] and the singularity type [3]. It is to be
expected that most of these noncommutative gadgets associated to dessins are
in fact Galois invariants.
### 2.1.
If we take commutative $\mathbb{F}_{1}$-algebras to be abelian groups, it make
sense to identify the category of all $\mathbb{F}_{1}$-algebras with groups
the category of all finite groups. Likewise, we have to extend Grothendieck’s
functor of points to all, that is including also noncommutative, algebras.
With these modifications we can extend Soulé’s definition to the
noncommutative world.
Define an affine noncommutative $\mathbb{F}_{1}$-scheme to be a covariant
functor
$X~{}:~{}{\text{\em groups}}\rTo{\text{\em sets}}$
from the category groups of all finite groups to sets. We require that there
is an affine $\mathbb{C}$-algebra $A$ and an evaluation natural transformation
$ev:X\rTo{\text{\em h}}_{A}=(A,-)_{\mathbb{C}}$, giving for every finite group
$G$ an evaluation map $X(G)\rTo(A,\mathbb{C}G)_{\mathbb{C}}$. Moreover, there
should be a ’best’ integral affine algebra $B$ with an inclusion of functors
$X\rInto{\text{\em h}}_{B}=(B,-)_{\mathbb{Z}}$.
That is, for every finite group $G$ we have an inclusion
$X(G)\rInto(B,\mathbb{Z}G)_{\mathbb{Z}}$. Here, ’best’ means that for every
$\mathbb{Z}$-algebra $C$ and every natural transformation $X\rTo{\text{\em
h}}_{C}=(C,-)_{\mathbb{Z}}$ and every $\mathbb{C}$-algebra morphism
$\psi:\mathbb{C}\otimes C\rTo A$ making the upper square in the diagram below
commute for every finite group $G$
$\textstyle{X(G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{ev}$$\scriptstyle{f}$$\scriptstyle{i}$$\textstyle{(A,\mathbb{C}G)_{\mathbb{C}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g=-\circ\psi}$$\textstyle{(C,\mathbb{Z}G)_{\mathbb{Z}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{-\otimes\mathbb{C}}$$\textstyle{(C\otimes\mathbb{C},\mathbb{C}G)_{\mathbb{C}}}$$\textstyle{(B,\mathbb{Z}G)_{\mathbb{Z}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\exists-\circ\phi}$$\scriptstyle{-\otimes\mathbb{C}}$$\textstyle{(B\otimes\mathbb{C},\mathbb{C}G)_{\mathbb{C}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
there exists a $\mathbb{Z}$-algebra morphism $\phi:C\rTo B$ making the entire
diagram commute.
### 2.2.
Our first example of a noncommutative F-un scheme is Grothendieck’s theory of
’dessins d’enfants’. Let $X_{\mathbb{C}}$ be a Riemann surface (projective
algebraic curve) defined over $\overline{\mathbb{Q}}$, then Belyi proved that
there is a degree $d$ map $\pi:C\rOnto\mathbb{P}^{1}_{\mathbb{C}}$ ramified
only in the points $\\{0,1,\infty\\}$. The open interval $]0,1[$ lifts to $d$
intervals on $C$. The endpoints of different lifts can be identified on $X$
indicating how the different sheets should be glued together in a neighborhood
of the ramification point. The resulting graph with $d$ edges on $C$ is then
called the dessin of $C$ and as the absolute Galois group
$Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ acts on the collection of all such
curves, it also acts on the dessins. Writing out this action allows one to
gain insight in the absolute Galois group. Hence it is a very important
problem to find new Galois invariants of dessins.
We will be particularly interested in modular dessins, that is such that the
preimages of $0$ all have valency 1 or 2 and the preimages of 1 all have
valency 1 or 3 in the graph. Alternatively, this means that the curve can be
viewed as the compactification of a quotient $C=\mathbb{H}/\Lambda$ of the
upper-halfplane under the action of a subgroup $\Lambda$ of finite index in
the modular group $\Gamma=PSL_{2}(\mathbb{Z})$. That is, modular dessins are
equivalent to finite dimensional permutation representations of the modular
group. Therefore, one is interested in the functor
$X~{}:~{}{\text{\em groups}}\rTo{\text{\em sets}}\qquad G\mapsto G_{(2)}\times
G_{(3)}$
sending a group to the set of all permutation representations of $\Gamma$
determined by elements of $G$. As $\Gamma\simeq C_{2}\ast C_{3}$ is the free
product of a cyclic group of order 2 with a cyclic group of order 3, this
functor sends a finite group $G$ to the set product of its elements of order 2
with the elements of order 3 : $G_{(2)}\times G_{(3)}$. This functor
determines a noncommutative affine $\mathbb{F}_{1}$-scheme as we can take as
the complex- and integral group-algebras
$A=\mathbb{C}\Gamma\quad\text{and}\quad B=\mathbb{Z}\Gamma$
of the modular group. As any $\mathbb{C}$-algebra morphism
$A=\mathbb{C}\Gamma\rTo\mathbb{C}G$ is determined by the images of the order
two (resp. three) generators $x$ and $y$ we can take as the evaluation and
inclusion maps
$ev~{}:~{}G_{(2)}\times
G_{(3)}\rTo(\mathbb{C}\Gamma,\mathbb{C}G)_{\mathbb{C}}\qquad(g_{2},g_{3})\mapsto\begin{cases}x\mapsto
g_{2}\\\ y\mapsto g_{3}\end{cases}$ $i~{}:~{}G_{(2)}\times
G_{(3)}\rInto(\mathbb{Z}\Gamma,\mathbb{Z}G)_{\mathbb{Z}}\qquad(g_{2},g_{3})\mapsto\begin{cases}x\mapsto
g_{2}\\\ y\mapsto g_{3}\end{cases}$
We can repeat the argument of 1.5 verbatim to prove that these data indeed
define a noncommutative $\mathbb{F}_{1}$-scheme using the fact that the
modular group $\Gamma$ satisfies condition $\bullet$.
### 2.3.
The second example is motivated by 2-dimensional TQFT. To a Riemann surface
$C$ of genus $g$ and any finite group $G$ one associates as topological
invariant $Z_{G}(C)$ the number of fields on $C$ with gauge group $G$, or
equivalently, the number of $G$-covers on $C$. By Frobenius-Schur this number
is equal to
$Z_{G}(C)=\sum_{\chi}(\frac{|G|}{dim~{}\chi})^{2g-2}$
where the sum runs over all irreducible representations $\chi$ of the finite
group $G$. As the number of $G$-covers is equal to the number of group-
morphisms $\pi_{1}(C)\rTo G$ from the fundamental group $\pi_{1}(C)=\langle
x_{1},\ldots,x_{g},y_{1},\ldots,y_{g}\rangle/(\prod
x_{i}y_{i}x_{i}^{-1}y_{i}^{-1})$, this motivates the functor
$X~{}:~{}{\text{\em groups}}\rTo{\text{\em sets}}\quad
G\mapsto\\{(a_{1},\ldots,a_{g},b_{1},\ldots,b_{g})\in G^{2g}~{}:~{}\prod
a_{i}b_{i}a_{i}^{-1}b_{i}^{-1}=1\\}$
This functor is again an affine noncommutative $\mathbb{F}_{1}$-scheme as we
can take the integral- and complex group-algebras $A=\mathbb{C}\pi_{1}(C)$ and
$B=\mathbb{Z}\pi_{1}(C)$ and the natural evaluation and inclusion maps. Once
again, the defining ”bestness” property is verified using the fact that
$\pi_{1}(C)$ satisfies condition $\bullet$.
Also in this example, the $\mathbb{F}_{1}$-info is given by all finite
permutation representations of the fundamental group $\pi_{1}(C)$. That is,
the F-un information is contained in the profinite completion
$\widehat{\pi_{1}(C)}$.
### 2.4.
These two examples illustrate that any discrete group $\Gamma$ satisfying
condition $\bullet$ determines a noncommutative affine
$\mathbb{F}_{1}$-scheme. The corresponding functor assigns to a group $G$ the
set of all groupmorphisms $\Gamma\rTo G$ and takes as the complex- and
integral algebras the complex and integral group-algebra of $\Gamma$.
As in the commutative case we do not require the full strength of the functor
of points ${\text{\em h}}_{A}:{\text{\em k-alg}}\rTo{\text{\em sets}}$ for a
given (not necessarily commutative) k-algebra $A$, but it suffices, for
applications to F-un geometry, to restrict to finite dimensional k-algebras
${\text{\em h}}^{\prime}_{A}~{}:~{}{\text{\em k-fd.alg}}\rTo{\text{\em
sets}}\qquad C\mapsto(A,C)_{{\text{\em k}}}$
If k is a field, the information contained in this restricted functor of
points is equivalent to that contained in the dual coalgebra $A^{o}$. For this
reason we want to associate noncommutative geometric data (say, a topological
space and function) to the dual $\mathbb{C}$-coalgebra $A^{o}$ where $A$ is
the complex algebra determining the evaluation natural transformation
$ev:X\rTo{\text{\em h}}^{\prime}_{A}$.
Observe that in [11] we initiated the description of the dual coalgebra of any
affine $\mathbb{C}$-algebra $A$ in terms of the $A_{\infty}$-structure on the
Yoneda space of all finite dimensional simple $A$-representations. For the
applications we have in mind here, that is, virtually free groups $G$ (such as
the modular group $\Gamma=PSL_{2}(\mathbb{Z})$), for which the group algebras
$\mathbb{C}G$ is formally smooth by [10], or 2-Calabi-Yau algebras such as
$\mathbb{C}\pi_{1}(C)$, we do not require the full power of
$A_{\infty}$-theory and can give, at least in principle, an explicit
description of the dual coalgebra.
The geometric space associated to an affine $\mathbb{C}$-algebra $A$ will be
the set of isomorphism classes of finite dimensional $A$-representations,
which as in the commutative case, is the set of direct summands of the
coradical of the dual coalgebra
${\text{\em simp}}(A)=corad(A^{o})$
In [11] we introduced a Zariski topology on ${\text{\em simp}}(A)$ in terms of
the measuring $A^{o}\otimes A\rTo\mathbb{C}$. Here we will follow a slightly
different approach based on noncommutative functions.
For a $\mathbb{C}$-algebra $A$ we define the noncommutative functions to be
the $\mathbb{C}$-vectorspace quotients
${\text{\em functions}}(A)=\mathfrak{g}_{A}=\frac{A}{[A,A]_{vect}}$
where $[A,A]_{vect}$ is the subvectorspace (and not the ideal) spanned by all
commutators in $A$. Note that in the classical case where $A=\mathbb{C}[X]$ is
the commutative coordinate ring of an affine variety $X$, there is nothing to
divide out and hence in this case we recover the coordinate ring
$\mathfrak{g}_{A}=\mathbb{C}[X]$. If $A=\mathbb{C}G$ the group-algebra of a
finite group $G$, then $\mathfrak{g}_{A}$ is the space dual to the space of
character-functions of $G$. Hence, in both cases the linear functionals
$\mathfrak{g}^{*}$ suffice to separate the points of $A$, that is ${\text{\em
simp}}(A)$. We will show that for a general affine $\mathbb{C}$-algebra $A$ we
do indeed have an embedding
${\text{\em simp}}(A)\rInto\mathfrak{g}^{*}$
Consider the (commutative) affine scheme ${\text{\em rep}}_{n}A$ of all
$n$-dimensional representations. A quick and dirty way to describe its
coordinate ring $\mathbb{C}[{\text{\em rep}}_{n}A]$ is to take a finite set of
algebra generators $\\{a_{1},\ldots,a_{m}\\}$ of $A$, consider a set of
$mn^{2}$ commuting variables $\\{x_{ij}(k):1\leq i,j\leq n,1\leq k\leq m\\}$
and consider the ideal $I_{n}(A)$ of the polynomial algebra
$\mathbb{C}[x_{ij}(k)~{}:~{}i,j,k]$ generated by all entries of all $n\times
n$ matrices $f(X_{1},\ldots,X_{m})$ where $f(a_{1},\ldots,a_{m})$ runs over
all relations holding in $A$ and where $X_{k}$ is the generic $n\times n$
matrix $(x_{ij}(k))_{i,j}$. Then,
$\mathbb{C}[{\text{\em
rep}}_{n}A]=\frac{\mathbb{C}[x_{ij}(k)~{}:~{}i,j,k]}{I_{n}(A)}$
On the affine scheme ${\text{\em rep}}_{n}A$ there is a natural action of
$GL_{n}$, the orbits of which correspond exactly to the isomorphism classes of
$n$-dimensional $A$-representations. Basic GIT-stuff tells us that one can
classify the closed orbits by points of the quotient-scheme ${\text{\em
iss}}_{n}A={\text{\em rep}}_{n}A/GL_{n}$ corresponding to the affine ring of
invariants
$\mathbb{C}[{\text{\em iss}}_{n}A]=\mathbb{C}[{\text{\em rep}}_{n}A]^{GL_{n}}$
and Artin proved that the closed orbits are precisely the isoclasses of semi-
simple representations.
Let us bring in our quotient $\mathfrak{g}_{A}=\frac{A}{[A,A]_{vect}}$. We can
evaluate its elements on all points of ${\text{\em rep}}_{n}A$ by taking
traces. That is, each $g\in\mathfrak{g}$ defines a function
${\text{\em rep}}_{n}A\rTo\mathbb{C}\qquad M\mapsto tr(g)(M)$
That is, lift $g$ to an element $a\in A$, write $a=f(a_{1},\ldots,a_{m})$ in
terms of its generators, then if $(m_{1},\ldots,m_{k})$ are the matrices
describing the $n$-dimensional representation $M$, then we define
$tr(g)(M)=Tr(f(m_{1},\ldots,m_{k}))$
where $Tr$ is the standard trace map on $M_{n}(\mathbb{C})$. Observe that this
does not depend on the chosen lift $a$ as all traces of elements from
$[A,A]_{vect}$ vanish. Observe that via this trace-trick we can view elements
of $\mathfrak{g}^{*}$ indeed as generalized characters as each representation
defines a linear functional
$\chi_{M}~{}:~{}\mathfrak{g}\rTo\mathbb{C}\qquad g\mapsto tr(g)(M)$
It is a classical result that the ring of invariants $\mathbb{C}[{\text{\em
rep}}_{n}A]^{GL_{n}}$ is generated by the invariant functions $tr(g)$ when $g$
runs over $\mathfrak{g}$. So, indeed, linear functionals on $\mathfrak{g}$ do
separate $n$-dimensional semi-simple representations (whence a fortiori also
simples). Actually, we only showed separation of simples for a fixed $n$, but
clearly one recovers the dimension from $tr(1)$. That is, we have proved that
for any affine $\mathbb{C}$-algebra $A$, the generalized character values give
an embedding
${\text{\em simp}}(A)\rInto\mathfrak{g}^{*}_{A}$
We will make the set ${\text{\em simp}}(A)$ into a topological space by taking
as the basic opens
$\mathbb{X}(g,\lambda)=\\{S\in{\text{\em
simp}}A~{}|~{}\chi_{S}(g)\not=\lambda\\}$
for all $g\in\mathfrak{g}_{A}$ and all $\lambda\in\mathbb{C}$. For example,
all simples of dimension $n$ form a closed subset. The obtained topology we
will call the Zariski topology on ${\text{\em simp}}(A)$.
Our use of this topology is to prove a denseness result similar to the fact
that roots of unity $\mu_{\infty}$ are Zariski dense in $\mathbb{C}^{*}$. Let
$G$ be a discrete group, as every finite dimensional $\hat{G}$ representation
factors over a finite group quotient of $G$ (and hence is semi-simple) we
deduce that the dual coalgebra $(\mathbb{C}G)^{o}$ is co-semi-simple and hence
${\text{\em
simp}}(\mathbb{C}\hat{G})=(\mathbb{C}G)^{o}=corad((\mathbb{C}G)^{o})$
We claim that when $G$ is a discrete group satisfying condition $\bullet$,
then
$\overline{{\text{\em simp}}(\mathbb{C}\hat{G})}={\text{\em
simp}}(\mathbb{C}G)$
That is, the subset of simple representations of the profinite completion is
Zariski dense in the noncommutative space ${\text{\em simp}}(\mathbb{C}G)$.
Observe that in the two examples given before, ${\text{\em
simp}}(\mathbb{C}\hat{G})$ is the image of the evaluation map determined by
the F-un geometry, hence this result is a direct generalization of the
commutative situation for the multiplicative group.
To prove this claim observe that the space of noncommutative functions
$\mathfrak{g}=\mathfrak{g}_{\mathbb{C}G}$ has as $\mathbb{C}$-basis the
conjugacy classes of elements of $G$. Hence, any linear functional
$\chi\in\mathfrak{g}^{*}$ is a linear combination
$\chi=\lambda_{1}\chi_{1}+\ldots+\lambda_{k}\chi_{k}$
where the $\chi_{i}$ are character functions corresponding to distinct
conjugacy classes of $G$. Vanishing of $\chi$ on the whole of ${\text{\em
simp}}(\mathbb{C}\hat{G})$ would imply that the characters
$\lambda_{1},\ldots,\lambda_{k}$ are linearly dependent on every finite
quotient $G/H$, which is impossible by the assumption on $G$.
### 2.5.
Let us recall briefly the main result of [11] describing the dual coalgebra
$A^{o}$ of a general affine $\mathbb{C}$-algebra $A$ and indicate the
geometric information contained in it. Let $Q$ be a possibly infinite quiver
and $\mathbb{C}Q$ the vectorspace spanned on all paths in $Q$ of positive
length. Then $\mathbb{C}Q$ is given a coalgebra structure (the path coalgebra)
$\Delta(p)=\sum_{p=p_{1}.p_{2}}p_{1}\otimes
p_{2}\qquad\epsilon(p)=\delta_{p,vertex}$
where $p_{1}.p_{2}$ is the concatenation of paths and the counit maps non-
vertex paths to zero.
Starting from $A$ we will construct a huge quiver $Q_{A}$ having as its
vertices the isoclasses of finite dimensional simple representations and with
the number of arrows between them
$\\#(S\rTo S^{\prime})=dim_{\mathbb{C}}~{}Ext^{1}_{A}(S,S^{\prime})$
We will now describe a certain subcoalgebra of the path coalgebra
$\mathbb{C}Q_{A}$ and as any coalgebra is the direct limit of its finite
dimensional subcoalgebras we may restrict attention to a finite collection of
simples and consider the semi-simple representation $M=S_{1}\oplus\ldots\oplus
S_{k}$ with restricted path-coalgebra $\mathbb{C}Q_{A}|M$. There is a natural
$A_{\infty}$-algebra structure on the Yoneda Ext-algebra
$Ext^{\bullet}_{A}(M,M)$, in particular there are higher multiplication maps
$m_{i}~{}:~{}\underbrace{Ext^{1}_{A}(M,M)\otimes\ldots\otimes
Ext^{1}_{A}(M,M)}_{i}\rTo Ext^{2}_{A}(M,M)$
defining a linear map, called the homotopy Maurer-Cartan map
$HMC_{M}=\oplus_{i}m_{i}~{}:~{}\mathbb{C}Q_{A}|M\rTo Ext^{2}_{A}(M,M)$
The main result of [11] asserts that the dual coalgebra $A^{o}$ is Morita-
Takeuchi equivalent to the largest subcoalgebra of $\mathbb{C}Q_{A}$ contained
in the kernel of $HMC_{M}$ for all semi-simple representations $M$.
We will now describe the geometric content of the dual coalgebra. Recall that
in the commutative case we had that the full linear dual of the dual coalgebra
$(\mathbb{C}[X]^{o})^{*}=\prod_{x}\hat{\mathcal{O}}_{X,x}$ gave us back all
the completed local rings at points of $X$. In the general case, assume as
above that $M=S_{1}\oplus\ldots\oplus S_{k}$ is a semi-simple representation
with all simple factors distinct.The action of $A$ on $M$ gives rise to an
epimorphism
$A\rOnto^{\pi_{M}}B_{M}=M_{n_{1}}(\mathbb{C})\oplus\ldots\oplus
M_{n_{k}}(\mathbb{C})$
and let us denote $\mathfrak{m}=Ker(\pi_{M})$. If $C_{M}$ is the maximal
subcoalgebra of $\mathbb{C}Q_{A}|M$ contained in the kernel of the $HMC_{M}$,
then we can generalize the commutative situation as follows. The
$\mathfrak{m}$-adic completion of $A$ is Morita equivalent to the full linear
dual of $C_{M}$
$\hat{A}_{\mathfrak{m}}\sim_{M}(C_{M})^{*}$
This means that all $\mathfrak{m}$-adic completion of $A$ can be computed from
the dual coalgebra $A^{o}$ and that each of them is a ring Morita equivalent
to (the completion of) a path algebra of the quiver $(Q_{A}|M)^{*}$ modulo
certain relations coming from the $A_{\infty}$-structure.
### 2.6.
Recall that a $\mathbb{C}$-algebra $A$ is said to be smooth if and only if the
kernel of the multiplication map
$\Omega^{1}_{A}=Ker(A\otimes A\rTo^{m}A)$
is a projective $A$-bimodule. Because $Ext^{2}_{A}(M,N)=0$ for all finite
dimensional $A$-representations when $A$ is smooth, we have from the above
general result that the $\mathfrak{m}$-adic completion
$\hat{A}_{\mathfrak{m}}$ is Morita-equivalent to the completion of the path
algebra $\mathbb{C}(Q_{A}|M)^{*}$ where we recall that this quiver depends
only on the dimensions of the ext-groups $Ext^{1}_{A}(S_{i},S_{j})$.
In fact, in this case we do not have to use the full strength of the general
result and deduce this fact from the formal neighborhood theorem for smooth
algebras due to Cuntz and Quillen [6, §6]. Note that
$Ker(\pi_{M})=\mathfrak{m}$ has a natural $B=B_{M}$-bimodule structure. In
analogy with the Zariski tangent space in the commutative case, we define
$T_{M}=\left(\frac{\mathfrak{m}}{\mathfrak{m}^{2}}\right)^{*}$
Because $B$ is a semi-simple algebra the simple $B$-bimodules are either of
the form $M_{n_{i}}(\mathbb{C})$ (with trivial action of the other components
of $B$) or $M_{n_{i}\times n_{j}}(\mathbb{C})$ with the component
$M_{n_{i}}(\mathbb{C})$ (resp. $M_{n_{j}}(\mathbb{C})$) acting by left (resp.
right) multiplication and all other actions being trivial. That is, there is a
natural one-to-one correspondence between
${\text{\em bimod}}~{}B\leftrightarrow{\text{\em quiver}}_{n}$
isoclasses of $B$-bimodules and quivers $n$ vertices (the number of simple
components). Under this correspondence, $B$-bimodule duals corresponds to
taking the opposite quiver. Hence, the tangent space $T_{M}$ can be identified
with a quiver on the vertices $\\{S_{1},\ldots,S_{n}\\}$ which we will now
show is the opposite quiver of $Q_{A}|M$.
By the formal tubular neighborhood theorem of Cuntz and Quillen [6, §6] (using
the fact that semi-simple algebras are formally smooth) we have an isomorphism
of completed algebras between the $\mathfrak{m}$-adic completion of $A$
$\hat{A}_{\mathfrak{m}}=\underset{\leftarrow}{lim}~{}A/\mathfrak{m}^{n}$
where $\mathfrak{m}=Ker(\pi)$ as above, and, the completion (with respect to
the natural gradation) of the tensor-algebra
$T_{B}(\mathfrak{m}/\mathfrak{m}^{2})$. That is, when we view $T_{M}$ as a
quiver, then there is a Morita-equivalence
$\hat{A}_{\mathfrak{m}}\underset{M}{\sim}\widehat{\mathbb{C}T_{M}^{\vee}}$
between the completion $\hat{A}_{\mathfrak{m}}$ and the completion (with
respect to the gradation giving all arrows degree one) of the path-algebra
$\mathbb{C}T_{M}^{\vee}$ of the opposite quiver $T_{M}^{\vee}$.
Under this Morita-equivalence the semi-simple
$\hat{A}_{\mathfrak{m}}$-representation $M=S_{1}\oplus\ldots\oplus S_{n}$
corresponds to the sum of the vertex-simples
$\mathbb{C}e_{1}\oplus\ldots\oplus\mathbb{C}e_{n}$, with the simple $S_{i}$
corresponding to the vertex-simple $\mathbb{C}e_{i}$ (the $e_{i}$ are the
vertex-idempotents in the path algebra). Hence, also by Morita-equivalence we
have an isomorphism
$Ext^{1}_{\hat{A}_{\mathfrak{m}}}(S_{i},S_{j})\simeq
Ext^{1}_{\widehat{\mathbb{C}T_{M}^{\vee}}}(\mathbb{C}e_{i},\mathbb{C}e_{j})$
Finally, because all ext-information is preserved under completions, and,
because we know from representation-theory that the dimension of the ext-space
between two vertex-simples for any quiver ${Q}$,
$dim_{\mathbb{C}}~{}Ext^{1}_{\mathbb{C}Q}(\mathbb{C}e_{i},\mathbb{C}e_{j})$ is
equal to the number of arrows starting in vertex $v_{i}$ and ending in vertex
$v_{j}$, we are done!
Clearly, computing all $Ext^{1}_{A}(S,S^{\prime})$ can still be a laborious
task. However, it was proved in [10] that all these dimensions follow often
from a finite set of calculations when $A$ is a smooth algebra. The component
semigroup ${\text{\em comp}}(A)$ is the set of all connected components of the
schemes ${\text{\em rep}}_{n}~{}A$, for all $n\in\mathbb{N}$, with addition
induced by the direct sum of finite dimensional representations.
The one quiver of $A$, ${\text{\em one}}(A)$ is a full subquiver of $Q_{A}$
with one simple representant for every component which is a generator of
${\text{\em comp}}(A)$ (note that such generators are determined by the fact
that the component consists entirely of simples). Now, if $S$ and $T$ are two
finite dimensional $A$-representations belonging to the connected components
$\alpha$ and $\beta$ in ${\text{\em comp}}(A)$ then we can write for certain
$a_{i},b_{i}\in\mathbb{N}$
$\alpha=\sum a_{i}g_{i}\quad\text{and}\quad\beta=\sum b_{i}g_{i}$
with the $g_{i}$ the generator components. Then, $\epsilon=(a_{i})_{i}$ and
$\eta=(b_{i})_{i}$ are dimension vectors for the one quiver. The main result
of [10] asserts now that
$dim_{\mathbb{C}}~{}Ext^{1}_{A}(S,T)=-\chi_{{\text{\em
one}}(A)}(\epsilon,\eta)$
so that all ext-dimensions, and hence all $\mathfrak{m}$-adic completions of
$A$ can be deduced from knowledge of the one quiver.
### 2.7.
We will now make all these calculations explicit in the case of prime interest
to us, which is the modular group $\Gamma=PSL_{2}(\mathbb{Z})$, that is, we
will describe the dual coalgebra $(\mathbb{C}\Gamma)^{o}$, at least in
principle. Because $\Gamma\simeq C_{2}\ast C_{3}$ we have that the group-
algebra is the free algebra product of two semi-simple group algebras
$\mathbb{C}\Gamma\simeq\mathbb{C}C_{2}\ast\mathbb{C}C_{3}$
and as such is a smooth algebra. In fact, a far more general result holds :
whenever $G$ is a virtually free group (that is $G$ contains a free subgroup
of finite index), then the group algebra $\mathbb{C}G$ is smooth by [10].
If $V$ is an $n$-dimensional $\Gamma$ representation, we can decompose it into
eigenspaces for the action of $C_{2}=\langle u\rangle$ and $C_{3}=\langle
v\rangle$ (let $\rho$ denote a primitive third root of unity) :
$V_{+}\oplus V_{-}=V_{1}\oplus
V_{2}=V\downarrow_{C_{2}}=V=V\downarrow_{C_{3}}=W_{1}\oplus W_{2}\oplus
W_{3}=W_{1}\oplus W_{\rho}\oplus W_{\rho^{2}}$
If the dimension of $V_{i}$ is $a_{i}$ and that of $W_{j}$ is $b_{j}$, we say
that $V$ is a $\Gamma$-representation of dimension vector
$\alpha=(a_{1},a_{2};b_{1},b_{2},b_{3})$. Choosing a basis $B_{1}$ of $V$ wrt.
the decomposition $V_{1}\oplus V_{2}$ and a basis $B_{2}$ wrt. $W_{1}\oplus
W_{2}\oplus W_{3}$, we can view the basechange matrix $B_{1}\rTo B_{2}$ as an
$\alpha$-dimensional representation $V_{Q}$ of the quiver ${Q}={Q}_{\Gamma}$
${Q}_{\Gamma}=\qquad\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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66.21547pt\raise-92.2873pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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89.2873pt\raise-92.2873pt\hbox{\hbox{\kern 3.0pt\raise
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For a general quiver ${Q}$ on $k$ vertices, a weight $\theta\in\mathbb{Z}^{k}$
acts on the dimension vectors via the usual (Euclidian) scalar inproduct. A
${Q}$-representation of dimension vector $\alpha\in\mathbb{N}^{k}$ is said to
be $\theta$-stable if and only if $\theta.\alpha=0$ and for every proper non-
zero subrepresentation $W\subset V$ of dimension vector $\beta<\alpha$ we have
that $\theta.\beta>0$.
Bruce Westbury [16] has shown that $V$ is an irreducible
$\Gamma$-representation if and only if $V_{Q}$ is a $\theta$-stable
$Q$-representation where $\theta=(-1,-1;1,1,1)$ and that the two notions of
isomorphism coincide. The Euler-form $\chi_{{Q}}$ of the quiver $Q$ is the
bilinear form on $\mathbb{Z}^{\oplus 5}$ determined by the matrix
$\chi_{{Q}}=\begin{bmatrix}1&0&-1&-1&-1\\\ 0&1&-1&-1&-1\\\ 0&0&1&0&0\\\
0&0&0&1&0\\\ 0&0&0&0&1\end{bmatrix}$
Westbury also showed that if there exists a $\theta$-stable
$\alpha$-dimensional $Q$-representation, then there is an
$1-\chi_{{Q}}(\alpha,\alpha)$ dimensional family of isomorphism classes of
such representations (and a Zariski open subset of them will correspond to
isomorphism classes of irreducible $\Gamma$-representations).
We will describe the one quiver ${\text{\em one}}(\mathbb{C}\Gamma)$. By the
above it follows that both the component semigroup ${\text{\em
comp}}~{}\mathbb{C}\Gamma$ and the semigroup of $\mathbb{Z}^{5}$ generated by
all $\theta$-stable ${Q}$-representations are generated by the following six
connected components, belonging to the dimension vectors
$g_{ij}=(\delta_{1i},\delta_{2i},\delta_{3i};\delta_{1j},\delta_{2j})$
and if we order and relabel these generators as
$a=g_{11},b=g_{22},c=g_{31},d=g_{12},e=g_{21},f=g_{32}$
we can compute from the Euler-form of ${Q}$ that the one-quiver of the modular
group algebra is the following hexagonal graph
${\text{\em one}}(\mathbb{C}\Gamma)=\qquad\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
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0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
47.28047pt\raise-76.88406pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}{\hbox{\kern
48.36153pt\raise-101.69698pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces$
which is the origin of a lot of hexagonal moonshine in the representation
theory of the modular group. In particular it follows from symmetry of the one
quiver that the quiver $Q_{\mathbb{C}\Gamma}$ is also symmetric!
### 2.8.
Recall that an affine $\mathbb{C}$-algebra $A$ is said to be 2-Calabi-Yau if
$gldim(A)=2$ and for any pair $S,T$ of finite dimensional $A$-representations,
there exists a natural duality
$Ext^{i}_{A}(S,T)\simeq(Ext^{2-i}_{A}(T,S))^{*}$
satisfying an additional sign condition. Raf Bocklandt [1] succeeded in
extending the results on smooth algebras recalled before to the setting of
2-Calabi-Yau algebras. From the duality condition it is immediate that the
quiver $Q_{A}$ is symmetric, that is, for every arrow $S\rTo^{a}T$ there is a
paired arrow in the other direction $T\rTo^{a^{*}}S$. Bocklandt’s result
asserts that the $\mathfrak{m}$-adic completion $\hat{A}_{\mathfrak{m}}$ with
$\mathfrak{m}=Ker(\pi_{M})$ is Morita equivalent to the completion of the path
algebra of the (dual) quiver $Q_{A}|M$ modulo the preprojective relation
$\sum_{a}[a,a^{*}]=0$
Further, he extends the idea of the one quiver to the 2-Calabi-Yau setting,
allowing to compute the quiver $Q_{A}$ often from a finite set of
calculations, using earlier results due to Crawley-Boevey [5].
The group algebra $\mathbb{C}\pi_{1}(C)$ of the fundamental group of a genus
$g$ Riemann surface is 2-Calabi-Yau by a result of Maxim Kontsevich. In [1,
§7.1] it is shown that the one-quiver of $\mathbb{C}\pi_{1}(C)$ consists of
one vertex, corresponding to any one-dimensional simple representation, and
$2g$ loops. From this and the results by Crawley-Boevey it follows that when
$M=S_{1}^{\oplus e_{1}}\oplus\ldots\oplus S_{k}^{\oplus e_{k}}$ is a semi-
simple $\mathbb{C}\pi_{1}(C)$-representation wit the simple factor $S_{i}$
having dimension $n_{i}$, then $\mathbb{C}Q_{\mathbb{C}\pi_{1}(C)}|M$ consists
of $k$ vertices (corresponding to the distinct simple components $S_{i}$),
such that the $i$-th vertex has exactly $2(g-1)n_{i}^{2}+2$ loops and there
are exactly $2n_{i}n_{j}(g-1)$ directed arrows from vertex $i$ to vertex $j$.
This information allows us then to compute all $\mathfrak{m}$-adic completions
of $\mathbb{C}\pi_{1}(C)$ as Morita equivalent to the completion of the path
algebra of this quiver modulo the preprojective relation.
### 2.9.
In 2.7 we described the structure of the path-coalgebra
$\mathbb{C}Q_{\mathbb{C}\Gamma}$ which is Morita-Takeuchi equivalent to the
dual complex coalgebra $(\mathbb{C}\Gamma)^{o}$. Describing the integral dual
coalgebra $(\mathbb{Z}\Gamma)^{o}$ is a lot more complicated and will involve
a good deal of knowledge of the integral (and modular) representation theory
of the modular group.
Observe that the calculations in 2.7 are valid for every algebraically closed
field, so we might as well describe the coalgebra
$(\overline{\mathbb{Q}}\Gamma)^{o}$ and study the action of the absolute
Galois group $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ on it, giving us an
handle on the rational dual coalgebra
$(\mathbb{Q}\Gamma)^{o}=((\overline{\mathbb{Q}}\Gamma)^{o})^{Gal(\overline{\mathbb{Q}}/\mathbb{Q})}$
which brings us closer to $(\mathbb{Z}\Gamma)^{o}$. But, as in the case of the
multiplicative group in the previous section, we do not require the full
structure of this dual coalgebra but rather the image of the F-un data in it.
As observed before, the $\mathbb{F}_{1}$-representation theory of $\Gamma$ is
equivalent to the study of all finite dimensional transitive permutation
representations of $\Gamma$ and hence to conjugacy classes of finite index
subgroups of $\Gamma$.
We will recall the combinatorial description of those, due to R. Kulkarni in
[9] in terms of generalized Farey symbols. Starting from this symbol we then
describe how to associate a dessin, its monodromy group an finally to derive
from it the modular content, that is the noncommutative gadget describing all
$\Gamma$-representations deforming to the given permutation representation. In
the next subsection we will give some interesting examples.
A generalized Farey sequence is an expression of the form
$\\{\infty=x_{-1},x_{0},x_{1},\ldots,x_{n},x_{n+1}=\infty\\}$
where $x_{0}$ and $x_{n}$ are integers and some $x_{i}=0$. Moreover, all
$x_{i}=\frac{a_{i}}{b_{i}}$ are rational numbers in reduced form and ordered
such that
$|a_{i}b_{i+1}-b_{i}a_{i+1}|=1\qquad\text{for all $1\leq i<n$}$
The terminology is motivated by the fact that the classical Farey sequence
$F(n)$, that is the ordered sequence of all rational numbers
$0\leq\frac{a}{b}\leq 1$ in reduced form with $b\leq n$, has this remarkable
property.
A Farey symbol is a generalized Farey sequence
$\\{\infty=x_{-1},x_{0},x_{1},\ldots,x_{n},x_{n+1}=\infty\\}$ such that for
all $-1\leq i\leq n$ we add one of the following symbols to two consecutive
terms
$\textstyle{x_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bullet}$$\textstyle{x_{i+1}}$
or
$\textstyle{x_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\circ}$$\textstyle{x_{i+1}}$
or
$\textstyle{x_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{k}$$\textstyle{x_{i+1}}$
where each of the occurring integers $k$ occur in pairs.
To connect Farey symbols with cofinite subgroups of the modular group $\Gamma$
we need to recall the Dedekind tessellation of the upper-half plane
$\mathbb{H}$. Recall that the extended modular group
$\Gamma^{*}=PGL_{2}(\mathbb{Z})$ acts on $\mathbb{H}$ via the natural action
of $\Gamma$ on it together with the extra symmetry $z\mapsto-\overline{z}$.
The Dedekind tessellation is the tessellation by fundamental domains for the
action of $\Gamma^{*}$ on $\mathbb{H}$. It splits every fundamental domain for
$\Gamma$ in two hyperbolic triangles, usually depicted as a black and a white
one. Here is a depiction of the upper part of the Dedekind tessellation
[110]-1201.5 (-1,0),(2,0)
red (0,0),(0,1.5) (-1,0),(-1,1.5) (1,0),(1,1.5) (2,0),(2,1.5)
[s](0,0),(-1,0),180 [s](1,0),(0,0),180 [s](2,0),(1,0),180
[s](-.5,0),(-1,0),180 [s](0,0),(-.5,0),180 [s](.5,0),(0,0),180
[s](1,0),(.5,0),180 [s](1.5,0),(1,0),180 [s](2,0),(1.5,0),180
blue (-.5,0.866),(-.5,1.5) (.5,0.866),(.5,1.5) (1.5,0.866),(1.5,1.5)
[s](-.5,0.866),(-1,0),60 [s](.5,0.866),(0,0),60 [s](1.5,0.866),(1,0),60
[s](0,0),(-.5,.866),60 [s](1,0),(.5,.866),60 [s](2,0),(1.5,.866),60
[s](-.5,.288),(-1,0),120 [s](.5,.288),(0,0),120 [s](1.5,.288),(1,0),120
[s](0,0),(-.5,.288),120 [s](1,0),(.5,.288),120 [s](2,0),(1.5,.288),120
(-.5,0),(-.5,.288) (.5,0),(.5,.288) (1.5,0),(1.5,.288)
black (-.5,.866),(-.5,.288) (.5,.866),(.5,.288) (1.5,.866),(1.5,.288)
[s](-.5,.866),(-1,1),30 [s](.5,.866),(0,1),30 [s](1.5,.866),(1,1),30
[s](0,1),(-.5,.866),30 [s](1,1),(.5,.866),30 [s](2,1),(1.5,.866),30
[s](1.6,.2),(3/2,0.288),30 [s](0.6,.2),(1/2,0.288),30
[s](-.4,.2),(-1/2,0.288),30 [s](-.5,0.288),(-.6,.2),30
[s](.5,0.288),(.4,.2),30 [s](1.5,0.288),(1.4,.2),30
Here, every red edge is a $\Gamma$-translate of the edge $[i,\infty]$, a blue
edge a $\Gamma$-translate of $[\rho,\infty]$ where $\rho$ is a primitive sixth
root of unity and every black edge is a $\Gamma$-translate of the circular arc
$[i,\rho]$. Observe that every hyperbolic triangle of this tessellation has
one edge of all three colors. Moving counterclockwise along the border of a
triangle we either have the ordering red-blue-black (in which case we call
this triangle a white triangle) or blue-red-black (and then we call it a black
triangle). Any pair of a white and black triangle make a fundamental domain
for the action of $\Gamma$.
Observe that any hyperbolic geodesic connecting two consecutive terms of a
generalized Farey sequence consists of two red edges (connected at an
intersection with black edges. We call these intersection points even points
(later in the theory of dessins they will be denoted by a $\bullet$). A point
where three blue edges come together with three black edges will be called an
odd point (later denoted by ).
A generalized Farey sequence therefore determines a hyperbolic polygonal
region of $\mathbb{H}$ bounded by the (red) full geodesics connecting
consecutive terms. The extra information contained in a Farey symbol tell us
how to identify sides of this polygon (as well as how to extend it slightly in
case of $\bullet$-connections) as follows :
* •
For
$\textstyle{x_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\circ}$$\textstyle{x_{i+1}}$
the two red edges making up the geodesic connecting $x_{i}$ with $x_{i+1}$ are
identified.
* •
For
$\textstyle{x_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{k}$$\textstyle{x_{i+1}}$
(with paired
$\textstyle{x_{j}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{k}$$\textstyle{x_{j+1}}$)
these two full geodesics (each consisting of two red edges) are identified.
* •
For
$\textstyle{x_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bullet}$$\textstyle{x_{i+1}}$
we extend the boundary of the polygon by adding the two triangles just outside
the full geodesic and identify the two blue edges forming the adjusted
boundary.
In this way, we associate to a Farey symbol a compact surface. Next, we will
construct a cuboid tree diagram out of it, that is, a tree embedded in
$\mathbb{H}$ such that all internal vertices are $3$-valent. Take as the
vertices all odd-points lying in the interior of the polygonal region together
with together with all even (red) and odd (blue) points on the boundary. We
connect these vertices with the black lines in the interior of the polygonal
region and add an involution on the red leaf-vertices determined by the side-
pairing information contained in the Farey-symbol.
Finally, we will also associate to it a bipartite cuboid graph (aka a ’dessin
d’enfants’). Start with the cuboid tree diagram and divide all edges in two
(that is, add also the even internal points connecting the two black edges
making up an edge in the tree diagram) and connect two red leaf-vertices when
they correspond to each other under the involution.
For example, consider the Farey symbol
$\textstyle{\infty\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bullet}$$\textstyle{\frac{1}{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bullet}$$\textstyle{\frac{1}{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bullet}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1}$$\textstyle{\infty}$
The boundary of the polygonal region determined by the symbol is indicated by
the slightly thicker red and blue edges. The vertices of the cuboid tree are
the red, blue and black points and the edges are the slightly thicker black
edges.
[110]-1201.5 .4 (-1,0),(2,0)
red (-1,0),(-1,1.5) 1.7 (0,0),(0,1.5) (1,0),(1,1.5) .4 (2,0),(2,1.5)
[s](0,0),(-1,0),180 [s](1,0),(0,0),180 [s](2,0),(1,0),180
[s](-.5,0),(-1,0),180 [s](0,0),(-.5,0),180 [s](.5,0),(0,0),180
[s](1,0),(.5,0),180 [s](1.5,0),(1,0),180 [s](2,0),(1.5,0),180
[s](1/3,0),(0,0),180 [s](1/2,0),(1/3,0),180
blue (-.5,0.866),(-.5,1.5) (.5,0.866),(.5,1.5) (1.5,0.866),(1.5,1.5)
[s](-.5,0.866),(-1,0),60 [s](.5,0.866),(0,0),60 [s](1.5,0.866),(1,0),60
[s](0,0),(-.5,.866),60 [s](1,0),(.5,.866),60 [s](2,0),(1.5,.866),60
[s](-.5,.288),(-1,0),120 [s](.5,.288),(0,0),120 [s](1.5,.288),(1,0),120
[s](0,0),(-.5,.288),120 [s](1,0),(.5,.288),120 [s](2,0),(1.5,.288),120
(-.5,0),(-.5,.288) (.5,0),(.5,.288) (1.5,0),(1.5,.288)
[s](.357,.123),(1/3,0),18
1.7 [s](0.268,0.0666),(0,0),152 [s](1/3,0),(0.268,0.0666),84
[s](0.394,0.046),(1/3,0),100 [s](1/2,0),(0.394,0.046),130
[s](0.642,.123),(1/2,0),95 [s](1,0),(0.642,.123),142
.4 black (-.5,.866),(-.5,.288) 1.7 (.5,.866),(.5,.288) .4
(1.5,.866),(1.5,.288) [s](-.5,.866),(-1,1),30 1.7 [s](.5,.866),(0,1),30 .4
[s](1.5,.866),(1,1),30
[s](0,1),(-.5,.866),30 1.7 [s](1,1),(.5,.866),30 .4 [s](2,1),(1.5,.866),30
[s](1.6,.2),(3/2,0.288),30 [s](0.6,.2),(1/2,0.288),30
[s](-.4,.2),(-1/2,0.288),30 [s](-.5,0.288),(-.6,.2),30
[s](.5,0.288),(.4,.2),30 [s](1.5,0.288),(1.4,.2),30
1.7 [s](.5,.288),(0.357,.123),45 [s](0.357,0.123),(0.269,0.0666),55
[s](0.394,0.046),(0.357,0.123),30 [s](.642,.123),(.5,.288),47
blue 2
red
black
[cc](.5,-.1)$\frac{1}{2}$ [cc](.333,-.1)$\frac{1}{3}$
[cc](.666,-.1)$\frac{2}{3}$ [cc](0,-.1)$0$ [cc](1,-.1)$1$
Because the two red leaf-vertices correspond to each other under the
involution, the corresponding bipartite cuboid diagram (or modular dessin) is
$\scriptstyle{{\bf 1}}$$\scriptstyle{\bullet}$$\scriptstyle{{\bf
2}}$$\scriptstyle{{\bf 3}}$$\scriptstyle{\bullet}$$\scriptstyle{{\bf
4}}$$\scriptstyle{{\bf 5}}$$\scriptstyle{\bullet}$$\scriptstyle{{\bf
6}}$$\scriptstyle{{\bf 7}}$$\scriptstyle{\bullet}$$\scriptstyle{{\bf
8}}$$\scriptstyle{{\bf 9}}$$\scriptstyle{\bullet}$$\scriptstyle{{\bf
10}}$$\scriptstyle{{\bf 11}}$$\scriptstyle{\bullet}$$\scriptstyle{{\bf 12}}$
Such a dessin encodes the data of a Belyi covering
$C\rOnto\mathbb{P}^{1}_{\mathbb{C}}$ ramified only in the points
$\\{0,1,\infty\\}$. The inverse images of $0$ will be represented by a
-vertex, those of $1$ by a $\bullet$-vertex. Of relevance for us are dessins
which are modular quilts meaning that every $\bullet$-vertex is $2$-valent and
every -vertex is $1$\- or $3$-valent.
Given a modular dessin, denote each of the edges by a different number between
$1$ and $d$ (the degree of $\pi$), then the monodromy group $G_{\pi}$ of $\pi$
is the subgroup of $S_{d}$ generated by the order three element $\sigma_{0}$
obtained by cycling round every -vertex counterclockwise and the order two
element $\sigma_{1}$ obtained by recording the two edges ending at every
$\bullet$-vertex. This defines an exact sequence of groups
$1\rTo G\rTo\Gamma\rTo G_{\pi}\rTo 1$
and the projective curve $C$ corresponding to the modular dessin can be
identified with a compactification of $\mathbb{H}/G$ where $\mathbb{H}$ is the
upper half-plane on which $G\subset\Gamma$ acts via Möbius transformations.
The $d$-dimensional permutation representation $M=\Gamma/G$ decomposes into
irreducible representations for the monodromy group $G_{\pi}$, say
$M=X_{1}^{\oplus e_{1}}\oplus\ldots\oplus X_{k}^{\oplus e_{k}}$
with every $X_{i}$ an irreducible $G_{\pi}$ and hence also irreducible
$\Gamma$-representation. The modular content of the dessin, or of the
permutation representation, is the quiver on $k$ vertices
$Q_{\pi}=Q_{\mathbb{C}\Gamma}|M$ together with the dimension vector
$\alpha_{\pi}=(e_{1},\ldots,e_{k})$ determined by the multiplicities of the
simples in the permutation representation.
Roughly speaking, the modular content $(Q_{\pi},\alpha_{\pi})$ encodes how
much the curve $C$, the dessin or the permutation representation ’sees’ of the
modular group. That is, the quotient variety ${\text{\em
iss}}_{\alpha_{\pi}}Q_{\pi}={\text{\em
rep}}_{\alpha_{\pi}}Q_{\pi}/GL(\alpha_{\pi})$ classifies all semi-simple
$d$-dimensional $\Gamma$-representations deforming to the permutation
representation $M$. As such, it is a new noncommutative gadget associated to a
classical object, the curve $C$. It would be interesting to know whether the
modular content is a Galois invariant of the dessin, or more generally, what
subsidiary information derived from it is a Galois invariant.
We now give an algorithm to compute the modular content, using the group-
theory program GAP, starting from the modular quilt $D$.
1. (1)
Determine the permutations $\sigma_{0},\sigma_{1}\in S_{d}$ described above,
that is obtained by walking around the $\bullet$-vertices (for $\sigma_{1}$)
and the -vertices (for $\sigma_{0}$) in $D$ and feed them to GAP as s0,s1.
2. (2)
Calculate the monodromy group $G_{\pi}$ via G:=Group(s0,s1) and determine its
character table via chars:=CharacterTable(G);)
3. (3)
Determine the $G_{\pi}$-character of the permutation representation by calling
ConjugacyClasses(G). This returns a list of $S_{d}$-permutations representing
the conjugacy classes of $G_{\pi}$. To determine the character-value we only
need to count the numbers missing in the cycle decomposition of the
permutation. Let $\chi$ be the obtained character which is the list chi.
4. (4)
Determine the irreducible components of $\chi$ and their multiplicities via
MatScalarProducts(chars,Irr(chars),[chi]);. The non-zero entries form the
dimension vector $\alpha_{\pi}$ and they determine the simple factors
$X_{1},\ldots,X_{k}$.
5. (5)
Determine the conjugacy classes of $\sigma_{0}$ and $\sigma_{1}$. For example,
the number of the conjugacy class in the character table is found by
FusionConjugacyClasses(Group(s0),G);. Alternatively, one can use
IsConjugate(G,s0,s); for s a suitable element representant obtained via
ConjugacyClasses(G);. Assume $\sigma_{0}$ (resp. $\sigma_{1}$) belongs to the
$a$-th (resp. $b$-th) conjugacy class.
6. (6)
From the character values of $X_{i}$ in the $a$-th and $b$-th column of
Display(chars); one deduces the dimension vector
$\alpha_{i}=(a_{1}(i),a_{2}(i);b_{1}(i),b_{2}(i),b_{3}(i))$ of the
${Q}_{\Gamma}$-representation corresponding to $X_{i}$.
7. (7)
Finally, the number of arrows (and loops) in the quiver ${Q}_{\pi}$ between
the vertices corresponding to $X_{i}$ and $X_{j}$ is given by
$\delta_{ij}-\chi_{{Q}_{\Gamma}}(\alpha_{i},\alpha_{j})$.
### 2.10.
As the modular content encodes all possible $\Gamma$-representation
deformations of the permutation representation, it is often a huge object
which makes it difficult to extract interesting deformations from it.
Sometimes though, a true gem reveals itself. In the previous subsection we
used the generalized Farey-symbol
$\textstyle{\infty\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bullet}$$\textstyle{\frac{1}{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bullet}$$\textstyle{\frac{1}{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bullet}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1}$$\textstyle{\infty}$
Note that it consists of half of the Farey-sequence $F(3)$ (those
$\leq\frac{1}{2}$). Generalizing this construction for all classical Farey
sequences leads to an intriguing class of examples. The $n$-th Iguanodon
Farey-symbol is the Farey symbol
$\textstyle{\infty\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bullet}$$\textstyle{\frac{1}{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bullet}$$\textstyle{\ldots}$$\textstyle{\frac{1}{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bullet}$$\scriptstyle{\bullet}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1}$$\textstyle{\infty}$
where the rational numbers occurring are precisely those Farey numbers in
$F(n)$ smaller or equal to $\frac{1}{2}$.
The terminology is explained by depicting the first few bipartite cuboid
diagrams associated to Farey sequences
$\scriptstyle{{\bf 3}}$$\scriptstyle{\bullet}$$\scriptstyle{{\bf
4}}$$\scriptstyle{{\bf 1}}$$\scriptstyle{\bullet}$$\scriptstyle{{\bf
2}}$$\scriptstyle{{\bf 50}}$$\scriptstyle{\bullet}$$\scriptstyle{{\bf
49}}$$\scriptstyle{{\bf 51}}$$\scriptstyle{\bullet}$$\scriptstyle{{\bf
52}}$$\scriptstyle{{\bf 42}}$$\scriptstyle{\bullet}$$\scriptstyle{{\bf
41}}$$\scriptstyle{{\bf 43}}$$\scriptstyle{\bullet}$$\scriptstyle{{\bf
44}}$$\scriptstyle{{\bf 30}}$$\scriptstyle{\bullet}$$\scriptstyle{{\bf
29}}$$\scriptstyle{{\bf 26}}$$\scriptstyle{\bullet}$$\scriptstyle{{\bf
25}}$$\scriptstyle{{\bf 31}}$$\scriptstyle{\bullet}$$\scriptstyle{{\bf
32}}$$\scriptstyle{{\bf 18}}$$\scriptstyle{\bullet}$$\scriptstyle{{\bf
17}}$$\scriptstyle{{\bf 27}}$$\scriptstyle{\bullet}$$\scriptstyle{{\bf
28}}$$\scriptstyle{{\bf 16}}$$\scriptstyle{\bullet}$$\scriptstyle{{\bf
15}}$$\scriptstyle{{\bf 19}}$$\scriptstyle{\bullet}$$\scriptstyle{{\bf
20}}$$\scriptstyle{{\bf 12}}$$\scriptstyle{\bullet}$$\scriptstyle{{\bf
11}}$$\scriptstyle{{\bf 13}}$$\scriptstyle{\bullet}$$\scriptstyle{{\bf
14}}$$\scriptstyle{{\bf 8}}$$\scriptstyle{\bullet}$$\scriptstyle{{\bf
7}}$$\scriptstyle{{\bf 9}}$$\scriptstyle{\bullet}$$\scriptstyle{{\bf
10}}$$\scriptstyle{{\bf 5}}$$\scriptstyle{\bullet}$$\scriptstyle{{\bf
6}}$$\scriptstyle{{\bf 54}}$$\scriptstyle{\bullet}$$\scriptstyle{{\bf
53}}$$\scriptstyle{{\bf 34}}$$\scriptstyle{\bullet}$$\scriptstyle{{\bf
33}}$$\scriptstyle{{\bf 22}}$$\scriptstyle{\bullet}$$\scriptstyle{{\bf
21}}$$\scriptstyle{{\bf 24}}$$\scriptstyle{\bullet}$$\scriptstyle{{\bf
23}}$$\scriptstyle{{\bf 56}}$$\scriptstyle{\bullet}$$\scriptstyle{{\bf
55}}$$\scriptstyle{{\bf 36}}$$\scriptstyle{\bullet}$$\scriptstyle{{\bf
35}}$$\scriptstyle{{\bf 46}}$$\scriptstyle{\bullet}$$\scriptstyle{{\bf
45}}$$\scriptstyle{{\bf 38}}$$\scriptstyle{\bullet}$$\scriptstyle{{\bf
37}}$$\scriptstyle{{\bf 40}}$$\scriptstyle{\bullet}$$\scriptstyle{{\bf
39}}$$\scriptstyle{{\bf 48}}$$\scriptstyle{\bullet}$$\scriptstyle{{\bf
47}}$$\scriptstyle{{\bf 58}}$$\scriptstyle{\bullet}$$\scriptstyle{{\bf
57}}$$\scriptstyle{{\bf 60}}$$\scriptstyle{\bullet}$$\scriptstyle{{\bf 59}}$
Here, the diagram corresponding to Farey sequence $F(n)$ is the full subfigure
on the first $m(n)$ (half)edges
$\begin{array}[]{c|cccccccc}n&2&3&4&5&6&7&8&9\\\ \hline\cr
m(n)&8&12&16&24&28&40&48&60\end{array}$
The monodromy groups corresponding to the $n$-th Iguanodon symbol are
$\begin{array}[]{c|cccccccc}n&2&3&4&5&6&7&8&9\\\
\hline\cr&L_{2}(7)&M_{12}&A_{16}&M_{24}&A_{28}&A_{40}&A_{48}&A_{60}\\\
\hline\cr\\\ \hline\cr n&10&11&12&13&14&15&16&17\\\
\hline\cr&A_{68}&A_{88}&A_{96}&A_{120}&A_{132}&A_{148}&A_{164}&A_{196}\end{array}$
This can be verified by hand (and GAP) using the above picture for $n\leq 9$
and by using the SAGE-package kfarey.sage for higher $n$. It is plausible that
the monodromy groups of the Iguanodon symbols are all simple groups and it is
quite remarkable that the Mathieu groups $M_{12}$ and $M_{24}$ appear in this
sequence of alternating groups.
Now, let us compute the modular content of these permutation representations.
The action of the monodromy group is clearly 2-transitive implying that as a
$\mathbb{C}G_{\pi}$-representation, the permutation representation splits into
two irreducibles, one of which being clearly the trivial representation. Note
also that the character of the generator of order $2$ is equal to zero as
there are no $\bullet$-end points. Further, $\circ$-endpoints appear in pairs
and add another 4 half-edges, that is 4 dimensions, to the permutation space.
By induction we see that the dimension of the permutation representation is
always of the form $4n$ with $\chi(\sigma_{1})=0$ and $\chi(\sigma_{0})=n$.
By the argument recalled in 2.7 it follows that the dimension vector of the
$Q_{\Gamma}$-quiver representation corresponding to the permutation
representation is
$\alpha_{4n}=\qquad\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
5.75058pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
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0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
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0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
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19.97694pt\raise-23.44856pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
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40.20331pt\raise-23.44856pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
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19.97694pt\raise-46.36241pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
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40.20331pt\raise-46.36241pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
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60.42967pt\raise-46.36241pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
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81.90604pt\raise-46.36241pt\hbox{\hbox{\kern
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n}$}}}}}{{{\hbox{\ellipsed@{5.75058pt}{4.6111pt}}}}\hbox{\kern-5.75058pt\raise-69.27626pt\hbox{\hbox{\kern
3.0pt\raise-1.61111pt\hbox{$\textstyle{\scriptscriptstyle
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0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{{{{}{}{}{}{}}}}\ignorespaces{{{{}{}{}{}{}}}}\ignorespaces{\hbox{\lx@xy@drawline@}}{{{{}{}{}{}{}}}}\ignorespaces{{{}{}{}{}{}}}{\hbox{\lx@xy@drawline@}}{\hbox{\kern
19.97694pt\raise-69.27626pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
40.20331pt\raise-69.27626pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
60.42967pt\raise-69.27626pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
83.40662pt\raise-69.27626pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern-3.0pt\raise-92.19011pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
19.97694pt\raise-92.19011pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
40.20331pt\raise-92.19011pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
60.42967pt\raise-92.19011pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
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81.90604pt\raise-92.19011pt\hbox{\hbox{\kern
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By 2-transitivity the dimension vectors of the two simple components $S$ and
$T$ are
$\alpha_{T}=\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
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0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
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0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
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3.0pt\raise-1.61111pt\hbox{$\textstyle{\scriptscriptstyle
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67.77368pt\raise-82.41248pt\hbox{\hbox{\kern
3.0pt\raise-1.61111pt\hbox{$\textstyle{\scriptscriptstyle
0}$}}}}}\ignorespaces}}}}\ignorespaces\qquad\text{and}\qquad\alpha_{S}=\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
7.83391pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
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0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
36.59575pt\raise-80.80833pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
53.97667pt\raise-80.80833pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{{{\hbox{\ellipsed@{4.50058pt}{4.07639pt}}}}\hbox{\kern
74.69092pt\raise-80.80833pt\hbox{\hbox{\kern
3.0pt\raise-1.07639pt\hbox{$\textstyle{\scriptscriptstyle
n}$}}}}}\ignorespaces}}}}\ignorespaces$
But then, by the algorithm we have that the modular content
$(Q_{\pi},\alpha_{\pi})$ of the permutation representation can be depicted as
$\textstyle{\scriptscriptstyle 1}$$\textstyle{\scriptscriptstyle
1}$$\scriptstyle{n^{2}}$
The $n^{2}$ loops in the vertex corresponding to the simple factor $S$
indicate that the moduli space of semi-stable $Q_{\Gamma}$-representations
$M_{\theta}^{ss}(Q_{\Gamma},\alpha_{S})$ is $n^{2}$-dimensional and as $S$ is
a smooth point in it, there is an $n^{2}$-dimensional family of simple
$\Gamma$-representations in the neighborhood of $S$. More interesting is the
fact that there is just one arrow in each direction between the two vertices.
This implies that the permutation representation is a smooth point in the
moduli space of semi-simple $\Gamma$-representations, a rare fact for higher
dimensional decomposable representations (see the paper [3] for more details
on singularities of quiver-representations). Further, this implies that there
is a unique (!) curve of simple $4n$-dimensional $\Gamma$-representations
degenerating to the given permutation representation! Certainly in the case of
the sporadic Mathieu groups it would be interesting to study these curves (and
their closures in the moduli space $M^{ss}_{\theta}(Q_{\Gamma},\alpha_{4n})$)
in more detail.
## References
* [1] Raf Bocklandt, Noncommutative tangent cones and Calabi-Yau algebras, arXiv:0711.0179 (2007)
* [2] Raf Bocklandt and Lieven Le Bruyn, Necklace Lie algebras and noncommutative symplectic geometry, Math. Z. 240 (2002) 141-167, arXiv:math/0010030
* [3] Raf Bocklandt, Lieven Le Bruyn and Geert Van de Weyer, Smooth order singularities, J. Alg. Appl. 2 (2003) 365-395, arXiv:math/0207250
* [4] Alain Connes and Katia Consani, On the notion of geometry over $\mathbb{F}_{1}$, arXiv:0809.2926
* [5] Bill Crawley-Boevey, Geometry of the moment map for representations of quivers, Compositio Math. 126 (2001) 257-293
* [6] Joachim Cuntz, Daniel Quillen, Algebra extensions and nonsingularity, Journal of AMS, v.8, no. 2 (1995) 251 289
* [7] Kazuo Habiro, Cyclotomic completions of polynomial rings, arXiv:0209324
* [8] Maxim Kontsevich and Yan Soibelman, Notes on $A_{\infty}$-algebras, $A_{\infty}$-categories and non-commutative geometry I, arXiv:math.RA/0606241 (2006)
* [9] Ravi S. Kulkarni, An arithmetic-geometric method in the study of the subgroups of the modular group, Amer. J. Math. 113 (1991) 1053-1133
* [10] Lieven Le Bruyn, Qurves and quivers, arXiv:math.RA/0406618 (2004), Journal of Algebra 290 (2005) 447-472
* [11] Lieven Le Bruyn, Noncommutative geometry and dual coalgebras, arXiv:0805.2377v1 (2008)
* [12] Yuri I. Manin, Cyclotomy and analytic geometry over $\mathbb{F}_{1}$, arXiv:0809.1564 (2008)
* [13] Matilde Marcolli, Cyclotomy and endomotives, arXiv:0901.3167 (2009)
* [14] Christophe Soul , Let variétés sur le corps à un élément, Moscow Math. J. 4 (2004) 217-244
* [15] Moss E. Sweedler, Hopf Algebras, monograph, W.A. Benjamin (New York) (1969)
* [16] Bruce Westbury, On the character varieties of the modular group, preprint Nottingham (1995)
* [17] Don Zagier, Vassiliev invariants and a strange identity related to the Dedekind eta-function, available as MPI-preprint
|
arxiv-papers
| 2009-09-14T12:09:07 |
2024-09-04T02:49:05.342054
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Lieven Le Bruyn",
"submitter": "Lieven Le Bruyn",
"url": "https://arxiv.org/abs/0909.2522"
}
|
0909.2587
|
# Phase Transition Signature Results from PHENIX
Brookhaven National Laboratory
E-mail and the PHENIX Collaboration
###### Abstract:
The PHENIX experiment has conducted searches for the QCD critical point with
measurements of multiplicity fluctuations, transverse momentum fluctuations,
event-by-event kaon-to-pion ratios, elliptic flow, and correlations.
Measurements have been made in several collision systems as a function of
centrality and transverse momentum. The results do not show significant
evidence of critical behavior in the collision systems and energies studied,
although several interesting features are discussed.
## 1 Introduction
Recent work with lattice gauge theory simulations indicate that the phase
diagram of Quantum Chromodynamics (QCD) may contain a first-order transition
line between the hadron gas phase and the strongly-coupled Quark-Gluon Plasma
(sQGP) phase that terminates at a critical point [1]. This property is
observed in many common liquids, including water. Near the QCD critical point,
several thermodynamic properties of the system will diverge with a power law
behavior in the variable $\epsilon=(T-T_{C})/T_{C}$, where $T_{C}$ is the
critical temperature. Here, several measurements made by the PHENIX experiment
at Brookhaven National Laboratory’s Relativistic Heavy Ion Collider that may
be sensitive to this critical behavior are discussed.
## 2 Multiplicity Fluctuations
In the Grand Canonical Ensemble, the variance and the mean of the particle
number, N, can be directly related to the compressibility, $k_{T}$:
$\omega_{N}=\frac{var(N)}{N}=k_{B}T\frac{N}{V}k_{T}$, where $k_{B}$ is
Boltzmann’s constant, T is the temperature, and V is the volume [2]. Near the
critical point, the compressibility diverges with a power law behavior with
exponent $\gamma$: $k_{T}\propto\epsilon^{-\gamma}$. The measurement of event-
by-event fluctuations in the multiplicity of charged hadrons may be sensitive
to critical behavior in the system. PHENIX has surveyed the behavior of
inclusive charged particle multiplicity fluctuations as a function of
centrality and transverse momentum in $\sqrt{s_{NN}}$=62.4 GeV and 200 GeV
Au+Au collisions, and in $\sqrt{s_{NN}}$=22.5, 62.4, and 200 GeV Cu+Cu
collisions.
Since multiplicity fluctuations are well described by Negative Binomial
Distributions (NBD) in both elementary [3] and heavy ion collisions [4], the
data for a given centrality and $p_{T}$ bin are fit to an NBD from which the
mean and variance are determined. Due to the finite width of each centrality
bin, there is a non-dynamic component of the observed fluctuations that is
present due to fluctuations in the impact parameter within a centrality bin.
The magnitude of this component is estimated using the HIJING event generator
[5], which well reproduces the mean multiplicity of RHIC collisions [6]. The
estimate is performed by comparing fluctuations from simulated events with a
fixed impact parameter to events with a range of impact parameters covering
the width of each centrality bin, as determined from Glauber model
simulations. The data are corrected to remove the impact parameter fluctuation
component.
Figure 1: Multiplicity fluctuations as a function of $N_{part}$ for Au+Au
collisions for $0.2<p_{T}<2.0$ GeV/c. Contributions from impact parameter
fluctuations have been removed. Shaded regions represent a 1$\sigma$ range of
the superposition model prediction derived from p+p data.
Baseline comparisons are made to the participant superposition model, in which
the total multiplicity fluctuations can be expressed in terms of the scaled
variance [7], $\omega_{N}=\omega_{\nu}+\mu_{WN}~{}\omega_{N_{part}}$, where
$\omega_{\nu}$ are the fluctuations from each individual source,
$\omega_{N_{part}}$ are the fluctuations of the number of sources, and
$\mu_{WN}$ is the mean multiplicity per wounded nucleon. The second term
includes non-dynamic contributions from impact parameter fluctuations along
with additional fluctuations in the number of participants for a fixed impact
parameter. Ideally, the second term is nearly nullified after applying the
previously described corrections, so the resulting fluctuations are
independent of centrality as well as collision species. Baseline comparisons
at 200 GeV are facilitated by PHENIX measurements of charged particle
multiplicity fluctuations in minimum bias 200 GeV p+p collisions with mean
$\mu$ = 0.32 $\pm$ 0.003, scaled variance $\omega$ = 1.17 $\pm$ 0.01, and NBD
fit parameter $k_{NBD}$ = 1.88 $\pm$ 0.01.
The scaled variance as a function of the number of participating nucleons,
$N_{part}$, over the $p_{T}$ range $0.2<p_{T}<2.0$ GeV/c is shown in Figure 1
for Au+Au collisions. For all centralities, the scaled variance values
consistently lie above the Poisson distribution value of 1.0. In all collision
systems, the minimum scaled variance occurs in the most central collisions and
then begins to increase as the centrality decreases. A similar centrality-
dependent trend of the scaled variance has also been observed at the SPS in
low energy Pb+Pb collisions at $\sqrt{s_{NN}}$=17.3 GeV, measured by
experiment NA49 [8], where the hard scattering contribution is expected to be
small. All of the data points are consistent with or below the participant
superposition model estimate. This suggests that the data do not show any
indications of the presence of a critical point, where the fluctuations are
expected to be much larger than the participant superposition model
expectation.
The clan model [9] has been developed to interpret the fact that Negative
Binomial Distributions describe charged hadron multiplicity distributions in
elementary and heavy ion collisions. In this model, hadron production is
modeled as independent emission of a number of hadronic clusters, $N_{c}$,
each with a mean number of hadrons, $n_{c}$. The independent emission is
described by a Poisson distribution with an average cluster, or clan,
multiplicity of $\bar{N_{c}}$. After the clusters are emitted, they fragment
into the final state hadrons. The measured value of the mean multiplicity,
$\mu_{\rm ch}$, is related to the cluster multiplicities by $\mu_{\rm
ch}=\bar{N_{c}}\bar{n_{c}}$. In this model, the cluster multiplicity
parameters can be simply related to the NBD parameters of the measured
multiplicity distribution as follows:
$\bar{N_{c}}=k_{\rm NBD}~{}log(1+\mu_{\rm ch}/k_{\rm NBD})$ (1)
and
$\bar{n_{c}}=(\mu_{\rm ch}/k_{\rm NBD})/log(1+\mu_{\rm ch}/k_{\rm NBD}).$ (2)
The results from the NBD fits to the data are plotted in Fig. 2 for all
collision species. Also shown are data from elementary and heavy ion
collisions at various collision energies. The individual data points from all
but the PHENIX data are taken from multiplicity distributions measured over
varying ranges of pseudorapidity, while the PHENIX data are taken as a
function of centrality. The characteristics of all of the heavy ion data sets
are the same. The value of $\bar{n_{c}}$ varies little within the range
1.0-1.1. The heavy ion data universally exhibit only weak clustering
characteristics as interpreted by the clan model. There is also no significant
variation seen with collision energy. However, $\bar{n_{c}}$ is consistently
significantly higher in elementary collisions. In elementary collisions, it is
less probable to produce events with a high multiplicity, which can reveal
rare sources of clusters such as jet production or multiple parton
interactions.
Figure 2: The correlation of the clan model parameters $\bar{n_{c}}$ and
$\bar{N_{c}}$ for all of the collision species measured as a function of
centrality. Also shown are results from pseudorapidity-dependent studies from
elementary collisions (UA5 [3], EMC [10], and NA22 [11]) and heavy ion
collisions (E802 [4] and NA35 [12]).
## 3 $\langle p_{T}\rangle$ Fluctuations
PHENIX has also completed a survey that expands upon previous measurements of
event-by-event transverse momentum fluctuations [13]. Here, the magnitude of
the $p_{T}$ fluctuations will be quoted using the variable $\Sigma_{p_{T}}$,
as described in [15]. $\Sigma_{p_{T}}$ is the mean of the covariance of all
particle pairs in an event, normalized by the inclusive mean $p_{T}$.
$\Sigma_{p_{T}}$ is related to the inverse of the heat capacity of the system
[16], which diverges with a power law behavior near the critical point:
$C_{V}\propto\epsilon^{-\alpha}$.
Figure 3 shows $\Sigma_{p_{T}}$ as a function of $N_{part}$ for all 5
collision systems measured over the $p_{T}$ range $0.2<p_{T}<2.0$ GeV/c. The
data is shown within the effective PHENIX azimuthal acceptance of 4.24
radians. The magnitude of $\Sigma_{p_{T}}$ exhibits little variation for the
different collision energies and does not scale with the jet cross section at
different energies, hence hard processes are not the primary contributor to
the observed fluctuations. Simulations show that elliptic flow contributes
little [13]. With the exception of the most peripheral collisions, all systems
exhibit a universal power law scaling as a function of $N_{part}$. The data
points for all systems are best described by the curve: $\Sigma_{p_{T}}\propto
N_{part}^{-1.02\pm 0.10}$. The observed scaling is independent of the $p_{T}$
range over which the measurement is made.
Figure 3: Event-by-event $p_{T}$ fluctuations for inclusive charged hadrons
within the PHENIX acceptance in the transverse momentum range $0.2<p_{T}<2.0$
GeV/c in terms of $\Sigma_{p_{T}}$ as a function of $N_{part}$.
## 4 K/$\pi$ Fluctuations
PHENIX has studied identified particle fluctuations by measuring the event-by-
event fluctuations of kaons to pions and protons to pions. One advantage of
particle ratio measurements is that contributions from volume fluctuations
cancel. Measurements are quoted in the variable $\nu_{dyn}$:
$\nu_{dyn}(K,\pi)=\frac{\langle\pi(\pi-1)\rangle}{\langle\pi\rangle^{2}}+\frac{\langle
K(K-1)\rangle}{\langle K\rangle^{2}}-2\frac{\langle K\pi\rangle}{\langle
K\rangle\langle\pi\rangle}.$ (3)
If only random fluctuations are present, $\nu_{dyn}$ is zero. Also,
$\nu_{dyn}$ is independent of acceptance.
The measurements for $\nu_{dyn}(K,\pi)$ for $0.34<p_{T}<1.05$ GeV/c are shown
in Figure 4. The measurements for $\nu_{dyn}(K,p)$ are shown in Figure 5. As
with the $p_{T}$ fluctuations, the fluctuations in $\langle
K\rangle/\langle\pi\rangle$ demonstrate a 1/$N_{part}$ dependence. This is not
seen in fluctuations of $\langle p\rangle/\langle\pi\rangle$, which instead
rise as centrality increases.
Figure 4: Event-by-event fluctuations of the kaon-to-pion ratio for inclusive
charged hadrons within the PHENIX acceptance in the transverse momentum range
$0.35<p_{T}<1.05$ GeV/c. The dashed line is a fit to the function
c+$N_{part}^{-1}$, where c is a constant.
Figure 5: Event-by-event fluctuations of the kaon-to-proton ratio for
inclusive charged hadrons within the PHENIX acceptance in the transverse
momentum range $0.35<p_{T}<1.05$ GeV/c.
## 5 Scaling of Elliptic Flow
One of the most striking RHIC results has been the observation of scaling
behavior in elliptic flow measurements below a transverse momentum of about 1
GeV that indicate that quark degrees of freedom are driving the dynamics of
the collision [14]. PHENIX measurements of the scaling behavior of elliptic
flow are compiled for various particle species and various collision systems
in Figure 6. Further measurements of this scaling behavior and the observation
of its breaking as a function of collision energy will be an important
ingredient in the search for a critical point.
Figure 6: PHENIX Preliminary elliptic flow $v_{2}$ normalized by the number of
quarks, collision eccentricity, and $N_{part}^{1/3}$ plotted as a function of
the transverse kinetic energy normalized by the number of quarks. Shown are
$v_{2}$ measurements of pions, kaons, and protons over a centrality range of
0-50% for 200 GeV Au+Au, 200 GeV Cu+Cu, and 62.4 GeV Au+Au.
## 6 Searching for a Critical Point with HBT Correlations
Near the critical point, correlation functions are also expected to be
described by a power law function with critical exponent $\eta$. This exponent
can be measured with Hanbury-Brown Twiss correlations in the $Q_{inv}$
variable [18]. Here, the $Q_{inv}$ correlations are fit with a Lévy function,
$C(Q_{inv})=\lambda exp(-|Rq/hc|^{-\alpha}),$ (4)
where R is the HBT radius and $\alpha$ is the Lévy index of stability. The
value of $\alpha$ is 1 for a Lorentzian source and 2 for a Gaussian source.
Since $\alpha$ equates to the exponent $\eta$, it is expected that its value
will approach the value expected for the universality class of QCD. If QCD
belongs to the 3d Ising model class, the value of $\eta$ would approach 0.5.
Figure 7 shows the results of the Lévy function fit to PHENIX HBT correlations
in 0-5% central 200 GeV Au+Au collisions as a function of transverse mass. The
fit results are inconsistent with the expected value of 0.5 in the vicinity of
a critical point. Analysis of the other PHENIX datasets is currently underway.
Figure 7: The Lévy index of stability $\alpha$ extracted from Lévy function
fits to $Q_{inv}$ correlations in 0-5% central 200 GeV Au+Au collisions as a
function of transverse mass.
## 7 Azimuthal Correlations at Low Transverse Momentum
Critical behavior may also be apparent in the width and shape of correlation
functions. PHENIX has measured azimuthal correlation functions of like-sign
pairs at low $p_{T}$ for several collision systems. The correlations isolate
the HBT peak in pseudorapidity by restricting $|\Delta\eta|<0.1$ for each
particle pair. Correlations are constructed for low $p_{T}$ pairs by
correlating all particle pairs in an event where both particles lie within the
$p_{T}$ range $0.2<p_{T,1}<0.4$ GeV/c and $0.2<p_{T,2}<0.4$ GeV/c. Note that
there is no trigger particle in this analysis. The correlation functions are
constructed using mixed events as follows:
$C(\Delta\phi)=\frac{dN/d\phi_{data}}{dN/d\phi_{mixed}}\frac{N_{events,mixed}}{N_{events,data}}$.
Confirmation of the HBT peak has been made by observing its disappearance in
unlike-sign pair correlations and by observing $Q_{invariant}$ peaks when
selecting this region.
Azimuthal correlation functions can be described by a power law function with
exponent $\eta$: $C(\Delta\phi)\propto\Delta\phi^{-(d-2+\eta)}$, where d is
the dimensionality of the system [2]. For all collision systems, including 200
GeV d+Au, the extracted value of the exponent $\eta$ is shown in Fig. 8. The
value of $\eta$ lies between -0.6 and -0.7 with d=3, independent of
centrality. Since $\eta$ is constant in heavy ion collisions, does not differ
from the d+Au system, and has a value that significantly differs from
expectations from a QCD phase transition (e.g. $\eta$=+0.5 for the 3-D Ising
model universality class [17]), it is unlikely that critical behavior is being
observed in the correlation functions measured thus far.
Near the critical point, it is also expected that the correlation length will
diverge with a power law behavior. The HBT peak of the correlation functions
with the estimated contribution from elliptic flow subtracted have been fit to
a Gaussian distribution. The standard deviation from the fit is shown in
Figure 9 for several collision species. There is no significant change in the
correlation widths between 200 GeV Au+Au and 62.4 GeV Au+Au collisions.
Figure 8: The exponent $\eta$ with d=3 extracted from the like-sign
correlation functions as a function of $N_{part}$.
Figure 9: The standard deviation of a Gaussian fit to the HBT peak in like-
sign correlation functions as a function of $N_{part}$ for several collision
species.
## 8 Conclusions
The fluctuation and correlation measures presented here do not provide a
significant indication of the existence of a critical point or phase
transition. This does not rule out the possibility that the critical point
exists. Further searches will be facilitated by the upcoming RHIC low energy
program.
## References
* [1] M. A. Stephanov, K. Rajagopal and E. V. Shuryak, Phys. Rev. Lett. 81, 4816 (1998).
* [2] H. Stanley, Introduction to Phase Transitions and Critical Phenomena (Oxford, New York and Oxford) 1971.
* [3] G. J. Alner et al. [UA5 Collaboration], Phys. Rept. 154, 247 (1987).
* [4] T. Abbott et al. [E-802 Collaboration], Phys. Rev. C 52, 2663 (1995).
* [5] X. N. Wang and M. Gyulassy, Phys. Rev. D 44, 3501 (1991).
* [6] S. S. Adler et al. [PHENIX Collaboration], Phys. Rev. C 71, 034908 (2005) [Erratum-ibid. C 71, 049901 (2005)].
* [7] H. Heiselberg, Phys. Rept. 351, 161 (2001).
* [8] C. Alt et al. [NA49 Collaboration], Phys. Rev. C 75, 064904 (2007).
* [9] A. Giovannini and L. Van Hove, Z. Phys. C 30, 391 (1986).
* [10] M. Arneodo et al. [European Muon Collaboration], Z. Phys. C 35, 335 (1987) [Erratum-ibid. C 36, 512 (1987)].
* [11] M. Adamus et al. [EHS/NA22 Collaboration], Z. Phys. C 37, 215 (1988).
* [12] J. Bachler et al. [NA35 Collaboration], Z. Phys. C 57, 541 (1993).
* [13] S. S. Adler et al. [PHENIX Collaboration], Phys. Rev. Lett. 93, 092301 (2004).
* [14] A. Adare et al. [PHENIX Collaboration], Phys. Rev. Lett. 98, 162301 (2007).
* [15] D. Adamova et al. [CERES Collaboration], Nucl. Phys. A 727, 97 (2003).
* [16] R. Korus, S. Mrowczynski, M. Rybczynski and Z. Wlodarczyk, Phys. Rev. C 64, 054908 (2001).
* [17] H. Reiger, Phys. Rev. B 52, 6659 (1995)
* [18] T. Csorgo et al. Acta. Phys. Pol. B36, 329 (2005).
|
arxiv-papers
| 2009-09-14T15:59:58 |
2024-09-04T02:49:05.354391
|
{
"license": "Public Domain",
"authors": "Jeffery T. Mitchell (PHENIX Experiment)",
"submitter": "Jeffery T. Mitchell",
"url": "https://arxiv.org/abs/0909.2587"
}
|
0909.2732
|
11institutetext: Fukui Prefectural University, 910-1195 Fukui, JAPAN
Probability theory Kinetic theory Kinetic and transport theory of gases
# Relativistic Equilibrium Distribution by Relative Entropy Maximization
Tadas K. Nakamura
###### Abstract
The equilibrium state of a relativistic gas has been calculated based on the
maximum entropy principle. Though the relativistic equilibrium state was long
believed to be the Jüttner distribution, a number of papers have been
published in recent years proposing alternative equilibrium states. However,
some of these papers do not pay enough attention to the covariance of
distribution functions, resulting confusion in equilibrium states. Starting
from a fully covariant expression to avoid this confusion, it has been shown
in the present paper that the Jüttner distribution is the maximum entropy
state if we assume the Lorentz symmetry.
###### pacs:
02.50.Cw
###### pacs:
05.20.Dd
###### pacs:
51.10.+y
## 1 Introduction
Little after the establishment of the theory of relativity, the equilibrium
particle distribution of a relativistic gas was investigated. The distribution
obtained, which is called Jüttner distribution [1, 2], has been long and
widely believed. However, relatively recent years a number of papers have been
published proposing equilibrium distribution functions other than the Jüttner
distribution ([3, 4, 5, 6, 7] and references therein). Dunkel and coworkers
[7, 8] have examined the discrepancy in the equilibrium distributions as the
maximum entropy state, and showed that the difference comes from the choice of
the reference measure.
The maximum entropy state cannot be uniquely determined when one naively
defines the entropy such as $S=-\int f(\mathbf{x},\mathbf{v})\ln
f(\mathbf{x},\mathbf{v})\,d\mathbf{x}d\mathbf{v}$ (symbols have conventional
meaning in the present paper unless otherwise stated). For instance, the
result would be different if we rewrite distribution function as a function of
momentum $\mathbf{p}$ instead of velocity $\mathbf{v}$. To overcome this
difficulty, it was proposed in Ref [7] to maximize the following relative
entropy
$S=-\int f(\mathbf{x},\mathbf{v})\ln
f(\mathbf{x},\mathbf{v})/\rho(\mathbf{x},\mathbf{v})\,d\mathbf{x}d\mathbf{v}\,,$
(1)
based on a given reference measure $\rho$. In the above expression $f$ is the
phase space distribution of particles and $\rho$ is the reference measure [7].
In this paper we denote a three vector by a bold font (e.g., $\mathbf{x}$) and
a four vector by an upper bar (e.g., $\bar{x}$). Each component of a vector is
represented by a subscript or a superscript (e.g., $x_{\mu}$ or $x^{\mu}$).
The equilibrium distribution is uniquely determined by maximizing the relative
entropy once the reference measure is given. The mathematical procedure in
this approach is essentially the same as the one utilized in Ref [2] to derive
the Jüttner distribution. What is called “a priori probability” in Ref [2]
plays the same role as the reference measure in Ref [7].
Two possibilities for the reference frame were suggested in Ref [7]. One is
the constant distributions a function of momentum, and the Jẗtner distribution
is obtained from this measure. This calculation is essentially the same as the
one in Ref[2]. Another possibility suggested in Ref [7] which is inversely
proportional to the energy. It was argued this measure is derived from the
Lorentz symmetry in Ref [7] and the result is the alternative equilibrium
distribution proposed in recent papers. However, as we will see in the present
paper, there is a confusion on the relativistic phase space density in this
argument. The Lorentz invariant reference measure is the same as the one in
Ref [2], i.e., the constant measure, which gives the Jüttner distribution.
There is a misleading point in defining a phase space density such as a
particle distribution in relativity. When we express a phase space density as
the time evolution of the density in a six (three space + three momentum)
dimensional phase space, it appears to be a Lorentz invariant. Actually, it
can be proved [9] (see also [2, 10]) that
$f(t,\mathbf{x},\mathbf{p})=f(t^{\prime},\mathbf{x^{\prime}},\mathbf{p}^{\prime})$
when the two sets of coordinates $(t,\mathbf{x},\mathbf{p})$ and
$(t^{\prime},\mathbf{x}^{\prime},\mathbf{p}^{\prime})$ are related by the
Lorentz transform, in other words, they are the same point in the spacetime
denoted by different reference coordinates. However, this does not mean
$f(t,\mathbf{x},\mathbf{p})d\mathbf{x}d\mathbf{p}=f(t^{\prime},\mathbf{x^{\prime}},\mathbf{p}^{\prime})d\mathbf{x}^{\prime}d\mathbf{p}^{\prime}$
because $\mathbf{x}$ and $\mathbf{x}^{\prime}$ do not belong to the same
spatial volume. In this sense, phase a space density in the form of
$f(t,\mathbf{x},\mathbf{p})$ is not covariant but frame dependent. It seems
that some of recent papers do not pay enough attention to this fact, resulting
confision in treating Lorentz transfrom.
In the present paper, we examine this confusing point by starting from the
fully covariant distribution function proposed by Hakim [11], and the result
shows the reference measure should be constant to satisfy the full Lorentz
symmetry; the one introduced in Ref [7] is invariant under the Lorentz
transform only in the momentum space. This result means the maximum entropy
state with Lorentz symmetry must be the Jüttner distribution.
## 2 Relativistic Phase Space Density
Let us suppose a relativistic gas as an example. The conservation law of its
particle number is expressed in the form of flux divergence in relativity:
$\frac{\partial}{\partial x_{\mu}}\,J_{\mu}=0\,,$ (2)
where
$J_{\mu}=n_{0}u_{\mu}\,.$ (3)
is the four flux derived from the proper number density $n_{0}$ and the four
velocity of the matter $u_{\mu}$. When we split the spacetime as
$t_{\Sigma}=x_{\Sigma 0}$ and $\mathbf{x_{\Sigma}}=(x_{\Sigma 1,}x_{\Sigma
2},x_{\Sigma 3})$ by choosing a specific reference frame $\Sigma$, the above
conservation is written as
$\frac{\partial}{\partial
t}n_{\Sigma}(t_{\Sigma},\mathbf{x}_{\Sigma})+\nabla\mathbf{J}_{\Sigma}(t_{\Sigma},\mathbf{x}_{\Sigma})=0\,.$
(4)
In the above expression, $n_{\Sigma}=J_{\Sigma 0}$ and
$\mathbf{J}_{\Sigma}=(J_{\Sigma 1},J_{\Sigma 2},J_{\Sigma 3})$ are the number
density and flux in the three dimensional space; the subscript $\Sigma$ is to
explicitly express the frame dependence.
When we decompose the spacetime in another reference frame $\Sigma^{\prime}$,
obviously $n_{\Sigma^{\prime}}$ is different from $n_{\Sigma}$. Moreover,
$n_{\Sigma}$ and $n_{\Sigma^{\prime}}$ cannot be related with a Jacobian
$\partial\mathbf{x}_{\Sigma}/\partial\mathbf{x}_{\Sigma^{\prime}}$ as
$n_{\Sigma}d\mathbf{x}_{\Sigma}=n_{\Sigma^{\prime}}\frac{\partial\mathbf{x}_{\Sigma}}{\partial\mathbf{x}_{\Sigma^{\prime}}}\,d\mathbf{x}_{\Sigma^{\prime}}\,,$
(5)
because $\mathbf{x}_{\Sigma}$ and $\mathbf{x}_{\Sigma}$ belong to different
spacelike volumes. There is no function to relate $\mathbf{x}_{\Sigma}$ and
$\mathbf{x}_{\Sigma^{\prime}}$ as
$\mathbf{x}_{\Sigma}=\mathbf{X}(\mathbf{x}_{\Sigma^{\prime}})$ where
$\mathbf{X}$ that does not depend on the time coordinate.
The above argument on the number density in a three dimensional space is also
valid for phase space densities in a six dimensional space. A phase space
density is often expressed as $f(t,\mathbf{x},\mathbf{p})$ and it should be
denoted in our notation as
$f_{\Sigma}(t_{\Sigma},\mathbf{x}_{\Sigma},\mathbf{p}_{\Sigma})$ because the
expression is based on a specific choice of the reference frame like
$n_{\Sigma}$ in (4). However, it is generally believed that the phase space
density is unchanged under the Lorentz transform. This is true in the sense
that the value of the phase space density is unchanged [2, 9, 10], but
$f_{\Sigma}(t_{\Sigma},\mathbf{x}_{\Sigma},\mathbf{p}_{\Sigma})$ is defined
only on a space volume in a specific reference frame, and not directly
applicable to other reference frames.
To correctly treat the phase space density, we derive the frame-dependent
phase space density
$f_{\Sigma}(t_{\Sigma},\mathbf{x}_{\Sigma},\mathbf{p}_{\Sigma})$ from the
fully covariant expression proposed by Hakim [11]. The relativistic particle
distribution $N(\bar{x},\bar{p})$ is defined such that $\bar{j}$ in the
following expression becomes the particle four-current:
$j_{\mu}(\bar{x})=\int
d_{4}p\,2mu_{\mu}N(\bar{x},\bar{p})\,\theta(p^{0})\delta(p^{\mu}p_{\mu}-m^{2})\,,$
(6)
where $\theta$ and $\delta$ are the theta and delta functions, and $m$ is the
particle rest mass.
In the above expression, $N(\bar{x},\bar{p})$ can be interpreted as the proper
density of the fluid element that has the four velocity $\bar{u}=\bar{p}/m$,
just like $n_{0}$ in (3). Thus its covariant form must be a four vector, which
is expressed as $N(\bar{x},\bar{p})\bar{u}$, like $\bar{J}$ in (3). The delta
function is due to the energy shell and the theta function is to discard the
negative energy solution. Hakim [11] has introduced the above expression for
the distribution of particle number, however, it is generally valid for a
conserved density flowing with the four velocity $\bar{u}$, therefore, it can
be applied to a probability distribution or a reference measure to calculate
entropy in the following.
When we pick up one reference frame $\Sigma$ and denote its unit vectors in
each coordinate direction as $(\bar{e}_{\Sigma t},\bar{e}_{\Sigma
x},\bar{e}_{\Sigma y},\bar{e}_{\Sigma z})$, an arbitrary point in the eight
dimensional phase space $(\bar{x},\bar{p})$ can be represented in this
reference frame as
$t_{\Sigma}=e_{\Sigma t}^{\mu}x_{\mu},~{}~{}\mathbf{x}_{\Sigma}=(e_{\Sigma
x}^{\mu}x_{\mu},e_{\Sigma y}^{\mu}x_{\mu},e_{\Sigma z}^{\mu}x_{\mu})\,,$ (7)
and
$E_{\Sigma}=e_{\Sigma t}^{\mu}p_{\mu},~{}~{}\mathbf{p}_{\Sigma}=(e_{\Sigma
x}^{\mu}p_{\mu},e_{\Sigma y}^{\mu}p_{\mu},e_{\Sigma z}^{\mu}p_{\mu})\,.$ (8)
A frame-dependent phase space density
$f_{\Sigma}(t_{\Sigma},\mathbf{x}_{\Sigma},\mathbf{p}_{\Sigma})$ is then
calculated from $N(\bar{x},\bar{p})$ as
$\displaystyle f_{\Sigma}(t_{\Sigma},\mathbf{x}_{\Sigma},\mathbf{p}_{\Sigma})$
$\displaystyle=$ $\displaystyle 2m\int e_{\Sigma
t}^{\mu}u_{\mu}N(\bar{X},\bar{P})\,\theta(p^{0})\delta(E_{\Sigma}^{2}-\mathbf{p}_{\Sigma}^{2}-m^{2})\,dE_{\Sigma}$
$\displaystyle=$ $\displaystyle\frac{me_{\Sigma
t}^{\mu}u_{\mu}}{E_{\Sigma}}\,N(\bar{X},\bar{P})=N(\bar{X},\bar{P})\,,$ (9)
where $\bar{u}=\bar{p}/m$, and $\bar{X}$ and $\bar{P}$ are the covariant
expression of the four dimensional position and momentum correspond to
$(t,\mathbf{x}_{\Sigma},\mathbf{p}_{\Sigma})$,
$\bar{X}=t_{\Sigma}\bar{e}_{\Sigma t}+x_{\Sigma}\bar{e}_{\Sigma
x}+y_{\Sigma}\bar{e}_{\Sigma y}+z_{\Sigma}\bar{e}_{\Sigma z}\,,$ (10)
and
$\bar{P}=\sqrt{\mathbf{p}^{2}+m^{2}}\,\bar{e}_{\Sigma t}+p_{\Sigma
x}\bar{e}_{\Sigma x}+p_{\Sigma y}\bar{e}_{\Sigma y}+p_{\Sigma
z}\bar{e}_{\Sigma z}\,.$ (11)
From (9) van Kampen [9] concluded that $f$ is unchanged under the Lorentz
transform (his derivation is different from ours, but the result is the same).
He considered the above result is purely kinematical. It is true in the sense
that no equation of motion is required for (9), however, it implicitly
includes kinetics in the expression of the energy shell. For example, if the
relativistic kinetics were such that the energy shell is expressed as
$4m^{3}\delta(E_{\Sigma}^{4}-\mathbf{p}_{\Sigma}^{4}-m^{4})$, (9) would be
$f_{\Sigma}(t_{\Sigma},\mathbf{x}_{\Sigma}\mathbf{p}_{\Sigma})=\frac{m^{3}e_{\Sigma
t}^{\mu}u_{\mu}}{E_{\Sigma}^{3}}\,N(\bar{X},\bar{P})=\frac{m^{2}}{E_{\Sigma}^{2}}\,N(\bar{X},\bar{P})\,,$
(12)
which means the value of $f$ changes under the Lorentz transform. This example
demonstrates the fact that $f_{\Sigma}$ is not identical to $N$, but should be
derived from $N$.
## 3 Lorentz Invariant Reference Frame
In (9) we assumed the spatial coordinates $(t_{\Sigma},\mathbf{x}_{\Sigma})$
and the momentum coordinates $(E_{\Sigma},\mathbf{p}_{\Sigma})$ are defined in
the same reference frame $\Sigma$. Mathematically the reference frames to
define spatial and momentum coordinates do not have to be the same; we may
have a phase space density whose spatial coordinates are defined in $\Sigma$
and momentum coordinates are in $\Sigma^{\prime}$ as in the following form:
$\displaystyle
f_{\Sigma\Sigma^{\prime}}(t_{\Sigma},\mathbf{x}_{\Sigma},\mathbf{p}_{\Sigma^{\prime}})$
$\displaystyle=$ $\displaystyle 2m\int e_{\Sigma
t}^{\mu}u_{\mu}N(\bar{X},\bar{P}^{\prime})\,\theta(p^{0})\delta(E_{\Sigma^{\prime}}^{2}-\mathbf{p}_{\Sigma^{\prime}}^{2}-m^{2})\,dE_{\Sigma^{\prime}}$
$\displaystyle=$ $\displaystyle\frac{me_{\Sigma
t}^{\mu}u_{\mu}}{E_{\Sigma^{\prime}}}\,N(\bar{X},\bar{P}^{\prime})\,,$ (13)
with
$\bar{P}^{\prime}=E_{\Sigma^{\prime}}\bar{e}_{\Sigma^{\prime}t}+p_{\Sigma^{\prime}x}\bar{e}_{\Sigma^{\prime}x}+p_{\Sigma^{\prime}y}\bar{e}_{\Sigma^{\prime}y}+p_{\Sigma^{\prime}z}\bar{e}_{\Sigma^{\prime}z}\,.$
(14)
From the above expression we understand that the factor of $e_{\Sigma
t}^{\mu}u_{\mu}$ comes from the spatial Lorentz transform whereas the factor
of $1/E_{\Sigma^{\prime}}$ is due to the transform in the momentum space. They
are canceled out when $\Sigma=\Sigma^{\prime}$ and $f_{\Sigma\Sigma}$ becomes
unchanged as seen in the previous section. This fact also indicates the phase
space density is not a covariant expression; if it were covariant,
$f_{\Sigma\Sigma^{\prime}}$ should be unchanged even when
$\Sigma\neq\Sigma^{\prime}$.
Since $f_{\Sigma\Sigma}$ and $f_{\Sigma\Sigma^{\prime}}$ are the densities
defined on a same spatial volume in $\Sigma$, we can relate them by
$\frac{1}{E_{\Sigma}}f_{\Sigma\Sigma}\,d\mathbf{p}_{\Sigma^{\prime}}d\mathbf{x}_{\Sigma}=\frac{1}{E_{\Sigma^{\prime}}}f_{\Sigma\Sigma^{\prime}}\,d\mathbf{p}_{\Sigma^{\prime}}d\mathbf{x}_{\Sigma}\,.$
(15)
When we apply the above result to the reference measure $\rho$ to calculate
the relative entropy, it has the same meaning as Equation (34) in Ref [7]. If
the measure $\rho$ is to be invariant under the transform of
$\rho_{\Sigma\Sigma}\rightarrow\rho_{\Sigma\Sigma^{\prime}}$, it must be
$\rho(p_{\Sigma})\propto\frac{1}{E_{\Sigma}}\,,$ (16)
which is suggested in Ref [7]. However, as seen from (15), the Lorentz
transform in this context is in the momentum space only and the spatial volume
to define the measure $\rho$ is unchanged.
The present paper proposes that the measure should have the Lorentz symmetry
under the transform both in space and momentum coordinates:
$\rho_{\Sigma\Sigma}\rightarrow\rho_{\Sigma^{\prime}\Sigma^{\prime}}$. Then we
have to choose the phase space density defined by (9) instead of (13) for the
reference measure. As discussed above, two phase space densities with
different reference frames $\Sigma$ and $\Sigma^{\prime}$ is not directly
connected with a equation such as (15). The Lorentz symmetry in this case
means the mathematical expression is unchanged under the transform, and this
is satisfied when $N(\bar{x},\bar{p})$ is constant. Therefore we obtain
$\rho(p_{\Sigma})=\textrm{constant}\,,$ (17)
in the reference frame $\Sigma$ instead of (16). Following the relative
entropy maximization procedure proposed in Ref [7] we obtain the Jüttner
distribution as
$\phi(p_{\Sigma})\propto\exp(-\beta E_{\Sigma})\,.$ (18)
by maximizing the relative entropy in (1).
## 4 Concluding Remarks
It has been shown in the present paper the maximum entropy state based on the
Lorentz symmetry is the Jüttner distribution. Recent years a number of papers
have been published claiming the relativistic equilibrium state is different
from the long believed Jüttner distribution. Dunkel and coworkers [7, 8] have
shed a light to this controversy by pointing the importance of the reference
measure in the maximum entropy approach.. They have shown that the difference
of the reference measure causes the difference of the equilibrium distribution
as the maximum entropy state.
Two typical reference measures were suggested in Ref [7]. One is constant as a
function of $p$ and the other is inversely proportional to the energy. In Ref
[7] it is conjectured the former is derived from the invariance of momentum
transition, and the latter comes from the Lorentz symmetry. However, as we
have seen in the present paper, the reference measure with Lorentz symmetry is
also found to be constant when we correctly formulate the covariance of
relativistic phase space density.
The constant reference measure we derived in this paper corresponds to the
constant “prior probability” employed by Synge [2]. The information theory was
developed long after the days of Synge, therefore, he did not know the modern
concepts such as relative entropy or reference measure. Nevertheless, his
calculation is quite similar to ours, and the result is the same Jüttner
distribution. (The author guesses his basic idea historically comes from the
probabilistic interpretation of the entropy by Boltzmann in his late years
[12].)
Therefore, the argument in the present paper might seem just another
interpretation of Synge’s result with information theory if one believes his
derivation. However, considerable number of papers have been published
recently against the Jüttner distribution and it is important to clarify the
foundation of the maximum entropy process based on information theory.
Moreover, it has become clear in the present paper what causes the confusion
of the reference measure (the difference of $\rho_{\Sigma^{\prime}\Sigma}$ and
$\rho_{\Sigma^{\prime}\Sigma^{\prime}}$).
The result in the present paper strongly suggests that the relativistic
equilibrium state is the Jüttner distribution. There are papers in favor of
the Jüttner distribution in the recent controversy. Debbasch [13] critically
reviewed the theories proposing alternatives to the Jüttner distribution. He
examined the relative entropy approach in Ref [7] and showed the result would
be inconsistent unless the reference measure is constant.
Also there is a result of numerical experiment that supports the Jüttner
distribution [14]. It was aregued in Ref [8] that the distribution measured in
Ref [14] is based on what they call “coordinate-time”, and and the modified
distribution would be obtained if it is defined with “proper-time”. In this
sense, what we examined in the present paper is the one with “coordinate-
time”, in agreement with the numerical experiment.
It has been known, but has not been well recognized, that the a conserved
quantity (energy-momentum, particle number etc.) distributed over a finite
volume is not a Lorentz invariant quantity because it belongs to a different
time slice of the volume’s world tube. Confusions on this point have caused
controversy on the relativistic thermodynamics ([15, 16] and references
therein).
To treat this point correctly any spatial density must be expressed by a flux
four vector as a covariant form. The density in the phase space is no
exception. However, the phase space density in the form of
$f(t,\mathbf{x},\mathbf{p})$ is often regarded as a covariant expression since
the value $f$ is unchanged under the Lorentz transform.
As discussed in Section 2, the expression of $f(t,\mathbf{x},\mathbf{p})$ is
frame dependent since it is defined on a three dimensional space volume in a
specific reference frame. It seems some of recent papers do not pay enough
attention to this point, and treat the phase space density in a confusing way.
In the present paper we start with the fully covariant expression of the phase
space density [11] to avoid this confusion. We have seen that the reference
measure with Lorentz symmetry is constant as a function of momentum.
Consequently the maximum entropy state with the Lorentz symmetry is the
Jüttner distribution.
It is known that the maximum entropy approach used in Ref [7] has the
mathematical structure almost parallel to the traditional ensemble approach
[17]. Therefore, the equilibrium distributions derived from ensemble approach
can be examined with the same basis. This means the result in the present
paper can be applicable to theories with the traditional approach.
## References
* [1] Jüttner F. Ann. Phys. (Leipzig) 341911856.
* [2] Synge J. L. The Relativistc Gas (Amsterdam: North-Holland) 1957.
* [3] Horwitz L. P., Schieve W. C. Piron C. Ann. Phys. 1371981306\.
* [4] Horwitz L. P., Shushoua S. Schieve W. C. Physica 1611989300\.
* [5] Lehmann E. J. Math Phys. 472006023303.
* [6] Schieve W. C. Found. Phys. 3520051359.
* [7] Dunkel J., Talkner P. Hänggi P. New J. Phys. 92007144\.
* [8] Cubero D. Dunkel J. arXiv:0902.4785 preprint, 2009.
* [9] van Kampen N. G. Physica 431969244.
* [10] Debbasch F., River J. P. van Leeuwen W. A. Physica 3012001181\.
* [11] Hakim R. J. Math. Phys. 819671315.
* [12] Boltzman L. Wiener Brichte 161878373.
* [13] Debbasch F. Physica 38720082443.
* [14] Cubero D., Casado-Pascual J., Dunkel J., Talkner P. Hänggi P. Phys. Rev. Lett. 992007170601.
* [15] Yuen C. K. Amer. J. Phys. 381970246.
* [16] Nakamura T. K. Phys. Lett. A 3522006175.
* [17] Rozenkrantz R. D. (Editor) Papers on probability statistics and statistical physics (Kluwer: Dordrecht) 1983.
|
arxiv-papers
| 2009-09-15T07:52:26 |
2024-09-04T02:49:05.360876
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Tadas K Nakamura",
"submitter": "Tadas Nakamura",
"url": "https://arxiv.org/abs/0909.2732"
}
|
0909.2893
|
# New Classes of Counterexamples to Hendrickson’s Global Rigidity Conjecture
Samuel Frank Department of Mathematics, Columbia University, New York, NY
10027 smf2147@columbia.edu and Jiayang Jiang Department of Mathematics,
Columbia University, New York, NY 10027 jj2333@columbia.edu
###### Abstract.
We examine the generic local and global rigidity of various graphs in
$\mathbb{R}^{d}$. Bruce Hendrickson showed that some necessary conditions for
generic global rigidity are ($d+1$)-connectedness and generic redundant
rigidity and hypothesized that they were sufficient in all dimensions. We
analyze two classes of graphs that satisfy Hendrickson’s conditions for
generic global rigidity, yet fail to be generically globally rigid. We find a
large family of bipartite graphs for $d>3$, and we define a construction that
generates infinitely many graphs in $\mathbb{R}^{5}$. Finally, we state some
conjectures for further exploration.
Special thanks to Dylan Thurston, Joe Ross, and Ina Petkova for their
immeasurable help. This work was partially supported by NSF RTG Grant
07-39392.
## 1\. Introduction and Preliminaries
A _framework_ consists of a graph whose vertices have been assigned
coordinates in $\mathbb{R}^{d}$. An important question is whether or not a
given framework is _locally rigid_ , that is, whether there is a way to
continuously deform the framework while maintaing its edge lengths. A related
question is whether or not the framework is _globally rigid_ , or whether any
other framework with the same underlying graph and the same edge lengths is
equivalent up to Euclidean motions (combinations of reflections, rotations,
and translations). For $d\leq 3$, this question has many important real-world
applications, such as analyzing the structural integrity of buildings or
determining molecular structure. However, the problem is not fully understood,
and only recently has it been explored in great detail. Some complete
bipartite graphs have the characteristic that most of their frameworks are not
globally rigid, but in a non-obvious way; these graphs have been well
characterized by Connelly [4], as well as Bolker and Roth [2]. In this paper,
we present more graphs with this characteristic.
A graph is defined by $G=(V,E)$ with $|V|=v$ and $|E|=e$, where $V$ is a set
of vertices and $E$ is composed of some $2$-element subsets of $V$ which
represent edges. A _realization_ is some
$p=(p_{1},p_{2},\ldots,p_{v})\in\mathbb{R}^{vd}$, where each $p_{i}$ is the
location of $v_{i}\in V$ in $\mathbb{R}^{d}$. This defines the framework
$G(p)$. For some framework $G(p)$, the half edge-length squared function is
$f_{G}(p):\mathbb{R}^{vd}\to\mathbb{R}^{e}$, where
$f_{G}(p)=\frac{1}{2}(\ldots,|p_{i}-p_{j}|^{2},\ldots)$ for $\\{i,j\\}\in E$.
Define a continuous _flexing_ of $G(p)$ as a differentiable one-parameter
family of realizations including $p$ such that for any $q$ in the family,
$f_{G}(q)=f_{G}(p)$. A framework is _locally rigid_ if all its flexings are
trivial (the Euclidean motions). A framework is _locally flexible_ if there
exists a non-trivial flexing.
The problem of determining the local rigidity of a framework is very
difficult. To simplify the problem, we will restrict our focus to _generic_
realizations, defined as realizations whose coordinates are algebraically
independent over the rationals. For generic realizations, this problem becomes
much easier and makes use of $df_{G}(p)$, which we will refer to as the
_rigidity matrix_. In this $e\times vd$ matrix, each row represents an edge,
and each column represents a coordinate of some vertex. In the row
representing the edge connecting $v_{i}$ and $v_{j}$, any given column will be
$0$ if it does not represent $v_{i}$ or $v_{j}$. If it represents the $k^{th}$
coordinate of $p_{i}$, the entry is the $k^{th}$ coordinate of $p_{j}$ minus
the $k^{th}$ coordinate of $p_{i}$.
Due to early results by Asimow and Roth [1], we know that local rigidity is a
generic property of the underlying graph, meaning that if it holds for one
generic framework, it holds for all generic frameworks. Thus, one can think of
generic local rigidity as an inherent property of the graph. The rank of the
rigidity matrix is closely related to the local rigidity of a generic
framework. For graphs with at least $d+1$ vertices, we say a framework $G(p)$
is _infinitesimally rigid_ if
$\operatorname{rank}df_{G}(p)=vd-\binom{d+1}{2}$. We say it is
_infinitesimally flexible_ if
$\operatorname{rank}df_{G}(p)<vd-\binom{d+1}{2}$.
###### Theorem 1 (Asimow and Roth [1]).
A graph $G$ with at least $d+1$ vertices is generically locally rigid in
$\mathbb{R}^{d}$ if and only if a generic realization is infinitesimally
rigid.
Note that the rank of the rigidity matrix cannot be greater than
$vd-\binom{d+1}{2}$, because the Euclidean motions are always in the kernel of
the rigidity matrix and there is a $\binom{d+1}{2}$-dimensional space of them.
This provides an algorithm to check if a graph is _generically locally rigid_
(GLR) [8, 7]: given a graph, randomize its coordinates, generate the rigidity
matrix modulo a large prime, and calculate its rank. With no false positives
and very few false negatives, this will decide the generic local rigidity of
the graph.
Next, for some graph $G$, let $K_{v}$ be the complete graph on the same set of
vertices, that is, the graph such that $E$ consists of all $2$-element subsets
of $V$. We will define a framework $G(p)$ as _globally rigid_ when
$f_{G}(p)=f_{G}(q)$ implies that $f_{K_{v}}(p)=f_{K_{v}}(q)$. This means that
a framework is globally rigid when, for any other framework with the same edge
lengths, all other pairwise distances are the same. Clearly, all globally
rigid frameworks are also locally rigid; however, not all locally rigid
frameworks are globally rigid. As an example, consider a generic realization
$p$ of a quadrilateral in $\mathbb{R}^{2}$ with an edge along one diagonal
[Figure 1]. This framework is locally rigid, since there is no non-trivial
continuous flexing. However, it is possible to reflect one part of the
framework over the diagonal to produce another realization $q$ such that
$f_{G}(p)=f_{G}(q)$ but $f_{K_{v}}(p)\neq f_{K_{v}}(q)$.
(a)
(b)
Figure 1. This framework is locally rigid in $\mathbb{R}^{2}$ but not globally
rigid.
One can test for global rigidity using stresses. A stress is some vector
$\omega=(\ldots,~{}\omega_{ij},\ldots)\in\mathbb{R}^{e}$ for all $\\{i,j\\}\in
E$. An _equilibrium stress_ is a stress such that, for all vertices $v_{i}\in
V$,
$\sum_{j|\\{i,j\\}\in E}\omega_{ij}\cdot(p_{j}-p_{i})=0.$
From now on, by _stress_ we mean equilibrium stress, unless otherwise
specified.
###### Proposition 2.
The space of stresses of a framework $G(p)$ is precisely
$\ker(df_{G}(p)^{T})$.
###### Proof.
Write the stress condition for each vertex $v_{i}$, and arrange them into a
matrix such that every vector in the kernel is a stress. It is not difficult
to show that this matrix is exactly $df_{G}(p)^{T}$. ∎
A _stress matrix_ $\Omega$ is a $v\times v$ matrix satisfying the following
conditions:
$\Omega_{ij}=\begin{cases}0&\text{if }\\{i,j\\}\not\in E\text{ and }i\neq j\\\
\omega_{ij}&\text{if }\\{i,j\\}\in E\\\ -\sum_{j^{\prime}\neq
i}\Omega_{ij^{\prime}}&\text{if }i=j\end{cases}$
Each of the coordinate projections is in the kernel of $\Omega$, as is the
vector $(1,\ldots,1)$. This means the dimension of the kernel is at least
$d+1$. Also as a consequence, across each row, the vectors
$p_{1},p_{2},\ldots,p_{v}$ fulfill an affine linear relation with the row’s
entries acting as coefficients.
###### Theorem 3 (Connelly [3], Gortler-Healy-Thurston[7]).
A graph with at least $d+2$ vertices is generically globally rigid if and only
if, for some generic realization, there is a stress matrix with nullity $d+1$.
Connelly showed that this condition is sufficient; Gortler, Healy and Thurston
showed that it is necessary as well, therefore implying that global rigidity
is a generic property of the graph. Furthermore, Gortler, Healy, and Thurston
proposed a randomized algorithm to efficiently check if a graph is
_generically globally rigid_ (GGR). Given a graph, randomize its coordinates
and create its rigidity matrix modulo a large prime. Due to Proposition 2, we
can find a random stress by selecting random vectors in $\ker(df_{G}(p)^{T})$.
Turn this stress into a stress matrix and check its rank; with no false
positives and very few false negatives, this returns whether or not the graph
is GGR. We used this algorithm, as well as the algorithm described before, to
experimentally check whether or not graphs were generically locally and
globally rigid.
However, this is not an intuitive way of determining generic global rigidity,
and a simpler process has eluded many mathematicians. Some necessary
conditions for generic global rigidity have been established by Hendrickson.
We define an edge of a framework as _redundant_ if one can remove it and be
left with a locally rigid framework. A framework is _redundantly rigid_ if all
of its edges are redundant. We say that a graph is _generically redundantly
rigid_ (GRR) if any of its generic frameworks are redundantly rigid.
###### Theorem 4 (Hendrickson [8]).
If a graph in $\mathbb{R}^{d}$ has at least $d+2$ vertices and is generically
globally rigid, then it is both generically redundantly rigid and vertex
$(d+1)$-connected.
From now on, when we use the term $k$-connected, we mean vertex $k$-connected.
In $\mathbb{R}^{1}$, the two conditions of Theorem 4 are equivalent to
$2$-connectedness, and they are also sufficient for generic global rigidity.
In $\mathbb{R}^{2}$, due to results from Connelly [3], Jackson and Jordán [9],
we know that the conditions are sufficient as well. Hendrickson conjectured
that they are sufficient in all dimensions. However, Connelly [4] found the
counterexample of $K_{5,5}$ in $\mathbb{R}^{3}$. He also generalized this into
a class of complete bipartite graphs.
###### Theorem 5 (Connelly [4]).
Any complete bipartite graph $K_{a,b}$ in $\mathbb{R}^{d}$ such that
$a+b=\binom{d+2}{2}$ and $a,b\geq d+2$ is $(d+1)$-connected and generically
redundantly rigid, but not generically globally rigid.
We denote all graphs that violate Hendrickson’s sufficiency conjecture and are
not GGR as _generically partially rigid_ (GPR). In addition to these complete
bipartite graphs, the process of coning can also create GPR graphs. _Coning_ a
graph $G$ is the process of adding a vertex to $G$ and connecting it to every
other vertex in $G$.
###### Theorem 6 (Connelly and Whiteley [6]).
For any graph $G$, coning preserves the generic local, redundant, and global
rigidity of $G$ from $\mathbb{R}^{d}$ to $\mathbb{R}^{d+1}$. It also transfers
$(d+1)$-connectedness to $(d+2)$-connectedness.
###### Corollary 7.
Coning a generically partially rigid graph in $\mathbb{R}^{d}$ creates a
generically partially rigid graph in $\mathbb{R}^{d+1}$.
However, so far the only documented graphs that are GPR are complete bipartite
graphs and their conings. In a very recent paper [5, 8.3], Connelly posed some
questions about the nature of GPR graphs in higher dimensions. We will present
two new classes of GPR graphs and answer two of Connelly’s questions. We find
two GPR graphs in $\mathbb{R}^{4}$ and infinitely many in $\mathbb{R}^{d}$ for
each $d\geq 5$.
In section $2$, we will present one of our main results for a class of graphs
called $k$-chains and present some simple proofs, including proving when these
graphs are GGR. In section $3$, we determine under what conditions these
graphs are GLR. In section $4$, we do the same for GRR and prove the main
result from section $2$. In section $5$, we introduce a new graph construction
and prove that it generates infinitely many GPR graphs. Finally, we present
some conjectures for further exploration in section $6$.
## 2\. Main Result for $k$-Chains
For positive integers $a_{1},a_{2},\ldots,a_{k}$, the _$k$ -chain_
$C_{a_{1},a_{2},\ldots,a_{k}}$ is the graph constructed as follows. The vertex
set $V$ is the union of $k$ disjoint sets of vertices $A_{1}$,
$A_{2}$,$\ldots$, $A_{k}$ such that $|A_{i}|=a_{i}$. For $1\leq i\leq k-1$,
there are edges between every vertex in $A_{i}$ and $A_{i+1}$, and the graph
has no other edges. Note that a 2-chain is simply a complete bipartite graph
and that the 3-chain $C_{a_{1},a_{2},a_{3}}$ is the complete bipartite graph
$K_{a_{1}+a_{3},a_{2}}$. In particular, Connelly’s GPR bipartite graphs can be
characterized as $3$-chains. We are interested in characterizing when a
$k$-chain is GPR.
###### Theorem 8.
A $k$-chain $C_{a_{1},a_{2},\cdots,a_{k}}$ with $k\geq 4$ and $\binom{d+2}{2}$
vertices is generically partially rigid if and only if it satisfies all of the
following conditions:
1. (1)
$a_{2},a_{3},\ldots,a_{k-1}\geq d+1$;
2. (2)
$a_{2},a_{k-1}\geq d+2$; and
3. (3)
there is no $i$ such that $a_{i}=a_{i+1}=d+1$.
The proof will occupy much of the rest of the paper. For $3$-chains with
$v=\binom{d+2}{2}$, one must add the additional condition that
$a_{1}+a_{3}\geq d+2$. Note that this condition holds for any $k$-chain with
$k\geq 4$ that fulfills the conditions of Theorem 8.
There are no $k$-chains satisfying the conditions of Theorem 8 in
$\mathbb{R}^{3}$. For $\mathbb{R}^{4}$, $v=\binom{6}{2}=15$, so the only GPR
examples in $\mathbb{R}^{4}$ are $C_{1,6,6,2}$ and $C_{1,6,7,1}$ [Figure 2].
(a)
(b)
Figure 2. These are the only GPR $k$-chains in $\mathbb{R}^{4}$ with
$\binom{d+2}{2}$ vertices.
###### Proposition 9.
A $k$-chain is $(d+1)$-connected if and only if it fulfills condition 1 of
Theorem 8.
###### Proof.
If $a_{2},\ldots,a_{k-1}\geq d+1$, then removing any $d$ vertices leaves at
least one vertex in each independent set, so the graph remains connected. If
not, then for some $i$, $2\leq i\leq k-1$, $a_{i}\leq d$, so one can remove
$A_{i}$, disconnecting the graph. ∎
###### Proposition 10.
Any $(d+1)$-connected $k$-chain with $k\geq 4$ and $\binom{d+2}{2}$ vertices
is not generically globally rigid in $\mathbb{R}^{d}$.
###### Proof.
This $k$-chain is the subgraph of some complete bipartite graph. Both
independent sets of this complete bipartite graph have more than $d+2$
vertices. Since the complete bipartite graph has $\binom{d+2}{2}$ vertices, by
Theorem 5 it is GPR, and thus it is not GGR. So, the $k$-chain is the subgraph
of a graph which is not GGR, and so is not GGR itself. ∎
It remains to be determined when these graphs are GLR and when they are GRR.
## 3\. Proof of Generic Local Rigidity
In this section we show that $k$-chains that are $(d+1)$-connected are GLR. We
assume $(d+1)$-connectedness and $k\geq 4$ throughout this section.
First note that $C_{a_{1},a_{2},\ldots,a_{k}}$ is a subgraph of the complete
bipartite graph $K_{a_{1}+a_{3}+\cdots,a_{2}+a_{4}+\cdots}$, which has
$\binom{d+2}{2}$ vertices with at least $d+2$ in each independent set. Due to
Bolker and Roth [2], we can calculate the dimension of the space of stresses
for a generic framework of this complete bipartite graph.
Let $A$ and $B$ be the independent sets of some complete bipartite graph. Let
$\Omega(A,B)$ be the space of stresses of a generic framework of the graph.
Additionally, for some set of vectors $X=\\{x_{1},x_{2},\ldots,x_{k}\\}$, let
$D(X)$ be the space of affine linear dependencies of $X$. Finally, for a
vector $v=(v_{1},\ldots,v_{n})$, let $\overline{v}$ be
$(v_{1},\ldots,v_{n},1)$. Then let $D^{2}(X)$ be the set of linear
dependencies of
$\\{\overline{x_{1}}\otimes\overline{x_{1}},\overline{x_{2}}\otimes\overline{x_{2}},\ldots,\overline{x_{k}}\otimes\overline{x_{k}}\\}$,
where $\otimes$ denotes the tensor product of two vectors.
###### Theorem 11 (Bolker and Roth [2]).
Given some complete bipartite graph $K_{A,B}$ such that $|A|,|B|\geq d+1$, let
$C=A\cup B$. Then for any generic realization, $\dim\Omega(A,B)=\dim
D(A)\cdot\dim D(B)+\dim D^{2}(C)$.
This is actually a specific instance of Bolker and Roth’s results. Bolker and
Roth provided a more general but more complicated formula for all frameworks,
but we are only interested in generic frameworks.
###### Remark 12.
For a generic set of points $X$,
$\displaystyle\dim D(X)$ $\displaystyle=\begin{cases}0&\text{if }|X|\leq
d+1\\\ |X|-d-1&\text{if }|X|>d+1\\\ \end{cases}$ $\displaystyle\dim D^{2}(X)$
$\displaystyle=\begin{cases}0&\text{if }|X|\leq\binom{d+2}{2}\\\
|X|-\binom{d+2}{2}&\text{if }|X|>\binom{d+2}{2}\\\ \end{cases}$
###### Corollary 13.
Suppose $v\leq\binom{d+2}{2}$. For any generic realization of $K_{A,B}$ with
$|A|,|B|\geq d+1$, $\dim\Omega(A,B)=(|A|-d-1)\cdot(|B|-d-1)$. If $|A|<d+1$ or
$|B|<d+1$, then $\dim\Omega(A,B)=0$.
From the corollary, it is possible to compute the dimension of the space of
stresses for the bipartite graph $K_{a_{1}+a_{3}+\cdots,a_{2}+a_{4}+\cdots}$.
If we let $k=a_{1}+a_{3}+\cdots$ and $l=a_{2}+a_{4}+\cdots$, then the
dimension is $(k-d-1)(l-d-1)$, and thus the rank of the rigidity matrix is
$kl-(k-d-1)(l-d-1)=(k+l)(d+1)-(d+1)^{2}=(k+l)d-\binom{d+1}{2}$
since $k+l=\binom{d+2}{2}$. Thus this complete bipartite graph is GLR, as also
indicated by Theorem 5.
When we remove some edges from a GLR graph, it is possible to determine
whether the new graph is GLR by examining the space of stresses.
###### Proposition 14.
Let G(p) be a generic, locally rigid framework, and let $e_{1},\ldots,e_{n}$
be some edges of G. Then $G\setminus\\{e_{1},\ldots,e_{n}\\}$ is generically
locally rigid if and only if, for any $a_{1},\ldots,a_{n}\in\mathbb{R}$, there
exists a stress on G(p) with values $a_{1},\ldots,a_{n}$ on
$e_{1},\ldots,e_{n}$.
###### Proof.
We use induction on $n$.
[Base Case $\Rightarrow$] For $n=1$, first suppose $G\setminus\\{e_{1}\\}$ is
GLR. Since $G(p)$ is locally rigid,
$\operatorname{rank}df_{G}(p)=\operatorname{rank}df_{G\setminus\\{e_{1}\\}}(p)$.
Adding $e_{1}$ to $G\setminus\\{e_{1}\\}$ does not increase the rank of the
rigidity matrix, so it increases the dimension of $\ker(df_{G}(p)^{T})$ by
$1$. This means adding $e_{1}$ adds a new dimension of stresses, which is only
possible if there is some stress with a non-zero value on $e_{1}$. By scaling
this stress, we can achieve any prescribed value on $e_{1}$.
[Base Case $\Leftarrow$] Assume there is some stress with a non-zero value on
$e_{1}$. Removing one edge can decrease the dimension of $\ker(df_{G}(p)^{T})$
by at most $1$. Moreover, there is a stress with a non-zero value on $e_{1}$,
and since this stress cannot exist without $e_{1}$, removing $e_{1}$ must
decrease $\dim\ker(df_{G}(p)^{T})$ by exactly $1$. But, the number of rows of
$df_{G}(p)$ also decreases by $1$, so $\operatorname{rank}df_{G}(p)$ stays the
same. Therefore,
$\operatorname{rank}df_{G}(p)=\operatorname{rank}df_{G\setminus\\{e_{1}\\}}(p)$,
and so $G\setminus\\{e_{1}\\}$ is GLR.
[Inductive Step $\Rightarrow$] Assume that for some $n$, when
$G\setminus\\{e_{1},\ldots,e_{n}\\}$ is GLR, there is a stress on $G(p)$ with
any $a_{1},\ldots,a_{n}$ on $e_{1},\ldots,e_{n}$. Then assume that
$G\setminus\\{e_{1},\ldots,e_{n+1}\\}$ is GLR. Let
$H=G\setminus\\{e_{1},\ldots,e_{n}\\}$. First, note that $H$ is also GLR, so
we can create a stress on a generic framework $G(p)$ with values
$a_{1},\ldots,a_{n}$ on $e_{1},\ldots,e_{n}$. Call the stress we create
$\omega$. Because $H\setminus\\{e_{n+1}\\}$ is GLR, we can create some stress
of $H(p)$ with any value we like on $e_{n+1}$ by Base Case $\Rightarrow$, and
we can artificially extend it to a stress of $G(p)$ with values of $0$ on
$e_{1},\ldots,e_{n}$. We give this stress the value on $e_{n+1}$ such that,
when we compose it with $\omega$, we create a stress with values
$a_{1},\ldots,a_{n+1}$ on $e_{1},\ldots,e_{n+1}$.
[Inductive Step $\Leftarrow$] Assume that for some $n$, if we can find a
stress with any value on $e_{1},\ldots,e_{n}$,
$G\setminus\\{e_{1},\ldots,e_{n}\\}$ is GLR. Then, suppose we can find some
stress with any value we want on $G\setminus\\{e_{1},\ldots,e_{n+1}\\}$. By
the inductive hypothesis, $H$ is GLR. If we set $a_{1},\ldots,a_{n}$ to all be
zero, then we can find a stress on $H$ with any value we wish on $e_{n+1}$.
So, by Base Case $\Leftarrow$, $G\setminus\\{e_{1},\ldots,e_{n+1}\\}$ is GLR.
∎
###### Corollary 15.
A graph $G$ is generically redundantly rigid in $\mathbb{R}^{d}$ if and only
if it is generically locally rigid in $\mathbb{R}^{d}$ and there is a non-zero
stress on every edge of $G$.
###### Proof.
If there is a non-zero stress on every edge of $G$, then by scaling, we can
find a stress of any value we want on any edge of $G$. Thus, by Proposition
14, each edge is redundant and $G$ is GRR.
On the other hand, if the only stress on some edge is the zero stress, we
cannot find a stress of any value on that edge. Thus, by Proposition 14, $G$
is not GRR. ∎
Now we need to show that a $(d+1)$-connected $k$-chain is GLR. Recall that the
$k$-chain is a subgraph of a complete bipartite graph, and by Theorem 5, that
complete bipartite graph is GLR. Therefore, it is sufficient to demonstrate
that the edges removed from the complete bipartite graph can take stresses of
any value. Pick any two vertices which are not connected in the $k$-chain, but
are connected in the complete bipartite graph. We will show that there exists
some stress with a non-zero value on the edge between these two vertices and
values of zero on all other removed edges.
Suppose the two vertices come from the sets $A_{i}$ and $A_{j}$, assuming
without a loss of generality that $i<j$. Note that $i-j$ is odd, since the
removed edges must come from different independent sets of the complete
bipartite graph. Furthermore, it is also evident that $i-j\geq 3$. Pick $d+1$
vertices from each of $A_{i+1},\ldots,A_{j-1}$. Use these vertices and the two
vertices in $A_{i}$ and $A_{j}$ to form $C_{1,d+1,\ldots,d+1,1}$, denoted by
$\Upsilon$. We will show that for generic realizations, $\Upsilon$ has a zero-
dimensional space of stresses and that the graph obtained by connecting the
two vertices at the ends, denoted by $\Upsilon^{\prime}$, has a
$1$-dimensional space of stresses.
Reorder the independent sets of $\Upsilon$ by
$A_{i},A_{i+2},\ldots,A_{j-1},A_{i+1},A_{i+3},\ldots,A_{j}$. Because
$\Upsilon$ is a bipartite graph, there are no edges between any two vertices
in $A_{i}\cup A_{i+2}\cup\ldots\cup A_{j-1}$; the same can be said of the
vertices in $A_{i+1}\cup A_{i+3}\cup\ldots\cup A_{j}$. Therefore, the upper-
left and bottom-right corners of the stress matrix of $\Upsilon$ have values
of zero on the non-diagonal entries. Moreover, Bolker and Roth [2]
demonstrated that the stress matrix has values of zero on the diagonal
entries. Furthermore, the stress matrix is symmetric across the diagonal, and
because each row fulfills an affine linear relation with the projection
vectors, so do the columns. Therefore, it is sufficient to examine the upper-
right corner of the matrix, keeping in mind the affine linear relations on
both the rows and the columns. We will use the following remark to analyze the
stress matrix.
###### Remark 16.
If $d+1$ generic vectors $v_{1},v_{2},\ldots,v_{d+1}\in\mathbb{R}^{d}$ satisfy
an affine linear relation, that is, for
$a_{1},a_{2},\ldots,a_{d+1}\in\mathbb{R}$
$a_{1}v_{1}+a_{2}v_{2}+\ldots+a_{d+1}v_{d+1}=0$ $a_{1}+a_{2}+\ldots+a_{d+1}=0$
Then $a_{1}=a_{2}=\cdots=a_{d+1}=0$. This can easily be seen by solving the
second equation for $a_{d+1}$ and substituting into the first equation. Then
we get a linear relation on $d$ generic vectors in $\mathbb{R}^{d}$, which
forces each of the coefficients to be $0$.
The upper-right corner of the stress matrix has the following shape.
| $A_{i+1}$ | $A_{i+3}$ | $A_{i+5}$ | $\cdots$ | $A_{j-2}$ | $A_{j}$
---|---|---|---|---|---|---
$A_{i}$ | $*_{1}$ | 0 | $\cdots$ | $\cdots$ | $\cdots$ | 0
$A_{i+2}$ | $*_{2}$ | * | 0 | $\cdots$ | $\cdots$ | 0
$A_{i+4}$ | 0 | * | * | 0 | $\cdots$ | 0
$\vdots$ | $\vdots$ | $\ddots$ | $\ddots$ | $\ddots$ | $\ddots$ | $\vdots$
$A_{j-3}$ | 0 | $\cdots$ | 0 | * | * | 0
$A_{j-1}$ | 0 | $\cdots$ | $\cdots$ | 0 | * | *
The asterisks represent all possible sets of non-zero entries, corresponding
to the edges of $\Upsilon$. All of the edges are between vertices in $A_{n}$
and $A_{n+1}$ for some $n$, causing the asterisks to form a “staircase”
pattern. Consider $*_{1}$, a $1$ by $d+1$ block of entries. These $d+1$
entries fulfill an affine linear relation among generic vectors across the
first row. By Remark 16, every entry in $*_{1}$ is therefore $0$. Next,
consider $*_{2}$, a $d+1$ by $d+1$ block of entries. Looking at the first
$d+1$ columns of the upper right corner of the stress matrix, $*_{2}$ must be
uniformly $0$ as well because the projection vectors fulfill an affine linear
relation on each column. Working down the “staircase” by alternately solving
for rows and columns, each of the asterisks must be uniformly $0$. Hence, the
only stress is the zero stress.
Now consider $\Upsilon^{\prime}$. Since $i-j$ is odd, let $i-j+1=2l$. The
graph $\Upsilon^{\prime}$ contains $(2l-2)(d+1)+2=2(l-1)(d+1)+2$ vertices and
$(2l-3)(d+1)^{2}+2(d+1)+1$ edges. $\Upsilon^{\prime}$ is a subgraph of some
complete bipartite graph with the same vertices. Each of the independent sets
of this complete bipartite graph has $(l-1)(d+1)+1$ vertices. The complete
bipartite graph has $[(l-1)(d+1)+1]^{2}$ edges, and so by Corollary 13, it has
a $[(l-1)(d+1)+1-d-1]^{2}=[(l-2)^{2}(d+1)^{2}+2(l-2)(d+1)+1]$-dimensional
space of stresses. $\Upsilon^{\prime}$ results from the removal of
$[(l-1)(d+1)+1]^{2}-[(2l-3)(d+1)^{2}+2(d+1)+1]=(l-2)^{2}(d+1)^{2}+2(l-2)(d+1)$
edges from the complete bipartite graph. Each edge removed reduces the rank of
the rigidity matrix by at most $1$, and so reduces the dimension of the space
of stresses by at most $1$. Therefore, after removing
$(l-2)^{2}(d+1)^{2}+2(l-2)(d+1)$ edges, the dimension of the space of stresses
is at least $1$.
Finally, since $\Upsilon$ has a zero-dimensional space of stresses and
$\Upsilon^{\prime}$ has a positive-dimensional space of stresses, there must
be a non-zero stress on the edge connecting the two vertices. In fact,
$\Upsilon^{\prime}$ has exactly a $1$-dimensional space of stresses, since
removing one edge forces the space of stresses to be zero-dimensional. This
implies that each of the removed edges can be written as a linear combination
of the remaining edges in the rigidity matrix, meaning that each of these
edges is responsible for a single independent dimension of stresses.
By composing the stresses of the subgraphs found above, we can obtain any
value we want on the removed edges of the complete bipartite graph. This leads
to the following result:
###### Lemma 17.
Any $(d+1)$-connected $k$-chain $C_{a_{1},a_{2},\cdots,a_{k}}$ with $k\geq 4$
and $\binom{d+2}{2}$ vertices is generically locally rigid in
$\mathbb{R}^{d}$.
###### Proof.
$C_{a_{1},a_{2},\cdots,a_{k}}$ is a subgraph of a complete bipartite graph
with the same vertices, which has already been proved to be GLR. Moreover, as
shown above, we can put arbitrary stresses on all of the edges that must be
removed to create $C_{a_{1},a_{2},\cdots,a_{k}}$. By Proposition 14, the
$k$-chain is GLR. ∎
## 4\. Proof of Generic Redundant Rigidity
Now that we know the $k$-chains in question are GLR if condition 1 of Theorem
8 is satisfied (which we will assume throughout the section), it remains to be
determined under what conditions they are GRR. According to Corollary 15, a
framework is redundantly rigid if and only if there is some stress with non-
zero entries on every edge. Consequently, we will find the space of stresses
of the $k$-chains.
By Proposition 2, the space of stresses is the kernel of the transpose of the
rigidity matrix. Because the graph is GLR, for any generic realization $p$,
the dimension of the space of stresses is
$e-\operatorname{rank}df_{G}(p)=e-vd+\binom{d+1}{2}$, where
$v=\binom{d+2}{2}$.
Now we consider all the $3$-chains $C_{a_{i},a_{i+1},a_{i+2}}$ that are
subgraphs of our $k$-chain. Call this set of $3$-chains the _$3$ -chain cover_
of the $k$-chain. Note that any stress of one of these $3$-chains is also a
stress of the entire $k$-chain. Moreover, we present the following lemma.
###### Lemma 18.
Let $C_{a_{1},a_{2},\cdots,a_{k}}$ be a $(d+1)$-connected $k$-chain with
$k\geq 4$ and $\binom{d+2}{2}$ vertices. Then the space of stresses of
$C_{a_{1},a_{2},\cdots,a_{k}}$ is precisely the space of stresses of the
$3$-chain cover of $C_{a_{1},a_{2},\cdots,a_{k}}$.
###### Proof.
To find the dimension of the space of stresses of the $3$-chain cover, use the
inclusion-exclusion principle. The overlap among the stresses stems from the
$2$-chains shared by adjacent $3$-chains. So, using Corollary 13 and some
simple algebra, the dimension of the space of stresses of the $3$-chain cover
is:
$\displaystyle\sum_{i=2}^{k-1}(a_{i-1}+a_{i+1}-d-1)(a_{i}-d-1)-\sum_{i=2}^{k-2}(a_{i}-d-1)(a_{i+1}-d-1)$
$\displaystyle=\sum_{i=1}^{k-1}a_{i}a_{i+1}-\left(\sum_{i=1}^{k}a_{i}\right)(d+1)+(d+1)^{2}$
$\displaystyle=e-v(d+1)+(d+1)^{2}$
If $v=\binom{d+2}{2}$, the reader can verify that this is also
$e-vd+\binom{d+1}{2}$. Since the stresses of the $3$-chain cover constitute a
subspace of the total space of stresses with equal dimension, they account for
the entire space of stresses of $C_{a_{1},a_{2},\cdots,a_{k}}$. ∎
Note that if a $3$-chain has a positive-dimensional space of stresses, we can
find some stress with non-zero values on every entry. To see this, first note
that a $3$-chain is a complete bipartite graph. If the graph has a positive-
dimensional space of stresses in $\mathbb{R}^{d}$, then some edge has a non-
zero stress on it. However, complete bipartite graphs are completely symmetric
across their edges with respect to the existence of non-zero stresses. If
there is a stress with entries of $0$ on some edge, we can use the symmetry of
the graph to find stresses with a non-zero value on that edge and then add the
stresses. Thus, it is possible to find a stress with non-zero values on every
edge.
Now we are equipped with all the tools necessary to examine redundant
rigidity. This leads to the following lemma.
###### Lemma 19.
A $(d+1)$-connected $k$-chain $C_{a_{1},a_{2},\cdots,a_{k}}$ with $k\geq 4$
and $\binom{d+2}{2}$ vertices is generically redundantly rigid if and only if
1. (1)
$a_{2},a_{k-1}\geq d+2$ and
2. (2)
there is no $i$ such that $a_{i}=a_{i+1}=d+1$.
###### Proof.
First, note that by Lemma 17, $C_{a_{1},a_{2},\cdots,a_{k}}$ is GLR. We will
apply Corollary 15 directly in the rest of this proof, so we only have to
determine whether the stresses of the graphs are non-zero on every edge.
[$\Rightarrow$] Suppose either condition $1$ or condition $2$ does not hold.
If $a_{2}<d+2$, then by Corollary 13, the $3$-chain $C_{a_{1},a_{2},a_{3}}$
(or bipartite graph $K_{a_{1}+a_{3},a_{2}}$) will have a zero-dimensional
space of stresses, and this is the only $3$-chain which includes the edges
connecting $A_{1}$ and $A_{2}$. Any stress on the $k$-chain will have entries
of 0 on these edges, meaning that they are not redundant and as a consequence,
the graph is not GRR. The same argument applies to $a_{k-1}$.
Moreover, suppose that for some $i$, $a_{i}=a_{i+1}=d+1$. Then by Corollary
13, both $C_{a_{i-1},a_{i},a_{i+1}}$ and $C_{a_{i},a_{i+1},a_{i+2}}$ have only
the zero stress. These are the only two $3$-chains that cover the edges
between $A_{i}$ and $A_{i+1}$, so by Lemma 18, any stress on the $k$-chain
will have entries of $0$ on these edges.
[$\Leftarrow$] Assume that $a_{2},a_{k-1}\geq d+2$ and there is no $i$ such
that $a_{i}=a_{i+1}=d+1$. Firstly, there is a non-zero stress covering the
edges between $A_{1}$ and $A_{2}$. To see this, consider
$C_{a_{1},a_{2},a_{3}}$, where each of $a_{1}+a_{3}$ and $a_{2}$ is at least
$d+2$, so this $3$-chain or bipartite graph has a non-zero space of stresses.
Hence, each edge between $A_{1}$ and $A_{2}$ has a non-zero stress covering
it. The same argument can be applied to the edges between $A_{k-1}$ and
$A_{k}$.
For the other edges, there are two cases to consider. In the first case, for
all $3\leq i\leq k-2$, $a_{i}\geq d+2$. In this case, there is obviously a
stress with non-zero values everywhere. Otherwise, there exists some $3\leq
i\leq k-2$ such that $a_{i}=d+1$. Then we know that $a_{i-1},a_{i+1}\geq d+2$,
so using Corollary 13 on $C_{a_{i-2},a_{i-1},a_{i}}$ and
$C_{a_{i},a_{i+1},a_{i+2}}$, we know that the edges between $A_{i-1}$ and
$A_{i}$ and the edges between $A_{i}$ and $A_{i+1}$ have non-zero stresses
covering them. Thus, the $k$-chain is GRR. ∎
We are now able to prove Theorem 8.
###### Proof.
[$\Rightarrow$] Suppose that the conditions do not all hold. If condition $1$
fails, then by Proposition 9, the $k$-chain is not $(d+1)$-connected, and
therefore not GPR. If either condition $2$ or condition $3$ fails, then by
Lemma 19, the $k$-chain is not GRR.
[$\Leftarrow$] Suppose that the conditions all hold. Since condition $1$
holds, by Proposition 9, the graph is $(d+1)$-connected, and by Lemma 17, it
is GLR. Since conditions $2$ and $3$ hold, by Lemma 19, the $k$-chain is GRR.
Finally, by Proposition 10, the graph is not GGR. Therefore, the $k$-chain is
GPR. ∎
## 5\. Graph Attachments in $\mathbb{R}^{5}$
Theorem 8 completely characterizes GPR $k$-chains with $\binom{d+2}{2}$
vertices. However, we also found a new class of GPR graphs which are not
necessarily bipartite. Here we present a specific case, which we expect can be
generalized in the future. Consider in $\mathbb{R}^{5}$ the $4$-chain
$C_{2,3,5,4}$, and another arbitrary graph $G=(V,E)$ with at least $6$
vertices. We _attach_ $C_{2,3,5,4}$ to $G$ by letting $A_{1}$ and $A_{4}$ be
disjoint $2$-element and $4$-element subsets of vertices in $V$, with none of
the vertices of $A_{2}$ and $A_{3}$ in $V$. The set of edges precisely
consists of all the edges in $G$ and $C_{2,3,5,4}$. Name the resulting graph
$G~{}{\cup}_{2,4}~{}C_{2,3,5,4}$.
###### Theorem 20.
Let $G$ be a generically redundantly rigid and $6$-connected graph in
$\mathbb{R}^{5}$ with at least $6$ vertices. Then
$G~{}{\cup}_{2,4}~{}C_{2,3,5,4}$ is generically partially rigid.
First we need to show that the new graph is GLR. To do this, we carefully
examine the graph $K_{6}~{}{\cup}_{2,4}~{}C_{2,3,5,4}$ [Figure
3LABEL:sub@fig:ring1].
(a)
(b)
Figure 3. Each node represents a set of vertices: the numbers represent
independent sets of the size indicated, and $K_{2}$ and $K_{4}$ represent
complete graphs. The lines represent edges between every combination of
vertices in the nodes connected. On left:
$K_{6}~{}{\cup}_{2,4}~{}C_{2,3,5,4}$. On right: Graph obtained from left by
deleting edges of $K_{2}$ and $K_{4}$.
###### Proposition 21.
The graph $K_{6}~{}{\cup}_{2,4}~{}C_{2,3,5,4}$ is generically locally rigid in
$\mathbb{R}^{5}$.
###### Proof.
We have not yet found a conceptual proof of this fact. However, using the
algorithm for testing generic local rigidity described earlier, we have found
one locally rigid realization, and the algorithm cannot return a false
positive for a graph being GLR, since the rank of the rigidity matrix can only
decrease due to non-generic realizations and special primes. This proves that
the graph is GLR. ∎
We want to know which edges of $K_{6}~{}{\cup}_{2,4}~{}C_{2,3,5,4}$ are
redundant. It is possible to do so by finding its stresses. The graph
$K_{6}~{}{\cup}_{2,4}~{}C_{2,3,5,4}$ has $56$ edges, and its rigidity matrix
has rank $55$. Hence, it has a $1$-dimensional space of stresses. We can
easily identify all the stresses of the graph.
Remove the edges of the $K_{2}$ and $K_{4}$ subgraphs [Figure
3LABEL:sub@fig:ring2]. The remaining graph is the bipartite graph $K_{7,7}$.
By Corollary 13, $K_{7,7}$ has a $1$-dimensional space of stresses, which is a
subspace of the $1$-dimensional space of stresses of
$K_{6}~{}{\cup}_{2,4}~{}C_{2,3,5,4}$. Hence, every stress in
$K_{6}~{}{\cup}_{2,4}~{}C_{2,3,5,4}$ must also be a stress of $K_{7,7}$.
Moreover, by symmetry, if there is a non-zero stress on one edge of $K_{7,7}$,
there is a non-zero stress on every edge. Therefore, all the edges of
$K_{6}~{}{\cup}_{2,4}~{}C_{2,3,5,4}$ have non-zero stresses except for the
edges on $K_{2}$ and $K_{4}$. In particular, by Corollary 15, each of the
edges of the subgraph $C_{2,3,5,4}$ are redundant. More generally, for any
$K_{i}~{}{\cup}_{2,4}~{}C_{2,3,5,4}$ with $i\geq 6$, we can find a non-zero
stress on the edges of $C_{2,3,5,4}$, making each of these edges redundant.
We will also use a very useful technique called a Hennenberg operation to
examine the rigidity of our graphs. One constructs a new graph with the
Hennenberg operation as follows: begin with a graph $G$ and some dimension
$d$, and pick any two vertices $v_{i}$ and $v_{j}$ with an edge between them.
Remove this edge, and add a vertex $v^{\prime}$ to $G$, connecting it to
$v_{i}$, $v_{j}$, and $d-1$ other vertices. This new graph, denoted by
$G^{\prime}$, is obtained from $G$ by a _Hennenberg operation_.
Connelly described the following theorem regarding Hennenberg operations.
###### Theorem 22 (Connelly [3]).
If $G$ is generically locally rigid in $\mathbb{R}^{d}$, and $G^{\prime}$ is
obtained from $G$ by an Hennenberg operation, then $G^{\prime}$ is generically
locally rigid in $\mathbb{R}^{d}$.
###### Lemma 23.
The graph $K_{n}~{}{\cup}_{2,4}~{}C_{2,3,5,4}$ is generically locally rigid in
$\mathbb{R}^{5}$ for all $n\geq 6$.
###### Proof.
We will prove the lemma by induction. For the base case, by Proposition 21,
$K_{6}~{}{\cup}_{2,4}~{}C_{2,3,5,4}$ is generically locally rigid in
$\mathbb{R}^{5}$.
For the inductive step, assume $K_{n}~{}{\cup}_{2,4}~{}C_{2,3,5,4}$ is
generically locally rigid in $\mathbb{R}^{5}$. Use a Hennenberg operation to
create a new graph, choosing $v_{i}$ and $v_{j}$ to be any vertices in $K_{n}$
and connecting $v^{\prime}$ to $4$ other vertices in $K_{n}$. This new graph
is GLR in $\mathbb{R}^{5}$ by the above theorem. Finally, we can add the edge
between $v_{i}$ and $v_{j}$, and connect $v^{\prime}$ to the rest of the
vertices in $K_{n}$. This constructs the graph
$K_{n+1}~{}{\cup}_{2,4}~{}C_{2,3,5,4}$, which is GLR because it is constructed
by adding edges to a GLR graph. ∎
The following important result arises from the previous lemma.
###### Lemma 24.
Let $G$ be a generically redundantly rigid graph in $\mathbb{R}^{5}$ with at
least $6$ vertices. Then $G~{}{\cup}_{2,4}~{}C_{2,3,5,4}$ is generically
redundantly rigid in $\mathbb{R}^{5}$.
###### Proof.
There are two cases to be examined. We can either remove an edge from $G$ or
from $C_{2,3,5,4}$. We want to show that each of the resulting graphs is GLR.
In the first case, we remove an edge from $G$ to get $G^{\prime}$, which is
still GLR. Suppose $G^{\prime}$ has $v$ vertices. $G^{\prime}$ is a subgraph
of $K_{v}$, so by Proposition 14 we can assign stresses for $K_{v}$ with any
values on the edges of $K_{v}\setminus G^{\prime}$. We know that
$K_{v}~{}{\cup}_{2,4}~{}C_{2,3,5,4}$ is GLR. By assigning zero stresses to the
edges of $C_{2,3,5,4}$, it is possible to assign stresses for
$K_{v}~{}{\cup}_{2,4}~{}C_{2,3,5,4}$ with any values on $K_{v}\setminus
G^{\prime}$. Thus, by Proposition 14,
$G^{\prime}~{}{\cup}_{2,4}~{}C_{2,3,5,4}$ is GLR.
In the second case, we remove an edge from $C_{2,3,5,4}$ and denote the
resulting graph $C^{\prime}_{2,3,5,4}$. Now suppose that $G$ has $v$ vertices.
The graph $K_{v}~{}{\cup}_{2,4}~{}C_{2,3,5,4}$ is GLR, and each of the edges
of the subgraph $C_{2,3,5,4}$ is redundant. Therefore, by Proposition 14,
$K_{v}~{}{\cup}_{2,4}~{}C^{\prime}_{2,3,5,4}$ is GLR. Finally, using the same
argument as before, it is possible to create stresses for $K_{v}$ with any
values on $K_{v}\setminus G$. These stresses will still exist on
$K_{v}~{}{\cup}_{2,4}~{}C^{\prime}_{2,3,5,4}$, so by Proposition 14,
$G~{}{\cup}_{2,4}~{}C^{\prime}_{2,3,5,4}$ is GLR. Consequently,
$G~{}{\cup}_{2,4}~{}C_{2,3,5,4}$ is GRR. ∎
###### Remark 25.
Let $G$ be any $6$-connected graph. Then $G~{}{\cup}_{2,4}~{}C_{2,3,5,4}$ is
still $6$-connected. Removing $5$ vertices from $G$ will not disconnect
$G~{}{\cup}_{2,4}~{}C_{2,3,5,4}$. It takes the removal of $7$ vertices to
isolate $A_{2}$, $7$ vertices to isolate $A_{3}$, and $6$ vertices to isolate
$C_{3,5}$.
To conclude our argument, we will use the following construction. Consider a
graph $G$ with a subgraph $H$, and suppose $H$ has $v$ vertices. Consider
another graph $H^{\prime}$ with at least $v$ vertices. We _replace_ $H$ with
$H^{\prime}$ as follows. Begin with $G$. Replace the vertices of $H$ with the
vertices of $H^{\prime}$. Create an injective mapping $\iota:H\to H^{\prime}$.
For each edge connecting vertex $g\in G$ to vertex $h\in H$, add an edge
connecting $g$ to $\iota(h)$. Finally, add all of the edges in $H^{\prime}$.
The new graph is the _replacement_ of $H$ with $H^{\prime}$. Intuitively,
replacing $H$ with $H^{\prime}$ consists of removing $H$ and placing
$H^{\prime}$ in its place.
###### Remark 26.
Given $K_{6}~{}{\cup}_{2,4}~{}C_{2,3,5,4}$ and some $G$ with at least $6$
vertices, replacing $K_{6}$ with $G$ is equivalent to creating
$G~{}{\cup}_{2,4}~{}C_{2,3,5,4}$.
###### Lemma 27.
Suppose $G$ is not generically globally rigid in $\mathbb{R}^{d}$ but contains
the subgraph $H$ which is generically globally rigid. Let $H^{\prime}$ be a
graph with at least as many vertices as $H$. Then replacing $H$ with
$H^{\prime}$ results in a graph that is not generically globally rigid in
$\mathbb{R}^{d}$.
###### Proof.
Since $G$ is not GGR, we can find a generic framework $G(p)$ and a non-
equivalent framework $G(q)$, up to Euclidean motions, in $\mathbb{R}^{d}$. Let
$p_{1},\ldots,p_{v},\allowbreak q_{1},\ldots,q_{v}$ represent the locations of
vertices in $H$, and $p_{v+1},\ldots,\allowbreak p_{w},\allowbreak
q_{v+1},\ldots,q_{w}$ represent the locations of vertices in $G\setminus H$.
It is possible to transform $G(q)$ into an equivalent framework
$G(q^{\prime})$ with $p_{i}=q^{\prime}_{i}$ for $i=1,\ldots,v$. First, through
translations, make $q^{\prime}_{1}=p_{1}$. Since $H$ is GGR, any other
realization of vertices in $H$ must be equivalent up to Euclidean motions.
Hence, one can reflect and rotate the entire framework to ensure
$q^{\prime}_{i}=p_{i}$ for $i=2,\ldots,n$. Finally, since $G(p)$ and $G(q)$
are non-equivalent frameworks, $p_{i}$ and $q^{\prime}_{i}$ are not all the
same for $i=v+1,\ldots,w$. We have shown that for any generic framework
$G(p)$, there exists a non-equivalent framework with the location of the
vertices in $H$ the same.
Now connect all the edges of $H$ for both frameworks, which has the same
effect as replacing $H$ with $K_{v}$ in both frameworks. Name this new graph
$G^{\prime}$. Since no edges are added between $G\setminus H$ and $H$, both
$p$ and $q^{\prime}$ preserve the edge lengths of $G^{\prime}$. Any generic
realization of $G^{\prime}$ is also a generic realization of $G$. Therefore,
for any generic framework $G^{\prime}(p)$, there is another non-equivalent
framework $G^{\prime}(q^{\prime})$, implying that $G^{\prime}$ is not GGR.
Next, replacing $H$ with any $K_{i}$ for $i\geq v$ results in a graph that is
not GGR. The case $i=v$ is already proved. The graph obtained by replacing $H$
with $K_{i}$, which we shall denote as $G^{\prime\prime}$, contains
$G^{\prime}$ as a subgraph. Starting with $G^{\prime}$, $p$ and $q^{\prime}$
as previously described, do the following to both frameworks: add $i-v$
vertices, connect them to all the vertices in $K_{v}$ only, project them onto
the same set of locations in $\mathbb{R}^{d}$ for both realizations, and
finally, ensure that $p$ is still generic. The new graph formed is precisely
$G^{\prime\prime}$. No edges are added between any of the points in $K_{i}$
and $G^{\prime\prime}\setminus K_{i}$, so $G^{\prime\prime}(q^{\prime})$ is a
non-equivalent realization of $G^{\prime\prime}(p)$. Finally, all generic
realizations of $G^{\prime\prime}$ must have the points in the subgraph
$G^{\prime}$ be generic as well, so a non-equivalent realization can be found
for any generic realization $G^{\prime\prime}(p)$. In this way,
$G^{\prime\prime}$ is not GGR for any $i\geq v$.
Finally, the replacement of $H$ with $H^{\prime}$ is a subgraph of the graph
obtained by replacing $H$ with $K_{i}$ for some $i$, so replacing $H$ with
$H^{\prime}$ results in a graph that is not generically globally rigid. ∎
###### Corollary 28.
Let $G$ be any graph in $\mathbb{R}^{5}$ with at least $6$ vertices. Then
$G~{}{\cup}_{2,4}~{}C_{2,3,5,4}$ is not generically globally rigid.
###### Proof.
We examine $K_{6}~{}{\cup}_{2,4}~{}C_{2,3,5,4}$. As discussed before, the only
stresses of this graph come from the subgraph $K_{7,7}$. Consequently,
$K_{6}~{}{\cup}_{2,4}~{}C_{2,3,5,4}$ has the same stress matrix, up to scale,
as $K_{7,7}$ for equivalent realizations. Both stress matrices have the same
nullity and as a consequence of Theorem 3,
$K_{6}~{}{\cup}_{2,4}~{}C_{2,3,5,4}$ is not GGR.
On the other hand, $K_{6}~{}{\cup}_{2,4}~{}C_{2,3,5,4}$ contains a subgraph
$K_{6}$ which is GGR. Replacing $K_{6}$ with any $G$ with at least $6$
vertices forms the graph $G~{}{\cup}_{2,4}~{}C_{2,3,5,4}$, and by Lemma 27,
this graph is not GGR. ∎
This completes the proof of Theorem 20. Specifically, Lemma 24 shows generic
redundant rigidity, Remark 25 shows $6$-connectedness, and Corollary 28 shows
lack of generic global rigidity. Now, we present some notable examples of $G$.
The graphs $K_{n}~{}{\cup}_{2,4}~{}C_{2,3,5,4}$, where $n\geq 7$, are GPR.
Note that for $K_{6}~{}{\cup}_{2,4}~{}C_{2,3,5,4}$, the edges of $K_{2}$ and
$K_{4}$ have no non-zero stresses, so it is not generically redundantly rigid.
Connelly [5, 8.3] recently asked the following: if a graph $G$ is
$(d+1)$-connected, GRR, and contains $K_{d+1}$ as a subgraph, is its Tutte
realization necessarily infinitesimally rigid? The concept of Tutte
realizations is outside of the scope of this paper. However, Connelly notes
that an affirmative answer would imply that $G$ is always GGR. The question is
answered in the negative, considering $K_{n}~{}{\cup}_{2,4}~{}C_{2,3,5,4}$,
where $n\geq 7$.
He also asked if, in any fixed dimension $d$, there are infinitely many GPR
graphs. Using the attachment construction described, we can find infinitely
many graphs in $\mathbb{R}^{5}$ which are GPR. Moreover, by the process of
coning [6], it is possible to preserve generic partial rigidity in these
graphs in higher dimensions. So, for any $d\geq 5$, this question has been
answered in the affirmative. It is still unknown for $d=3$ and $d=4$.
We can also let $G$ be a $3$-chain with $a_{1}=2$ and $a_{3}=4$. The
$3$-chains $C_{2,k,4}$ with $k\geq 16$ are equivalent to $K_{6,k}$, and can
easily be shown to be GRR and 6-connected using the algorithms described
earlier in this paper, or finding the space of stresses. This makes
$C_{2,k,4}~{}{\cup}_{2,4}~{}C_{2,3,5,4}$ GPR. This class of graphs is
especially notable since they form a $5$-ring, that is, a graph made from a
$5$-chain by adding all edges between $A_{1}$ and $A_{5}$. It is intriguing
that the size of one of the independent sets can be arbitrarily large. Also, a
$5$-ring cannot be expressed as the subgraph of a complete bipartite graph. We
also remark that it is possible to have $G$ be some $4$-chain, creating a
$6$-ring, which can be expressed as a subgraph of a complete bipartite graph.
Using Gortler, Healy and Thurston’s algorithm [7], we have proven that
$C_{2,15,4}~{}{\cup}_{2,4}~{}C_{2,3,5,4}$ is GPR as well. However,
$C_{2,15,4}$ is $K_{6,15}$ and is not GRR. Knowing that $G$ is not GRR is not
enough to say whether $G~{}{\cup}_{2,4}~{}C_{2,3,5,4}$ is GPR. Its properties
rely on the individual characteristics of $G$. We have seen an example
($K_{6}~{}{\cup}_{2,4}~{}C_{2,3,5,4}$) that is not GPR, and another
($C_{2,15,4}~{}{\cup}_{2,4}~{}C_{2,3,5,4}$) that is.
In addition to $C_{2,3,5,4}$, the $4$-chain $C_{3,4,5,3}$ in $\mathbb{R}^{5}$
can also act as an attachment. The proof is analogous. There are more
$4$-chains and also greater $k$-chains that we found in higher dimensions, but
we have not found any $3$-chains, or any $k$-chains in lower dimensions. We
have not completely categorized which $k$-chains can act as attachments.
It is unknown whether graphs other than $k$-chains can act as attachments.
However, it would be extremely interesting to find a graph that could act as
an attachment in $\mathbb{R}^{3}$, as right now there is only one GPR example
in $\mathbb{R}^{3}$, and such an example would generate infinitely many graphs
that are GPR. We leave this question as an open problem.
## 6\. Further Exploration
The $k$-chains with $\binom{d+2}{2}$ vertices have been fully explored in this
paper. Additionally, for $v<\binom{d+2}{2}$, simple calculations show that the
$k$-chain is a subgraph of a complete bipartite graph that is not GLR.
Experimental evidence suggests the following conjecture for
$v>\binom{d+2}{2}$:
###### Conjecture 29.
Any $(d+1)$-connected $k$-chain in $\mathbb{R}^{d}$ with more than
$\binom{d+2}{2}$ vertices is generically globally rigid.
Figure 4. Every irreducible graph with at most $4$ vertices.
We have found a class of GPR subgraphs of GPR complete bipartite graphs. The
more general question is to characterize which subgraphs of complete bipartite
graphs are GPR. This is a very difficult question to answer generally, as we
have also found examples of GPR graphs that are subgraphs of non-GPR complete
bipartite graphs, as evidenced by the $6$-rings.
The $k$-chains and $k$-rings that we have found can be characterized as part
of a larger family of graphs. Given some initial connected graph $G$, replace
each of the vertices with independent sets and completely connect the new
vertices according to $G$. When will this produce a graph that is GPR?
There exist many congruences among the initial graphs $G$. If there are two
vertices in $G$ that connect to the exact same set of vertices, then they can
be combined into one independent set. Call a graph $G$ _irreducible_ if there
do not exist vertices that can be combined this way. Using this fact, we have
identified $1$ irreducible connected graph with $2$ vertices, $1$ with $3$
vertices, $3$ with $4$ vertices, and $11$ with $5$ vertices [Figure 4]. From
there the number seems to grow exponentially. Experimentally, we have found
GPR graphs made from every irreducible graph with at most $5$ vertices. We
have also proved that we can make a GPR graph from every $k$-chain and
$k$-ring with $k\geq 2$. Hence we suggest the following bold conjecture.
###### Conjecture 30.
For any connected graph $G$ with $v>1$, there exists some
$a_{1},a_{2},\ldots,a_{v}$ and some $d$ such that if we replace each $v_{i}$
with an independent set of size $a_{i}$ and connect them accordingly, the
resulting graph is GPR in $\mathbb{R}^{d}$.
Remember that coning a graph that is GPR in $\mathbb{R}^{d}$ creates a graph
that is GPR in $\mathbb{R}^{d+1}$. For virtually all of the initial graphs
with $4$ or $5$ vertices, the GPR graph was obtained from a previous graph
that was GPR, either by coning or by coning and removing some edges. This may
help to explain why the conjecture might be true. On the other hand, the
$k$-chains and $k$-rings which are GPR are not obtained by coning, so there
might be other types of graphs that resist coning.
## References
* [1] L. Asimow and B. Roth, _The rigidity of graphs_ , Trans. Amer. Math. Soc. 245 (1978), 279-89.
* [2] E. Bolker and B. Roth, _When is a bipartite graph a rigid framework?_ , Pacific J. Math. 90 (1980), 27-44.
* [3] R. Connelly, _Generic global rigidity_ , Discrete Comput. Geom 33 (2005), no. 4, 549-563.
* [4] R. Connelly, _On generic global rigidity_ , DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 4 (1991), 147-155.
* [5] R. Connelly, _Questions, conjectures and remarks on globally rigid tensegrities_ , preprint, 2009, http://www.math.cornell.edu/~connelly/09-Thoughts.pdf.
* [6] R. Connelly and W. Whiteley, _Global rigidity: the effect of coning_ , preprint, 2009, http://www.math.cornell.edu/~connelly/GlobalTechniquesConing-09V2.pdf.
* [7] S. Gortler, A. Healy and D. Thurston, _Characterizing generic global rigidity_ , 2008, arXiv:0710.0926v3.
* [8] B. Hendrickson, _Conditions for unique graph realizations_ , SIAM J. Comput. 21 (1992), no. 1, 65-84.
* [9] B. Jackson and T. Jordán, _Connected rigidity matroids and unique realizations of graphs_ , J. Combin. Theory Ser. B 94 (2005), no. 1, 1-29.
* [10] W, Whiteley, _La division de sommet dans les charpentes isostatiques. [Vertex splitting in isostatic frameworks]_ Dual French-English text. Structural Topology No. 16 (1990), 23-30.
|
arxiv-papers
| 2009-09-15T22:16:40 |
2024-09-04T02:49:05.368390
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Samuel Frank and Jiayang Jiang",
"submitter": "Jiayang Jiang",
"url": "https://arxiv.org/abs/0909.2893"
}
|
0909.2937
|
# Neutrino mass from a hidden world and its phenomenological implications
Seong Chan Park Kai Wang Tsutomu T. Yanagida Institute for the Physics and
Mathematics of the Universe, University of Tokyo, Kashiwa, Chiba 277-8568,
JAPAN
###### Abstract
We propose a model of neutrino mass generation in extra dimension. Allowing a
large lepton number violation on a distant brane spatially separated from the
standard model brane, a small neutrino mass is naturally generated due to an
exponential suppression of the messenger field in the 5D bulk. The model
accommodates a large Yukawa coupling with the singlet neutrino ($n_{R}$) which
may change the standard Higgs search and can simultaneously accommodate
visible lepton number violation at the electroweak scale, which leads to very
interesting phenomenology at the CERN Large Hadron Collider.
††preprint: IPMU-09-0104
## I Introduction
Enormous experimental evidence clearly indicates that neutrinos have tiny but
non-zero masses, and understanding the origin of neutrino masses is one of the
most pressing problems in particle physics. The minimal Higgs boson model with
additional right-handed neutrinos provides a simple solution to all the
fermion masses including neutrino masses by generating all of them through
Yukawa interactions. However, the $10^{12}$ order hierarchy between
dimensionless Yukawa coupling $y_{t}$ and neutrino Yukawa coupling $y_{\nu}$
suggests that neutrino masses may arise from an additional source besides the
electroweak symmetry breaking. Their electric neutrality allows for the
possibility of neutrinos being Majorana fermions. Within the Higgs boson model
framework, the seesaw mechanism GRSY ; M is an elegant proposal accounting
for the tiny neutrino masses. It is crucial that the small neutrino masses
arise as a consequence of the grand unification at ultra high energy scales
GRSY ; so10 . The mechanism is based on the presence of singlet heavy Majorana
neutrinos of mass $m_{R}$,
$y\overline{l_{L}}n_{R}\tilde{H}+m_{R}\overline{n^{c}_{R}}n_{R}~{},$ (1)
which is impossible to probe directly at the collider experiments. However,
there have recently been several proposals to show how to test the origin of
neutrino mass directly at the coming CERN Large Hadron Collider (LHC) lhc . In
the seesaw mechanism, the tiny neutrino masses arise as a consequence of
lepton number violation at the ultra high energy scale while if the new
physics responsible for neutrino mass is accessible at the LHC, additional
tuning may be needed.
In the effective theory language, the Majorana neutrino mass after integrating
out new physics at a scale $\Lambda$ is due to the term
$y^{2}l_{L}l_{L}HH{S^{n}\over\Lambda^{n+1}}~{},$ (2)
where $S$ is a dimension one scale and $y$ is a dimensionless Yukawa coupling.
Therefore, without tuning $y$ in Eq. (2), there are only two approaches to
generate a tiny Majorana neutrino mass. One approach is to impose discrete
(gauge) symmetries which generate a large value of $n$ and tiny neutrino
masses arise as high dimensional operators discrete . Another approach is to
have a small scale $S$. For instance, in the “inverse seesaw” models inverse
or “TeV triplet” models typeii , to obtain the correct neutrino mass, a keV
order scale needs to be introduced, and this small scale may be identified as
a soft breaking of Lepton symmetry 'tHooft:1979bh . Within the framework of
minimal type-I seesaw GRSY ; M , where $n=0$ in Eq.(2), if the heavy Majorana
neutrinos are of the electroweak scale, the only way to generate the correct
neutrino mass is to require the Yukawa coupling $y$ to be of
$\mathcal{O}(10^{-6})$. Due to the tiny Yukawa coupling, the production of
$n_{R}$ can only be enhanced through new gauge interactions such as with a
$U(1)_{\rm B-L}$ gauge boson $Z_{\rm B-L}$ or $SU(2)_{R}$ gauge boson $W_{R}$
zbl .
In this paper, we propose a new model based on extra dimension extra_d . Here,
neutrino mass is generated by a cooperation with a right-handed singlet field
$n_{R}$ on “visible brane” and a lepton number violation at a distant brane,
“hidden brane,” spatially separated in the extra dimension communicating
through a messenger field in the 5D bulk (See Fig. 1). With a messenger field
whose zero-mode wave function has an exponential suppression at the hidden
brane, even if the lepton number violation on the hidden brane is as large as
the electroweak scale, the resultant neutrino mass remains naturally small.
Differently from existing models in extra dimension models_extrad our model
predicts two phenomenologically interesting features (i) a large Yukawa
coupling with $n_{R}$, which can completely change Higgs phenomenology (ii) a
sizable lepton number violation through mixings with Kaluza-Klein excitation
modes of the messenger field, which can be tested by future collider
experiments such as the CERN Large Hadron Collider (LHC).
Figure 1: Five dimensional setup of the present model. All the standard model
particles plus one singlet neutrino $n_{R}$ are localized on the visible brane
located at $y=\pi R$. The lepton violating sector is on a distant brane
located at $y=0$ and has only small overlap with the zero-mode of a messenger
field ($\psi_{L}^{(0)}$), which induces a small lepton number violation
transmuted to the visible sector and results a small neutrino mass.
## II Model
The space-time in the model is a five dimensional spacetime with an orbifold
extra dimension $S^{1}/Z_{2}$ whose fixed points are located at $y=0$ and
$y=\pi R$. Two branes are introduced at the end points: “hidden brane” at
$y=0$ and “visible brane” at $y=\pi R$. All the standard model particles
including leptons ($l_{L}=(\nu_{L},e_{L})^{T},e_{R}$) and Higgs field ($H$)
are localized on the visible brane with an additional singlet right-handed
fermion ($n_{R}$), hence the particle spectrum is the same as the one in the
conventional $SO(10)$ GUT model. Lepton number is a good symmetry on the
visible brane. However a large lepton number violation is allowed on the
hidden brane and it can communicate with the standard model sector through a
messenger field, $\Psi(x,y)=\psi_{L}+\psi_{R}$, in the 5D bulk. Here,
$\psi_{L/R}=(1\pm\gamma_{5})\Psi/2$. As the minimal spinor representation in
5D is vector-like so that a 5D bulk field has both of chiralities. Imposing
lepton number $+1$ for the messenger field the violation of lepton number is
effectively parameterized by the localized Majorana mass of the messenger
field ($m_{M}$). Furthermore, the $Z_{2}$ transformation of the messenger
field is defined as
$Z_{2}:\Psi(x,y)\rightarrow\Psi(x,-y)=\gamma_{5}\Psi(x,y)$
so that $\psi_{L}$ has even parity satisfying Neuman boundary conditions at
the end points ($y=0$ and $\pi R$) and conversely $\psi_{R}$ has odd parity
satisfying Dirichlet boundary conditions thus vanishes at the end points.
Accordingly, only $\psi_{L}$ can have a non-vanishing Majorana mass term,
$\overline{\psi_{L}^{c}}\psi_{L}\delta(y)$ at $y=0$. $\psi_{R}$, on the other
hand, could have neither boundary localized Majorana mass term
$\overline{\psi^{c}_{R}}\psi_{R}\delta(y)$ nor couplings with the SM Higgs
field at $y=\pi R$. One should also notice that only the left-chiral state
($\psi_{L}^{(0)}\sim e^{m_{\Psi}y}$) has a zero-mode. KK excitation modes
consist of massive Dirac spinors ($\psi_{L}^{(n>0)},\psi_{R}^{(n>0)}$) having
Kaluza-Klein masses $m_{n}^{2}=m_{\Psi}^{2}+n^{2}/R^{2}$.
The action of the model is given as:
$\displaystyle S_{5}$ $\displaystyle=\int d^{4}x\int_{0}^{\pi
R}dy\,\,\overline{\Psi}i\Gamma^{M}D_{M}\Psi-m_{\Psi}\overline{\Psi}\Psi$ (3)
$\displaystyle+\delta(y-\pi R)\left({\cal L}_{\rm
SM}+y\overline{l_{L}}\tilde{H}n_{R}+m_{D}\overline{\psi}_{L}n_{R}\right)$
$\displaystyle+\delta(y)\left(m_{M}\overline{\psi_{L}^{c}}\psi_{L}+H.C.\right).$
Notice that lepton number is violated only on the hidden brane and the amount
of violation is effectively parametrized by a Majorana mass parameter $m_{M}$.
After the Kaluza-Klein decomposition
$\psi_{L/R}(x,y)=\sum_{n}\psi^{(n)}_{L/R}(x)f_{L/R}^{(n)}(y)$ and integrating
out the fifth dimension, we get the 4D effective Lagrangian with a tower of
Kaluza-Klein states as
$\displaystyle{\cal L}_{\rm
eff}\ni\delta\bar{\nu}_{L}n_{R}+\Delta_{0}\bar{\psi}_{L}^{(0)}n_{R}+\sum_{n,m\geq
0}\epsilon_{nm}\bar{\psi^{c}}_{L}^{(n)}\psi_{L}^{(m)}$
$\displaystyle+\sum_{n>0}\Delta_{n}\bar{\psi}_{L}^{(n)}n_{R}+m_{n}(\bar{\psi}_{L}^{(n)}\psi_{R}^{(n)}+\bar{\psi}_{R}^{(n)}\psi_{L}^{(n)})~{}.$
(4)
Here, we have introduced convenient parameters $\delta=y\langle
H\rangle,\epsilon_{nm}=\epsilon_{mn}=m_{M}f_{L}^{(n)}(0)f_{L}^{(m)}(0)$ ,
$\Delta_{n}=m_{D}f_{L}^{(n)}(\pi R)$ where $f_{L}^{(n)}$ is the wave function
of the $n$-th Kaluza-Klein mode $\psi_{L}^{(n)}.$ One should notice that
$\epsilon_{00}$ is much smaller than $\epsilon_{nm}$ with a non-zero $n$
and/or $m$:
$\epsilon_{00}\ll\epsilon_{0n>0}\ll\epsilon_{n>0,m>0}$
due to the large exponential suppression $f_{L}^{(0)}(0)^{2}\sim e^{-2\pi
Rm_{\Psi}}$ as we have assumed $m_{\Psi}>1/R$. This exponential suppression is
the very essence ensuring a small neutrino mass at the end.
In the basis of
$(\nu_{L},\psi_{L}^{(0)},n_{R}^{c},\psi_{L}^{(1)}\psi_{R}^{(1)c},\psi_{L}^{(2)},\psi_{R}^{(2)c}\cdots)$
the mass matirx is given as
$\displaystyle{\cal M}=\left(\begin{array}[]{ c c c c c c c c
}0&0&\delta&0&0&0&0&\cdots\\\
0&\epsilon_{00}&\Delta_{0}&\epsilon_{01}&0&\epsilon_{02}&0&\cdots\\\
\delta&\Delta_{0}&0&\Delta_{1}&0&\Delta_{2}&0&\cdots\\\
0&\epsilon_{10}&\Delta_{1}&\epsilon_{11}&m_{1}&\epsilon_{12}&0&\cdots\\\
0&0&0&m_{1}&0&0&0&\cdots\\\
0&\epsilon_{20}&\Delta_{2}&\epsilon_{21}&0&\epsilon_{22}&m_{2}&\cdots\\\
0&0&0&0&0&m_{2}&0&\cdots\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots\end{array}\right).$
(13)
Interestingly, the determinant of this mass matrix can be easily calculated
thanks to lots of zeroes in the matrix coming from lepton number conservation
on the visible brane, the Dirichlet boundary condition for $\psi_{R}^{(n)}$
and the orthogonality of KK states, as
$\displaystyle\det{\cal M}=\epsilon_{00}\delta^{2}\Pi_{i}m_{i}^{2}.$ (14)
Obviously, if $\delta$ or $\epsilon_{00}$ vanishes this determinant vanishes
thus the neutrino remains exactly massless. Since $\epsilon_{00}$ is
exponentially small in our model the smallness of neutrino mass is achieved.
## III Phenomenology
The key feature of the model is that an exponential suppression of the zero-
mode wave function of the messenger field leads to the small Majorana mass
$\epsilon_{0}$ of $\psi^{0}_{L}$ while the small $\epsilon_{0}$ naturally
leads to the small neutrino masses even when the Dirac mass parameter $m_{D}$
or Yukawa coupling constant $y$ ($\delta=y\langle H\rangle$) of $n_{R}$ are
sizable. Consequently, the model predictions can then be directly tested or
constrained by experiments.
Assuming all the mass parameters are in the electroweak scale, $1/R$ controls
the lepton number violation in the heavy neutrino states and may lead to
totally different collider signature based on different choices of the
parameter region. We first consider the case when the compactification scale
is similar to the weak scale ($1/R\sim M_{W}$). In this case KK states
directly come into play in low energy phenomenology. To illustrate the
feature, we choose a set of model parameters as an example as
$(1/R,m_{\Psi},m_{M},m_{D},\delta)=(200,800,1000,500,20)\,{\rm GeV},$
then the mass spectrum is given by
$\displaystyle(m_{\nu},m_{-},m_{+},m_{4},m_{5},\cdots)$
$\displaystyle=(2.4\times 10^{-10},181,193,360,859,\cdots)\,{\rm GeV}.$
a sizable mass splitting between the lightest two heavy states arises. The
splitting of the Majorana masses will lead to visible Lepton number violation
in the heavy neutrino states. The direct search of such heavy neutrinos at the
LHC are widely studied in lhc in completely model independent
phenomenological approach. If the KK states are not decoupled, the current
model becomes an explicit model realization for this signature.
If the compactification scale is much higher than the electroweak scale
($1/R\gg M_{W}$), only three light modes,
$(\nu_{L},\psi^{(0)}_{L},n^{c}_{R})$, are directly relevant in low energy
phenomenology. The neutrino mass matrix is reduced to a simple $3\times 3$
matrix
$\displaystyle{\cal M}\rightarrow{\cal M}_{\rm
eff}\simeq\left(\begin{array}[]{ c c c}0&0&\delta\\\
0&\epsilon_{0}&\Delta_{0}\\\ \delta&\Delta_{0}&0\end{array}\right)~{}.$ (18)
In this limit, the model essentially reduces to conventional “inverse seesaw”
models in 4D inverse where the smallness of $\epsilon_{0}$ can be argued as a
soft breaking of Lepton symmetry 'tHooft:1979bh . In this model, the small
Majorana mass ($\epsilon_{0}$) is guaranteed by the higher dimensional nature
of the model in a natural way. At ${\cal O}(\epsilon_{0})$, the eigenstates
are one light state ($\nu$) and two nearly degenerate heavy states
($N_{\pm}$):
$\displaystyle\nu$ $\displaystyle\approx$
$\displaystyle\frac{\Delta_{0}}{\sqrt{\Delta_{0}^{2}+\delta^{2}}}\nu_{L}-\frac{\delta}{\sqrt{\Delta_{0}^{2}+\delta^{2}}}\psi^{(0)}_{L},$
(19) $\displaystyle N_{\pm}$ $\displaystyle\approx$
$\displaystyle\frac{1}{\sqrt{m_{\pm}^{2}+\Delta_{0}^{2}+\delta^{2}}}(m_{\pm}n_{R}^{c}+\delta\nu_{L}+\Delta_{0}\psi^{(0)}_{L})$
(20)
with corresponding mass eigenvalues
$\displaystyle
m_{\nu}\approx\epsilon_{0}\frac{\delta^{2}}{\delta^{2}+\Delta_{0}^{2}},\,\,m_{\pm}\approx\sqrt{\delta^{2}+\Delta_{0}^{2}}\pm{\cal
O}(\epsilon_{0}).$ (21)
The Lepton number violation effects are suppressed as
$\epsilon_{0}/\sqrt{\delta^{2}+\Delta_{0}^{2}}$. The heavy neutrinos are
dominant by their Dirac component and direct search of heavy Dirac neutrino
has been studied in dirac in the framework of Type-III seesaw.
We want to emphasize that the Yukawa coupling $y$ can be of $\mathcal{O}(1)$
in the model. Therefore, one may expect immediate implication in Higgs
phenomenology since Higgs can decay into neutrino states as
$H\rightarrow\overline{\nu}N_{\pm},$ (22)
provided these decays are allowed kinematically. Then, $N_{\pm}$ states can
decay into $\nu b\bar{b}$ through a virtual Higgs.
Before going to details of Higgs study, we will discuss the physical states
and mixing constraints first. The above mixings contribute to $\mu$-decay by
the effective muon decay constants as
$\displaystyle G_{\mu}\simeq
G_{F}(1-\frac{1}{2}\frac{\delta_{\mu}^{2}}{\Delta_{0}^{2}})(1-\frac{1}{2}\frac{\delta_{e}^{2}}{\Delta_{0}^{2}}).$
(23)
As a consequence, all the observables which depend on the Fermi constant will
be affected by the mixing angles or $\delta_{e}/\Delta_{0}$ and
$\delta_{\mu}/\Delta_{0}$. Also if $\Delta_{0}<M_{Z}$, the invisible decay
rate of $Z^{0}$ can be reduced by
$(1-\frac{1}{6}\sum_{l}\delta_{l}^{2}/\Delta_{0}^{2})$ with respect to the
standard model one. Taking $\mu-e$ universality, CKM unitarity and the
invisible decay rate of $Z^{0}$ one can get a results at $90\%$ C.L. bound1 ;
bound2 ; bound3 as
$\displaystyle\frac{\delta_{e,\mu,\tau}^{2}}{\Delta_{0}^{2}}<(6.5,5.9,18)\times
10^{-3}.$ (24)
If Higgs is light as $m_{H}\lesssim 140$ GeV, it dominantly decays into
$b\bar{b}$ due to the limited allowed phase-space. However the bottom Yukawa
coupling is only $y_{b}=m_{b}/\langle H\rangle\simeq 1.7\times 10^{-2}$. If
heavy neutrino Yukawa coupling $y$ is sufficient large and the Higgs has a
decay phase space, the Higgs decays into neutrino, $H\rightarrow\nu N$, may
siginificantly changes the standard Higgs search procedure. The partial decay
width is
$\Gamma(H\rightarrow\nu^{i}_{L}\bar{N^{j}})=\frac{|{y}_{ij}V_{ij}|^{2}}{8\pi}m_{H}\left(1-{m^{2}_{N}\over
m^{2}_{H}}\right)^{2},$ (25)
where $m_{H}$, $m_{N}$ are masses of Higgs and heavy neutrino respectively and
$V_{ij}$ is the mixing fraction of the neutrino state $n^{i}_{R}$ in the
physical states $N^{j}$. To simplify the discussion, we focus on the “inverse
seesaw” limit where $N_{+}$ and $N_{-}$ are nearly degenerate and
$n_{R}\simeq(N_{+}+N_{-})/\sqrt{2}~{}~{};m_{N_{+}}\simeq m_{N_{-}}~{}.$ (26)
Then, the total width for Higgs decaying into heavy neutrino states is given
by
$\displaystyle\sum\Gamma(H\rightarrow\nu N_{\pm})$ (27) $\displaystyle={1\over
2|V_{R+}|^{2}}\Gamma(H\rightarrow\nu N_{+})+{1\over
2|V_{R-}|^{2}}\Gamma(H\rightarrow\nu N_{-}).$
As discussed previously, there are strong bounds on the heavy neutrino mixing
both from direct search experiments and precision test of electroweak
interactions. For $m_{N}<$100 GeV, LEP experiments L3 and DELPHI provided
bounds as $\delta_{l}/\Delta_{0}<10^{-2}$ bound:LEP and hence $y\sim
m_{N}/\langle H\rangle\times\delta_{l}/\Delta_{0}<4\times 10^{-3}$. The phase
space suppression in $H\rightarrow\nu N$ is mostly larger than in
$H\rightarrow b\bar{b}$. We don’t expect the standard model Higgs decay
branching fraction to have visible change in this case. If $m_{N}$ is large
enough to escape from LEP bounds, the precision test on electroweak
interaction put less stringent bound on
$\delta_{\tau}^{2}/\Delta_{0}^{2}<0.018$. $y$ can then be as large as
$\mathcal{O}(10^{-2})$. Even though the $y$ can be a few times bigger than
$y_{b}$, the heavy neutrino channel has a much larger decay phase space
suppression. When the mass difference between $m_{H}$ and $m_{N}$ are
sufficiently large of $30\sim 40$ GeV, however, the new channel will
significantly reduce the BR($H\rightarrow b\bar{b}$) and
BR($H\rightarrow\gamma\gamma$).
To illustrate the qualitative feature, we assume there is only one generation
heavy neutrino state that have large Yukawa coupling and plot the Higgs decay
BR figure for $m_{N}=105$ GeV, $y=6\times 10^{-2}$ in Fig. 2.
Figure 2: Modified Decay BR of the SM Higgs for $y=6\times 10^{-2}$,
$m_{N}$=105 GeV
The dash-lines and solid-lines correspond to the original Higgs decay BR
without $H\rightarrow\nu N$ and the modified BR of Higgs with $H\rightarrow\nu
N$ decay, respectively. In principle, we will need three generations of heavy
neutrino states and this can significantly increase the partial width of Higgs
decaying to heavy neutrinos.
The $H\rightarrow\nu N$ channel will be very challenging to be identified
since there is always a missing neutrino. Then, it is impossible to fully
reconstruct the Higgs. Since the Higgs decay may not be dominantly into heavy
neutrinos, the conventional searching channels are still available. However,
notice that some decay BRs are significantly changed as in Fig. 2. Due to the
new decaying channel, for instance Br($H\rightarrow\gamma\gamma$) can be
reduced by a factor more than of 50% while as argued in Low:2009di .
## IV Conclusion and Remarks
In this paper we suggest a new model of neutrino which only involves TeV scale
masses. The lightness of neutrinos is guaranteed by an exponential suppression
in the zero-mode wave function of a messenger field in 5D bulk which mediates
lepton number violation taking place on a distant brane separated from the
brane where all the standard model leptons are confined. Depending on
parameter choice of $1/R$, the model can accommodate both the electroweak
scale Majorana neutrino lhc models and the “inverse seesaw” type model
inverse . The model naturally accommodates a large Yukawa coupling in
$\bar{l}n_{R}H$ and may lead to interesting phenomenology in the Higgs search.
In the end, we want to add another remark regarding on the sizable Lepton
number violation effects at the electroweak scale. Even if one allows a large
$m_{R}\overline{n^{c}_{R}}n_{R}$ term on the visible brane, the smallness of a
light neutrino is guaranteed by the tiny Majorana mass $\epsilon_{0}$. Heavy
states ($N_{\pm}$), on the other hand, have a sizable mass splitting
$\displaystyle m_{\pm}$
$\displaystyle\approx\sqrt{\delta^{2}+\Delta_{0}^{2}+\frac{m_{R}^{2}}{4}}\pm\frac{m_{R}}{2}\pm{\cal
O}(\epsilon_{0}),$ (28)
and the lepton number violation effects are now of order
$m_{R}/\sqrt{\delta^{2}+\Delta^{2}+\frac{m_{R}^{2}}{4}}$. Thus, if $m_{R}$ is
as large as the electroweak scale, one will expect significant lepton number
violation effects at the LHC.
## Acknowledgement
The work supported by the World Premier International Research Center
Initiative (WPI Initiative), MEXT, Japan.
## References
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|
arxiv-papers
| 2009-09-16T11:38:14 |
2024-09-04T02:49:05.377029
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Seong Chan Park, Kai Wang and Tsutomu T. Yanagida (Tokyo U., IPMU)",
"submitter": "Seong Chan Park",
"url": "https://arxiv.org/abs/0909.2937"
}
|
0909.3013
|
# Crossing the phantom divide
Hongsheng Zhang111Electronic address: hongsheng@kasi.re.kr Shanghai United
Center for Astrophysics (SUCA), Shanghai Normal University, 100 Guilin Road,
Shanghai 200234,China Korea Astronomy and Space Science Institute, Daejeon
305-348, Korea Department of Astronomy, Beijing Normal University, Beijing
100875, China
###### Abstract
The cosmic acceleration is one of the most significant cosmological
discoveries over the last century. Following the more accurate data a more
dramatic result appears: the recent analysis of the observation data
(especially from SNe Ia) indicate that the time varying dark energy gives a
better fit than a cosmological constant, and in particular, the equation of
state parameter $w$ (defined as the ratio of pressure to energy density)
crosses $-1$ at some low redshift region. This crossing behavior is a serious
challenge to fundamental physics. In this article, we review a number of
approaches which try to explain this remarkable crossing behavior. First we
show the key observations which imply the crossing behavior. And then we
concentrate on the theoretical progresses on the dark energy models which can
realize the crossing $-1$ phenomenon. We discuss three kinds of dark energy
models: 1. two-field models (quintom-like), 2. interacting models (dark energy
interacts with dark matter), and 3. the models in frame of modified gravity
theory (concentrating on brane world).
###### pacs:
95.36.+x 04.50.+h
## I The universe is accelerating
Cosmology is an old and young branch of science. Every nation had his own
creative idea about this subject. However, till the 1920s about the unique
observation which had cosmological significance was a dark sky at night. On
the other hand, we did not prepare a proper theoretical foundation till the
construction of general relativity. Einstein’s 1917 paper is the starting
point of modern cosmology e1 . The next mile stone was the discovery of cosmic
expansion, that was the recession of galaxies and the recession velocity was
proportional to the distance to us.
Except some rare cases, our researches are always based on the cosmological
principle, which says that the universe is homogeneous and isotropic. In the
early time, this is only a supposition to simplify the discussions. Now we
have enough evidences that the universe is homogeneous and isotropic at the
scale larger than 100 Mpc. The cosmological principle requires that the metric
of the universe is FRW metric,
$\displaystyle ds^{2}=-dt^{2}+a^{2}(t)(dr^{2}+r^{2}d\Omega_{2}^{2});$ (1)
$\displaystyle ds^{2}=-dt^{2}+a^{2}(t)(dr^{2}+\sin(r)^{2}d\Omega_{2}^{2});$
(2) $\displaystyle
ds^{2}=-dt^{2}+a^{2}(t)(dr^{2}+\sinh(r)^{2}d\Omega_{2}^{2}),$ (3)
depending on the spatial curvature, which can be Euclidean, spherical or
pseudo-spherical. Here, $t$ is the cosmic time, $a$ denotes the scale factor,
$r$ represents the comoving radial coordinate of the maximal symmetric
3-space, and $d\Omega_{2}^{2}$ stands for a 2-sphere. Which geometry serves
our space is decided by observations. FRW metric describes the kinetic
evolution of the universe. To describe the dynamical evolution of the
universe, that is, the function of $a(t)$, we need the gravity theory which
ascribes the space geometry to matter. The present standard gravity theory is
general relativity. In 1922 and 1924, Friedmann found that there was no static
cosmological solution in general relativity, that is to say, the universe is
either expanding or contracting f1 . To get a static universe, Einstein
introduce the cosmological constant. However, even in the Einstein universe,
where the contraction of the dust is exactly counteracted by the repulsion of
the cosmological constant, the equilibrium is only tentative since it is a
non-stationary equilibrium. Any small perturbation will cause it to contract
or expand. Hence, in some sense we can say that general relativity predicts an
expanding (or contracting) universe, which should be regarded as one of the
most important prediction of relativity.
In almost 70 years since the discovery of the cosmic expansion in 1929 h1 ,
people generally believe that the universe is expanding but the velocity is
slowing down. People try to understand via observation that the universe will
expand forever or become contracting at some stage. A striking result appeared
in 1998, which demonstrated that the universe is accelerating rather than
decelerating. Now we show how to conclude that our universe is accelerating.
We introduce the standard general relativity,
$G_{\mu\nu}+\Lambda g_{\mu\nu}=8\pi Gt_{\mu\nu},$ (4)
where $G_{\mu\nu}$ is Einstein tensor, $G$ is (4-dimensional) Newton constant,
$t_{\mu\nu}$ denotes the energy-momentum tensor, $g_{\mu\nu}$ stands for the
metric of a spacetime, $\mu,\nu$ run from 0 to 3. Throughout this article, we
take a convention that $c=\hbar=1$ without special notation. Define a new
energy momentum $T_{\mu\nu}$,
$8\pi GT_{\mu\nu}=8\pi Gt_{\mu\nu}-\Lambda g_{\mu\nu}.$ (5)
$T_{\mu\nu}$ has included the contribution of the cosmological constant, whose
effect can not be distinguished with vacuum if we only consider gravity.
The $00$ component of Einstein equation (4) is called Friedmann equation,
$H^{2}+\frac{k}{a^{2}}=\frac{8\pi G}{3}\rho,$ (6)
where $H=\frac{\dot{a}}{a}$ is the Hubble parameter, an overdot stands for the
derivative with respect to the cosmic time, $k$ is the spatial curvature of
the FRW metric, for (1), $k=0$; for (2), $k=1$; for (3), $k=-1$,
$\rho=-T^{0}_{0}$. Throughout this article, we take the signature
$(-,+,...,+)$. The spatial component of Einstein equation can be replaced by
the continuity equation, which is much more convenient,
$\dot{\rho}+3H(\rho+p)=0,$ (7)
where $p=T_{1}^{1}=T_{2}^{2}=T_{3}^{3}$. Here we use a supposition that the
source of the universe $T_{\mu}^{\nu}$ is in perfect fluid form. Using (6) and
(7), we derive the condition for acceleration,
$\frac{\ddot{a}}{a}=-\frac{4\pi G}{3}(\rho+3p).$ (8)
We see that the universe is accelerating when $\rho+3p<0$. We know that the
galaxies and dark matter inhabit in the universe long ago. They are dust
matters with zero pressure. Thus, if the universe is accelerating, there must
exist an exotic matter with negative pressure or we should modify general
relativity. The cosmological constant is a far simple candidate for this
exotic matter, or dubbed dark energy. For convenience we separate the
contribution of the cosmological constant (dark energy) from other sectors in
the energy momentum, the Friedmann equation becomes,
$\frac{H^{2}}{H^{2}_{0}}=\Omega_{\Lambda
0}+\Omega_{m0}(1+z)^{3}+\Omega_{k0}(1+z)^{2},$ (9)
where $z$ is the redshift, the subscript 0 denotes the present value of a
quantity,
$\Omega_{\Lambda 0}=\frac{\Lambda}{8\pi H_{0}^{2}G},$ (10)
$\Omega_{m0}=\frac{\rho_{m0}}{8\pi H_{0}^{2}G},$ (11)
$\Omega_{k0}=-\frac{k}{H_{0}^{2}a_{0}^{2}}.$ (12)
The type Ia supernova is a most powerful tool to probe the expanding rate of
the universe. In short, type Ia supernova is a supernova which just reaches
the Chandrasekhar limit (1.4 solar mass) and then explodes. Hence they have
the same local luminosity since they have roughly the same mass and the same
exploding process. They are the standard candles in the unverse. We can get
the distance of a type Ia supernova through its apparent magnitude. A sample
of type Ia supernovae will generates a diagram of Hubble parameter versus
distance, through which we get the information of the expanding velocity in
the history of the universe. In 1998, two independent groups found that the
universe is accelerating using the observation data of supernovae acce . After
that the data accumulate fairly quickly. The famous sample includes Gold04
gold04 , Gold06 gold06 , SNLS snls , ESSENCE essence , Davis07 davis07 , Union
union , Constitution constitution . Here, we show some results of one of the
most recent sample, Union union , which is plotted by $\chi^{2}$ statistics.
Figure 1: The figure displays the counter constrained by SNe Ia (in blue), by
BAO (in green), by CMB (in yellow), and the joint constraint by all the three
kinds of observation (in black and white). $\Omega_{\Lambda}$ is
$\Omega_{\Lambda 0}$ in (10), $\Omega_{m}$ is $\Omega_{m0}$ in (11). This
figure is borrowed from union .
From fig 1, we see that: 1. the universe is almost spatially flat, that is the
curvature term $\Omega_{k0}$ is very small. 2. the present universe is
dominated by cosmological constant (dark energy), whose partition is
approximately 70%, and the partition of dust is 30%. We introduce a
dimensionless parameter, the deceleration parameter $q$,
$q=-\frac{\ddot{a}a}{\dot{a}^{2}}.$ (13)
A negative deceleration parameter denotes acceleration. In a universe with
dust and cosmological constant (which is called $\Lambda$CDM model), by
definition
$q=\frac{1}{2}\Omega_{m}-\Omega_{\Lambda},$ (14)
whose present value $q_{0}=-0.55$. Hence the present universe is accelerating.
The previous result depends on a special cosmological model, $\Lambda$CDM
model in frame of general relativity. How about the conclusion if we only
consider the kinetics of the universe?
Figure 2: Kinetic universe vs dynamic universe. The fitting results of sudden
transition model, linear expansion model, and $\Lambda$CDM model. The solid
line is the best-fit of the sudden transition model (the deceleration
parameter jumps at some redshift); the long-dashed line denotes the best-fit
of linear expansion model ($q=q_{0}+q_{1}z$, $q_{1}$ is a constant) gold04 ;
the short-dashed line represents the best-fit of $\Lambda$CDM model. From sha
.
The simplest kinetic model is a sudden transition model, in which the
deceleration parameter is a constant in some high redshift region and jumps to
another constant at a critical redshift. The other simple choice is that the
deceleration parameter is a linear function of $z$. We show the two kinetic
models with the dynamic model $\Lambda$CDM in fig 2. It is clear that the
universe accelerates in the present epoch in all the three models. A more
rigorous analyze shows that the evidence for an accelerating universe is
fairly strong (more than 5 $\sigma$) sha . So we should investigate it
seriously.
$\Lambda$CDM is the most simple model for the acceleration, which is a
concordance model of several observations. As we shown in fig 1, the counters
of CMB, BAO, and SNe Ia have cross section, which almost laps over result of
the joint fittings. However, $\Lambda$CDM has its own theoretical problems.
Furthermore, it is found that a dynamical dark energy model fits the
observation data better. Especially, there are some evidences that the
equation of state (EOS) of dark energy may cross $-1$, which is a serious
challenge to the foundation of theoretical physics.
In the next section we shall study some problems of $\Lambda$CDM model and
display that a dynamical dark energy model is favored by observations. We’ll
focus on the crossing behavior implied by the observation. In section III, we
study 3 kinds of models with a crossing phantom divide dark energy. In section
IV, we present the conclusion and more references of this topic.
## II A dark energy with crossing $-1$ EOS is slightly favored by
observations
### II.1 The problems of $\Lambda$CDM
$\Lambda$CDM has two famous theoretical problems.
The first is the finetune problem. The effect of the vacuum energy can not be
distinguished from the cosmological constant in gravity theory. We can
calculate the vacuum energy by a well-constructed theory, quantum field theory
(QFT), which says that the vacuum energy should be larger than the observed
value by 122 orders of magnitude, if QFT works well up to the Planck scale. In
supersymmetric (SUSY) theory, the vacuum energy of the Bosons exactly
counteracts the vacuum energy of Fermions, such that we obtain a zero vacuum
energy. However, SUSY must break at the electro-weak scale. At that scale, the
vacuum energy is still large than the observed value by 60 orders of
magnitude. So for getting a vacuum energy we observed, we should introduce a
bare cosmological constant $\Lambda_{\rm bare}$. The effective vacuum energy
$\rho_{\rm effect}$ then becomes,
$\rho_{\rm effect}=\frac{1}{8\pi G}\Lambda_{\rm bare}+\rho_{\rm vacuum}.$ (15)
$\frac{1}{8\pi G}\Lambda_{\rm bare}$ and $\rho_{\rm vacuum}$ have to almost
counteract each other but do not exactly counteract each other, leaving a tiny
tail which is smaller than the $\rho_{\rm vacuum}$ by 60 orders of magnitude.
Which mechanism can realize such a miraculous counteraction?
The second problem is coincidence problem, which says that the cosmological
constant keeps a constant while the density of the dust evolves as $(1+z)^{3}$
in the history of the universe, then why do they approximately equal each
other at “our era”? Different from the first problem, the second problem says
the present ratio of dark energy and dark matter is sensitively depends on the
initial conditions. Essentially, the coincidence problem is the problem of an
unnatural initial condition. The densities of different species in the
universe redshift with different rate in the evolution of the universe, so if
their densities coincidence in $our~{}era$, their density ratio must be a
specific, tiny number in the $early~{}universe$. It is also a finetune
problem, but a finetune problem of the initial condition.
Except the above theoretical problems, $\Lambda$CDM also suffers from
observation problem, especially when faced to the fine structure of the
universe, including galaxies, clusters and voids. Some specific observations
differ from the predictions of $\Lambda$CDM (with standard partitions of dust
and cosmological constant) at a level of 2$\sigma$ or higher. Six observations
are summarized in problem lcdm : 1\. scale velocity flows is much larger than
the prediction of $\Lambda$CDM, 2\. Type Ia Supernovae (Sne Ia) at High
Redshift are brighter than what $\Lambda$CDM indicates, 3. the void seems more
empty than what $\Lambda$CDM predicts, 4\. the cluster haloes look denser than
what $\Lambda$CDM says, 5\. the density function of galaxy haloes is smooth,
while $\Lambda$CDM indicates a cusp in the core, 6\. there are too much disk
galaxies than the prediction of $\Lambda$CDM. We do not fully understand the
dynamics and galaxies and galaxy clusters, that is, the gravitational
perturbation theory at the small scale. The agreement may approve when we
advance our perturbation theory with a cosmological constant and the
simulation methods at the small scale. However, in the cosmological scale,
there are also some evidences that the dark energy is dynamical, including no.
2 of the previous 6 problems.
### II.2 crossing $-1$
With data accumulation, observations which favor dynamical dark energy become
more and accurate. Now we loose the condition that $p=-\rho$ for the exotic
matter (dark energy) which accelerates the universe. We go beyond the
$\Lambda$CDM model. We permit that the EOS of dark energy is not exactly equal
to $-1$, but still a constant. The fitting results by different samples of SNe
Ia are displayed in fig 3. We see that although a cosmological constant is
permitted, the dark energy whose EOS $<-1$ is favored by SNe Ia. The essence
whose EOS is less than $-1$ is called phantom, which can be realized by a
scalar field with negative kinetic term. The action for phantom $\psi$ is
$S_{\rm ph}=\int
d^{4}x\sqrt{-g}\left(\frac{1}{2}\partial_{\mu}\psi\partial^{\mu}\psi-U(\psi)\right),$
(16)
where $\sqrt{-g}$ is the determinate of the metric, the lowercase Greeks run
from 0 to 3, $U(\psi)$ denotes the potential of the phantom. In an FRW
universe, the density and pressure of the phantom (16) reduce to
$\rho=-\frac{1}{2}\dot{\psi}^{2}+U,$ (17) $p=-\frac{1}{2}\dot{\psi}^{2}-U,$
(18)
respectively. Now the EOS of the phantom $w$ is given by,
$w=\frac{p}{\rho}=\frac{-\frac{1}{2}\dot{\psi}^{2}-U}{-\frac{1}{2}\dot{\psi}^{2}+U},$
(19)
which is always less than $-1$ for a positive $U$. It seems that phantom is
proper candidate for the dark energy whose EOS less than $-1$ phantom . It is
very famous that a phantom field is unstable when quantized since the energy
has no lower bound. It will transit to a lower and lower energy state. In this
article we have no time to discuss this important topic for phantom dark
energy. We would point out the basic idea for this issue, which requires the
life time of the phantom is much longer than the age of the universe such that
we still have no chance to observe the decay of the phantom, though it is
fundamentally unstable. For references, see stablephan . A completely regular
quantum stress with $w<-1$ is suggested in phanquan .
Figure 3: The EOS of dark energy fitted by SNe Ia in a spatially flat
universe, the contours display 68.3 %, 95.4 % and 99.7% confidence level on
$w$ and $\Omega_{m0}$ ($\Omega_{m}$ in the figure). The results of the Union
set are shown as filled contours. The empty contours, from left to right, show
the results of the Gold sample, Davis 07, and the Union without SCP nearby
data. From union .
If the dark energy really behaves as phantom at some low redshift region, it
is an unusual discovery. But the dark energy may be more fantastic. In some
model-independent fittings, the EOS of dark energy crosses $-1$, which is a
really remarkable property and a serious challenge to our present theory of
fundamental physics.
Pioneer results of the crossing $-1$ of EOS of dark energy appeared in cross0
; cross1 . Fig 4 illuminates that the EOS of dark energy may cross $-1$ in
some low redshift. In fig 4, the Gold04 data are applied, a uniform prior of
$0.22\leq\Omega_{m0}\leq 0.38$ is assumed, and a spatially flat universe is
the working frame.
Figure 4: Panel (a): Uncorrelated band-power estimates of the EOS $w(z)$ of
dark energy by SNe Ia (Gold set gold04 ). Vertical error bars show the 1 and
2-$\sigma$ error bars (in blue and green, respectively). The horizontal error
bars denote the data bins used in cross1 . Panel (b): The window functions for
each bin from low redshift to high redshift. Panel (c): the likelihoods of
$w(z)$ in the bins from low redshift to high redshift. From cross1 .
The perturbation of the dark energy will growth if its EOS is not exactly $-1$
in the evolution history of the universe. Hence to fit a model with dynamical
dark energy with observation, the perturbation of the dark matter should be
considered in principle. Such a study was presented in cross2 , in which a
parametrization of the EOS of the dark energy with two constant
$w_{0},~{}w_{1}$ was applied,
$w=w_{0}+w_{1}\frac{z}{1+z}.$ (20)
The result is shown in fig 5.
Figure 5: Constraints on the EOS of w(z) by WMAP3 wmap3 and Gold04 gold04 .
The light grey region denotes 2 $\sigma$ constraint, while the dark grey for
1$\sigma$ constraint. The left panel shows the constraint with dark energy
perturbation, while the right displays the result without dark energy
perturbation. From cross2 .
We see from fig 5 that there is a mild tendency that the EOS of the dark
energy cross $-1$. For a more general parametrization of EOS for dark energy,
see zhupara .
With more and accurate data, the possibility of crossing $-1$ (phantom divide)
seems a little more specific, see for example crossmany . This crossing
behavior is a significant challenge for theoretical physics. It was proved
that the EOS of dark energy can not cross the phantom divide if 1. a dark
energy component with an arbitrary scalar-field Lagrangian, which has a
general dependence on the field itself and its first derivatives, 2\. general
relativity holds and 3. the spatially flat Friedmann universe nogo , for a
more detailed proof, wee the appendix of nogo2 . Thus realizing such a
crossing is not a trivial work. In the next section we investigate the
theoretical progresses for this extraordinary phenomenon.
## III three roads to cross the phantom divide
To cross the phantom divide, we must break at least one of the conditions in
nogo . Now that the dark energy behaves as quintessence at some stage , while
evolves as phantom at the other stage, a natural suggestion is that we should
consider a 2-field model, a quintessence and a phantom. The potential is
carefully chosen such that the quintessence dominates the universe at some
stage while the phantom dominates the universe at the other stage. It was
invented a name for such 2-field model, “quintom” . There are also some
varieties of quintom, such as hessence. We introduce these 2-field models in
the first subsection. The next road is to consider an interacting model, in
which the dark energy interacts with dark matter. The interaction can realize
the crossing behavior which is difficult for independent dark energy. We shall
study the interacting models in subsection B. The other possibility is that
general relativity fails at the cosmological scale. The ordinary dark energy
candidates, such as quintessence or phantom, can cross the phantom divide in a
modified gravity theory. We investigate this approach in subsection C.
### III.1 2-field model
A typical 2-field model is the quintom model, which was proposed in quintom1 ,
and was widely investigated later quintom2 . Generally , the action of a
universe with quintom dark energy $S$ is
$S=\int d^{4}x\sqrt{-g}\left(\frac{R}{16\pi G}+{\cal L_{\rm stuff}}\right),$
(21)
where $R$ is the Ricci scalar, $\cal L_{\rm stuff}$ encloses all kinds of the
stuff in the universe, for instance the dust matter, radiation, and quintom.
At the late universe, the radiation can be negligible. So, often we only
consider the dark energy, here quintom $\cal L_{\rm quintom}$, and dust matter
$\cal L_{\rm dm}$,
${\cal L_{\rm
quintom}}=-\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi+\frac{1}{2}\partial_{\mu}\psi\partial^{\mu}\psi-W(\phi,~{}\psi).$
(22)
In (22) the first term is the kinetic term of an ordinary scalar, the second
term is the kinetic term of a phantom, and $W(\phi,~{}\psi)$ is an arbitrary
function of $\phi$ and $\psi$. In an FRW universe, the density and pressure of
the quintom are
$\rho=\frac{1}{2}\dot{\phi}^{2}-\frac{1}{2}\dot{\psi}^{2}+W,$ (23)
$p=\frac{1}{2}\dot{\phi}^{2}-\frac{1}{2}\dot{\psi}^{2}-W.$ (24)
Hence, the EOS of the quintom $w$ is
$w=\frac{\frac{1}{2}\dot{\phi}^{2}-\frac{1}{2}\dot{\psi}^{2}-W}{\frac{1}{2}\dot{\phi}^{2}-\frac{1}{2}\dot{\psi}^{2}+W}.$
(25)
$w=-1$ requires
$\dot{\phi}^{2}=\dot{\psi}^{2}.$ (26)
We see that in a quintom model, we do not require a static field (a field with
zero kinetic term or a field at ground state) to get a cosmological constant.
We only need that $\psi$ and $\phi$ evolves in the same step. $w<-1$ implies,
$\dot{\phi}^{2}-\dot{\psi}^{2}<0,$ (27)
if
$\frac{1}{2}\dot{\phi}^{2}-\frac{1}{2}\dot{\psi}^{2}+W>0;$ (28)
and
$\dot{\phi}^{2}-\dot{\psi}^{2}>0,$ (29)
if
$\frac{1}{2}\dot{\phi}^{2}-\frac{1}{2}\dot{\psi}^{2}+W<0.$ (30)
(30) yields an unnatural physical result,that is, the density of dark energy
is negative. However, this is not as serious as the first glance, since we
have little knowledge of the dark energy besides its effect of gravitation.
Several evidences imply that we should go beyond the standard model of the
particle physics when we describe dark energy. There are a few dark energy
models permit density of the dark energy, or a component of it is negative (at
the same time keep the total density positive), for example, see negden1 ;
negden2 . But, for a model with only two components, a dust and a quintom, it
is difficult to set a negative density dark energy. In that case we need too
much dust than we observed or a big curvature term. In the following text of
this section, we only consider a dark energy with positive density. So $w>-1$
implies,
$\dot{\phi}^{2}-\dot{\psi}^{2}>0.$ (31)
In summary, if the kinetic term of the quintessence dominates that of phantom,
the quintom behaves as quintessence; else it behaves as phantom. We should
select a proper potential to make quintessence and phantom dominate
alternatively such that we can realize the crossing behavior.
A simple choice of the potential is that the quintessence and the phantom do
not interact with each other, which requires, $W(\phi,\psi)=V(\phi)+U(\psi)$.
The exponential potential is an important example which can be solved exactly
in the quintessence model (a toy universe only composed by quintessence). In
addition, we know that such exponential potentials of scalar fields occur
naturally in some fundamental theories such as string/M theories. We introduce
a model with such potentials in quintomguo , in which the potential
$V(\phi,\psi)$ is given by
$W(\phi,\psi)=V(\phi)+U(\psi)=A_{\phi}e^{-\lambda_{\phi}\kappa\phi}+A_{\psi}e^{-\lambda_{\psi}\kappa\psi},$
(32)
where $A_{\phi}$ and $A_{\psi}$ are the amplitude of the potentials,
$\kappa^{2}=8\pi G$, $\lambda_{\phi}$ and $\lambda_{\psi}$ are two constants.
Since there is no direct couple between the quintessence and the phantom, the
equations of motion of the quintessence and the phantom are two independent
equations,
$\ddot{\phi}+3H\dot{\phi}+\frac{dV}{d\phi}=0,$ (33)
$\ddot{\psi}+3H\dot{\psi}-\frac{dU}{d\psi}=0.$ (34)
The continuity equation of the dust reads,
$\rho_{\rm dust}+3H\rho_{\rm dust}=0,$ (35)
where $\rho_{\rm dust}$ denotes the density of the dust. The method of
dynamical system has been widely used in cosmology. This method can offer a
clear history of the cosmic evolution, especially the final states of the
university. For applying this method, first we define the following
dimensionless variables,
$\displaystyle x_{\phi}\equiv\frac{\kappa\dot{\phi}}{\sqrt{6}H}$ ,
$\displaystyle\quad y_{\phi}\equiv\frac{\kappa\sqrt{V_{\phi}}}{\sqrt{3}H},$
$\displaystyle x_{\psi}\equiv\frac{\kappa\dot{\psi_{i}}}{\sqrt{6}H}$ ,
$\displaystyle\quad y_{\psi}\equiv\frac{\kappa\sqrt{V_{\psi}}}{\sqrt{3}H},$
(36) $\displaystyle z\equiv\frac{\kappa\sqrt{\rho_{\rm dust}}}{\sqrt{3}H}$ ,
the evolution equations (33)-(35) become,
$\displaystyle x^{\prime}_{\phi}$ $\displaystyle=$
$\displaystyle-3x_{\phi}\left(1+x_{\phi}^{2}-x_{\psi}^{2}-\frac{1}{2}z^{2}\right)+\lambda_{\phi}\frac{\sqrt{6}}{2}y_{\phi}^{2}\,,$
(37) $\displaystyle y^{\prime}_{\phi}$ $\displaystyle=$ $\displaystyle
3y_{\phi}\left(-x_{\phi}^{2}+x_{\psi}^{2}+\frac{1}{2}z^{2}-\lambda_{\phi}\frac{\sqrt{6}}{6}x_{\phi}\right),$
(38) $\displaystyle x^{\prime}_{\psi}$ $\displaystyle=$
$\displaystyle-3x_{\psi}\left(1+x_{\phi}^{2}-x_{\psi}^{2}-\frac{1}{2}z^{2}\right)-\lambda_{\psi}\frac{\sqrt{6}}{2}y_{\psi}^{2}\,,$
(39) $\displaystyle y^{\prime}_{\psi}$ $\displaystyle=$ $\displaystyle
3y_{\psi}\left(-x_{\phi}^{2}+x_{\psi}^{2}+\frac{1}{2}z^{2}-\lambda_{\psi}\frac{\sqrt{6}}{6}x_{\psi}\right),$
(40) $\displaystyle z^{\prime}$ $\displaystyle=$ $\displaystyle
3z\left(-x_{\phi}^{2}+x_{\psi}^{2}+\frac{1}{2}z^{2}-\frac{1}{2}\right),$ (41)
in which a prime denotes derivative with respect to $\ln a$. Generally, $z$ in
the above set will not be confused with redshift. The five equations in this
system are not independent. They are constrained by Fridemann equation,
$H^{2}=\frac{\kappa^{2}}{3}\left(\frac{1}{2}\dot{\phi}^{2}+V-\frac{1}{2}\dot{\psi}^{2}+U+\rho_{\rm
dust}\right),$ (42)
which becomes
$x_{\phi}^{2}+y_{\phi}^{2}-x_{\psi}^{2}+y_{\psi}^{2}+z^{2}=1.$ (43)
with the dimensionless variables defined before. The critical points dwell at
$x_{\phi}^{\prime}=y_{\phi}^{\prime}=x_{\psi}^{\prime}=y_{\psi}^{\prime}=z^{\prime}=0$.
We present the result in table 1.
Label | $x_{\psi}$ | $y_{\psi}$ | $x_{\phi}$ | $y_{\phi}$ | z | Stability
---|---|---|---|---|---|---
$K$ | $-x_{\psi}^{2}+x_{\phi}^{2}=1$ | 0 | | 0 | 0 | unstable
$P$ | $-\frac{\lambda_{\psi}}{\sqrt{6}}$ | $\sqrt{(1+\frac{\lambda_{\psi}^{2}}{6})}$ | 0 | 0 | 0 | stable
$S$ | 0 | 0 | $\frac{\lambda_{\phi}}{\sqrt{6}}$ | $\sqrt{(1-\frac{\lambda_{\phi}^{2}}{6})}$ | 0 | unstable
$F$ | 0 | 0 | 0 | 0 | 1 | unstable
$T$ | 0 | 0 | $\frac{3}{\sqrt{6}\lambda_{\phi}}$ | $\frac{\sqrt{{3}}}{\lambda_{\phi}}$ | $\sqrt{1-\frac{3}{\lambda_{\phi}^{2}}}$ | unstable
Table 1: The critical points, from quintomguo
For detailed discussion of the critical points, see quintomguo . We would like
to show a numerical example in which the EOS of the quintom crosses the
phantom divide. Fig 6 illuminates that the EOS crosses $-1$.
Figure 6: The evolution of the effective equation of state of the phantom and
normal scalar fields with $W(\phi,\sigma)$ for the case $\lambda_{\phi}=1$.
From quintomguo .
The previous quintom model includes two fields, which are completely
independent and rather arbitrary. We can impose some symmetry in the quintom
model. An interesting model with an internal symmetry between the two fields
which work as dark energy is hessence hessence . Rather than two uncorrelated
fields, we consider one complex scalar field with internal symmetry between
the real and the imaginary parts,
$\Phi=\phi_{1}+i\phi_{2},$ (44)
with a Lagrangian density
${\cal L}_{\rm
hess}=-\frac{1}{4}\left[(\partial_{\mu}\Phi)^{2}+(\partial_{\mu}\Phi^{*})^{2}\right]-V(\xi,\Phi^{\ast})=-\frac{1}{2}\left[\,(\partial_{\mu}\xi)^{2}-\xi^{2}(\partial_{\mu}\theta)^{2}\,\right]-V(\xi),$
(45)
which is invariant under the transformation,
$\displaystyle\phi_{1}\to\phi_{1}\cos\alpha-i\phi_{2}\sin\alpha,$ (46)
$\displaystyle\phi_{2}\to-i\phi_{1}\sin\alpha+\phi_{2}\cos\alpha,$ (47)
if the potential is only a function of $\Phi^{2}+(\Phi^{*})^{2}$. For
convenience, in (45) we have introduced two new variables $(\xi,\theta)$,
$\phi_{1}=\xi\cosh\theta,~{}~{}~{}~{}~{}~{}~{}\phi_{2}=\xi\sinh\theta,$ (48)
which are defined by
$\xi^{2}=\phi_{1}^{2}-\phi_{2}^{2},~{}~{}~{}~{}~{}~{}~{}\coth\theta=\frac{\phi_{1}}{\phi_{2}}.$
(49)
The equations of motion of $\xi$ and $\theta$ are
$\ddot{\xi}+3H\dot{\xi}+\xi\dot{\theta}^{2}+\frac{dV}{d\xi}=0,$ (50)
$\xi^{2}\ddot{\theta}+(2\xi\dot{\xi}+3H\xi^{2})\dot{\theta}=0.$ (51)
Clearly, $\xi$ and $\theta$ couple to each other. The pressure and density of
the hessence read,
$p_{\rm
hess}=\frac{1}{2}\left(\dot{\xi}^{2}-\xi^{2}\dot{\theta}^{2}\right)-V(\xi),$
(52) $\rho_{\rm
hess}=\frac{1}{2}\left(\dot{\xi}^{2}-\xi^{2}\dot{\theta}^{2}\right)+V(\xi),$
(53)
respectively. The EOS of hessence, playing as dark energy,
$w=\frac{\frac{1}{2}\left(\dot{\xi}^{2}-\xi^{2}\dot{\theta}^{2}\right)-V(\xi)}{\frac{1}{2}\left(\dot{\xi}^{2}-\xi^{2}\dot{\theta}^{2}\right)+V(\xi)}.$
(54)
Qualitatively, hessence evolves as quintessence when
$\dot{\xi}^{2}\geq\xi^{2}\dot{\theta}^{2}$, while as phantom when
$\dot{\xi}^{2}<\xi^{2}\dot{\theta}^{2}$. The Lagrangian (45) does not include
$\theta$, hence the canonical momentum $\pi_{\theta}^{\mu}$ corresponding to
the cyclic coordinate $\theta$ are conserved quantities,
$\pi_{\theta}^{\mu}=\frac{\partial({\cal L}_{\rm
hess}\sqrt{-g})}{\partial(\partial_{\mu}\theta)}.$ (55)
In an FRW universe, only $\pi_{\theta}^{0}$ exists. We define a conserved
quantity $Q$ which is proportional to $\pi_{\theta}^{0}$,
$Q=a^{3}\xi^{2}\dot{\theta}.$ (56)
With this conserved quantity, the EOS becomes,
$w=\frac{\frac{1}{2}\dot{\xi}^{2}-\frac{Q^{2}}{2a^{6}\xi^{2}}-V(\xi)}{\frac{1}{2}\dot{\xi}^{2}-\frac{Q^{2}}{2a^{6}\xi^{2}}+V(\xi)},$
(57)
which is only a function of $\xi$. The Friedmann equations read as
$\displaystyle H^{2}=\frac{8\pi G}{3}\left[\rho_{\rm
dust}+\frac{1}{2}\left(\dot{\xi}^{2}-\xi^{2}\dot{\theta}^{2}\right)+V(\xi)\right],$
(58)
where $\rho_{\rm dust}$ is the energy density of dust. The continuity equation
of dust is (35). The continuity equations of hessence are identical to the
equations of motion (50) and (51). Then the system is closed and we present a
numerical example in fig 7. Evidently, the EOS of hessence, playing the role
of dark energy, crosses $-1$ at about $a=0.95$ ($z=0.06$).
Figure 7: The EOS of hessence $w$ as a function of scale factor with the
potential $V(\xi)=\lambda\xi^{4}$. The parameters for this plot are as
follows: $\Omega_{m0}=\rho_{m0}/(3H_{0}^{2})=0.3$, $\lambda=5.0$, $Q=1.0$,
$a_{0}=1$ and the unit $8\pi G=1$. From hessence .
After the presentation of hessence model, several aspects of this model have
been investigated, including to avoid the big rip wei2 , attractor solutions
for general hessence hess3 , reconstruction of hessence by recent observations
hess4 , dynamics of hessence in frame of loop quantum cosmology hess5 , and
holographic hessence modelhess6 .
### III.2 interacting model
Two-field model is a natural and obvious construction to realize the crossing
$-1$ behavior of dark energy. However, there are two many parameters in the
set-up, though we can impose some symmetries to reduce the parameters to a
smaller region. One symmetry decrease one parameter, but we have little clue
to impose the symmetries since we have no evidence in the ground labs.
Interaction is a universal phenomenon in the physics world. An interaction
term is helpful to cross the phantom divide. To illuminate this point, we
first carefully analyze the previous observations which imply the crossing.
Both the results of cross1 and cross2 , which are shown in fig 4 and fig 5
respectively, are derived with a presupposition, that is, the dark energy
evolves freely. In fact, what we observed is the effective EOS of the dark
energy in the sense of gravity at the cosmological scale. When we suppose it
evolves freely, we find that its EOS may cross the phantom divide. We can
demonstrate for an essence with (local) EOS$<-1$, the cosmological effective
EOS can cross $-1$ by aids of an interacting term. For the case with
interaction, the continuity equation for dark energy becomes,
$\dot{\rho}_{\rm de}+3H(\rho_{\rm de}+p_{\rm de})=-\Gamma,$ (59)
or
$\dot{\rho}_{\rm de}+3H(\rho_{\rm de}+p_{\rm de}+\frac{\Gamma}{3H})=0.$ (60)
Here $\rho_{\rm de}$ is the density of dark energy, $p_{\rm de}$ denotes the
local pressure measured in the lab (if we can measure), $\Gamma$ stands for
the interaction term, and $p_{\rm eff}=p_{\rm de}+\frac{\Gamma}{3H}$ is the
effective pressure in the cosmological sense. In a universe without expanding
or contracting, $H=0$, the interaction does no effect on the continuity
equation, or energy conservation law, and thus does not yield surplus pressure
222One may think that $\Gamma/H$ is meaningless when $H=0$. But in fact, in
most realistic cases, we always assume that $\Gamma$ is proportional to $H$..
Two special cases are interesting: 1\. $\frac{\Gamma}{3H}$ is a constant,
under which the interaction term contributes a constant pressure throughout
the history of the universe. 2. $\frac{\Gamma}{3H\rho_{de}}$ is a constant,
under which the interaction term contributes a constant EOS in the history of
the universe. In frame of a quintessence or phantom dark energy, the
interaction term $\frac{\Gamma}{3H\rho_{de}}$ only shifts the EOS up or down
by a constant distance in the $w-z$ plane, without changing the profile of the
curve of $w$. While the term $\frac{\Gamma}{3H}$ shifts the pressure, which
can change the EOS significantly since the density $\rho_{\rm de}$ is a
variable in the history of the universe.
If the dark energy can couple to some stuff of the universe, the dark matter
is the best candidate. Although non-minimal coupling between the dark energy
and ordinary matter fluids is strongly restricted by the experimental tests in
the solar system will , due to the unknown nature of the dark matter as part
of the background, it is possible to have non-gravitational interactions
between the dark energy and the dark matter components, without conflict with
the experimental data. The continuity equation for dust-like dark matter
reads,
$\dot{\rho}_{\rm dm}+3H\rho_{\rm dm}=\Gamma.$ (61)
Based on the previous discussion, we assume a most simple case
$\Gamma=H\delta\rho_{\rm dm},$ (62)
where $\delta$ is constant guointer ; weiinter ; amen . This interaction term
shifts a constant to the EOS of $dark~{}matter$, that is, it is no longer
evolving as $(1+z)^{3}$. We uniformly deal with quintessence and phantom,
which are often labeled by $X$, with a constant EOS $w_{X}$. So, the
continuity equation of dark energy can be written as,
$\dot{\rho}_{\rm X}+3H(\rho_{\rm X}+w_{X}\rho_{\rm X})=-H\delta\rho_{\rm dm}.$
(63)
Integrating (61), we derive
$\rho_{\rm dm}=\rho_{\rm dm0}a^{-3+\delta}=\rho_{\rm dm0}(1+z)^{3-\delta}.$
(64)
Substituting to (63), we reach
$\rho_{X}=\rho_{X0}(1+z)^{3(1+w_{X})}+\rho_{dm0}\frac{\delta}{\delta+3w_{X}}\left[(1+z)^{3(1+w_{X})}-(1+z)^{3-\delta}\right].$
(65)
Only from the above equation, we can extract the effective EOS of the dark
energy. To see this point, we make a short discussion. In a dynamical universe
with interaction, the effective EOS of dark energy reads,
$w_{de}=\frac{p_{\rm eff}}{\rho_{\rm de}}=\frac{p_{\rm
de}+\Gamma/3H}{\rho_{\rm de}}=-1+\frac{1}{3}\frac{d\ln\rho_{de}}{d\ln(1+z)}.$
(66)
Clearly, if $\frac{d\ln\rho_{de}}{d\ln(1+z)}$ is greater than 0, dark energy
evolves as quintessence; if $\frac{d\ln\rho_{de}}{d\ln(1+z)}$ is less than 0,
it evolves as phantom; if $\frac{d\ln\rho_{de}}{d\ln(1+z)}$ equals 0, it is
just cosmological constant. In a more intuitionistic way, if $\rho_{de}$
decreases and then increases with respect to redshift (or time), or increases
and then decreases, which implies that EOS of dark energy crosses phantom
divide. So, some time we directly use the evolution of density of dark energy
to describe the EOS of it. There is a more important motivation to use the
density directly: the density is more closely related to observables, hence is
more tightly constrained for the same number of redshift bins used wangyun .
The derivative of $\rho_{X}$ with respect to (1+z) reads,
$\frac{d\rho_{X}}{d(1+z)}=3(1+w_{X})\rho_{X0}(1+z)^{2+3w_{X}}+\rho_{dm0}\frac{\delta}{\delta+3w_{X}}\left[3(1+w_{X})(1+z)^{2+3w_{X}}-(3-\delta)(1+z)^{2-\delta}\right].$
(67)
If $\frac{d\rho_{X}}{d(1+z)}=0$ at some redshift $z=z_{c}$, the effective EOS
crosses $-1$. The result is illuminated by fig 8, in which we set $z_{c}=0.3$
as an example. This figure displays the corresponding $w_{X}$ when one fixes a
$\delta$, or vice versa if we require the EOS crosses $-1$ at $z_{c}=0.3$.
This is an original figure plotted for this review article.
Figure 8: $w_{X}$ vs $\delta$ under the condition
$\frac{d\rho_{X}}{d(1+z)}=0$.
Then the Friedmann equation reads,
$\frac{H^{2}}{H_{0}^{2}}=\Omega_{X0}(1+z)^{3(1+w_{X})}+\frac{1-\Omega_{X0}}{\delta+3w_{X}}\left[\delta(1+z)^{3(1+w_{X})}+3w_{X}(1+z)^{3-\delta}\right],$
(68)
where $\Omega_{\rm X0}=\kappa^{2}\rho_{\rm X0}/(3H_{0}^{2})$, and we have used
$\Omega_{\rm dm0}+\Omega_{\rm X0}=1$. Thus we need to constrain the three
parameters $\delta,\Omega_{\rm X0},w_{\rm X}$. The constraint result by SNLS
data is shown in fig 9. $\delta=0$ and $w_{X}=-1$ are indicated by the
horizontal and vertical dashed lines, which represent the non-interacting XCDM
model and interacting $\Lambda$CDM model, respectively.
Figure 9: Constraints of ($w_{X},\delta$) by SNLS data at 68.3%, 95.4% and
99.7% confidence levels marginalized over $\Omega_{X0}$ with priors
$\Omega_{X0}=0.72\pm 0.04$ and $\delta<3$. From guointer
From fig 8 and 9, we see that the observations leave enough space for the
parameters ($\delta$, $w_{X}$) to cross the phantom divide.
In the previous interacting model, we consider a phenomenological interaction,
which is put in “by hand”. We should find a more sound physical foundation for
the interactions. We will deduce an interaction term from the low energy limit
of string/M theory in the scenario of the interacting Chaplygin gas model
negden1 .
The Chaplygin gas model was suggested as a candidate of a unified model of
dark energy and dark matter cp . The Chaplygin gas is characterized by an
exotic equation of state
$p_{ch}=-A/\rho_{ch},$ (69)
where $A$ is a positive constant. The above equation of state leads to a
density evolution in the form
$\rho_{ch}=\sqrt{A+\frac{B}{a^{6}}},$ (70)
where $B$ is an integration constant. The attractive feature of the model is
that it naturally unifies both dark energy and dark matter. The reason is
that, from (70), the Chaplygin gas behaves as dust-like matter at early stage
and as a cosmological constant at later stage.
Though Chaplygin gas has such a nice property, it is a serious flaw when one
studies the fluctuation growth in Chaplygin gas model. It is found that
Chaplygin gas produces oscillations or exponential blowup of the matter power
spectrum, which is inconsistent with observations antiudm . So we turn to a
model that the Chaplygin gas only plays the role of dark energy. To cross the
phantom divide we consider a model in which the Chaplygin gas couples to dark
mater.
Although non-minimal coupling between the dark energy and ordinary matter
fluids is strongly restricted by the experimental tests in the solar system
will , due to the unknown nature of the dark matter as part of the background,
it is possible to have non-gravitational interactions between the dark energy
and the dark matter components, without conflict with the experimental data.
Thus, the observation constrain the only proper candidate to be coupled to
Chaplygin gas is dark matter.
We consider the original Chaplygin gas, whose pressure and energy density
satisfy the relation, $p_{ch}=-A/\rho_{ch}$. By assuming the cosmological
principle the continuity equations are written as
$\dot{\rho}_{ch}+3H\gamma_{ch}\rho_{ch}=-\Gamma,$ (71)
and
$\dot{\rho}_{dm}+3H\gamma_{dm}\rho_{dm}=\Gamma,$ (72)
where the subscript $dm$ denotes dark matter, and $\gamma$ is defined as
$\gamma=1+\frac{p}{\rho}=1+w,$ (73)
in which $w$ is the parameter of the state of equation, and $\gamma_{dm}=1$
throughout the evolution of the universe, whereas $\gamma_{ch}$ is a variable.
$\Gamma$ is the interaction term between Chaplygin gas and dark matter. Since
there does not exist any microphysical hint on the possible nature of a
coupling between dark matter and Chaplygin gas (as dark energy), the
interaction terms between dark energy and dark matter are rather arbitrary in
literatures inter . Here we try to present a possible origin from fundamental
field theory for $\Gamma$.
Whereas we are still lack of a complete formulation of unified theory of all
interactions (including gravity, electroweak and strong), there at present is
at least one hopeful candidate, string/M theory. However, the theory is far
away from mature such that it is still not known in a way that would enable us
to ask the questions about space-time in a general manner, say nothing of the
properties of realistic particles. Instead, we have to either resort to the
effective action approach which takes into account stringy phenomena in
perturbation theory, or we could study some special classes of string
solutions which can be formulated in the non-perturbative regime. But the
latter approach is available only for some special solutions, most notably the
BPS states or nearly BPS states in the string spectrum: They seems to have no
relation to our realistic Universe. Especially, there still does not exist a
non-perturbative formulation of generic cosmological solutions in string
theory. Hence nearly all the investigations of realistic string cosmologies
have been carried out essentially in the effective action range. Note that the
departure of string-theoretic solutions away from general relativity is
induced by the presence of additional degrees of freedom which emerge in the
massless string spectrum. These fields, including the scalar dilaton field,
the torsion tensor field, and others, couple to each other and to gravity non-
minimally, and can influence the dynamics significantly. Thus such an
effective low energy string theory deserve research to solve the dark energy
problem. There a special class of scalar-tensor theories of gravity is
considered to avoid singularities in cosmologies in st . The action is written
below,
$\displaystyle S_{st}=\int d^{4}x\sqrt{-g}\left[\frac{1}{16\pi
G}R-\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi+\frac{1}{q(\phi)^{2}}L_{\rm
dm}(\xi,\partial\xi,q^{-1}g_{\mu\nu})\right],$ (74)
where $G$ is the Newton gravitational constant, $\phi$ is a scalar field,
$L_{\rm dm}$ denotes Lagrangian of matter , $\xi$ represents different matter
degrees of matter fields, $q$ guarantees the coupling strength between the
matter fields and the dilaton. With action (74), the interaction term can be
written as follow st ,
$\Gamma=H\rho_{\rm dm}\frac{d\ln q^{\prime}}{d\ln a}.$ (75)
Here we introduce new variable $q(a)^{\prime}\triangleq q(a)^{(3w_{n}-1)/2}$,
where $a$ is the scale factor in standard FRW metric. By assuming
$q^{\prime}(a)=q_{0}e^{3\int c(\rho_{\rm dm}+\rho_{\xi})/\rho_{\rm dm}d\ln
a},$ (76)
where $\rho_{\rm dm}$ and $\rho_{\xi}$ are the densities of dark matter and
the scalar field respectively, one arrive at the interaction term,
$\Gamma=3Hc(\rho_{\rm dm}+\rho_{\phi}).$ (77)
With this interaction form we study the equation set (71) and (72). Set
$s=-\ln(1+z)$, $\Gamma=3Hc(\rho_{ch}+\rho_{dm})$,
$u=(3H_{0}^{2})^{-1}(3\mu^{2})^{-1}\rho_{dm}$,
$v=(3H_{0}^{2})^{-1}(3\mu^{2})^{-1}\rho_{ch}$,
$A^{\prime}=A(3H_{0}^{2})^{-2}(3\mu^{2})^{-2}$, where $c$ is a constant
without dimension. Using these variables, (71) and (72) reduce to
$\frac{du}{ds}=-3u+3c(u+v),$ (78) $\frac{dv}{ds}=-3(v-A^{\prime}/v)-3c(u+v).$
(79)
We note that the variable time does not appear in the dynamical system (78)
and (79) because time has been completely replaced by redshift $s=-ln(1+z)$.
The critical points of dynamical system (78) and (79) are given by
$\frac{du}{ds}=\frac{dv}{ds}=0.$ (80)
The solution of the above equation is
$\displaystyle u_{c}=\frac{c}{1-c}v_{c},$ (81) $\displaystyle
v_{c}^{2}=(1-c)A^{\prime}.$ (82)
We see the final state of the model contains both Chaplygin gas and dark
matter of constant densities if the singularity is stationary. The final state
satisfies perfect cosmological principle: the universe is homogeneous and
isotropic in space, as well as constant in time. Physically $\Gamma$ in (72)
plays the role of matter creation term $C$ in the theory of steady state
universe at the future time-like infinity. Recall that $c$ is the coupling
constant, may be positive or negative, corresponds the energy to transfer from
Chaplygin gas to dark matter or reversely. $A^{\prime}$ must be a positive
constant, which denotes the final energy density if $c$ is fixed. Also we can
derive an interesting and simple relation between the static energy density
ratio
$c=\frac{r_{s}}{1+r_{s}},$ (83)
where
$r_{s}=\lim_{z\to-1}\frac{\rho_{dm}}{\rho_{ch}}.$ (84)
To investigate the properties of the dynamical system in the neighbourhood of
the singularities, impose a perturbation to the critical points,
$\displaystyle\frac{d(\delta u)}{ds}=-3\delta u+3c(\delta u+\delta v),$ (85)
$\displaystyle\frac{d(\delta v)}{ds}=-3(\delta
v+\frac{A^{\prime}}{v_{c}^{2}}\delta v)-3c(\delta u+\delta v).$ (86)
The eigen equation of the above linear dynamical system $(\delta u,\delta v)$
reads
$(\lambda/3)^{2}+(2+\frac{1}{1-c})\lambda/3+2-2c^{2}=0,$ (87)
whose discriminant is
$\Delta=[(1-c)^{4}+(3/2-c)^{2}]/(1-c)^{2}\geq 0.$ (88)
Therefore both of the two roots of eigen equation (87) are real, consequently
centre and focus singularities can not appear. Furthermore only
$r_{s}\in(0,\infty)$, such that $c\in(0,1)$, makes physical sense. Under this
condition it is easy to show that both the two roots of (87) are negative.
Hence the two singularities are stationary. However it is only the property of
the linearized system (85) and (86), or the property of orbits of the
neighbourhoods of the singularities, while global Poincare-Hopf theorem
requires that the total index of the singularities equals the Euler number of
the phase space for the non-linear system (78) and (79). So there exists other
singularity except for the two nodes. In fact it is a non-stationary saddle
point at $u=0,~{}v=0$ with index $-1$. This singularity has been omitted in
solving equations (78) and (79). The total index of the three singularities is
$1$, which equals the Euler number of the phase space of this plane dynamical
system. Hence there is no other singularities in this system. From these
discussions we conclude that the global outline of the orbits of this non-
linear dynamical system (78) and (79) is similar to the electric fluxlines of
two negative point charges. Here we plot figs 10 and 11 to show the properties
of evolution of the universe controlled by the dynamical system (78) and (79).
As an example we set $c=0.2,~{}~{}A^{\prime}=0.9$ in figs 10-12.
Figure 10: The plane v versus u. (a) left panel: We consider the evolution of
the universe from redshift $z=e^{2}-1$. The initial condition is taken as
$u=0,~{}v=400$; $u=50,~{}v=350$; $u=100,~{}v=300$; $u=120,~{}v=280$ on the
four orbits, from the left to the right, respectively. It is clear that there
is a stationary node, which attracts most orbits in the first quadrant. At the
same time the orbits around the neibourhood of the singularity is not shown
clearly. (b) right panel: Orbit distributions around the node
$u_{c}=v_{c}c/(1-c),~{}v_{c}=\sqrt{(1-c)A^{\prime}}$. From negden1
Figure 11: The plane v versus u. (a) left panel: To show the global properties
of dynamical system (78) and (79) we have to include some “unphysical ”
initial conditions, such as $u=-100,~{}v=-300$, except for physical initial
conditions which have been shown in figure 10. (b) right panel: Orbits
distributions around the nodes. The two nodes
$u_{c}=v_{c}c/(1-c),~{}v_{c}=\sqrt{(1-c)A^{\prime}}$ and
$u_{c}=v_{c}c/(1-c),~{}v_{c}=-\sqrt{(1-c)A^{\prime}}$ keep reflection symmetry
about the original point. Just as we have analyzed, we see that the orbits of
this dynamical system are similar to the electric fluxlines of two negative
point charges. From negden1
Further, to compare with observation data we need the explicit forms of $u(x)$
and $v(x)$, especially $v(x)$. We need the properties of $\gamma_{ch}$ in our
model, which is contained in $v(x)$, to compare with observations. Eliminate
$u(x)$ by using (78) and (79) we derive
$\displaystyle\frac{1}{3c}\frac{d^{2}v}{ds^{2}}+[1+(1+A^{\prime}/v^{2})/c]\frac{dv}{ds}+3cv+3(1-c)\left\\{v+\left[\frac{dv}{ds}+3(v-A^{\prime}/v)\right]/(3c)\right\\}=0,$
(89)
which has no analytic solution. We show some numerical solutions in figure 12.
We find that for proper region of parameter spaces, the effective equation of
state of Chaplygin gas crosses the phantom divide successfully.
Figure 12: v versus s. The evolution of $v$ with different initial conditions
$u(-2)=0,~{}v(-2)=400$; $u(-2)=50,~{}v(-2)=350$; $u(-2)=100,~{}v(-2)=300$
reside on the blue, red, and yellow curves, respectively. Obviously the energy
density of Chaplygin gas rolls down and then climbs up in some low redshift
region. So the Chaplygin gas dark energy can cross the phantom divide $w=-1$
in a fitting where the dark energy is treated as an independent component to
dark matter. From negden1 , this figure has been re-plotted.
Up to now all of our results do not depend on Einstein field equation. They
only depend on the most sound principle in physics, that is, the continuity
principle, or the energy conservation law. Different gravity theories
correspond to different constraints imposed on our previous discussions. Our
improvements show how far we can reach without information of dynamical
evolution of the universe.
(78) illuminates that the dark matter in this interacting model does not
behaves as dust. Qualitatively, the dark matter gets energy from dark energy
for a positive $c$, and becomes soft, ie, its energy density decreases slower
than $(1+z)^{3}$ in an expanding universe. The parameter which carries the
total effects of cosmic fluids is the deceleration parameter $q$. From now on
we introduce the Friedmann equation of the standard general relativity. As a
simple case we study the evolution of $q$ in a spatially flat universe. So $q$
reads
$q=-\frac{\ddot{a}a}{\dot{a}^{2}}=\frac{1}{2}\left(\frac{u+v-3A^{\prime}/v^{2}}{u+v}\right),$
(90)
and density of Chaplygin gas $u$ and density of dark matter $v$ should satisfy
$u(0)+v(0)=1.$ (91)
And then Friedmann equation ensures the spatial flatness in the whole history
of the universe. Before analyzing the evolution of $q$ with redshift, we first
study its asymptotic behaviors. When $z\to\infty$, $q$ must go to $1/2$
because both Chaplygin gas and dark matter behave like dust , while when
$z\to-1$ $q$ is determined by
$\lim_{z\to-1}q=\frac{1}{2}\left(\frac{u_{c}+v_{c}-3A^{\prime}/v_{c}^{2}}{u_{c}+v_{c}}\right).$
(92)
One can finds the parameters $c=0.2,~{}A^{\prime}=0.9$ are difficult to
content the previous constraint Friedmann constraint (91). Here we carefully
choose a new set of parameter which satisfies Friedmann constraint (91), say,
$A^{\prime}=0.4,~{}c=0.06$. Therefore we obtain
$\lim_{z\to-1}q=-1.95,$ (93)
by using (81) and (82). Then we plot figure 13 to clearly display the
evolution of $q$. One can check
$u(0)=0.25,~{}v(0)=0.75;~{}u(0)=0.28,~{}v(0)=0.72;~{}u(0)=0.3,~{}v(0)=0.7$,
respectively on the curves $v(-2)=273;~{}v(-2)=250;~{}v(-2)=233$. One may find
an interesting property of the deceleration parameter displayed in fig 13: the
bigger the proportion of the dark energy, the smaller the absolute value of
the deceleration parameter. The reason roots in the extraordinary state of
Chaplygin gas (69), in which the pressure $p_{ch}$ is inversely proportional
to the energy density $\rho_{ch}$.
Figure 13: q versus s. The evolution of $q$ with different initial conditions
$u(-2)=0,~{}v(-2)=273$; $u(-2)=15,~{}v(-2)=250$; $u(-2)=25,~{}v(-2)=233$,
reside on the blue, red, and yellow curves, respectively. Evidently the
deceleration parameter $q$ of Chaplygin gas rolls down and crosses $q=0$ in
some low redshift region. The transition from deceleration phase to
acceleration phase occurs at $z=0.18;~{}z=0.21;~{}z=0.23$ to the curves
$u(-2)=0,~{}v(-2)=273$; $u(-2)=15,~{}v(-2)=250$; $u(-2)=25,~{}v(-2)=233$,
respectively. One finds $-q\thickapprox 0.5\sim 0.6$ at $z=0$, which is well
consistent with observations. From negden1 , this figure has been re-plotted.
Also we note that maybe an FRW universe with non-zero spatial curvature fits
deceleration parameter better than spatially flat FRW universe. This point
deservers to research further.
After the presentation of the original interacting Chaplygin gas model, there
are several generalizations. For details of these generalizations, see
generalcp .
### III.3 model in frame of modified gravity
The judgement that there exists an exotic component with negative pressure, or
dark energy, which accelerates the universe, is derived in frame of general
relativity. The validity of general relativity has been well tested from the
scale of millimeter to the scale of the solar system. Beyond this scale, the
evidences are not so sound. So we should not be surprised if general
relativity fails at the scale of the Hubble radius. Surely, any new gravity
theory must reduce to general relativity at the scale between millimeter to
the solar system. In frame of the new gravity theories, the cosmic
acceleration may be a natural result even we only have dust in the universe.
There are various suggestions on how to modify general relativity. In this
brief review we concentrates on the brane world theory. Inspired by the
developments of string/M theory, the idea that our universe is a 3-brane
embedded in a higher dimensional spacetime has received a great deal of
attention in recent years. In this brane world scenario, the standard model
particles are confined on the 3-brane, while the gravitation can propagate in
the whole space. In this picture, the gravity field equation gets modified at
the left hand side (LHS) in (4), while the dark energy is a stuff put at the
right hand side (RHS) in (4). In the modified gravity model, the surplus
geometric terms respective to the Einstein tensor play the role of the dark
energy in general relativity.
We consider a 3-brane imbedded in a 5-dimensional bulk. The action includes
the action of the bulk and the action of the brane,
$S=S_{\rm bulk}+S_{\rm brane}.$ (94)
Here
$S_{\rm bulk}=\int_{\cal M}d^{5}X\sqrt{-{g_{5}}}{\cal L}_{\rm bulk},$ (95)
where $X=(t,z,x^{1},x^{2},x^{3})$ is the bulk coordinate, $x^{1},x^{2},x^{3}$
are the coordinates of the maximally symmetric space. ${\cal M}$ denotes the
bulk manifold. The bulk Lagrangian can be
${\cal L}_{\rm bulk}={1\over 2\kappa_{5}^{2}}\left[R_{5}+\alpha
F(R_{5})\right]+{\cal L}_{\rm m}+\Lambda_{5},$ (96)
where ${g_{5}}$, $\kappa_{5}$, $R_{5}$, ${\cal L}_{\rm m}$, denote the bulk
manifold, the determinant of the bulk metric, the 5-dimensional Newton
constant, the 5-dimensional Ricci scalar, and the bulk matter Lagrangian,
respectively. $F(R_{5})$ denotes the higher order term of scalar curvature
$R_{5}$, the Ricci curvature $R_{5{\rm AB}}$, the Riemann curvature $R_{5{\rm
ABCD}}$.
There are too much possibilities and rather arbitrary to choose the higher
order terms. Generally the resulting equations of motion of such a term give
more than second derivatives of metric and the resulting theory is plagued by
ghosts. However there exists a combination of quadratic terms, called Gauss-
Bonnet term, which generates equation of motion without the terms more than
second derivatives of metric and the theory is free of ghosts earlygb .
Another important property of Gauss-Bonnet term is that, just like Hilbert
Lagrangian is a pure divergence in 2 dimensions and Einstein tensor identifies
zero in 1 and 2 dimensions, we have that in 4 or less dimension the Gauss-
Bonnet Lagrangian is a pure divergence. We see the dilemma of quadratic term
in 4 dimensional theory: if we include it with non pure divergence we shall
confront ghosts; if we want to remove ghosts we get a pure divergence term. So
only in theories in more than 4 dimensional Gauss-Bonnet combination provides
physical effects. Moreover the Gauss-Bonnet term also appears in both low
energy effective action of Bosonic string theory stringgb1 and low energy
effective action of Bosonic modes of heterotic and type II super string theory
stringgb2 . An investigation into the effects of a Gauss-Bonnet term in the 5
dimensional bulk of brane world models is therefore well motivated. The Gauss-
Bonnet term in 5 dimension reads,
$F(R_{5})=R_{5}^{2}-4R_{5{\rm AB}}R_{5}^{\rm AB}+R_{5{\rm ABCD}}R_{5}^{\rm
ABCD}.$ (97)
The action of the brane can be written as,
$S_{\rm brane}=\int_{M}d^{4}x\sqrt{-g}\left({\kappa_{5}^{-2}}K+L_{\rm
brane}\right),$ (98)
where $M$ indicates the brane manifold, $g$ denotes the determinant of the
brane metric, $L_{\rm brane}$ stands for the Lagrangian confined to the brane,
and $K$ marks the trace of the second fundamental form of the brane.
$x=(\tau,x^{1},x^{2},x^{3})$ is the brane coordinate. Note that $\tau$ is not
identified with $t$ if the the brane is not fixed at a position in the extra
dimension $z=$constant. We will investigate the cosmology of a moving brane
along the extra dimension $z$ in the bulk, and such that $\tau$ is different
from $t$.
We set the Lagrangian confined to the brane as follows,
$L_{\rm brane}=\frac{1}{16\pi G}R-\lambda+L_{\rm m},$ (99)
where $\lambda$ is the brane tension and $L_{\rm m}$ denotes the ordinary
matter, such as dust and radiation, located at the brane. $R$ denotes the 4
dimensional scalar curvature term on the brane, which is an important one
except a Gauss-Bonnet term in the bulk. This induced gravity correction arises
because the localized matter fields on the brane, which couple to bulk
gravitons, can generate via quantum loops a localized four-dimensional world-
volume kinetic term for gravitons dgp .
Assuming there is a mirror symmetry in the bulk, we have the Friedmann
equation on the brane combinecos , see also cov ,
$\displaystyle{4\over r_{c}^{2}}\left[1+\frac{8}{3}\alpha\left(H^{2}+{k\over
a^{2}}+{U\over 2}\right)\right]^{2}\left(H^{2}+{k\over
a^{2}}-U\right)=\left(H^{2}+{k\over a^{2}}-\frac{8\pi
G}{3}(\rho+\lambda)\right)^{2},$ (100)
where
$U=-\frac{1}{4\alpha}\pm\frac{1}{4\alpha}\sqrt{1+4\alpha\left(\frac{\Lambda_{5}}{6}+\frac{M\kappa_{5}^{2}}{4\pi
a^{4}}\right)},$ (101) $r_{c}=\kappa_{5}^{2}\mu^{2}.$ (102)
Here $M$ is a constant, standing for the mass of bulk black hole. For various
limits of (100), see selfgbde .
For convenience, we introduce the following new variables and parameters,
$\displaystyle
x\equiv\frac{H^{2}}{H_{0}^{2}}+\frac{k}{a^{2}H_{0}^{2}}=\frac{H^{2}}{H_{0}^{2}}-\Omega_{k0}(1+z)^{2},$
$\displaystyle u\equiv\frac{8\pi\
^{(4)}G}{3H_{0}^{2}}(\rho+\lambda)=\Omega_{m0}(1+z)^{3}+\Omega_{\lambda},$
$\displaystyle m\equiv\frac{8}{3}\alpha H_{0}^{2},$ $\displaystyle
n\equiv\frac{1}{H_{0}^{2}r_{c}^{2}},$ $\displaystyle
y\equiv\frac{1}{2}UH_{0}^{-2}=\frac{1}{3m}\left(-1+\sqrt{1+\frac{4\alpha\Lambda_{5}}{6}+{\frac{8\alpha
M{}^{(5)}G}{a^{4}}}}\right)$
$\displaystyle~{}~{}~{}=\frac{1}{3m}\left(-1+\sqrt{1+m\Omega_{\Lambda_{5}}+m\Omega_{M0}(1+z)^{4}}\right),$
(103)
and we have assumed that there is only pressureless dust in the universe. As
before, we have used the following notations
$\Omega_{k0}=-\frac{k}{a_{0}^{2}H_{0}^{2}},~{}~{}\Omega_{m0}=\frac{8\pi
G}{3}\frac{\rho_{m0}}{H_{0}^{2}},~{}~{}\Omega_{\lambda}=\frac{8\pi
G}{3}\frac{\lambda}{H_{0}^{2}},~{}~{}\Omega_{\Lambda_{5}}=\frac{3\Lambda_{5}}{8H_{0}^{2}},~{}~{}\Omega_{M0}=\frac{3M\kappa_{5}^{2}}{8\pi
a_{0}^{4}H_{0}^{2}}.$ (104)
With these new variables and parameters, (100) can be rewritten as
$4n(x-2y)[1+m(x+y)]^{2}=(x-u)^{2}.$ (105)
This is a cubic equation of the variable $x$. According to algebraic theory it
has 3 roots. One can explicitly write down three roots. But they are too
lengthy and complicated to present here. Instead we only express those three
roots formally in the order given in Mathematica
$\displaystyle x_{1}=x_{1}(y,u|m,n),$ $\displaystyle x_{2}=x_{2}(y,u|m,n),$
$\displaystyle x_{3}=x_{3}(y,u|m,n),$ (106)
where $y$ and $u$ are two variables, $m$ and $n$ stand for two parameters. The
root on $x$ of the equation (105) gives us the modified Friedmann equation on
the Gauss-Bonnet brane world with induced gravity. From the solutions given in
(III.3), this model seems to have three branches. In addition, note that all
parameters introduced in (III.3) and (104) are not independent of each other.
According to the Friedmann equation (III.3), when all variables are taken
current values, for example, $z=0$, the Friedmann equation will give us a
constraint on those parameters,
$1=f(\Omega_{k0},~{}\Omega_{m0},~{}\Omega_{M0},~{}\Omega_{\Lambda_{5}},~{}\Omega_{\lambda},~{}m,~{}n).$
(107)
To compare with observation, we introduce the concept “equivalent dark energy”
or “virtual dark energy” in the modified gravity models, since almost all the
properties of dark energy are deduced in the frame of general relativity with
a dark energy.
The Friedmann equation in the four dimensional general relativity can be
written as
$H^{2}+\frac{k}{a^{2}}=\frac{8\pi G}{3}(\rho+\rho_{de}),$ (108)
where the first term of RHS of the above equation represents the dust matter
and the second term stands for the dark energy. Generally speaking the Bianchi
identity requires,
$\frac{d\rho_{de}}{dt}+3H(\rho_{de}+p_{de})=0,$ (109)
we can then express the equation of state for the dark energy as
$w_{de}=\frac{p_{de}}{\rho_{de}}=-1-\frac{1}{3}\frac{d\ln\rho_{de}}{dlna}.$
(110)
Note that we can rewrite the Friedmann equation (III.3) in the form of (108)
as
$xH_{0}^{2}=\frac{8\pi G}{3}\rho+\left(H_{0}^{2}x(y,u|m,n)-\frac{8\pi
G}{3}\rho\right)=\frac{8\pi G}{3}(\rho+Q),$ (111)
where $\rho$ is the energy density of dust matter on the brane and the term
$Q\equiv\frac{3H_{0}^{2}}{8\pi G}x(y,u|m,n)-\rho$ (112)
corresponds to $\rho_{de}$ in (108).
In fig 14 we show the equation of state for the virtual dark energy when we
take $m=1.036$ and $n=0.04917$. In this case, from the constraint equation
(128), one has $\Omega_{M0}=2.08$. From the figure we see that $w_{eff}<-1$ at
$z=0$ and
$\left.\frac{dw_{eff}}{dz}\right|_{z=0}<0.$
Therefore the equation of state for the virtual dark energy can indeed cross
the phantom divide $w=-1$ near $z\sim 0$.
Figure 14: The equation of state $w_{eff}$ with respect to the red shift
$1+z$, with $\Omega_{m0}=0.28$ and $\Omega_{A}=2.08$. From selfgbde .
Fig 14 illuminates that the behavior of the virtual dark energy seems rather
strange. However we should remember that it is only virtual dark energy, not
actual stuff. The whole evolution of the universe is described by the Hubble
parameter. We plot the Hubble parameter $H$ corresponding to fig 14 in fig 15.
Figure 15: $H^{2}/H_{0}^{2}$ versus $1+z$, with $\Omega_{m0}=0.28$ and
$\Omega_{A}=2.08$. From selfgbde .
Fig 15 displays that the universe will eventually becomes a de Sitter one. For
more figures with different parameters, see selfgbde . The constraint of this
brane model with induced scalar term on the brane and Gauss-Bonnet term in the
bulk has been investigated in DGPGBcon .
The above is an example of “pure geometric” dark energy. We can also consider
some mixed dark energy model, ie, the cosmic acceleration is driven by an
exotic matter and some geometric effect in part. Why such an apparently
complicated suggestion? There are many interesting models are proposed to
explain the cosmic acceleration, including dark energy and modified gravity
models. However, several influential and hopeful models, such as quintessence
and DGP model, fundamentally can not account for the crossing $-1$ behavior of
dark energy. By contrast, some hybrid model of the dark energy and modified
gravity may realize such a crossing. As an example we study the quintessence
and phantom in frame of DGP self3 .
Our starting point is still action (94). In a DGP model with a scalar, (96)
becomes a pure Einstein-Hilbert action,
${\cal L}_{\rm bulk}={1\over 2\kappa_{5}^{2}}R_{5},$ (113)
and we add a scalar term in (99),
$L_{\rm brane}=\frac{1}{16\pi G}R+L_{\rm m}+L_{\rm scalar}.$ (114)
Here the scalar term can be ordinary scalar (quintessence) or phantom (scalar
with negative kinetic term). The Lagrangian of a quintessence reads,
$L_{\phi}=-\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-V(\phi),$ (115)
and for phantom,
$L_{\psi}=\frac{1}{2}\partial_{\mu}\psi\partial^{\mu}\psi-U(\psi).$ (116)
In an FRW universe we have
$\displaystyle\rho_{\phi}=\frac{1}{2}\dot{\phi}^{2}+V(\phi),$ (117)
$\displaystyle p_{\phi}=\frac{1}{2}\dot{\phi}^{2}-V(\phi).$ (118)
The exponential potential is an important example which can be solved exactly
in the standard model. Also it has been shown that the inflation driven by a
scalar with exponential potential can exit naturally in the warped DGP model
hs1 . It is therefore quite interesting to investigate a scalar with such a
potential in late time universe on a DGP brane. Here we set
$V=V_{0}e^{-\lambda_{1}\frac{\phi}{\mu}}.$ (119)
Here $\lambda_{1}$ is a constant and $V_{0}$ denotes the initial value of the
potential.
The Friedmann equation (100) becomes
$\displaystyle
H^{2}+\frac{k}{a^{2}}=\frac{1}{3\mu^{2}}\left[\rho+\rho_{0}+\theta\rho_{0}(1+\frac{2\rho}{\rho_{0}})^{1/2}\right],$
(120)
where
$\rho_{0}=\frac{6\mu^{2}}{r_{c}^{2}}.$ (121)
Similar to the previous case, we derive the virtual dark energy by comparing
(120) and (108),
$\rho_{de}=\rho_{\phi}+\rho_{0}+\theta\rho_{0}\left[\rho+\rho_{0}+\theta\rho_{0}(1+\frac{2\rho}{\rho_{0}})^{1/2}\right].$
(122)
From (110), we calculate the derivation of effective density of dark energy
with respective to $\ln(1+z)$ for a ordinary scalar,
$\displaystyle\frac{d\rho_{de}}{d\ln(1+z)}=3[\dot{\phi}^{2}+\theta(1+\frac{\dot{\phi}^{2}+2V+2\rho_{dm}}{\rho_{0}})^{-1/2}(\dot{\phi}^{2}+\rho_{dm})].$
(123)
If $\theta=1$, both terms of RHS are positive, hence it never goes to zero at
finite time. But if $\theta=-1$, the two terms of RHS carry opposite sign,
therefore it is possible that the EOS of dark energy crosses phantom divide.
In a scalar-driven DGP, we only consider the case of $\theta=-1$.
For convenience, we define some dimensionless variables,
$\displaystyle y_{1}$ $\displaystyle\triangleq$
$\displaystyle\frac{\dot{\phi}}{\sqrt{6}\mu H},$ (124) $\displaystyle y_{2}$
$\displaystyle\triangleq$ $\displaystyle\frac{\sqrt{V}}{\sqrt{3}\mu H},$ (125)
$\displaystyle y_{3}$ $\displaystyle\triangleq$
$\displaystyle\frac{\sqrt{\rho_{m}}}{\sqrt{3}\mu H},$ (126) $\displaystyle
y_{4}$ $\displaystyle\triangleq$
$\displaystyle\frac{\sqrt{\rho_{0}}}{\sqrt{3}\mu H}.$ (127)
The Friedmann equation (120) becomes
$y_{1}^{2}+y_{2}^{2}+y_{3}^{2}+y_{4}^{2}-y_{4}^{2}\left(1+2\frac{y_{1}^{2}+y_{2}^{2}+y_{3}^{2}}{y_{4}^{2}}\right)^{1/2}=1.$
(128)
The stagnation point, that is, $d\rho_{de}/d\ln(1+z)=0$ dwells at
$\frac{y_{4}}{\sqrt{2}+y_{4}}\left(2+\frac{y_{3}^{2}}{y_{1}^{2}}\right)=2,$
(129)
which can be derived from (123) and (128).
One concludes from the above equation that a smaller $r_{c}$, a smaller
$\Omega_{m0}$ (Recall that it is defined as the present value of the energy
density of dust matter over the critical density), or a larger $\Omega_{ki}$
(which is defined as the present value of the kinetic energy density of the
scalar over the critical density) is helpful to shift the stagnation point to
lower redshift region. We show a concrete numerical example of this crossing
behaviours in fig 16. For convenience we introduce the dimensionless density
and rate of change with respect to redshift of dark energy as below,
$\displaystyle\beta=\frac{\rho_{de}}{\rho_{c}}=\frac{\Omega_{r_{c}}}{b^{2}}\left[y_{1}^{2}+y_{2}^{2}+y_{4}^{2}-y_{4}^{2}(1+2\frac{y_{1}^{2}+y_{2}^{2}+y_{3}^{2}}{y_{4}^{2}})^{1/2}\right],$
(130)
where $\rho_{c}$ denotes the present critical density of the universe, and
$\displaystyle\gamma$ $\displaystyle=$
$\displaystyle\frac{1}{\rho_{c}}\frac{y_{4}^{2}}{\Omega_{r_{c}}}\frac{d\rho_{de}}{ds}$
(131) $\displaystyle=$ $\displaystyle
3\left[(1+2\frac{y_{1}^{2}+y_{2}^{2}+y_{3}^{2}}{y_{4}^{2}})^{-1/2}(2y_{1}^{2}+y_{3}^{2})-2y_{1}^{2}\right].$
A significant parameters from the viewpoint of observations is the
deceleration parameter $q$, which carries the total effects of cosmic fluids.
We plot $q$ in these figures for corresponding density curve of dark energy.
In the fig 16 we set $\Omega_{m}=0.3$. $\Omega_{r_{c}}$ is defined as the
present value of the energy density of $\rho_{0}$ over the critical density
$\Omega_{r_{c}}={\rho_{0}}/{\rho_{c}}$.
Figure 16: For this figure, $\Omega_{ki}=0.01$, $\Omega_{r_{c}}=0.01$,
$\lambda_{1}=0.5$. (a) The left panel: $\beta$ and $\gamma$ as functions of
$s$, in which $\beta$ resides on the solid line, while $\gamma$ dwells at the
dotted line. The EOS of dark energy crosses $-1$ at about $s=-0.22$, or
$z=0.25$. (b) The right panel: the corresponding deceleration parameter, which
crosses 0 at about $s=-0.40$, or $z=0.49$. From self3 .
Now, we turn to the evolution of a universe with a phantom (116) in DGP. In an
FRW universe the density and pressure of a phantom can be written as (17),
(18).
To compare with the results of the ordinary scalar, here we set a same
potential as before,
$U=U_{0}e^{-\lambda_{2}\frac{\psi}{\mu}}.$ (132)
The ratio of change of density of virtual dark energy with respective to
$\ln(1+z)$ becomes,
$\displaystyle\frac{d\rho_{de}}{d\ln(1+z)}=3[-\dot{\psi}^{2}+\theta(1+\frac{-\dot{\psi}^{2}+2U+2\rho_{dm}}{\rho_{0}})^{-1/2}(-\dot{\psi}^{2}+\rho_{dm})].$
(133)
To study the behaviour of the EOS of dark energy, we first take a look at the
signs of the terms of RHS of the above equation. $(-\dot{\psi}^{2}+\rho_{dm})$
represents the total energy density of the cosmic fluids, which should be
positive. The term $(1+\frac{-\dot{\psi}^{2}+2U+2\rho_{dm}}{\rho_{0}})^{-1/2}$
should also be positive. Hence if $\theta=-1$, both terms of RHS are negative:
it never goes to zero at finite time. Contrarily, if $\theta=1$, the two terms
of RHS carry opposite sign: the EOS of dark energy is able to cross phantom
divide. In the following of the present subsection we consider the branch of
$\theta=1$.
Now the Friedmann constraint becomes
$-y_{1}^{2}+y_{2}^{2}+y_{3}^{2}+y_{4}^{2}+y_{4}^{2}\left(1+2\frac{-y_{1}^{2}+y_{2}^{2}+y_{3}^{2}}{y_{4}^{2}}\right)^{1/2}=1.$
(134)
Again, one will see that in reasonable regions of parameters, the EOS of dark
energy crosses $-1$, but from below $-1$ to above $-1$.
The stagnation point of $\rho_{de}$ inhabits at
$\frac{y_{4}}{\sqrt{2}-y_{4}}\left(-2+\frac{y_{3}^{2}}{y_{1}^{2}}\right)=2,$
(135)
which can be derived from (133) and (134). One concludes from the above
equation that a smaller $r_{c}$, a smaller $\Omega_{m}$, or a larger
$\Omega_{ki}$ is helpful to shift the stagnation point to lower redshift
region, which is the same as the case of an ordinary scalar. Then we show a
concrete numerical example of the crossing behaviour of this case in fig 17.
The dimensionless density and rate of change with respect to redshift of dark
energy become,
$\displaystyle\beta=\frac{\rho_{de}}{\rho_{c}}=\frac{\Omega_{r_{c}}}{b^{2}}\left[-y_{1}^{2}+y_{2}^{2}+y_{4}^{2}+y_{4}^{2}(1+2\frac{-y_{1}^{2}+y_{2}^{2}+y_{3}^{2}}{y_{4}^{2}})^{1/2}\right],$
(136)
and
$\gamma=3\left[-(1+2\frac{-y_{1}^{2}+y_{2}^{2}+y_{3}^{2}}{y_{4}^{2}})^{-1/2}(-2y_{1}^{2}+y_{3}^{2})+2y_{1}^{2}\right].$
(137)
Similarly, the deceleration parameter is plotted in the figure 17 for
corresponding density curve of dark energy. In this figures we also set
$\Omega_{m}=0.3$.
Figure 17: For this figure, $\Omega_{ki}=0.01$, $\Omega_{r_{c}}=0.01$,
$\lambda=0.01$. (a) The left panel: $\beta$ and $\gamma$ as functions of $s$,
in which $\beta$ resides on the solid line, while $\gamma$ dwells at the
dotted line. The EOS of dark energy crosses $-1$ at about $s=-1.25$, or
$z=1.49$. (b) The right panel: The corresponding deceleration parameter, which
crosses 0 at about $s=-0.50$, or $z=0.65$. From self3 .
Fig 17 explicitly illuminates that the EOS of virtual dark energy crosses
$-1$, as expected. At the same time the deceleration parameter is consistent
with observations.
## IV summary
The recent observations imply that the EOS of dark energy may cross $-1$. This
is a remarkable phenomenon and attracts much theoretical attention.
We review three typical models for the crossing behavior. They are two-field
model, interacting model, and modified gravity model.
There are several other interesting suggestions in or beyond the three
categories mentioned above. We try to list them here for the future
researches. We are apologized for this incomplete reference list on this
topic.
Almost in all dark energy models the dark energy is suggested as scalar.
However, 3 orthogonal vectors can also play this role. For the interacting
vector dark energy and phantom divide crossing, see weivector . For the
suggestion of crossing the phantom divide with a spinor, see spin . Multiple
k-essence sources are helpful to fulfil the condition for phantom divide
crossing multike . Phantom divide crossing can be realized by non-minimal
coupling and Lorentz invariance violation lorevio . An exact solution of a
two-field model for this crossing has been found in exacttwo .
For previous interacting $X$ (quintessence or phantom) models with crossing
$-1$, see internew . The cosmology of interacting $X$ in loop gravity has been
studied in Xloop .
Interacting holographic dark energy is a possible mechanism for the phantom
divide crossing ihd . And the thermodynamics of interacting holographic dark
energy with phantom divide crossing is investigated in thermalihd . An
explicit model of $F(R)$ gravity in which the dark energy crosses the phantom
divide is reconstructed in f(R) . The phantom-like effects in a DGP-inspired
$F(R,\phi)$ gravity model is investigated in phandgp . Based on the recent
progress in studies of source of Taub space taubsource , a new braneworld in
the sourced-Taub background is proposed taubbrane , while the previous brane
world models are imbedded in AdS (RS) or Minkowski (DGP). In this model the
EOS for the virtual dark energy of a dust brane in the source region can cross
the phantom divide. For other suggestions in brane world model, see branecorss
.
Similar to the coincidence problem of dark energy, we can ask why the EOS
crosses $-1$ recently? This problem is studied in wei2coin .
On the observational side, the present data only mildly favor the crossing
behavior. We need more data to confirm or exclude it.
Theoretically, we should find more natural model which has less parameters. We
must go beyond the standard model of particle physics. The problem cosmic
acceleration is a pivotal problem to access new physics. To study the problem
of crossing $-1$ EOS will impel the investigation to the new Laws of nature.
Acknowledgments We thank to all the original authors who permit us to use
their figures. After submission of this invited review, Our special
thankfulness goes to Y. Cai, who informed me that their review on quintom
cosmology would appear soon. And Y Cai give us several beneficial suggestions.
Hence this article is slightly different from the published version.
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|
arxiv-papers
| 2009-09-16T13:53:48 |
2024-09-04T02:49:05.384583
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Hongsheng Zhang",
"submitter": "Hongsheng Zhang",
"url": "https://arxiv.org/abs/0909.3013"
}
|
0909.3089
|
# Non-stationary heat conduction in one-dimensional chains with conserved
momentum.
Oleg V. Gendelman Faculty of Mechanical Engineering, Technion – Israel
Institute of Technology, Haifa, Israel Alexander V. Savin Semenov Institute
of Chemical Physics, Russian Academy of Sciences, Moscow 117977, Russia
###### Abstract
The Letter addresses the relationship between hyperbolic equations of heat
conduction and microscopic models of dielectrics. Effects of the non-
stationary heat conduction are investigated in two one-dimensional models with
conserved momentum: Fermi-Pasta-Ulam (FPU) chain and chain of rotators (CR).
These models belong to different universality classes with respect to
stationary heat conduction. Direct numeric simulations reveal in both models a
crossover from oscillatory decay of short-wave perturbations of the
temperature field to smooth diffusive decay of the long-wave perturbations.
Such behavior is inconsistent with parabolic Fourier equation of the heat
conduction. The crossover wavelength decreases with increase of average
temperature in both models. For the FPU model the lowest order hyperbolic
Cattaneo-Vernotte equation for the non-stationary heat conduction is not
applicable, since no unique relaxation time can be determined.
###### pacs:
44.10.+i; 05.45.-a; 05.60.-k; 05.70.Ln
It is well-known that parabolic Fourier equation of heat conduction implies
infinite speed of the signal propagation and thus is inconsistent with
causality p1 ; p2 ; p3 ; p4 ; p5 . Numerous modifications were suggested to
recover the hyperbolic character of the heat transport equation p2 . Perhaps,
the most known is the lowest-order approximation known as Cattaneo-Vernotte
(CV) law p1 ; p2 . In its one-dimensional version it is written as
$(1+\tau\frac{\partial}{\partial t})\vec{q}=-\kappa\nabla T$ (1)
where $\kappa$ is standard heat conduction coefficient and $\tau$ is
characteristic relaxation time of the system. The latter can be of macroscopic
order p5 . Importance of the hyperbolic heat conduction models for description
of a nanoscale heat transfer has been recognized p6 ; p7 .
Only few papers dealt with numeric verification of such laws from the first
principles p8 . As it is well-known now from numerous numeric simulations and
few analytic results, the relationship between the microscopic structure and
applicability of the Fourier law for description of the stationary heat
conduction is highly nontrivial and depends both on size and dimensionality of
the model p9 . In particular, the heat conduction coefficient can diverge in
the thermodynamic limit. Hyperbolic equations describing the non-stationary
heat conduction inevitably include more empiric constants and therefore their
relationship to the microscopic models can be even less trivial. To the best
of our knowledge, no conclusive data exist in this respect.
This Letter deals with a study of spatial and temporal peculiarities of the
non-stationary heat conduction in two simple one-dimensional models with
conserved momentum – Fermi-Pasta-Ulam (FPU) chain and chain of rotators (CR).
From the viewpoint of the stationary heat conduction, these two systems are
known to belong to different universality classes. Namely, in the FPU chain
the heat conduction coefficient diverges with the size of the system p10 ,
whereas in the CR model it converges to a finite value p11 ; p12 ; p13 . So,
it is interesting to check whether other differences between these models
models will reveal themselves in the problem of non-stationary heat
conduction.
In order to investigate this process, one should choose the parameters to
measure. This question is not easy, since the situation in this problem is
different from the stationary heat conduction, where only one commonly
accepted macroscopic equation exists and only one empiric parameter should be
computed. The simplest CV law already has two independent coefficients,
whereas more elaborate approximations can include even more parameters. Just
because many different empiric equations exist, it is not desirable to pick
one of them ab initio and to fit the data to find particular set of constants.
Instead, it seems reasonable to look for some quantity which will characterize
the process of the non-stationary conduction and can be measured from the
simulations without relying on particular approximate equation. For this sake,
we choose the characteristic length which characterizes the scale at which the
nonstationarity effects are significant.
In order to explain the appearance of this scale, let us refer to 1D version
of the CV equation for the temperature:
$\tau\frac{\partial^{2}T}{\partial t^{2}}+\frac{\partial T}{\partial
t}=\alpha\frac{\partial^{2}T}{\partial x^{2}}$ (2)
where $\alpha$ is the temperature conduction coefficient.
Let us consider the problem of non-stationary heat conduction in a one-
dimensional specimen with periodic boundary conditions $T(L,t)=T(0,t)$, where
$T(x,t)$ is the temperature distribution, $L$ is the length of the specimen,
$t\geq 0$. If it is the case, one can expand the temperature distribution to
Fourier series:
$T(x,t)=\sum_{n=-\infty}^{\infty}a_{n}(t)\exp(2\pi inx/L)$ (3)
with $a_{n}(t)=a_{-n}^{*}(t)$, since $T(x,t)$ is real function.
Substituting (2) to (3), one obtains the equations for time evolution of the
modal amplitudes:
$\tau\ddot{a}_{n}+\dot{a}_{n}+4\pi^{2}n^{2}\alpha a_{n}/L^{2}=0.$ (4)
Solutions of Eq. (4) are written as:
$\begin{array}[]{l}{a_{n}(t)=C_{1n}\exp(\lambda_{1}t)+C_{2n}(\lambda_{2}t)}\\\
{\lambda_{1,2}=\left(-1\pm\sqrt{1-{16\pi^{2}n^{2}\alpha\tau}/{L^{2}}}\right)/2}\end{array}$
(5)
where $C_{1n}$ and $C_{2n}$ are constants determined by the initial
distribution.
¿From (5) it immediately follows that for sufficiently short modes the
temperature profile will relax in oscillatory manner:
$\displaystyle n>L/4\pi\sqrt{\alpha\tau},$ $\displaystyle
a_{n}(t)\sim\exp(-t/2\tau)\exp(i\omega_{n}t),$ (6)
$\displaystyle\omega_{n}=\left(\sqrt{{16\pi^{2}n^{2}\alpha\tau}/{L^{2}}-1}\right)/2\tau$
If the specimen is rather long ($L>>4\pi\sqrt{\alpha\tau}$) then for small
wavenumbers (acoustic modes):
$\lambda_{1}\approx-1/\tau,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \lambda_{2}\approx-{4\pi^{2}n^{2}\alpha}/{L^{2}}.$ (7)
The first eigenvalue describes fast initial transient relaxation, and the
second one corresponds to stationary slow diffusion and, quite naturally, does
not depend on $\tau$. So, we can conclude that there exists a critical length
of the mode
$l^{*}=4\pi\sqrt{\alpha\tau},$ (8)
which separates between two different types of the relaxation: oscillatory and
diffusive. The oscillatory behavior is naturally related to the hyperbolicity
of the system. Existence of this critical scale characterizes the deviance of
the system from parabolic Fourier law.
Figure 1: Relaxation of initial periodic thermal profile in the chain of
rotators, $Z=64$, $L=1024$, (a) $T_{0}=0.2$, $A=0.05$ (oscillatory decay) and
(b) $T_{0}=0.5$, $A=0.15$ (smooth decay of the initial thermal profile).
This critical wavelength scale $l^{*}$ can be measured directly from the
numeric simulation without relying on any particular empiric equation of the
non-stationary heat conduction. The numeric experiment should be designed in
order to simulate the relaxation of thermal profile to its equilibrium value
for different spatial modes of the initial temperature distribution. We
simulate the periodic chain of particles with conserved momentum with
Hamiltonian
$H=\sum_{n=1}^{N}\frac{1}{2}\dot{u}_{n}^{2}+V(u_{n+1}-u_{n})\leavevmode\nobreak\
.$ (9)
In order to obtain the initial nonequilibrium temperature distribution, all
particles in the chain were embedded in the Langevin thermostat. For this
sake, the following system of equations was simulated:
$\displaystyle\ddot{u}_{n}$ $\displaystyle=$ $\displaystyle
V^{\prime}(u_{n+1}-u_{n})-V^{\prime}(u_{n}-u_{n-1})-\gamma_{n}\dot{u}_{n}+\xi_{n}$
$\displaystyle n$ $\displaystyle=$ $\displaystyle 1,...,N$ (10)
where $\gamma_{n}$ is the relaxation coefficient of the $n$-th particle and
the white noise $\xi_{n}$ is normalized by the following conditions:
$\left\langle\xi_{n}\right\rangle=0,\left\langle\xi_{n}(t_{1})\xi_{k}(t_{2})\right\rangle=2\sqrt{\gamma_{n}\gamma_{k}}T_{n}\delta_{nk}\delta(t_{1}-t_{2}),$
(11)
where $T_{n}$ is the prescribed temperature of the $n$-th particle. The
numeric integration has been performed for $\gamma_{n}=0.1$ for every $n$ and
within time interval $t=250$. After that, the Langevin thermostat was switched
off and relaxation of the system to a stationary temperature profile was
studied for various initial distributions $T_{n}$ for two particular choices
of the nearest-neighbor interaction described above (FPU and chain of
rotators):
$V_{1}(x)=x^{2}/2+x^{4}/4,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ V_{2}(x)=1-\cos x.$
Separate analysis of individual spatial modes will provide insight into the
global behavior of the system only if these modes are, at least approximately,
not interacting. For CV equation (2) this is the case since it is linear.
However, we do not rely on it a priori and the absence of interaction between
different spatial relaxation modes should be checked numerically. We simulate
the relaxation of the initial thermal profile comprising five different modes:
$T_{n}=T_{0}+\sum_{i=4}^{8}A_{i}\cos[2\pi(n-1)/2^{i}]$ (12)
in cyclic chain of $N=2^{10}$ particles, with average temperature $T_{0}=1$
and modal amplitudes $A_{4}=...=A_{8}=0.02$ for the CR potential, and with
$T_{0}=20$, $A_{4}=...=A_{8}=0.4$ for the FPU potential. Both simulations
demonstrated almost complete lack of interaction between the modes. No other
modes were excited with visible amplitudes. Each mode in the collective
excitation relaxed similarly to the profile obtained when it was excited
individually. So, it is possible to conclude that for given values of the
parameters the equation of the non-stationary heat conduction should be
approximately linear and separate analysis of spatial relaxation modes is
justified.
In order to study the relaxation of different spatial modes of the initial
temperature distribution, its profile has been prescribed as
$T_{n}=T_{0}+A\cos[2\pi(n-1)/Z]$ (13)
where $T_{0}$ is he average temperature, $A$ – amplitude of the perturbation,
$Z$ – the length of the mode (number of particles). The overall length of the
chain $L$ has to be multiple of $Z$ in order to ensure the periodic boundary
conditions. The results were averaged over $10^{6}$ realizations of the
initial profile in order to reduce the effect of fluctuations.
Figure 2: (Color online) Evolution of the relaxation profile in the chain of
rotators with change of the mode length $Z$. Time dependence of the mode
maximum $T(1+Z/2)$ (red lines) and minimum $T(1)$ (blue lines) are depicted
with average temperature $T_{0}=0.4$ and $Z=16\times 2^{k-1}$, $k=1,...,5$,
scaling time $t_{k}=2^{k-1}$. For all simulations length of chain $L=1024$.
Typical result of the simulation is presented at Fig. 1. The chain of rotators
of the same length $N=1024$ and the same modal wavelength $Z=64$ demonstrates
qualitatively different relaxation behavior for different temperatures – the
oscillatory one for lower temperature and the smooth decay – for higher
temperature. This observation suggests that the critical wavelength mentioned
above, if it exists, should decrease with the temperature increase. However,
its existence should be checked for constant temperature and varying
wavelength.
Such simulations are presented at Fig. 2 (for the CR) and Fig. 3 (for the FPU
chain). In both models one observes oscillatory decay for the short
wavelengths and smooth exponential decay for relatively long waves. It means
that for both models there exists some critical wavelength $l^{*}$ which
separates two types of the decay and thus the effect of the non-stationary
heat conduction is revealed.
Figure 3: (Color online) Evolution of the relaxation profile in the FPU chain
with change of the mode length $Z$. Time dependence of the mode maximum
$T(1+Z/2)$ (red lines) and minimum $T(1)$ (blue lines) are depicted with
average temperature $T_{0}=20$ and $Z=16\times 2^{k-1}$, $k=1,...,7$, scaling
time $t_{k}=2^{k-1}$, length $L=1024$.
Results presented at Fig. 3 allow one to conclude that the critical wavelength
for the FPU chain for given temperature may be estimated as $512<l^{*}<1024$.
The interpretation of Fig. 2 is not that straightforward. It is clear that
$32<l^{*}<128$, but for $Z=64$ the result is not clear. Within the accuracy of
the simulation, it seems that only finite number of the oscillations is
observed. It is possible to speculate that such behavior is not consistent
with the lowest order CV equation, since expressions (6), (7) suggest either
infinite number of the oscillations, or at most single crossing of the average
temperature or no crossing at all. Possible interpretation may be that if the
modal wavelength is close enough to the critical, the second-order CV model is
not sufficient any more and the nonlocal effects of higher order should be
taken into account. Still, these conclusions should be verified by more
detailed simulations in the vicinity of the crossover wavelength.
Figure 4: (Color online) Exponential decay of the normalized oscillation
amplitude in the chain of rotators $A(t)=(T(1+Z/2)(t)-T_{0})/A)$ for the
average temperature $T_{0}=0.3$, initial amplitude $A=0.05$ and different
periods of the thermal profile $Z=16\times 2^{k-1}$, $k=1,2,3$. The straight
lines illustrates the decay of the maximum envelope according to
$A=\exp(-\lambda t)$ with universal value $\lambda=0.015$ for all three
simulations. Figure 5: (Color online) Exponential decay of the normalized
oscillation amplitude in the FPU chain $a(t)=(T(1+Z/2)(t)-T_{0})/A$ for the
average temperature $T_{0}=10$, initial amplitude $A=0.05$ and different
periods of the thermal profile $Z=16\times 2^{k-1}$, $k=1,...,5$. The straight
lines illustrates the decay of the maximum envelope according to
$A=\exp(-\lambda t)$ with values $\lambda=0.0015$, 0.003, 0.008, 0.024 and
0.06 for $k=1$, 2, 3, 4 and 5.
The latter observation has motivated us to check whether the data of numeric
simulations in these one-dimensional models offer a support for the CV
macroscopic equation. For this sake, one can check another prediction of this
equation – the independence of the amplitude decrement of the relaxation
profile on the wavelength in the oscillatory regime (6). The results of
simulation are presented at Fig. 4 (CR) and Fig. 5 (FPU). One can see that for
the chain of rotators the above prediction more or less corresponds to the
simulation results. For the FPU chain the decrement is strongly dependent on
the wavelength, at odds with the CV equation. In this latter case, no unique
relaxation time exists.
To summarize, we reveal the hyperbolicity effects of the non-stationary heat
conduction in one-dimensional models of dielectrics without relying on any
particular empiric equation. There exists a critical modal wavelength $l^{*}$
which separates between oscillating and diffusive relaxation of the
temperature field; such crossover (actually, the oscillatory decay of the
temperature field perturbations) is inconsistent with parabolic Fourier
equation. So, if the size of the system is close to this critical scale, more
exact macroscopic equations should be used for description of the non-
stationary heat conduction. In both models studied the critical size decreases
with the temperature increase. As for the CV equation itself, in the FPU chain
this equation clearly contradicts the simulations for the short-wave
perturbations of the temperature field. In the chain of rotators it seems to
be inconsistent with the simulations in the vicinity of the critical
wavelength, however is more or less justified for longer and shorter modes.
One can speculate that this difference between two models is related to their
difference with respect to the stationary heat conduction – saturating versus
size dependent behavior of the heat conduction coefficient p9 ; p10 ; p11 ;
p12 ; p13 .
The authors are very grateful to Israel Science Foundation for financial
support. The authors also thank the Joint Supercomputer Center of the Russian
Academy of Sciences for using computer facilities.
## References
* (1) P. Vernotte, C. R. Acad. Sci. 246, 3154 (1958).
* (2) C. Cattaneo, C. R. Acad. Sci. 247, 431 (1958).
* (3) D.S. Chandrasekhararaiah, Appl. Mech. Rev., 39, 355 (1986).
* (4) D.S. Chandrasekhararaiah Appl. Mech. Rev., 51, 705 (1998).
* (5) C.I. Christov and P.M. Jordan, Phys. Rev. Lett., 94, 154301 (2005).
* (6) P. Heino, Journal of Comput. and Theor. Nanoscience, 4, 896 (2007).
* (7) J. Shiomi and S. Maruyama, Phys. Rev B 73, 205420 (2006).
* (8) S. Volz et al, Phys. Rev. B, 54, 340 (1996).
* (9) S. Lepri, R. Livi and A. Politi, Phys. Reports, 377, 1 (2003).
* (10) S. Lepri, R. Livi and A. Politi, Phys. Rev. Lett. 78 1896 (1997).
* (11) O.V. Gendelman and A.V. Savin, Phys. Rev. Lett. 84 2381 (2000).
* (12) C. Giardina, R. Livi, A. Politi and M. Vassalli, Phys. Rev. Lett. 84 2144 (2000).
* (13) A.V. Savin and O.V. Gendelman, Fiz. Tverd. Tela (Leningrad) 43, 341 (2001) [Sov. Phys. Solid State 43, 355 (2001)].
|
arxiv-papers
| 2009-09-16T19:24:12 |
2024-09-04T02:49:05.395637
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Oleg V. Gendelman, Alexander V. Savin",
"submitter": "Alexander V. Savin",
"url": "https://arxiv.org/abs/0909.3089"
}
|
0909.3116
|
# Maximal mixing as a ‘sum’ of small mixings
Joydeep Chakrabortty joydeep@hri.res.in Harish-Chandra Research Institute,
Allahabad 211 019, India Anjan S. Joshipura anjan@prl.res.in Theory Group,
Physical Research Laboratory, Ahmedabad 380 009, India Poonam Mehta
poonam@rri.res.in Theoretical Physics Group, Raman Research Institute,
Bangalore 560 080, India Sudhir K. Vempati vempati@cts.iisc.ernet.in Centre
for High Energy Physics, Indian Institute of Science, Bangalore 560 012, India
###### Abstract
In models with two sources of neutrino masses, we look at the possibility of
generating maximal/large mixing angles in the total mass matrix, where both
the sources have only small mixing angles. We show that in the two generation
case, maximal mixing can naturally arise only when the total neutrino mass
matrix has a quasi-degenerate pattern. The best way to demonstrate this is by
decomposing the quasi-degenerate spectrum in to hierarchial and inverse-
hierarchial mass matrices, both with small mixing. Such a decomposition of the
quasi-degenerate spectra is in fact very general and can be done irrespective
of the mixing present in the mass matrices. With three generations, and two
sources, we show that only one or all the three small mixing angles in the
total neutrino mass matrix can be converted to maximal/large mixing angles.
The decomposition of the degenerate pattern in this case is best realised in
to sub-matrices whose dominant eigenvalues have an alternating pattern. On the
other hand, it is possible to generate two large and one small mixing angle if
either one or both of the sub-matrices contain maximal mixing. We present
example textures of this. With three sources of neutrino masses, the results
remain almost the same as long as all the sub-matrices contribute equally. The
Left-Right Symmetric model where Type I and Type II seesaw mechanisms are
related provides a framework where small mixings can be converted to large
mixing angles, for degenerate neutrinos.
###### pacs:
14.60.Pq, 14.60.St, 11.30.Hv
††preprint: RECAPP-HRI-2009-016
## I Introduction
While neutrino masses have been thoroughly established experimentally
conchamaltoni , the question of how they attain their masses still needs to be
understood. Perhaps, the most elegant mechanism of generating neutrino masses
is through the seesaw mechanism seesaw . Here one trades the tininess of the
neutrino masses with high scale Majorana masses for right-handed neutrinos
introduced for this purpose. While the original seesaw mechanism dealt only
with right-handed heavy neutrino states, in recent years, it has been realized
that there could be other heavy triplet scalars seesaw2 or even triplet
fermions seesaw3 ; goran which could play the same role as right-handed
neutrinos in the original seesaw mechanism. These mechnaisms are named as Type
I, Type II and Type III seesaw mechanisms respectively (for recent reviews
please see strumiareview ; nirdavidsonreview ). While one of the three seesaw
mechanisms suffices to generate non-zero neutrino masses, it is interesting to
note that in most Grand Unified Theory (GUT) models, there is more than one
seesaw mechanism at work. For example, in SO(10) models both Type I and Type
II seesaw mechanisms are simultaneously present as soon as one considers
representations of the type $\overline{126}$ babumohapatra . In Left-Right
Symmetric (LRS) models, the Type I and Type II seesaw mechanisms are not just
present, but they are also related to each other akhmedov . Similarly, Type I
and Type III mechanisms co-exist in SU(5) model with an adjoint fermion
representation goran . In most of these investigations, typically one
considers one of them to be dominant while the other to be subdominant.
One of the crucial features of seesaw mechanism was its ability to generate
large or maximal mixing even though the mixing present in the Dirac neutrino
Yukawa couplings is small like in the hadronic sector. In fact this is what
typically happens in a SO(10) GUT smirnovaf , where neutrino Dirac Yukawa
couplings have the same structure as the top Yukawa couplings; even in such
cases large mixing in the neutrino sector is possible. However this would
require large hierarchies in the masses of the right-handed neutrinos which is
in conflict with thermal leptogenesis in these models thermallepto 111This is
true when the mixing angles in neutrino Dirac Yukawa are exactly like CKM
angles.. In the present work, we look for an alternative method to generate
large/maximal mixings instead of using the ‘seesaw-effect’. We will use the
fact that most models GUT models like SO(10) have more than one seesaw
mechanism at work. However, instead of restricting ourselves to any particular
GUT model or the seesaw mechanism, we analyze the general situation where
there are two sources for neutrino masses and both of these contain small
neutrino mixing.
Our analysis shows that the total neutrino mass matrix which is given by the
sum of the two neutrino sources can have large or maximal mixing only if the
resulting pattern of the neutrino masses is of the quasi-degenerate form. A
crucial condition which needs to be satisfied to reach this conclusion is that
the large eigenvalues of the sub-matrices do not cancel in the total mass
matrix. This results in the sub-matrices taking the form of hierarchial and
inverse-hierarchial matrices whose sum leads to the quasi-degenerate form.
Given that the decomposition of the degenerate spectrum in to hierarchial and
inverse-hierarchial mass matrices is quite generic, as we will demonstrate
here, one can enumerate the possible forms the individual sub-matrices can
take. It should be noted that the decomposition itself is independent of the
actual mechanism responsible for generating neutrino masses i.e, doesn’t
depend on whether there is a seesaw mechanism at work or not. It is well known
that the quasi-degenerate pattern for neutrino masses can be achieved both
with joshipura and without marajasekaran seesaw mechanism. However, as will
demonstrate later, the model dependence enters, if one wants to realise the
decomposition in terms of independent Lagrangian parameters which for example
is possible in Type I seesaw mechanism.
The simple example where our scheme of things can be realised is the LRS model
where both Type I and Type II seesaw mechanisms are simultaneously present. We
will explicitly present the conditions on the LRS parameters required in order
to realize the mechanism. The paper is organised as follows. In Sec. II, we
analyse the two generation case and show how only when the quasi-degeneracy is
satisfied in the final matrix, one can have large or maximal mixing. We also
describe all the possible decompositions of the degenerate spectra. We further
discuss how this scheme can be incorporated within the LRS models. In Sec.
III, we consider two cases (a) with two seesaw mechanisms or two sources, and
(b) with three sources. We then demonstrate the decomposition of the quasi-
degenerate spectrum and discuss the subtleties which arise in this case. We
also determine the required parameter values within the LRS model for both the
cases. We close with summary and outlook in Sec. IV. Generalisation of our
result to the case of $n$ sources of neutrino masses is given in Appendix A.
## II Large mixing as sum of small mixing angles
Consider a model for neutrino masses in which the total neutrino mass matrix
is given by
${\mathbb{M}}_{\nu}={\mathbb{M}}_{\nu}^{(1)}+{\mathbb{M}}_{\nu}^{(2)},$ (1)
where ${\mathbb{M}}_{\nu}^{(1)}$ and ${\mathbb{M}}_{\nu}^{(2)}$ can be thought
of as two individual sources of neutrino mass. For example,
${\mathbb{M}}_{\nu}^{(1)}$ could have its origin in Type I seesaw whereas
${\mathbb{M}}_{\nu}^{(2)}$ could have its origin in Type II seesaw mechanism
in a model like SO(10) where both these mechanisms are simultaneously present
222In fact, in most models of neutrino masses, one of them, say
${\mathbb{M}}_{\nu}^{(1)}$ could correspond to zeroth order mass while the
other ${\mathbb{M}}_{\nu}^{(2)}$ could correspond to perturbations required to
make contact with the experimental results.. Irrespective of their origin, let
us assume that both ${\mathbb{M}}_{\nu}^{(1)}$ and ${\mathbb{M}}_{\nu}^{(2)}$
contain only small mixing angles. We now ask the question whether it is
possible to have in the total mass matrix ${\mathbb{M}}_{\nu}$ (a) maximal or
large mixing, and (b) a reasonable $\Delta\mbox{m}^{2}$ without fine-tuning.
By this we mean, that the $\Delta m^{2}$ is determined in terms of the
dominant eigenvalues of ${\mathbb{M}}_{\nu}^{(i)}$ (where $i=1,2$). To make
the discussion concrete, we will stick to two generation case in the present
section. Denoting
${\mathbb{M}}_{\nu}^{(1)}=\left(\begin{array}[]{cc}m_{ee}^{(1)}&m_{e\mu}^{(1)}\\\
m_{e\mu}^{(1)}&m_{\mu\mu}^{(1)}\end{array}\right),\;\;\;\;\;\;{\mathbb{M}}_{\nu}^{(2)}=\left(\begin{array}[]{cc}m_{ee}^{(2)}&m_{e\mu}^{(2)}\\\
m_{e\mu}^{(2)}&m_{\mu\mu}^{(2)}\end{array}\right),$ (2)
we can easily derive the following relations :
$\displaystyle\tan 2\theta$ $\displaystyle=$
$\displaystyle{2m_{e\mu}^{(1)}+2m_{e\mu}^{(2)}\over
m_{\mu\mu}^{(2)}+m_{\mu\mu}^{(1)}-m_{ee}^{(2)}-m_{ee}^{(1)}}$ (3)
$\displaystyle=$ $\displaystyle\tan 2\theta^{(1)}{1\over(1+d)}+\tan
2\theta^{(2)}{d\over(1+d)}~{},$ (4)
where $d=(m_{\mu\mu}^{(2)}-m_{ee}^{(2)})/(m_{\mu\mu}^{(1)}-m_{ee}^{(1)})$ and
$\theta^{(1)}$ and $\theta^{(2)}$ are the mixing angles of
${\mathbb{M}}_{\nu}^{(1)}$ and ${\mathbb{M}}_{\nu}^{(2)}$ respectively. From
this expression, it is obvious that when both the mixing angles,
$\theta^{(1)}$ and $\theta^{(2)}$ are small, the only region where $\theta$
would be maximal is when $d=-1$. Notice that the small mixing in
${\mathbb{M}}_{\nu}^{(i)}$ would mean (a) 2
$m_{e\mu}^{(i)}\ll|m_{\mu\mu}^{(i)}-m^{(i)}_{ee}|$, and (b)
$m^{(i)}_{\mu\mu}\neq m^{(i)}_{ee}$ for $i=(1,2)$ i.e, the splitting in the
diagonal entries is much larger than the off-diagonal entry such that the
mixing remains small. Assuming at least one of the diagonal entries in each of
the matrix ${\mathbb{M}}_{\nu}^{(i)}$ is large, we have the following three
solutions for $d=-1$
1. (A) $m^{(2)}_{\mu\mu}=-m^{(1)}_{\mu\mu}$ ,
2. (B) $m^{(2)}_{ee}=-m^{(1)}_{ee}$ , and
3. (C) $m^{(2)}_{ee}=m^{(1)}_{\mu\mu}$ or $m^{(2)}_{\mu\mu}=m^{(1)}_{ee}$ .
The solution of the type (A) would represent the case in which both the
matrices ${\mathbb{M}}_{\nu}^{(i)}$ are of the hierarchial form with one
dominant diagonal element (the $\mu\mu$ entry). However in the total mass
matrix ${\mathbb{M}}_{\nu}$ this entry gets cancelled. To illustrate this,
consider the following textures for ${\mathbb{M}}_{\nu}^{(i)}$
${\mathbb{M}}_{\nu}=m_{1}\left(\begin{array}[]{cc}z&x\\\
x&1+z^{\prime}\end{array}\right)+m_{2}\left(\begin{array}[]{cc}0&y\\\
y&-1\end{array}\right)~{},$ (5)
where $x,y,z$ are the small entries compared to
$m_{\mu\mu}^{(1)}/m_{1}\equiv(1+z^{\prime})$ and
$m_{\mu\mu}^{(2)}/m_{2}\equiv-1$. Notice that the dominant eigenvalues of
${\mathbb{M}}_{\nu}^{(i)}$ have opposite CP parities as the maximal mixing
requirement condition is now given as $m_{1}\approx m_{2}\approx m$. In this
limit, the total mass matrix has the form
${\mathbb{M}}_{\nu}=m\left(\begin{array}[]{cc}z&x+y\\\
x+y&z^{\prime}\end{array}\right)~{}.$ (6)
The total mass matrix here has no trace of the dominant element of the
$\mathcal{O}(m)$ which was present in the sub-matrices. It has been cancelled
in such a way that the condition
$m^{(2)}_{\mu\mu}+m^{(1)}_{\mu\mu}=m^{(1)}_{ee}+m^{(2)}_{ee}$ is satisfied,
which is the same as the full condition of $d=-1$ which would mean
$z^{\prime}=z$, rather than the sub-condition, (A)
$m^{(2)}_{\mu\mu}=-m^{(1)}_{\mu\mu}$, which would instead mean $z^{\prime}=0$.
The role of the large element of the $\mathcal{O}(m)$ has only been to
generate the small mixing in the respective ${\mathbb{M}}_{\nu}^{(i)}$. Thus,
at the level of total mass matrix ${\mathbb{M}}_{\nu}$, the properties are
determined by the small entries of the original sub-matrices. The mass-squared
splitting $\Delta\mbox{m}^{2}$ of the total mass matrix in terms of the
elements of ${\mathbb{M}}_{\nu}^{(i)}$ is given by
$\Delta\mbox{m}^{2}=(m^{(1)}_{ee}+m^{(1)}_{\mu\mu}+m^{(2)}_{ee}+m^{(2)}_{\mu\mu})\sqrt{(m^{(1)}_{\mu\mu}+m^{(2)}_{\mu\mu}-m^{(1)}_{ee}-m^{(2)}_{ee})^{2}+4(m^{(1)}_{e\mu}+m^{(2)}_{e\mu})^{2}}~{},$
(7)
which reduces in the present case to
$\Delta\mbox{m}^{2}=m^{2}(z+z^{\prime})\sqrt{4(x+y)^{2}+(z-z^{\prime})^{2}}~{}.$
(8)
In the limit $z^{\prime}\rightarrow 0$, the mixing $\tan 2\theta=2(x+y)/z$
would depend on the relative magnitudes of $x,y$ and $z$ with large mixing
being possible as long as $x+y\gg z$. Similarly, in the limit
$z^{\prime}\approx z$, maximal/large mixing is possible, and a hierarchial
pattern for the neutrinos can arise if $x,y\sim z,z^{\prime}$. Solutions of
the class (B) also lead to similar results with the dominant entries of the
sub-matrices ${\mathbb{M}}_{\nu}^{(i)}$ being cancelled in the total mass
matrix. We do not find these solutions attractive as large mixing can only
come when the dominant elements (‘$ee$’ elements in this case) cancel
precisely to such an extent to be equal to the sum of the other diagonal
elements (‘$\mu\mu$’ elements). We now go on to discuss the solutions (C)
which we find more natural.
The solutions of the type (C) are given by $m^{(2)}_{ee}=m^{(1)}_{\mu\mu}$ or
$m^{(2)}_{\mu\mu}=m^{(1)}_{ee}$. The condition now requires that the opposite
diagonal elements of the sub-matrices are equal. This naturally sets the
${\mathbb{M}}_{\nu}^{(i)}$ to have an opposite ordering of their eigenvalues
i.e, one with normal hierarchy and the other has inverse hierarchy. For
illustration, let us consider the (sub)-case with
$m^{(2)}_{\mu\mu}=m^{(1)}_{ee}$. This can be represented as
$\displaystyle{\mathbb{M}}_{\nu}={\mathbb{M}}^{\rm{(1)}}_{\nu}+{\mathbb{M}}^{\rm{(2)}}_{\nu}$
$\displaystyle=$ $\displaystyle m_{1}\begin{pmatrix}0&x\\\
x&1\end{pmatrix}+m_{2}\begin{pmatrix}1&-y\\\ -y&0\end{pmatrix}$ (9)
$\displaystyle=$ $\displaystyle\begin{pmatrix}m_{2}&m_{1}x-m_{2}y\\\
m_{1}x-m_{2}y&m_{1}\end{pmatrix}~{},$
where $x,y$ are small entries with $m_{\mu\mu}^{(1)}\equiv m_{1}$ and
$m_{ee}^{(2)}\equiv m_{2}$. Note that here too as in the earlier case the
mixing angles in the individual sub-matrices are small, $\theta\simeq$ $x$ or
$y$, where as the total mixing matrix is given by
$\tan 2\theta={2(m_{1}x-m_{2}y)\over m_{1}-m_{2}}~{}.$ (10)
In the limit of exact degeneracy between $m_{1}$ and $m_{2}$, the mixing is
maximal as is evident. However, an important assumption is that both the
$m_{1}$ and $m_{2}$ carry the same sign or equivalently have the same CP
parity 333If the CP parities are opposite the mixing will remain small.. In a
more general situation, say when the zeros of the matrices on the RHS are of
Eq. (9) are filled with small entries (‘$ee$’ element in
${\mathbb{M}}_{\nu}^{(1)}$ and ‘$\mu\mu$’ element in
${\mathbb{M}}_{\nu}^{(2)}$), the condition for the large mixing is given by
$|m_{1}-m_{2}|<2(m_{1}x-m_{2}y)$. Thus, the splitting in the diagonal entries
should be much smaller than the off-diagonal elements. The spectrum of the
total mass matrix points towards a quasi-degenerate pattern. The eigenvalues
are given by :
$\lambda_{1,2}={\displaystyle\frac{1}{2}}\left[m_{1}+m_{2}\mp\sqrt{(m_{1}-m_{2})^{2}+4(m_{1}x-m_{2}y)^{2}}\right],$
(11)
which in the limit $m_{1}\approx m_{2}\approx m$ take the form
$m-\epsilon,m+\epsilon$, with $\epsilon=m(x-y)$ being the order of the off-
diagonal entry. The $\Delta\mbox{m}^{2}=4m\epsilon$. The other solution of
(C), $m^{(2)}_{ee}=m^{(1)}_{\mu\mu}$, corresponds to an interchange of $m_{1}$
and $m_{2}$ and would lead to similar conclusions.
Finally, let us consider a class of solutions with two large diagonal entries
in each of the ${\mathbb{M}}_{\nu}^{(i)}$. However given that the mixing in
each of them is small, as per the discussion above, the splitting between the
diagonal elements should be larger than the off-diagonal entry. This can be
parameterised by the following set of matrices :
$\displaystyle{\mathbb{M}}_{\nu}={\mathbb{M}}^{\rm{(1)}}_{\nu}+{\mathbb{M}}^{\rm{(2)}}_{\nu}$
$\displaystyle=$ $\displaystyle m_{1}\begin{pmatrix}1+\rho&x\\\
x&1\end{pmatrix}+m_{2}\begin{pmatrix}1&x^{\prime}\\\
x^{\prime}&1+\rho^{\prime}\end{pmatrix}$ (12) $\displaystyle=$
$\displaystyle\begin{pmatrix}m_{1}(1+\rho)+m_{2}&m_{1}x+m_{2}x^{\prime}\\\
m_{1}x^{\prime}+m_{2}y&m_{1}+m_{2}(1+\rho^{\prime})\end{pmatrix},$
where $x,x^{\prime}$ are small entries compared to one as before and
$\rho,\rho^{\prime}$ are chosen such that $2|x/\rho|\ll 1$ and
$2|x^{\prime}/\rho^{\prime}|\ll 1$ to keep the mixing small in
${\mathbb{M}}_{\rm{i}}^{\nu}$. This would mean a relative hierarchy of the
elements in the individual matrices, $m_{ee}^{(1)}\gg m_{\mu\mu}^{(1)}\gg
m_{e\mu}^{(1)}$ in ${\mathbb{M}}_{\rm{1}}^{\nu}$ and $m_{\mu\mu}^{(2)}\gg
m_{ee}^{(2)}\gg m_{e\mu}^{(2)}$ in ${\mathbb{M}}_{\rm{2}}^{\nu}$, which is
very similar to the case of solutions (C) with a large $m_{ee}$ ($m_{\mu\mu}$)
in ${\mathbb{M}}^{\rm{(1)}}_{\nu}$ (${\mathbb{M}}^{\rm{(2)}}_{\nu}$).
Qualitatively, these could form a different class of solutions compared to
type (C) with each of the sub-matrices here forming a quasi-degenerate pair
with small mixing. However, notice that the total mixing is now given by $\tan
2\theta\approx(x+x^{\prime})/(\rho^{\prime}-\rho)$ would remain small as
$x,x^{\prime}\ll\rho,\rho^{\prime}$ unless $\rho=\rho^{\prime}$. With this
additional condition, this class of solutions again falls in to the class (C)
i.e, $m^{(2)}_{ee}=m^{(1)}_{\mu\mu}$ or $m^{(2)}_{\mu\mu}=m^{(1)}_{ee}$.
However, to distinguish from the solutions in Eq. (9), we will call the class
of solutions represented by Eq. (12) as type (C1) 444Solutions with three
large entries in each sub-matrix violate the small mixing assumption..
In summary, the sum of two mass matrices with small mixing angles would
naturally lead to a degenerate spectrum with maximal/large mixing provided we
insist there are no cancellations of the large eigenvalues of the individual
sub-matrices. The individual sub-matrices could be (a) ordered as hierarchial
+ inverse-hierarchial with small mixing (solutions of type (C)) or (b) be
quasi-degenerate themselves but with small mixing (C1). However as we have
seen, solutions of the type (C1) require further precise cancellation in the
differences of their large diagonal elements. For this reason, we consider
solutions of the type (C) i.e, matrices as parameterised in Eq. (9) to be the
most natural. Thus to convert one small mixing angle in two matrices to one
maximal mixing in the total matrix, we would require a pair of
(quasi)-degenerate eigenvalues with the same CP parities, ordered oppositely
in the sub-matrices. This count would be useful when we extend this degeneracy
induced large mixing to three generations.
### II.1 Decomposition of the Degenerate Spectrum
In the previous section we have seen that a quasi-degenerate pattern naturally
emerges if two mass matrices of small mixing are added and we demand large
mixing in the total mass matrix. One can instead reverse the argument and
might say that the quasi-degenerate spectrum with large mixing can be
decomposed in to two matrices with small mixing. In fact, the decomposition of
the quasi-degenerate spectrum in to two matrices is more generic and is
independent of the mixing present in them. This can be easily be demonstrated
by considering zeroth order neutrino mass matrices in the flavour basis. Let
us denote the neutrino mass matrix in the flavour basis by
$\displaystyle{\mathbb{M}}_{\nu}={\mathbb{U}}_{PMNS}{\mathbb{M}}_{diag}{\mathbb{U}}_{PMNS}^{\dagger}~{},$
(13)
where ${\mathbb{U}}_{PMNS}={\mathbb{U}}_{l}^{\dagger}{\mathbb{U}}_{\nu}$ is
the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) unitary leptonic mixing matrix.
${\mathbb{U}}_{PMNS}={\mathbb{U}}_{\nu}$ in a basis in which charged lepton
mass matrix is diagonal, i.e, ${\mathbb{U}}_{l}={\mathbb{I}}_{n\times n}$. In
Table 1, we have listed the zeroth order mass matrices for hierarchal,
inverse-hierarchal and degenerate spectra for the case of small mixing and
maximal mixing. In writing down these textures, we have followed Altarelli and
Feruglio afreview method, where each of these (zeroth order) mass matrices
has to be multiplied by a mass scale $m$ representing the heaviest eigenvalue
of the mass matrix. From Table 1 we can see that, as we go along each column,
the degenerate mass matrices $\mathbb{C}_{i}$ can be expressed as a sum of
hierarchal, $\mathbb{A}$ and inverse hierarchal, $\mathbb{B}$ matrices. For
example, $\mathbb{C}_{0}=\mathbb{A}+\mathbb{B}$,
$\mathbb{C}_{1}=\mathbb{A}-\mathbb{B}$,
$\mathbb{C}_{2}=\mathbb{B}-\mathbb{A}$. Note that the mass scale $m$
multiplying $\mathbb{C}_{i}$ now multiplies both $\mathbb{A}$ and
$\mathbb{B}$. These equations hold irrespective of the mixing being small or
maximal. Thus every degenerate mass matrix can be expressed a sum (or
difference) of a hierarchial and inverse-hierarchial mass matrices, but with
common mass scale given by the degenerate mass $m$, which is an obvious
observation if one just sees the diagonal eigenvalues of each mass matrix in
the first column.
Mixing $\Rightarrow$ | Small | Maximal
---|---|---
$\mathbb{M}_{diag}$ | ${\mathbb{X}}_{\epsilon}$ | ${\mathbb{X}}_{M}$
Hierarchial | |
${\mathbb{A}}$: Diag[0,1] | $\begin{pmatrix}0&\epsilon\\\ \epsilon&1\end{pmatrix}$ | $\begin{pmatrix}1/2&1/2\\\ 1/2&1/2\end{pmatrix}$
Inverse hierarchial | |
${\mathbb{B}}$: Diag[1,0] | $\begin{pmatrix}1&-\epsilon\\\ -\epsilon&0\\\ \end{pmatrix}$ | $\begin{pmatrix}1/2&-1/2\\\ -1/2&1/2\end{pmatrix}$
Degenerate | |
${\mathbb{C}}_{0}$: Diag[1,1] | $\begin{pmatrix}1&0\\\ 0&1\end{pmatrix}$ | $\begin{pmatrix}1&0\\\ 0&1\end{pmatrix}$
${\mathbb{C}}_{1}$: Diag[-1,1] | $\begin{pmatrix}-1&2\epsilon\\\ 2\epsilon&1\\\ \end{pmatrix}$ | $\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}$
${\mathbb{C}}_{2}$: Diag[1,-1] | $\begin{pmatrix}1&-2\epsilon\\\ -2\epsilon&-1\\\ \end{pmatrix}$ | $\begin{pmatrix}0&-1\\\ -1&0\end{pmatrix}$
Table 1: Zeroth order textures for small and maximal mixing (setting $m_{1}$
and $m_{2}$ as dimensionless quantities which are either zero or one depending
on the different cases listed) for the two-generation case.
Let us now turn to the question of mixing for the degenerate cases mentioned
above. The mixing in $\mathbb{C}_{0}=\mathbb{A}+\mathbb{B}$ in undetermined as
it is proportional to the identity matrix. This is also the exact degeneracy
limit. This situation arises if the mixing angles of $\mathbb{A}$ and
$\mathbb{B}$ are not only small, but are also equal. On the other hand, the
mixing in $\mathbb{C}_{1}=\mathbb{A}-\mathbb{B}$ can be maximal again as we
explained above in the previous section. The mixing in $\mathbb{C}_{1}$ and
$\mathbb{C}_{2}$ will remain small as they have opposite CP parities. An
important exception to generate large mixing in terms of small mixing angles
through quasi-degeneracy is the pseudo-Dirac pattern. The pseudo-Dirac pair
can come as a sum (difference) of two sub-matrices both with maximal mixing,
one hierarchal and the other inverse-hierarchal. This is clearly evident from
the last column of Table 1. We see the pseudo-Dirac pairs
$\mathbb{C}_{1}=\mathbb{A}-\mathbb{B}$ and
$\mathbb{C}_{2}=\mathbb{B}-\mathbb{A}$ with both $\mathbb{A}$ and $\mathbb{B}$
containing maximal mixing.
The decomposition of quasi-degenerate spectra can easily be incorporated
within models of neutrino masses. For example, in the Type I seesaw mechanism
(with two generations) the mass matrix is given by
$\displaystyle-{\mathbb{M}}^{\rm{I}}_{\nu}$ $\displaystyle=$ $\displaystyle
v^{2}\begin{pmatrix}h_{ee}^{D}&h_{\mu e}^{D}\\\
h_{e\mu}^{D}&h_{\mu\mu}^{D}\end{pmatrix}\begin{pmatrix}1/M_{R1}&0\\\
0&1/M_{R2}\end{pmatrix}\begin{pmatrix}h_{ee}^{D}&h_{e\mu}^{D}\\\ h_{\mu
e}^{D}&h_{\mu\mu}^{D}\end{pmatrix}$ (18) $\displaystyle=$ $\displaystyle
m_{1}\left(\begin{array}[]{cc}(h_{ee}^{D})^{2}&h_{ee}^{D}h_{e\mu}^{D}\\\
h_{ee}^{D}h_{e\mu}^{D}&(h_{e\mu}^{D})^{2}\end{array}\right)+m_{2}\left(\begin{array}[]{cc}(h_{\mu
e}^{D})^{2}&h_{\mu e}^{D}h_{\mu\mu}^{D}\\\ h_{\mu
e}^{D}h_{\mu\mu}^{D}&(h_{\mu\mu}^{D})^{2}\end{array}\right),$
where $m_{1}$ and $m_{2}$ are given as $v^{2}/(M_{R_{1}})$ and
$v^{2}/(M_{R_{2}})$ respectively. Each of these sub-matrices is result of a
seesaw mechanism with one right-handed neutrino. Comparing the above with Eq.
(9), we can determine the parameter regions required for quasi-degeneracy and
large mixing. For, $M_{R_{1}}=M_{R_{2}}$, we see that for the Yukawa
parameters, there are two choices where the mixing in the sub-matrices is
small
$\displaystyle
h_{e\mu}^{D}~{}\sim~{}\mathcal{O}(1)~{},\;\;h_{ee}^{D}~{}\sim~{}x~{},\;\;h^{D}_{\mu\mu}~{}\sim~{}-y~{},\;\;h_{\mu
e}^{D}~{}\sim~{}\mathcal{O}(1)~{},~{}~{}~{}\mbox{or}$ $\displaystyle
h_{ee}^{D}~{}\sim~{}\mathcal{O}(1)~{},\;\;h_{e\mu}^{D}~{}\sim~{}-x~{},\;\;h^{D}_{\mu
e}~{}\sim~{}y~{},\;\;h_{\mu\mu}^{D}~{}\sim~{}\mathcal{O}(1)~{}.$ (19)
Thus each right-handed neutrino couples with the Standard Model neutrinos with
only small mixing angles, but total mass matrix ensures maximal mixing angles
for the above choice of parameters. These conditions are already known in the
literature for some time afreview . So this is an alternative approach of
arriving at these conditions. The interesting aspect of Type I seesaw
mechanism is that the decomposition at the neutrino mass matrix level can be
realised at the Lagrangian level in terms of independent parameters with
‘independent mass’ scales for the the individual sub-matrices, for instance
the sub-matrices have mass scales $v^{2}/M_{R_{1}}$ and $v^{2}/M_{R_{2}}$.
Such a realisation might not be possible in other models for degenerate
neutrinos like in Type II seesaw mechanism. A further interesting possibility
would be to consider the case when there are two independent seesaw mechanisms
at work.
### II.2 Left-Right Symmetric Model
The simplest model where the above mechanism can be realised is the LRS model.
In the recent years, this model has been thoroughly analyzed for its duality
properties akhmedov . The LRS model naturally contains both Type I and Type II
seesaw contributions, which can be thought of as two sub-matrices discussed
above. Further more, these models are characterized by a common Yukawa
coupling to both the left-handed and right-handed Majorana mass matrices
$\mathcal{L}_{M}=-{f\over
2}\left(\overline{\nu_{L}^{c}}\nu_{L}\Delta^{0}_{L}+\overline{\nu_{R}^{c}}\nu_{R}\Delta^{0}_{R}\right)+h.c.~{},$
(20)
where $\Delta_{L(R)}$ is the triplet Higgs field whose neutral component
attains a vacuum expectation value (vev) giving rise to the Majorana mass to
the left (right) handed neutrino fields. In addition the Dirac neutrino Yukawa
coupling is also present
$\mathcal{L}_{D}=-Y\overline{\nu}_{L}\nu_{R}\phi^{0}+h.c.$ (21)
In the limit where $v_{R}\gg v$, the Type I seesaw mechanism becomes operative
and the total neutrino mass matrix is now given as
${\mathbb{M}}_{\nu}=fv_{L}-{v^{2}\over v_{R}}Yf^{-1}Y^{T}~{}.$ (22)
Along the lines of the discussion we had for the two-generation case, Eq. (9),
we can assume the contribution (first term on the RHS of Eq. (22)), due to
Type II to be hierarchial with small mixing and second part due to the Type I
contribution inverse hierarchial with small mixing. The appropriate choice of
the Yukawa textures in this case are as follows
$f=\left(\begin{array}[]{cc}0&x\\\ x&1\\\
\end{array}\right)\;\;,\;\;Y=\left(\begin{array}[]{cc}1&y\\\ y&0\\\
\end{array}\right)~{}.$ (23)
With this choice the total mass matrix takes the form
$\displaystyle{\mathbb{M}}_{\nu}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}0&m_{1}x\\\ m_{1}x&m_{1}\\\
\end{array}\right)+{m_{2}\over x^{2}}\left(\begin{array}[]{cc}1-2xy&y(1-xy)\\\
y(1-xy)&y^{2}\\\ \end{array}\right)$ (28) $\displaystyle=$
$\displaystyle{1\over
x^{2}}\left(\begin{array}[]{cc}m_{2}(1-2xy)&m_{1}x^{3}+m_{2}y(1-xy)\\\
m_{1}x^{3}+m_{2}y(1-xy)&m_{1}x^{2}+m_{2}y^{2}\\\ \end{array}\right)~{},$ (31)
where $m_{1}=v_{L}$ and $m_{2}=v^{2}/v_{R}$. The mixing angle in the above
mass matrix is given by
$\tan
2\theta={2[m_{1}x^{3}+m_{2}y(1-xy)]\over[m_{1}x^{2}+m_{2}y^{2}]-m_{2}(1-2xy)}~{}.$
(32)
From the above it is clear that the degeneracy requirement $m_{1}x^{2}\approx
m_{2}$ automatically leads to large mixing, $\tan 2\theta\sim{\cal
O}(\frac{1}{2y})$. A rough idea of how stable this mixing would be under
radiative corrections can be obtained by considering the modification of the
neutrino mass matrix below the seesaw scale. The modification is set by the
matrix ${\mathbb{P}}=\mbox{Diag}\\{1,1+\delta_{\mu}\\}$ and is given as
${\mathbb{M}}_{\nu}={\mathbb{P}}{\mathbb{M}}_{\nu}{\mathbb{P}}$. In this case,
the mixing angle now takes the form
$\tan
2\theta={2[m_{1}x^{3}+m_{2}y(1-xy)](1+\delta_{\mu})\over(1+\delta_{\mu})^{2}[m_{1}x^{2}+m_{2}y^{2}]-m_{2}(1-2xy)}~{},$
(33)
where $\delta_{\mu}=c~{}h_{\mu}^{2}/(16\pi^{2})\log(M_{X}/M_{W})$ specifies
the size of the radiative corrections induced by the Yukawa coupling of the
$\mu$, $h_{\mu}$. Here $c$ is a constant depending on whether the theory is
supersymmetric or not and $M_{X}$ is the high scale just below the seesaw
scale antusch . The condition for large mixing case now gets modified as
$[m_{1}x^{2}+m_{2}y^{2}](1+\delta_{\mu})^{2}\approx m_{2}$. Note that this
same condition is also required to keep the degeneracy stable even after
radiative corrections. Of course, the splitting of the degeneracy can come
from the radiative effects. A more detailed analysis of radiative corrections
will be presented elsewhere.
## III Extension to Three Generations
Let us extend the analysis of the previous section to the case of three
generations. Here we will consider two cases - (a) Case I: two seesaw
mechanisms or two sources of neutrino masses (Sec. III.1), and (b) Case II:
three seesaw mechanisms or three sources of neutrino masses (Sec. III.2).
### III.1 Case I: Two Seesaw Mechanisms
As before, let us consider two $3\times 3$ mass matrices each with a small
mixing angle and one large eigenvalue,
${\mathbb{M}}_{\nu}={\mathbb{M}}_{\nu}^{(1)}+{\mathbb{M}}_{\nu}^{(2)}$.
Instead of representing them as general mass matrices as we have done for the
case of two generations, we will represent them by using
$\mathbb{M}_{\nu}^{(i)}=[{\mathbb{U}}_{\mbox{mix}}^{(i)}]^{T}\cdot\mbox{Diag}[\mathbb{M}_{\nu}^{(i)}]\cdot{\mathbb{U}}_{\mbox{mix}}^{(i)}~{},$
(34)
where
$\mbox{Diag}[\mathbb{M}_{\nu}^{(i)}]=\mbox{Diag}[\\{m_{1}^{(1)},m_{2}^{(2)},m_{3}^{(3)}\\}]$,
the eigenvalues of the mass matrices and ${\mathbb{U}}_{\mbox{mix}}^{(i)}$
represents the mixing present in each of the mass matrices, with $i=1,2$.
Given that the mixing angles in $\mathbb{M}_{\nu}^{(i)}$ are small, we can
expand ${\mathbb{U}}_{\mbox{mix}}^{(i)}$ in terms of small parameters
$\cos\theta_{m}^{(i)}\approx 1$,
$\sin\theta_{m}^{(i)}\approx\epsilon^{(i)}_{m}$, where $m=\\{12,23,13\\}$
labels the three angles. The total mass matrix now takes the form
$\mathbb{M}_{\nu}=\left(\begin{array}[]{ccc}m_{1}^{(1)}+m_{1}^{(2)}&(m_{2}^{(1)}-m_{1}^{(1)})\epsilon^{(1)}_{12}+(m_{2}^{(2)}-m_{1}^{(2)})\epsilon^{(2)}_{12}&(m_{3}^{(1)}-m_{1}^{(1)})\epsilon^{(1)}_{13}+(m_{2}^{(2)}-m_{1}^{(2)})\epsilon^{(2)}_{13}\\\
&m_{2}^{(1)}+m_{2}^{(2)}&(m_{3}^{(1)}-m_{2}^{(1)})\epsilon^{(1)}_{23}+(m_{3}^{(2)}-m_{2}^{(2)})\epsilon^{(2)}_{23}\\\
&*&m_{3}^{(1)}+m_{3}^{(2)}\end{array}\right),$ (35)
where the symmetric elements of the matrix have been represented by $*$. We
can determine the mixing present in the total mass matrix by diagonalising the
above matrix. We have
$\mathbb{M}^{\prime}_{\nu}={\mathbb{U}}_{23}^{T}\mathbb{M}_{\nu}{\mathbb{U}}_{23}~{},$
(36)
where
${\mathbb{U}}_{23}=\left(\begin{array}[]{ccc}1&0&0\\\
0&\cos\theta_{23}&\sin\theta_{23}\\\
0&-\sin\theta_{23}&\cos\theta_{23}\end{array}\right)~{},$
with
$\tan
2\theta_{23}=2~{}{(m_{3}^{(1)}-m_{2}^{(1)})\epsilon^{(1)}_{23}+(m_{3}^{(2)}-m_{2}^{(2)})\epsilon^{(2)}_{23}\over
m_{3}^{(1)}+m_{3}^{(2)}-m_{2}^{(1)}-m_{2}^{(2)}}~{}.$ (37)
For this mixing to be maximal the condition would be
$(m_{3}^{(1)}-m_{2}^{(1)})=-(m_{3}^{(2)}-m_{2}^{(2)})$. This condition is
similar to the one we have seen earlier for the two generation case and as
argued in that case, the only natural solution is to have
$m_{3}^{(1)}=m_{2}^{(2)}$ with $m_{2}^{(1)},m_{3}^{(2)}$ negligible or
$m_{2}^{(1)}=m_{3}^{(2)}$ with 555 More precisely, we should have
$m_{3}^{(1)}-m_{2}^{(2)}\approx{\cal
O}(m_{3}^{(1)}(\epsilon_{23}^{1}-\epsilon_{23}^{2}))$and
$m_{2}^{(1)},m_{2}^{(2)}$ much smaller compared to them.
$m_{3}^{(1)},m_{2}^{(2)}$ negligible. We now proceed to show that if we accept
either of these two solutions, it would not be possible to have one another
large mixing angle in $\mathbb{M}_{\nu}$, if they have to satisfy the
naturalness criteria that the large eigenvalues of the individual matrices
should not cancel in the total mass matrix. Defining
${\mathbb{U}}_{13}=\left(\begin{array}[]{ccc}\cos\theta_{13}&0&\sin\theta_{13}\\\
0&1&0\\\ -\sin\theta_{13}&0&\cos\theta_{13}\end{array}\right)~{},$ (38)
we have
$\mathbb{M}^{{}^{\prime\prime}}_{\nu}={\mathbb{U}}_{13}^{T}\mathbb{M}^{\prime}_{\nu}{\mathbb{U}}_{13}~{}.$
(39)
$\tan 2\theta_{23}$ in the limit where the solution for maximal mixing of the
$23$ angle, $m_{3}^{(1)}=m_{2}^{(2)}=\bar{m}$ with
$m_{2}^{(1)},m_{3}^{(2)}\sim 0$ is taken is given by
$\tan
2\theta_{13}\approx{m_{1}^{(1)}(\epsilon^{(1)}_{12}+\epsilon^{(1)}_{13})-\bar{m}(\epsilon^{(1)}_{13}+\epsilon^{(2)}_{12})+m_{1}^{(2)}(\epsilon^{(2)}_{12}+\epsilon^{(2)}_{13})\over\sqrt{2}(m_{1}^{(1)}+m_{1}^{(2)}-\bar{m}(1+\epsilon^{(1)}_{23}-\epsilon^{(2)}_{23}))}~{}.$
(40)
From the above we realize the following conditions for (a) small mixing :
$|m_{1}^{(1)}+m_{1}^{(2)}-\bar{m}|\gg 0$, and (b) maximal mixing :
$|m_{1}^{(1)}+m_{1}^{(2)}-\bar{m}|=0$. Finally, defining
${\mathbb{U}}_{12}=\left(\begin{array}[]{ccc}\cos\theta_{12}&\sin\theta_{12}&0\\\
-\sin\theta_{12}&\cos\theta_{12}&0\\\ 0&0&1\end{array}\right)~{},$ (41)
we have
$\mathbb{M}^{{}^{\prime\prime\prime}}_{\nu}={\mathbb{U}}_{12}^{T}\mathbb{M}^{{}^{\prime\prime}}_{\nu}{\mathbb{U}}_{12}~{}.$
(42)
$\tan 2\theta_{12}$ has the following form in the limiting case when
$\theta_{13}$ is very small
$\tan
2\theta_{12}\approx{m_{1}^{(1)}(\epsilon_{12}^{(1)}-\epsilon_{13}^{(1)})+\bar{m}(\epsilon_{13}^{(1)}-\epsilon_{12}^{(2)})+m_{1}^{(1)}(\epsilon^{(2)}_{12}-\epsilon^{(2)}_{13})\over\sqrt{2}(m_{1}^{(1)}+m_{1}^{(2)}-\bar{m}(1-\epsilon_{23}^{(1)}+\epsilon_{23}^{(2)}))}+\mathcal{O}(\theta_{13})~{}.$
(43)
From the above we see that the conditions for the mixing are the same for both
$\theta_{12}$ and $\theta_{13}$ in this limit. Thus either both become
maximal/large or both remain small. Finally, in the limit of maximal
$\theta_{13}$ mixing, the expression for $\tan 2\theta_{12}$ becomes
${(m_{1}^{(1)}(\epsilon_{12}^{(1)}-\epsilon_{13}^{(1)})+\bar{m}(\epsilon_{13}^{(1)}-\epsilon_{12}^{(2)})+m_{1}^{(2)}(\epsilon_{12}^{(2)}-\epsilon_{13}^{(2)}))\over
m_{1}^{(1)}+m_{1}^{(2)}+\bar{m}(-1+3\epsilon_{23}^{(1)}-3\epsilon^{(2)}_{23})+\sqrt{2}(m_{1}^{(1)}(\epsilon_{12}^{(1)}+\epsilon_{13}^{(1)})-\bar{m}(\epsilon_{13}^{(1)}+\epsilon_{12}^{(2)})+m_{1}^{(2)}(\epsilon^{(2)}_{12}+\epsilon_{13}^{(2)}))}~{},$
(44)
which is also automatically maximal/large within the small
$\epsilon_{ij}^{(k)}$ limit. Before we proceed, a few comments are in order
regarding the ordering of the eigenvalues. In the case where there is only one
maximal/large mixing, the sub matrices can have hierarchal and inverse-
hierarchal patterns, with the hierarchal sub-matrix containing one large
eigenvalue and the inverse-hierarchal containing two large eigenvalues. The
only condition is on their CP parties; the eigenvalues taking part in the
enhancement of the mixing should have the same CP parities. The list of
possible forms the sub-matrices can take is discussed in the subsection
III.1.2 where decomposition of the degenerate spectrum is considered in three
generation case.
On the other hand, for the case with all the three large/maximal mixing case,
as per our arguments earlier, i.e, the large eigenvalues of the individual
matrices should not cancel in the total matrix, the present solution
necessarily favours an alternating pattern for the eigenvalues for the
individual mass matrices 666The zeroth order textures for alternating pattern
of neutrino mass matrices are given in Table 4.
$\displaystyle\mbox{Diag}[\mathbb{M}_{\nu}^{(1)}]=\mbox{Diag}[\\{m_{1}^{(1)},0,m_{3}^{(1)}\\}]$
,
$\displaystyle\mbox{Diag}[\mathbb{M}_{\nu}^{(2)}=\mbox{Diag}[\\{0,m_{2}^{(2)},0\\}]$
$\displaystyle\mbox{Diag}[\mathbb{M}_{\nu}^{(1)}]=\mbox{Diag}[\\{0,m_{2}^{(1)},0\\}]$
,
$\displaystyle\mbox{Diag}[\mathbb{M}_{\nu}^{(2)}]=\mbox{Diag}[\\{m_{1}^{(2)},0,m_{3}^{(2)}\\}]~{}.$
(45)
In this case, the mixing pattern corresponds to the truly maximal mixing
matrix of Cabibbo and Wolfenstein cabwolf along with the degeneracy condition
$m_{1}\approx m_{2}\approx m_{3}$. In the recent years, the truly maximal
mixing matrix has been achieved from $A_{4}$ symmetry by Ma and Rajasekaran
also for degenerate case marajasekaran . As it stands this matrix is not
phenomenologically viable as all the three mixing angles it predicts are
large. However there could be other corrections to this mass matrix depending
on the model which would rectify this situation vallemababu and make the mass
matrix phenomenologically viable.
#### III.1.1 Two Equivalent Textures
From our arguments above, it appears that we can generate only one large
mixing angle in the case when there are only two sub-matrices, because of the
important constraint that the third mixing angle ($\theta_{13}$) must not be
large 777This would be case in models where there are no large radiative
corrections effecting the mixing angles strongly.. Given that we can only
generate one large mixing from the small mixing using the degenerate
conditions, we will have to assume that at least one of the sub-matrices has
intrinsically one maximal/large mixing angle. However, the presence of this
mixing should not disturb the smallness of $\theta_{13}$ angle in the total
mass matrix. In the following, we will consider one of the sub-matrices to
have pseudo-Dirac structure and other one to have one large eigenvalue and all
the three mixing angles small. This is because the pseudo-Dirac structure not
only gives maximal mixing but also has the eigenvalues with opposite CP
parities.
${\mathbb{M}}_{\nu}=m_{1}\left(\begin{array}[]{ccc}x^{2}&x&y^{2}\\\ x&0&1\\\
y^{2}&1&0\end{array}\right)+m_{2}\left(\begin{array}[]{ccc}1&z&t^{3}\\\
z&z^{3}&t^{3}\\\ t^{3}&t^{3}&z^{3}\end{array}\right)~{},$ (46)
where $x,y,z,t$ are small entries compared to $m_{1},m_{2}$. We will
diagonalise this matrix in the following manner. Rotating by
${\mathbb{O}}_{23}$ on both sides, we have
${\mathbb{O}}_{23}^{T}{\mathbb{M}}_{\nu}{\mathbb{O}}_{23}=\left(\begin{array}[]{ccc}m_{2}+m_{1}x^{2}&m_{1}x\cos\theta_{23}+\widetilde{m}_{12}&-m_{1}x\sin\theta_{23}+\widetilde{m}_{13}\\\
m_{1}x\cos\theta_{23}+\widetilde{m}_{12}&m_{1}\sin
2\theta_{23}+\widetilde{m}_{22}&0\\\
-m_{1}x\sin\theta_{23}+\widetilde{m}_{13}&0&-m_{1}\sin
2\theta_{23}+\widetilde{m}_{33}\end{array}\right)~{},$ (47)
where ${\mathbb{O}}_{23}$ is defined as
${\mathbb{O}}_{23}=\left(\begin{array}[]{ccc}1&0&0\\\
0&\cos\theta_{23}&-\sin\theta_{23}\\\
0&\sin\theta_{23}&\cos\theta_{23}\end{array}\right)~{},$ (48)
with the angle $\theta_{23}$ given by
$\theta_{23}={1\over
2}\tan^{-1}\left[{2(m_{1}+m_{2}t^{3})\over(m_{2}z^{3}-m_{2}z^{3})}\right]={\pi\over
4}~{}.$ (49)
The explicit forms for $\widetilde{m}_{ij}$ can be easily deduced. A crucial
point to note is that the diagonal elements of the matrix in Eq. (47) carry
opposite sign for the dominant element ($m_{1}$). This would have the
consequence of keeping the $13$ mixing small, while making $23$ mixing large,
when the degeneracy condition $m_{1}\approx m_{2}\approx m$ is imposed. The
total mixing matrix is given by
${\mathbb{O}}={\mathbb{O}}_{12}{\mathbb{O}}_{13}{\mathbb{O}}_{23}$ with the
angles $\theta_{13}$ and $\theta_{12}$ defined as
$\displaystyle\theta_{13}$ $\displaystyle=$ $\displaystyle{1\over
2}\tan^{-1}\left[{2(-m_{1}x\sin\theta_{23}+\widetilde{m}_{13})\over-m_{1}\sin
2\theta_{23}+\widetilde{m}_{33}-m_{1}x^{2}-{m}_{2}}\right]~{},$
$\displaystyle\theta_{12}$ $\displaystyle=$ $\displaystyle{1\over
2}\tan^{-1}\left[{2\widetilde{m}^{\prime}_{12}\over\widetilde{m}^{\prime}_{22}-\widetilde{m}^{\prime}_{11}}\right]~{},$
(50)
where the explicit form of $\widetilde{m}^{\prime}_{ij}$ can easily be
deduced. From the above, we can see that the degeneracy induced large mixing
mechanism works for the $12$ mixing, while it does not generate large
(maximal) mixing for the $13$ mixing angle. This is due to the choice of
having $\tau\tau$ element with opposite sign (loosely speaking CP parity)
compared to the $\mu\mu$ element.
The above Yukawa matrices can be easily incorporated in the LRS model by
choosing $f$ and $Y$ of Eq. (22) (at the leading order) as
$f=\left(\begin{array}[]{ccc}x^{2}&x&y^{2}\\\ x&0&1\\\
y^{2}&1&0\end{array}\right)\;\;\;Y=\left(\begin{array}[]{ccc}1&0&0\\\ 0&0&0\\\
0&0&0\end{array}\right)~{}.$ (51)
Notice that it reproduces the Eq. (46) at the zeroth order. From the
discussion in the previous section, we also know that the pseudo-Dirac mass
matrix can be decomposed in to maximally mixing sub-matrices. Thus another
texture which could equally give the same results is given by
${\mathbb{M}}_{\nu}=m_{1}\left(\begin{array}[]{ccc}x^{2}&x&y^{2}\\\ x&{1\over
2}&{1\over 2}\\\ y^{2}&{1\over 2}&{1\over
2}\end{array}\right)+m_{2}\left(\begin{array}[]{ccc}1&z&t^{3}\\\ z&{1\over
2}&-{1\over 2}\\\ t^{3}&-{1\over 2}&{1\over 2}\end{array}\right)~{},$ (52)
The first of the matrices has only one large eigenvalue in a hierarchial
pattern with maximal mixing, whereas the second one has two large eigenvalues
with one maximal mixing and two small mixings with inverted hierarchy. Lets
emphasize once more that one needs opposite eigenvalues $m_{1}\approx-m_{2}$
to obtain the large atmospheric mixing in this case.
#### III.1.2 Decomposition of the Degenerate Spectrum
For three generations the decomposition of the degenerate spectrum in to
hierarchal and inverse-hierarchal mass patterns is straight forward. In Table
9, we present the zeroth order mass matrices for the three generation case.
Mixing $\Rightarrow$ | Small | Single maximal | Bimaximal | Tribimaximal
---|---|---|---|---
$\mathbb{M}_{diag}$ | ${\mathbb{X}}_{\epsilon}$ | ${\mathbb{X}}_{SM}$ | ${\mathbb{X}}_{BM}$ | ${\mathbb{X}}_{TBM}$
Hierarchial | | | |
${\mathbb{A}}$: Diag[0,0,1] | $\begin{pmatrix}0&0&\epsilon_{13}\\\ 0&0&\epsilon_{23}\\\ \epsilon_{13}&\epsilon_{23}&1\end{pmatrix}$ | $\begin{pmatrix}0&0&0\\\ 0&\frac{1}{2}&\frac{1}{2}\\\ 0&\frac{1}{2}&\frac{1}{2}\end{pmatrix}$ | $\begin{pmatrix}0&0&0\\\ 0&\frac{1}{2}&\frac{1}{2}\\\ 0&\frac{1}{2}&\frac{1}{2}\end{pmatrix}$ | $\begin{pmatrix}0&0&0\\\ 0&\frac{1}{2}&\frac{1}{2}\\\ 0&\frac{1}{2}&\frac{1}{2}\end{pmatrix}$
Inverse hierarchial | | | |
${\mathbb{B}}_{1}$: Diag[1,-1,0] | $\begin{pmatrix}1&-2\epsilon_{12}&-\epsilon_{13}\\\ -2\epsilon_{12}&-1&\epsilon_{23}\\\ -\epsilon_{13}&\epsilon_{23}&0\end{pmatrix}$ | $\begin{pmatrix}1&0&0\\\ 0&-\frac{1}{2}&\frac{1}{2}\\\ 0&\frac{1}{2}&-\frac{1}{2}\end{pmatrix}$ | $\begin{pmatrix}0&-\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\\\ -\frac{1}{\sqrt{2}}&0&0\\\ \frac{1}{\sqrt{2}}&0&0\end{pmatrix}$ | $\begin{pmatrix}\frac{1}{3}&-\frac{2}{3}&\frac{2}{3}\\\ -\frac{2}{3}&-\frac{1}{6}&\frac{1}{6}\\\ \frac{2}{3}&\frac{1}{6}&-\frac{1}{6}\end{pmatrix}$
${\mathbb{B}}_{2}$: Diag[1,1,0] | $\begin{pmatrix}1&0&-\epsilon_{13}\\\ 0&1&-\epsilon_{23}\\\ -\epsilon_{13}&-\epsilon_{23}&0\end{pmatrix}$ | $\begin{pmatrix}1&0&0\\\ 0&\frac{1}{2}&-\frac{1}{2}\\\ 0&-\frac{1}{2}&\frac{1}{2}\end{pmatrix}$ | $\begin{pmatrix}1&0&0\\\ 0&\frac{1}{2}&-\frac{1}{2}\\\ 0&-\frac{1}{2}&\frac{1}{2}\end{pmatrix}$ | $\begin{pmatrix}1&0&0\\\ 0&\frac{1}{2}&-\frac{1}{2}\\\ 0&-\frac{1}{2}&\frac{1}{2}\end{pmatrix}$
Degenerate | | | |
${\mathbb{C}}_{0}$: Diag[1,1,1] | $\begin{pmatrix}1&0&0\\\ 0&1&0\\\ 0&0&1\end{pmatrix}$ | $\begin{pmatrix}1&0&0\\\ 0&1&0\\\ 0&0&1\end{pmatrix}$ | $\begin{pmatrix}1&0&0\\\ 0&1&0\\\ 0&0&1\end{pmatrix}$ | $\begin{pmatrix}1&0&0\\\ 0&1&0\\\ 0&0&1\end{pmatrix}$
${\mathbb{C}}_{1}$: Diag[-1,1,1] | $\begin{pmatrix}-1&2\epsilon_{12}&2\epsilon_{13}\\\ 2\epsilon_{12}&1&0\\\ 2\epsilon_{13}&0&1\end{pmatrix}$ | $\begin{pmatrix}-1&0&0\\\ 0&1&0\\\ 0&0&1\end{pmatrix}$ | $\begin{pmatrix}0&\frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{2}}\\\ \frac{1}{\sqrt{2}}&\frac{1}{2}&\frac{1}{2}\\\ -\frac{1}{\sqrt{2}}&\frac{1}{2}&\frac{1}{2}\end{pmatrix}$ | $\begin{pmatrix}-\frac{1}{3}&\frac{2}{3}&-\frac{2}{3}\\\ \frac{2}{3}&\frac{2}{3}&\frac{1}{3}\\\ -\frac{2}{3}&\frac{1}{3}&\frac{2}{3}\end{pmatrix}$
${\mathbb{C}}_{2}$: Diag[1,-1,1] | $\begin{pmatrix}1&-2\epsilon_{12}&0\\\ -2\epsilon_{12}&-1&2\epsilon_{23}\\\ 0&2\epsilon_{23}&1\end{pmatrix}$ | $\begin{pmatrix}1&0&0\\\ 0&0&1\\\ 0&1&0\end{pmatrix}$ | $\begin{pmatrix}0&-\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\\\ -\frac{1}{\sqrt{2}}&\frac{1}{2}&\frac{1}{2}\\\ \frac{1}{\sqrt{2}}&\frac{1}{2}&\frac{1}{2}\end{pmatrix}$ | $\begin{pmatrix}\frac{1}{3}&-\frac{2}{3}&\frac{2}{3}\\\ -\frac{2}{3}&\frac{1}{3}&\frac{2}{3}\\\ \frac{2}{3}&\frac{2}{3}&\frac{1}{3}\end{pmatrix}$
${\mathbb{C}}_{3}$: Diag[1,1,-1] | $\begin{pmatrix}1&0&-2\epsilon_{13}\\\ 0&1&-2\epsilon_{23}\\\ -2\epsilon_{13}&-2\epsilon_{23}&-1\end{pmatrix}$ | $\begin{pmatrix}1&0&0\\\ 0&0&-1\\\ 0&-1&0\end{pmatrix}$ | $\begin{pmatrix}1&0&0\\\ 0&0&-1\\\ 0&-1&0\end{pmatrix}$ | $\begin{pmatrix}1&0&0\\\ 0&0&-1\\\ 0&-1&0\end{pmatrix}$
Table 2: Different standard textures (zeroth order) for different combinations
of mixings (setting $m_{1}$, $m_{2}$ and $m_{3}$ as dimensionless quantities
which are either zero or one depending on the different cases listed)
consistent with data 999In Ref. afreview, , only two cases (single and
bimaximal mixing) were considered and they used
${\mathbb{M}}_{\nu}={\mathbb{U}}_{PMNS}^{\dagger}{\mathbb{M}}_{diag}{\mathbb{U}}_{PMNS}$,
which is different from our definition (Eq. (13)). .
Note that the present notation has been previously used in the literature
afreview and all the matrices present in this table have been used previously
to describe the neutrino mass matrix at the zeroth order. After adding small
perturbations to these matrices they can explain the neutrino data. However,
as before we are interested in only decomposing the degenerate mass matrix in
terms of the hierarchal and inverse-hierarchal mass matrices. As before, from
each of the columns, we can see that each degenerate case can be constructed
as a sum of hierarchal and inverse hierarchal textures. For example,
$\mathbb{C}_{0}$ can be considered as $\mathbb{A}~{}+~{}\mathbb{B}_{2}$.
Similarly, $\mathbb{C}_{1}$ can be considered as
$-\mathbb{B}_{1}~{}+~{}\mathbb{A}$ and so on. And this is true as we go along
each of the columns, i.e for all kinds of mixing angles. This simple
observation can be restated as every degenerate neutrino mass matrix can be
thought of a sum of hierarchal and inverse hierarchal sub-mass matrices while
the converse is not generally true.
Mixing $\Rightarrow$ | Small | Single maximal | Bimaximal | Tribimaximal
---|---|---|---|---
$\mathbb{M}_{diag}$ | ${\mathbb{X}}_{\epsilon}$ | ${\mathbb{X}}_{SM}$ | ${\mathbb{X}}_{BM}$ | ${\mathbb{X}}_{TBM}$
$\widetilde{\mathbb{A}}_{1}$: Diag[0,1,1] | $\begin{pmatrix}0&\epsilon_{12}&\epsilon_{13}\\\ \epsilon_{12}&1&0\\\ \epsilon_{13}&0&1\end{pmatrix}$ | $\begin{pmatrix}0&0&0\\\ 0&1&0\\\ 0&0&1\end{pmatrix}$ | $\begin{pmatrix}\frac{1}{2}&\frac{1}{2\sqrt{2}}&-\frac{1}{2\sqrt{2}}\\\ \frac{1}{2\sqrt{2}}&\frac{3}{4}&\frac{1}{4}\\\ -\frac{1}{2\sqrt{2}}&\frac{1}{4}&\frac{3}{4}\end{pmatrix}$ | $\begin{pmatrix}\frac{1}{3}&\frac{1}{3}&-\frac{1}{3}\\\ \frac{1}{3}&\frac{5}{6}&\frac{1}{6}\\\ -\frac{1}{3}&\frac{1}{6}&\frac{5}{6}\end{pmatrix}$
$\widetilde{\mathbb{A}}_{2}$: Diag[0,1,-1] | $\begin{pmatrix}0&\epsilon_{12}&-\epsilon_{13}\\\ \epsilon_{12}&1&-2\epsilon_{23}\\\ -\epsilon_{13}&-2\epsilon_{23}&-1\end{pmatrix}$ | $\begin{pmatrix}0&0&0\\\ 0&0&-1\\\ 0&-1&0\end{pmatrix}$ | $\begin{pmatrix}\frac{1}{2}&\frac{1}{2\sqrt{2}}&-\frac{1}{2\sqrt{2}}\\\ \frac{1}{2\sqrt{2}}&-\frac{1}{4}&-\frac{3}{4}\\\ -\frac{1}{2\sqrt{2}}&-\frac{3}{4}&-\frac{1}{4}\end{pmatrix}$ | $\begin{pmatrix}\frac{1}{3}&\frac{1}{3}&-\frac{1}{3}\\\ \frac{1}{3}&-\frac{1}{6}&-\frac{5}{6}\\\ -\frac{1}{3}&-\frac{5}{6}&-\frac{1}{6}\end{pmatrix}$
$\widetilde{\mathbb{B}}$: Diag[1,0,0] | $\begin{pmatrix}1&-\epsilon_{12}&-\epsilon_{13}\\\ -\epsilon_{12}&0&0\\\ -\epsilon_{13}&0&0\end{pmatrix}$ | $\begin{pmatrix}1&0&0\\\ 0&0&0\\\ 0&0&0\end{pmatrix}$ | $\begin{pmatrix}\frac{1}{2}&-\frac{1}{2\sqrt{2}}&\frac{1}{2\sqrt{2}}\\\ -\frac{1}{2\sqrt{2}}&\frac{1}{4}&-\frac{1}{4}\\\ \frac{1}{2\sqrt{2}}&-\frac{1}{4}&\frac{1}{4}\end{pmatrix}$ | $\begin{pmatrix}\frac{2}{3}&-\frac{1}{3}&\frac{1}{3}\\\ -\frac{1}{3}&\frac{1}{6}&-\frac{1}{6}\\\ \frac{1}{3}&-\frac{1}{6}&\frac{1}{6}\end{pmatrix}$
Table 3: Novel textures (leading order) for different mixing scenarios which
by themselves need not be consistent with data. These cases are useful when we
consider adding two different textures to obtain the degenerate cases. The
labels with tilde sign are new textures by taking into account the fact that
hierarchy or inverse hierarchy can appear in either 1-2 sector or the 2-3
sector respectively. The standard textures considered degeneracy in 1-2 sector
and hierarchy or inverse hierarchy only in the 2-3 sector.
In three generations, the above set of decomposition which is based on
neutrino data is not exhaustive. This is essentially because the constraints
of the neutrino data are not on the individual sub-matrices but on the total
mass matrix. In such a case, the normal and inverse hierarchial sub matrices
can take other possible forms ${\mathbb{A}}$ and ${\mathbb{B}}_{i}$ than those
listed in Table 9. From Table 3, it is easy to see that the combinations of
$\widetilde{\mathbb{A}}_{i}$ and $\widetilde{\mathbb{B}}$ would produce one of
the degenerate textures ${\mathbb{C}}_{i}$ of the original Table 9. However,
even this list is not exhaustive for the degenerate case. We could have
textures which are not traditionally ordered as either hierarchial or inverse
hierarchial in the three generation case. These cases are listed in Table 4
and we call them as alternating textures (see Eq. (III.1)). Thus in summary,
we have covered all possible ways of ordering the three degenerate eigenvalues
in to two sub-matrices, which are not degenerate themselves.
Mixing $\Rightarrow$ | Small | Single maximal | Bimaximal | Tribimaximal
---|---|---|---|---
$\mathbb{M}_{diag}$ | ${\mathbb{X}}_{\epsilon}$ | ${\mathbb{X}}_{SM}$ | ${\mathbb{X}}_{BM}$ | ${\mathbb{X}}_{TBM}$
${\mathbb{T}}_{1}$: Diag[0,1,0] | $\begin{pmatrix}0&\epsilon_{12}&0\\\ \epsilon_{12}&1&-\epsilon_{23}\\\ \epsilon_{13}&-\epsilon_{23}&0\end{pmatrix}$ | $\begin{pmatrix}0&0&0\\\ 0&\frac{1}{2}&-\frac{1}{2}\\\ 0&-\frac{1}{2}&\frac{1}{2}\end{pmatrix}$ | $\begin{pmatrix}\frac{1}{2}&\frac{1}{2\sqrt{2}}&-\frac{1}{2\sqrt{2}}\\\ \frac{1}{2\sqrt{2}}&\frac{1}{4}&-\frac{1}{4}\\\ -\frac{1}{2\sqrt{2}}&-\frac{1}{4}&\frac{1}{4}\end{pmatrix}$ | $\begin{pmatrix}\frac{1}{3}&\frac{1}{3}&-\frac{1}{3}\\\ \frac{1}{3}&\frac{1}{3}&-\frac{1}{3}\\\ -\frac{1}{3}&-\frac{1}{3}&\frac{1}{3}\end{pmatrix}$
${\mathbb{T}}_{2}$: Diag[1,0,1] | $\begin{pmatrix}1&-\epsilon_{12}&0\\\ -\epsilon_{12}&0&\epsilon_{23}\\\ 0&\epsilon_{23}&1\end{pmatrix}$ | $\begin{pmatrix}1&0&0\\\ 0&\frac{1}{2}&\frac{1}{2}\\\ 0&\frac{1}{2}&\frac{1}{2}\end{pmatrix}$ | $\begin{pmatrix}\frac{1}{2}&-\frac{1}{2\sqrt{2}}&\frac{1}{2\sqrt{2}}\\\ -\frac{1}{2\sqrt{2}}&\frac{3}{4}&\frac{1}{4}\\\ \frac{1}{2\sqrt{2}}&\frac{1}{4}&\frac{3}{4}\end{pmatrix}$ | $\begin{pmatrix}\frac{2}{3}&-\frac{1}{3}&\frac{1}{3}\\\ -\frac{1}{3}&\frac{2}{3}&\frac{1}{3}\\\ \frac{1}{3}&\frac{1}{3}&\frac{2}{3}\end{pmatrix}$
Table 4: Alternating textures (leading order) for different mixing scenarios.
### III.2 Case II: Three Sources
For more than two seesaw mechanisms at work, the generalisation is straight
forward. Lets consider the case where there are three sources of neutrino
masses. The total mass matrix in this case is given by
$\mathbb{M}_{\nu}=\mathbb{M}_{\nu}^{(1)}+\mathbb{M}_{\nu}^{(2)}+\mathbb{M}_{\nu}^{(3)}~{},$
(53)
where each of the sub matrices can be thought of having independent origin
through a seesaw mechanism or any other scheme to generate non-zero neutrino
masses. As with the two-generation case, we will now consider the case where
all the mixings present in each of the sub matrices are taken to be small and
each sub-matrix is assumed to have only one large eigenvalue. The second
assumption is a direct consequence of assuming that all the three sources
contribute equally and there are no cancellations between the dominant
eigenvalues of the sub-matrices. With these assumptions, the total mass matrix
can now be written in terms of the individual mass matrices as
$\mathbb{M}_{\nu}=m_{1}\left(\begin{array}[]{ccc}\epsilon_{13}^{2}&\epsilon_{13}\epsilon_{23}&\epsilon_{13}\\\
\epsilon_{13}\epsilon_{23}&\epsilon_{23}^{2}&\epsilon_{23}\\\
\epsilon_{13}&\epsilon_{23}&1\end{array}\right)+m_{2}\left(\begin{array}[]{ccc}\epsilon_{12}^{{}^{\prime}2}&\epsilon^{\prime}_{12}&\epsilon^{\prime}_{12}\epsilon^{\prime}_{23}\\\
\epsilon^{\prime}_{12}&1&-\epsilon^{\prime}_{23}\\\
\epsilon^{\prime}_{12}\epsilon^{\prime}_{23}&-\epsilon^{\prime}_{23}&\epsilon_{23}^{{}^{\prime}2}\end{array}\right)+m_{3}\left(\begin{array}[]{ccc}1&-\epsilon^{\prime\prime}_{12}&-\epsilon^{\prime\prime}_{13}\\\
-\epsilon^{\prime\prime}_{12}&\epsilon_{12}^{{}^{\prime\prime}2}&\epsilon^{\prime\prime}_{12}\epsilon^{\prime\prime}_{13}\\\
-\epsilon^{\prime\prime}_{13}&\epsilon^{\prime\prime}_{12}\epsilon^{\prime\prime}_{13}&\epsilon_{13}^{{}^{\prime\prime}2}\end{array}\right)~{},$
(54)
where $\epsilon_{ij},\epsilon^{\prime}_{ij},\epsilon^{\prime\prime}_{ij}$
($i,j=1,2,3$) are small entries corresponding to small mixing angles in
$U^{(i)}_{mix}$. This total matrix can be diagonalised by an orthogonal matrix
${\mathbb{O}}\equiv{\mathbb{O}}_{23}{\mathbb{O}}_{13}{\mathbb{O}}_{12}$, such
that
${\mathbb{O}}^{T}{\mathbb{M}}_{\nu}{\mathbb{O}=\mbox{Diag}[\mathbb{M}}_{\nu}]$.
${\mathbb{O}}_{ij}$ represents a rotation in the ${ij}^{th}$ plane. For
example
${\mathbb{O}}_{23}=\left(\begin{array}[]{ccc}1&0&0\\\
0&\cos\theta_{23}&\sin\theta_{23}\\\
0&-\sin\theta_{23}&\cos\theta_{23}\end{array}\right)~{}.$ (55)
$\displaystyle\theta_{23}$ $\displaystyle\approx$ $\displaystyle{1\over
2}\tan^{-1}\left[{2(m_{1}\epsilon_{23}-m_{2}\epsilon^{\prime}_{23}+m_{3}\epsilon^{\prime\prime}_{12}\epsilon^{\prime\prime}_{13})\over
m_{1}(1-\epsilon_{23}^{2})-m_{2}(1-\epsilon_{23}^{{}^{\prime}2})+m_{3}(\epsilon_{13}^{{}^{\prime\prime}2}-\epsilon_{12}^{"2})}\right]~{},$
$\displaystyle\theta_{13}$ $\displaystyle\approx$ $\displaystyle{1\over
2}\tan^{-1}\left[{2(\widetilde{m}_{13})\over\widetilde{m}_{33}-\widetilde{m}_{11}}\right]~{},$
$\displaystyle\theta_{12}$ $\displaystyle\approx$ $\displaystyle{1\over
2}\tan^{-1}\left[{2(\widetilde{m}^{\prime}_{12})\over\widetilde{m}^{\prime}_{22}-\widetilde{m}^{\prime}_{11}}\right]~{},$
(56)
where
$\displaystyle\widetilde{m}_{13}$ $\displaystyle=$ $\displaystyle
s_{23}(m_{1}\epsilon_{13}\epsilon_{23}+m_{2}\epsilon^{\prime}_{12}-m_{3}\epsilon^{\prime\prime}_{12})+c_{23}(m_{1}\epsilon_{13}+m_{2}\epsilon^{\prime}_{12}\epsilon^{\prime}_{13}-m_{3}\epsilon^{\prime\prime}_{13})~{},$
$\displaystyle\widetilde{m}_{33}$ $\displaystyle=$ $\displaystyle
s_{23}[s_{23}(m_{2}+m_{1}\epsilon_{23}^{2}+m_{3}\epsilon^{\prime\prime}_{12}\epsilon^{\prime\prime}_{13})+c_{23}(m_{1}\epsilon_{23}-m_{2}\epsilon^{\prime}_{23}+m_{3}\epsilon^{\prime\prime}_{12}\epsilon^{\prime\prime}_{13})]$
$\displaystyle+$ $\displaystyle
c_{23}[s_{23}(m_{1}\epsilon_{23}-m_{2}\epsilon^{\prime}_{23}+m_{3}\epsilon^{\prime\prime}_{12}\epsilon^{\prime\prime}_{13})+c_{23}(m_{1}+m_{2}\epsilon^{{}^{\prime}2}_{23}+m_{3}\epsilon^{{}^{\prime\prime}2}_{13})]~{},$
$\displaystyle\widetilde{m}_{11}$ $\displaystyle=$ $\displaystyle
m_{3}+m_{1}\epsilon_{13}^{2}+m_{2}\epsilon_{12}^{{}^{\prime}2}~{},$
$\displaystyle\widetilde{m}^{\prime}_{12}$ $\displaystyle=$ $\displaystyle
c_{13}\widetilde{m}_{12}=c_{13}[c_{23}(m_{1}\epsilon_{13}\epsilon_{23}+m_{2}\epsilon^{\prime}_{12}-m_{3}\epsilon^{\prime\prime}_{12})-s_{23}(m_{1}\epsilon_{13}+m_{2}\epsilon^{\prime}_{12}\epsilon^{\prime}_{13}-m_{3}\epsilon^{\prime\prime}_{13})]~{},$
$\displaystyle\widetilde{m}^{\prime}_{22}$ $\displaystyle=$
$\displaystyle\widetilde{m}_{22}=c_{23}[c_{23}(m_{2}+m_{1}\epsilon_{23}^{2}+m_{3}\epsilon^{\prime\prime}_{12}\epsilon^{\prime\prime}_{13})-s_{23}(m_{1}\epsilon_{23}-m_{2}\epsilon^{\prime}_{23}+m_{3}\epsilon^{\prime\prime}_{12}\epsilon^{\prime\prime}_{13})]$
$\displaystyle-$ $\displaystyle
s_{23}[c_{23}(m_{1}\epsilon_{23}-m_{2}\epsilon^{\prime}_{23}+m_{3}\epsilon^{\prime\prime}_{12}\epsilon^{\prime\prime}_{13})-s_{23}(m_{1}+m_{2}\epsilon_{23}^{{}^{\prime}2}+m_{3}\epsilon_{13}^{{}^{\prime\prime}2})]~{},$
$\displaystyle\widetilde{m}^{\prime}_{11}$ $\displaystyle=$ $\displaystyle
c_{13}(\widetilde{m}_{11}c_{13}-\widetilde{m}_{13}s_{13})-s_{13}(\widetilde{m}_{13}c_{13}-\widetilde{m}_{33}s_{13})~{}.$
(57)
Notice that all the three mass eigenvalues are of the same CP parity in the
above and the degeneracy induced mixing thus works for the all the three
mixing angles. Thus all the three mixing angles are large. One can then ask
the question whether choosing one of the mass eigenvalues with a negative CP
parity would help in keeping one of the mixing angles small. The answer is
negative, choosing one of the eigenvalues to have CP parity negative leads to
at least two of the mixing angles to remain small as the degeneracy induced
large mixing mechanism is no longer operative for two of the mixing angles.
Thus we are back to the case of two seesaw mechanisms which we have seen in
the previous subsection.
While it is possible to visualise GUT models where there are three seesaw
mechanisms at work, it much easier to suitably split a single Type I seesaw
mass matrix into three sub-matrices. In this case, we can extend Eq. (18) to
three generations as
$\displaystyle-{\mathbb{M}}^{\rm{I}}_{\nu}$ $\displaystyle=$ $\displaystyle
m_{1}\left(\begin{array}[]{ccc}(h_{ee}^{D})^{2}&h_{ee}^{D}h_{e\mu}^{D}&h_{ee}^{D}h^{D}_{e\tau}\\\
h_{ee}^{D}h_{e\mu}^{D}&(h_{e\mu}^{D})^{2}&h^{D}_{e\mu}h^{D}_{e\tau}\\\
h_{ee}^{D}h^{D}_{e\tau}&h^{D}_{e\mu}h^{D}_{e\tau}&(h^{D}_{e\tau})^{2}\end{array}\right)+m_{2}\left(\begin{array}[]{ccc}(h_{\mu
e})^{2}&h_{\mu e}^{D}h_{\mu\mu}^{D}&h_{\mu e}^{D}h^{D}_{\mu\tau}\\\ h_{\mu
e}^{D}h_{\mu\mu}^{D}&(h_{\mu\mu}^{D})^{2}&h^{D}_{\mu\mu}h^{D}_{\mu\tau}\\\
h_{\mu
e}^{D}h^{D}_{\mu\tau}&h^{D}_{\mu\mu}h^{D}_{\mu\tau}&(h^{D}_{\mu\tau})^{2}\end{array}\right)$
(64) $\displaystyle+$ $\displaystyle m_{3}\left(\begin{array}[]{ccc}(h_{\tau
e})^{2}&h_{\tau e}^{D}h_{\tau\mu}^{D}&h_{\tau e}^{D}h^{D}_{\tau\tau}\\\
h_{\tau
e}^{D}h_{\tau\mu}^{D}&(h_{\tau\mu}^{D})^{2}&h^{D}_{\tau\mu}h^{D}_{\tau\tau}\\\
h_{\tau
e}^{D}h^{D}_{\tau\tau}&h^{D}_{\tau\mu}h^{D}_{\tau\tau}&(h^{D}_{\tau\tau})^{2}\end{array}\right)~{}.$
(68)
Comparing this with Eq. (54), we see that we will have three possible
solutions for the Yukawa couplings in this case. The first solution is
$\displaystyle
h_{ee}^{D}~{}\sim~{}\epsilon_{13}~{},\;\;h_{e\mu}^{D}~{}\sim~{}\epsilon_{23}~{},\;\;h^{D}_{e\tau}~{}\sim~{}\mathcal{O}(1)~{},\;\;$
$\displaystyle h_{\mu
e}^{D}~{}\sim~{}\epsilon^{\prime}_{12}~{},\;\;h_{\mu\mu}^{D}~{}\sim~{}\mathcal{O}(1)~{},\;\;h_{\mu\tau}^{D}~{}\sim~{}-\epsilon^{\prime}_{23}~{},\;\;$
$\displaystyle h_{\tau
e}^{D}~{}\sim~{}\mathcal{O}(1)~{},\;\;h_{\tau\mu}^{D}~{}\sim~{}-\epsilon^{\prime\prime}_{12}~{},\;\;h_{\tau\tau}^{D}~{}\sim~{}-\epsilon^{\prime\prime}_{13}~{}.$
(69)
There are two more possibilities given by
$\displaystyle h_{\mu
e}^{D}~{}\sim~{}\epsilon_{13}~{},\;\;h_{\mu\mu}^{D}~{}\sim~{}\epsilon_{23}~{},\;\;h^{D}_{\mu\tau}~{}\sim~{}\mathcal{O}(1)~{},\;\;$
$\displaystyle
h_{ee}^{D}~{}\sim~{}\epsilon^{\prime}_{12}~{},\;\;h_{e\mu}^{D}~{}\sim~{}\mathcal{O}(1)~{},\;\;h_{e\tau}^{D}~{}\sim~{}-\epsilon^{\prime}_{23}~{},\;\;$
$\displaystyle h_{\tau
e}^{D}~{}\sim~{}\mathcal{O}(1)~{},\;\;h_{\tau\mu}^{D}~{}\sim~{}-\epsilon^{\prime\prime}_{12}~{},\;\;h_{\tau\tau}^{D}~{}\sim~{}-\epsilon^{\prime\prime}_{13}~{},$
(70)
or
$\displaystyle h_{\tau
e}^{D}~{}\sim~{}\epsilon_{13}~{},\;\;h_{\tau\mu}^{D}~{}\sim~{}\epsilon_{23}~{},\;\;h^{D}_{\tau\tau}~{}\sim~{}\mathcal{O}(1)~{},\;\;$
$\displaystyle h_{\mu
e}^{D}~{}\sim~{}\epsilon^{\prime}_{12}~{},\;\;h_{\mu\mu}^{D}~{}\sim~{}\mathcal{O}(1)~{},\;\;h_{\mu\tau}^{D}~{}\sim~{}-\epsilon^{\prime}_{23}~{},\;\;$
$\displaystyle
h_{ee}^{D}~{}\sim~{}\mathcal{O}(1)~{},\;\;h_{e\mu}^{D}~{}\sim~{}-\epsilon^{\prime\prime}_{12}~{},\;\;h_{e\tau}^{D}~{}\sim~{}-\epsilon^{\prime\prime}_{13}~{}.$
(71)
From the above we see that even if each of the leptonic generation couples
minimally with each of the right-handed neutrino, the total mixing can be
maximal, purely due to the degeneracy requirement. The comments at the end of
subsection III.1 regarding maximally symmetric leptonic mixing matrix hold in
this case too. Finally, note that each set of these solutions is related by
$S_{3}$ symmetry to the other set.
## IV Summary
In the present work, we have concentrated on the case with two seesaw
mechanisms at work which occurs naturally in many examples like LRS models,
SO(10) based GUT models etc. We have shown that if both these seesaw
mechanisms result in mass matrices which only have small mixing in them, then
the only pattern of mass eigenvalues which is naturally consistent with
maximal/large mixing is the quasi-degenerate pattern for the total mass
matrix.
All the arguments presented in the present work are independent of the details
of the sources of neutrino masses. However, depending on the specifics of the
model, there could be radiative corrections which could significantly modify
the mixing angles. For example, if one has Type I + Type II seesaw mechanism
operating at the high scale, radiative corrections could significantly modify
the mixing angles at the weak scale. These effects should be taken in to
account when applying the results of the present work to any particular model.
The impact of radiative corrections, models and implications for leptogenesis
within this class of hybrid degenerate models are being studied for a future
publication ourupcoming .
## Appendix A Generalization of the result for $n$ sources
If there are $n$ sources of neutrino masses in a particular model such that
the total mass matrix is given by
${\mathbb{M}}_{\nu}={\mathbb{M}}^{{(1)}}_{\nu}+{\mathbb{M}}^{{(2)}}_{\nu}+\ldots+{\mathbb{M}}^{{(n)}}_{\nu}~{}.$
(72)
And further each of the ${\mathbb{M}}^{{(i)}}_{\nu}$ have one dominant
diagonal element proportional to its largest eigenvalue $m_{i}$, and rest of
the entries to be tiny (all the mixing angles in all the
${\mathbb{M}}^{{(i)}}_{\nu}$ are small); ${\mathbb{M}}^{{(i)}}_{\nu}$ are
ordered in such a way that the ${ii}^{th}$ element is dominant. There are $n$
possible orderings of ${\mathbb{M}}^{{(i)}}_{\nu}$. Then the total mass matrix
would naturally have a quasi-degenerate pattern with maximal/large mixing
depending on the number of pairs of eigenvalues which have the same CP parity,
if $m_{1}\approx m_{2}\approx\ldots\approx m_{n}$. If there are $l$
eigenvalues with the same CP parity 101010And if the splitting between
relevant $m_{i}$ is smaller than the tiny off-diagonal entries., then
${}^{n}{\cal{C}}_{2}+^{n-l}{\cal{C}}_{2}$ (if $(n-l)>2$) angles will be large
or maximal and the remaining will be small. An important exception to the
above is the pseudo-Dirac pattern of degenerate masses, which can only result
from a ‘sum’ of two mass matrices both containing maximal mixing and equal
eigenvalues with opposite ordering in hierarchy. Conversely, at the zeroth
order a $n\times n$ quasi-degenerate matrix with eigenvalues
$m_{1},m_{2},\ldots m_{i}\ldots m_{n}$ (by definition $m_{1}\approx
m_{2}\approx\ldots\approx m_{n}$) can be decomposed in to $n$ sub-matrices
${\mathbb{M}}^{{(n)}}_{\nu}$, with eigenvalues distributed as
$\displaystyle{\mathbb{M}}_{\nu}$ $\displaystyle=$
$\displaystyle{\mathbb{M}}^{{(1)}}_{\nu}+{\mathbb{M}}^{{(2)}}_{\nu}+\ldots+{\mathbb{M}}^{{(n)}}_{\nu}$
$\displaystyle\begin{pmatrix}m_{1}&&&&\\\ &m_{2}&&&\\\ &&\ddots&&\\\
&&&&m_{n}\end{pmatrix}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}m_{1}&&&&\\\ &0&&&\\\ &&\ddots&&\\\
&&&&0\end{pmatrix}+\begin{pmatrix}0&&&&\\\ &m_{2}&&&\\\ &&\ddots&&\\\
&&&&0\end{pmatrix}+\ldots+\begin{pmatrix}0&&&&\\\ &0&&&\\\ &&\ddots&&\\\
&&&&m_{n}\end{pmatrix}~{}.$ (73)
This holds true irrespective of the mixing present in the total mass matrix
${\mathbb{M}}_{\nu}$.
###### Acknowledgements.
J.C. thanks A. Raychaudhuri for encouragement and discussions. He also
acknowledges support from the Neutrino Project and RECAPP under the XIth plan
of Harish-Chandra Research Institute. J.C. further acknowledges the
hospitality and support from CHEP, IISc., Bangalore where part of the work was
carried out. P.M. acknowledges the kind hospitality received from the Institut
für Theoretische Physik und Astrophysik, Universität Würzburg; Institut für
Theoretische Physik E, RWTH Aachen; CFTP, Instituto Superior Técnico -
Universidade Técnica de Lisboa as well as the organisers of “Workshop towards
neutrino technologies” at ICTP, Italy and “Lepton Photon 2009” in Hamburg
during the final stages of this work.
## References
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* (16) N. Cabibbo, Phys. Lett. B 72, 333 (1978); L. Wolfenstein, Phys. Rev. D 18, 958 (1978); and C. W. Kim and A. Pevsner, Neutrinos in physics and astrophysics, Harwood Academic Publishers, Chur, Switzerland (1993).
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* (18) J. Chakrabortty et. al, in prepartion.
|
arxiv-papers
| 2009-09-17T16:59:11 |
2024-09-04T02:49:05.401693
|
{
"license": "Public Domain",
"authors": "Joydeep Chakrabortty, Anjan S. Joshipura, Poonam Mehta and Sudhir K.\n Vempati",
"submitter": "Sudhir Vempati",
"url": "https://arxiv.org/abs/0909.3116"
}
|
0909.3266
|
# Controlling the carrier concentration of the high temperature superconductor
Bi2Sr2CaCu2O8+δ in Angle Resolved Photoemission Spectroscopy (ARPES)
experiments
A. D. Palczewski Ames Laboratory and Department of Physics and Astronomy,
Iowa State University, Ames, IA 50011, USA T. Kondo Ames Laboratory and
Department of Physics and Astronomy, Iowa State University, Ames, IA 50011,
USA J. S. Wen Condensed Matter Physics and Materials Science Department,
Brookhaven National Laboratory, Upton, New York 11973, USA G. Z. J. Xu
Condensed Matter Physics and Materials Science Department, Brookhaven National
Laboratory, Upton, New York 11973, USA G. Gu Condensed Matter Physics and
Materials Science Department, Brookhaven National Laboratory, Upton, New York
11973, USA A. Kaminski Ames Laboratory and Department of Physics and
Astronomy, Iowa State University, Ames, IA 50011, USA
###### Abstract
We study the variation of the electronic properties at the surface of a high
temperature superconductor as a function of vacuum conditions in angle
resolved photoemission spectroscopy (ARPES) experiments. Normally, under less
than ideal vacuum conditions the carrier concentration of Bi2Sr2CaCu2O8+δ
(Bi2212) increases with time due to the absorption of oxygen from CO2 and CO
molecules that are prime contaminants present in ultra high vacuum (UHV)
systems. We find that in a high quality vacuum environment at low
temperatures, the surface of Bi2212 is quite stable (the carrier concentration
remains constant), however at elevated temperatures the carrier concentration
decreases due to the loss of oxygen atoms from the Bi-O layer. These two
effects can be used to control the carrier concentration in-situ. Our finding
opens the possibility of studying the electronic properties of the cuprates as
a function of doping across the phase diagram on the same piece of sample
(i.e. with the same impurities and defects). We envision that this method
could be utilized in other surface sensitive techniques such as scanning
tunneling microscopy/spectroscopy.
###### pacs:
74.70.Dd, 71.18.+y, 71.20.-b, 71.27.+a
## I Introduction
Surface techniques have played an important role in understanding the
properties of the high temperature superconductors. They have revealed a
number of fascinating phenomena such as the direct observation of the
superconducting gapOLSON and its anisotropySHENSC ; HONGSC , confirmation of
the d-wave symmetry of the order parameter, direct observation of the
pseudogap and its anisotropyHONGPG ; LOESERPG ; MIKEPG , discovery of spatial
inhomogeneitiesDAVIS ; YAZDANI ,unusual spatial ordering,DAVISCHECKER nodal
quasiparticlesKAMINSKIQP , renormalization effectsVALLA ; BOGDANOV ;
KAMINSKIKINK and many othersSHENREVIEW ; JCREVIEW . The success of these
techniques rely on the fact that the layers in some cuprates are very weakly
bonded via the Van der Waals interaction. In such cases the bulk properties
and surface properties are essentially identical, since there is no charge
exchange between the layers. The samples in such cases can be thought of as a
stack of very weakly electrically coupled 2-dimensional conducting surfaces
rather than a 3-dimentional object. Two of the most commonly studied materials
with this property are Bi2Sr2CaCu2O8+δ (Bi2212) and Bi2Sr2CuO6+δ (Bi2201).
There is however one important aspect that needs to be carefully considered,
namely the stability of the cleaved samples under ultra high vacuum (UHV)
conditions. UHV is a rather broad term and refers to pressures lower than
1$\times$10-9 Torr. Quite often such conditions are not sufficient to
guarantee the stability of the surface, particularly in the case of non-
stoichiometric materials such as the cuprates. These problems were recognized
early onSHEN1 , and subsequent measurements revealed significant changes in
the electronic properties as a function of time after cleaving. This issue was
not carefully examined following these first measurements, and it is likely an
important source of data discrepancies among the various groups SHENREVIEW ;
JCREVIEW .
Here we present a systematic study of the electronic properties of Bi2212 as a
function of vacuum conditions. We demonstrate that under poor vacuum
conditions increased carrier concentration arises due to the breakup of CO and
CO2 molecules by exposure to vacuum ultra-violet (VUV) photons and the
subsequent adsorption of oxygen into the BiO layers. We show that with a UHV
leak a sample can increase it carrier concentration just by sitting in the
vaccum. This observation confirms that bilayer splitting is only observed in
over-doped Bi2212. When the partial pressure of active gases is kept at low
levels, the lifetime of cleaved surface of Bi2212 can be as long as a few
weeks at low temperatures (T$<$150K). At elevated temperatures (T$>$200K) the
sample surface loses oxygen, which results in the reduction of carrier
concentration. This second effect is most likely responsible for the recently
reported non-monotonic temperature dependence of the pseudogapA. A. Kordyuk
2008 , where at elevated temperatures the sample surface becomes underdoped
and therefore develops a pseudogap. We demonstrate that these two effects (in-
situ absorption and desorption of oxygen) can be utilized to control the
carrier concentration of the sample surface. This approach enables one to
study the intrinsic electronic properties (i.e. without changing the
impurities and defects) of the cuprates across the phase diagram.
## II Experimental Details
The ARPES data was acquired using a laboratory-based Scienta 2002 electron
analyzer and high intensity Gammadata UV4050 UV source with custom designed
optics. The photocurrent at the sample was approximately 1 $\mu A$, which
corresponds to roughly 1013 photons/sec at 0.05% of the bandwidth. The energy
resolution was set at 10 meV and momentum resolution at 0.12∘ and 0.5∘ along a
direction parallel and perpendicular to the analyzer slits, respectively.
Samples were mounted on a variable temperature cryostat (10-300K) cooled by a
closed cycle refrigerator. The precision of the sample positioning stage was
1$\mu m$. The partial pressure of the active gases was at the detection limit
of the Residual Gas Analyzer (RGA) and the pressure of hydrogen was below
3$\times$10-11 Torr. Excellent vacuum conditions were achieved by strict
adherence to good vacuum practices, use of UHV compatible materials and a
cumulative bake-out time of the system in excess of 6 months. The typical
lifetime of the optimally doped Bi2212 surfaces was greater than two weeks
after cleaving, defined as less than 5% change of the superconducting gap (2
meV) at 40K. The core-level spectra was acquired on the Hermon beam-line at
the Synchrotron Radiation Center using a Scienta 2002 end-station. The photon
energy was set at 500 eV and energy resolution at 200 meV.
## III Increasing carrier concentration
It has been known for some time that aging (increased surface doping) in the
cuprates is caused by less than ideal UHV conditionsSHEN1 . Aging is usually
detected by measuring the superconducting gap (the energy gap as defined by
the difference between the peak position of a Bi2212 spectrum and the chemical
potential measured by a polycrystalline gold sample) as a function of time. If
the gap shifts to a lower binding energy the sample has aged.H. Ding 1997 ; P.
Schwaller 2000 . Fig. 1 (a)-(b) shows an example of this where a freshly
cleaved Bi2212 single crystal was scanned in a relatively poor vacuum to see
how the spectrum changed over time. A shift to lower binding energy as well as
a peak suppression was detected showing the sample was aging. In Fig. 1 (c)
the size of the superconducting gap is shown as a function of time. Only when
there were VUV photons on the sample did the sample age. While there we no VUV
photons on the sample, from 5 hour to 21 hours, the aging stopped. If the non-
VUV scanning time is taken out Fig. 1 (d), the magnitude of the gap shows an
exponential decay (blue line). While it is know that surface aging of Bi2212
happens in a poor UHV system, only when there were VUV photons on the sample
does did sample actually age, signaling that aging is directly related to
having VUV photons on the sample, not just the vacuum conditions.
Figure 1: Color online) ARPES specta of Bi2212 taken under poor vacuum
conditions (a) sample EDC (energy distribution curves) taken at the anti-node
where the band crosses the Fermi energy at 5 different times, (b) narrow view
of (a), (c) time evolution of Bi2212’s superconducting gap as a function of
tine, (d) the time evolution of Bi2212’s superconducting gap under VUV photons
(red) fitted with an exponential decay (blue), the temperature for (a)-(d) was
set to 20K, (e) C 1s, Sr 3P 1/2, Sr 3p 3/2 core level data from Bi2212 showing
carbon deposits some time after cleaving and after while cooling. Figure 2:
(Color online) (a) ARPES intensity map of freshly cleaved optimally doped
Bi2212 at ($\pi$, 0) showing no bilayer splitting, (b) ARPES intensity maps on
the same sample and the same location as in (a) only oxygen aged (in-situ
overdoing) in a UHV system with a leak showing bilayer splitting and a peak
shift location of the Fermi momentum, with the black line as a guide to the
eye.
n absence of leaks, a reasonable UHV system has normally undetectable levels
of oxygen. However in stainless steel vessels CO and CO2 are always present.
These oxide molecules can adhere to clean sample surfaces especially at low
temperatures. When the molecules are exposed to VUV photons above 6 eV they
break into carbon and oxygen M. M. Halmann ; the oxygen can then be
incorporated into BiO layer as dopant, while the carbon atoms remain on the
surface. The proof of this scenario is in Fig. 1 (e) where the core-level
spectrum of Bi2212 at 300K and 40K are shown. As the sample cooled more CO and
CO2 molecules adhered to the surface of the sample. Since there are carbon
deposits some time after cleaving and even more after cooling, it is likely
the oxygen accompanied the carbon to the surface. This oxygen can then change
the doping of the sample after it is dissociated from the carbon.
Figure 3: (Color online) (a)-(c) symmetrized ARPES EDC’s for Bi2212 taken at
three points near ($\pi$,0) showing the time evolution of the spectrum at
280K.
In the presence of a leak a UHV system can have detectable amounts of oxygen.
Under these conditions a Bi2212 sample can age even without the breakdown of
CO and CO2. One of the trademarks of an over-doped (aged) Bi2212 sample is the
appearance of bi-layer band splitting at the antinode ($\pi$, 0). While there
has been a relatively active discussion on whether Bi2212 contains bilayer
band splitting all the time or just in an over-doped state; bilayer splitting
has only been seen in over-doped samples when using a helium discharge lamp
Y.-D. Chaung 2004 ; S. V. Borisenko 2004 ; S. V. Borisenko 2006 ; A. A.
Kordyuk 2004 . An example of this is shown in FIG. 2 where a fresh Bi2212
sample was scanned and then allowed to sit in the leaky UHV system overnight
before scanning again. Even though the sample was kept a 20 K, bilayer band
splitting was detected after the break, signaling that the sample aged because
of oxygen absorption.
## IV Decreasing carrier concentration
Figure 4: (Color online) (a) EDC at the Fermi momentum close to the anti-node
before (green circles) and after (solid red squares) annealing at 280K over 28
hours with their respective superconducting gaps $\Delta$, (b)-(c) momentum
intensities maps taken across Fermi momentum close to ($\pi$, 0) before and
after annealing.
While in a reasonable vacuum system there can be enough CO2/CO to change the
surface doping of a sample over time; in an ultra clean UHV system samples can
live for many weeks without surface degradation or a change in doping
(assuming the sample is kept at low temperature). Yet, when the sample is
annealed above 200K an interesting thing happens to the Bi2212’s doping level;
the sample doping level is reduced (the opposite of aging). This is seen in
Fig. 3 (a)-(c) where the time evolution of Bi2212’s EDCs at three locations at
or near ($\pi$,0) with the sample at 280K is shown. The sample actually
changes doping moving towards lower doping (signified by a larger spectral
gap). Fig. 4 (a) shows the energy distribution curve (EDC) at the anti-nodal
Fermi momentum from the same sample before and after annealing at 280K for 28
hours. The superconducting gap clearly shifts from 33 meV to 41 meV and the
peak is suppressed, signaling that the doping has changed from a slightly over
doped sample to a more under doped sampleT. Sato 2001 . The momentum color
maps from Fig. 4 (a) are shown in FIG. 4 (b)-(c); after annealing the gap
shifts to higher binding energy, there is also a shift in the location of the
Fermi momentum. This momentum shift comes from a change in the chemical
potential, which moves lower in a ridged-band-like fashion upon doping.M.
Hashimoto 2008
Another way to see if a samples carrier concentration has decreased is to look
at the pseudogap. Fig. 5 (a) shows the EDC at the Fermi momentum before and
after annealing at 280K for 28 hours. The pseudogap shifts from 30 meV to 50
meV. As Bi2212 goes to lower doping levels the pseudogap becomes bigger and
the temperature at which the pseudogap remains (T*) becomes higherH. Ding 1996
. Fig. 5 (b)-(c) demonstrates that before annealing T* is below 140K with the
pseudogap disappearing and after annealing T* is above 200K. The pseudogap
after annealing is above 200K, which guarantees that the sample is at a lower
doping level.
Figure 5: (Color online) (a) 100 K symmetrized ARPES data taken at the Fermi
momentum before and after annealing at 280 K for 28 hours, (b) ARPES
intensities at 140 K before annealing, (c) ARPES intensities at 200 K after
annealing.
Until now we have only shown the lowering of doping on Bi2212 at elevated
temperature. While we still haven’t shown if the doping change is caused by
the elevated temperature or a combination of elevated temperature and VUV
photons. This was tested by scanning the sample just after cleaving and again
after the sample sat under UHV for 16 days at 100K. This data is shown in
figure Fig. 6 (a). The spectrum barely changed over the two weeks. While in
Fig. 6 (b) we show the 280K spectrum just after cleaving, and again after the
sample sat under UHV for 8 days at 280K. Most of the spectral weight has
shifted to higher energies and the Fermi edge has all but disappeared,
signifying an almost completely insulating sample. From Fig. 6 we can conclude
that the lowering of the samples doping is only caused by the elevated
temperatures.
Figure 6: (Color online) Bi2212 EDC at the Fermi momentum close to ($\pi$,0)
(a) just after cleaving at 100K (red circles) and again after sitting at 100K
for 16 days (solid blue squares), (b) just after cleaving at 280K (red
circles) and again after sitting at 280K for 8 days (solid blue squares).
Figure 7: (Color online) ARPES intensity plots at the Fermi energy from the
same sample at three different times all at 12K: (a) just after cleaving, (b)
after a couple of days of VUV aging at low temperature, (c) after annealing at
280K overnight; the upper right hand corner of (a)-(c) are the zoomed in
images from the bottom left hand corner, the red and dotted yellow curves are
from a tight binding fit for optimally doped Bi2212 as a guide to the eye,
(d)-(f) the size of the superconducting gap as a function of angle $\phi$ from
(a)-(c) respectively.
The greatest consequence of this study is that Bi2212’s doping can be change
from over doped all the way down to insulating in a systematic fashion on a
single crystal. To this point the data presented has been either over doped by
aging or under doped by annealing on different samples. Fig. 7 demonstrates
how a single sample can be over-doped by aging and then under-doped by
annealing to move across the phase diagram. An optimally doped Bi2212 sample
was cleaved, the Fermi surface and superconducting gap values as a function of
angle $\phi$ (angle clockwise from the line ($\pi$,-$\pi$) to (2
$\pi$,-$\pi$)) was scanned Fig. 7 (a) $\&$ (d). Aging was detected after a
couple of days of scanning Fig. 7 (b) $\&$ (e). The sample was then annealed
overnight at 280K to remove the aging Fig. 7 (c) $\&$ (f).
## V Conclusion
We have presented a systematic study of the electronic properties at the
surface of Bi2212 as a function of vacuum conditions. The results confirm that
under poor vacuum conditions there is an increase in carrier concentration due
to the breakup of CO and CO2 molecules by exposure to vacuum ultra-violet
(VUV) photons and a subsequent adsorption of oxygen into the BiO layers. We
also show that with a UHV leak a sample can increase its carrier concentration
just by sitting in the vaccum. This observation confirms that bilayer
splitting only occurs in over-doped Bi2212. We then show that at elevated
temperatures (T$>$200K) the sample surface loses oxygen, which results in a
reduction of the carrier concentration. These two effects (in-situ absorption
and desorption of oxygen) can be utilized in order to control the carrier
concentration of Bi2212. This approach enables one to study the intrinsic
electronic properties (i.e. without changing the impurities and defects) of
the cuprates across the phase diagram in ARPES as well as other surface
sensitive techniques on a single sample.
## VI Acknowledgments
This work was supported by Director Office for Basic Energy Sciences, US DOE.
Work at Ames Laboratory was supported by the Department of Energy - Basic
Energy Sciences under Contract No. DE-AC02-07CH11358. The work at BNL was
supported by Department of Energy - Basic Energy Sciences under Contract No.
DE-AC02-98CH10886. Synchrotron Radiation Center is supported by the National
Science Foundation under award No. DMR-0537588.
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* (27) H. Ding, T. Yokoya, J. C. Campuzano, T. Takahashi, M. Randeria, M. R. Norman, T. Mochikuparallel, K. Kadowakiparallel J. Giapintzakis, Nature 382, 51 (1996).
|
arxiv-papers
| 2009-09-17T16:43:33 |
2024-09-04T02:49:05.412032
|
{
"license": "Public Domain",
"authors": "A. D. Palczewski, T. Kondo, J. S. Wen, G. Z. J. Xu, G. Gu, A. Kaminski",
"submitter": "Adam Kaminski",
"url": "https://arxiv.org/abs/0909.3266"
}
|
0909.3422
|
# Highly Selective Terahertz Bandpass Filters Based on Trapped Mode Excitation
Oliver Paul paul@physik.uni-kl.de Department of Physics and Research Center
OPTIMAS, University of Kaiserslautern, Germany René Beigang Department of
Physics and Research Center OPTIMAS, University of Kaiserslautern, Germany
Fraunhofer Institute for Physical Measurement Techniques IPM, Freiburg,
Germany Marco Rahm Department of Physics and Research Center OPTIMAS,
University of Kaiserslautern, Germany Fraunhofer Institute for Physical
Measurement Techniques IPM, Freiburg, Germany
###### Abstract
We present two types of metamaterial-based spectral bandpass filters for the
terahertz (THz) frequency range. The metamaterials are specifically designed
to operate for waves at normal incidence and to be independent of the field
polarization. The functional structures are embedded in films of
benzocyclobutene (BCB) resulting in large-area, free-standing and flexible
membranes with low intrinsic loss. The proposed filters are investigated by
THz time-domain spectroscopy and show a pronounced transmission peak with over
80 % amplitude transmission in the passband and a transmission rejection down
to the noise level in the stopbands. The measurements are supported by
numerical simulations which evidence that the high transmission response is
related to the excitation of trapped modes.
###### pacs:
42.79.-e; 42.79.Ci; 07.57.Hm; 42.70.-a
## I Introduction
In the last ten years, metamaterials have emerged to be powerful tools for the
manipulation of light on the subwavelength scale. The scientific interest has
been primarily driven by the possibility of creating materials with new
electromagnetic properties not occurring in nature such as e. g. negative
index materials veselago1968 ; smith2000a , invisibility cloaks and
transformation optics pendry2006 ; pendry2008 . However, metamaterials are not
only of scientific interest for their exotic properties. In the frequency
range between 0.1 and 10 THz, which is usually referred to as the THz gap, the
lack of electromagnetic response of most natural materials has substantially
obstructed the development of functional components. For the THz technology,
metamaterials can play a crucial role for the conception of artificial optical
components since their electromagnetic properties can be exactly designed to
match the functionality of an envisioned optical component. In this context,
several optical elements as e. g. wave plates averitt2009 , THz amplitude
padilla2006b and phase modulators chen2009 and spatial modulators chan2009
have already been successfully demonstrated.
Fig. 1: Microscope pictures of (a) the cross-slot structure and (b) the two
layers of wire-and-plate structure. (c) Resulting metamaterial membrane with a
functional area of 9$\times$9 mm2.
Recently, the excitation of so-called trapped modes, i.e. modes that are
weakly coupled to an external electromagnetic field, has been observed in
metamaterials fedotov2007 ; zhang2008 ; Papasimakis2008 ; tassin2009 ; liu2009
. These modes show analogies to the electromagnetically induced transparency
(EIT) of atomic systems harris1990 ; liu2001 like a sharp phase dispersion of
the transmitted radiation and a narrow transmission band within a broad
stopband. Such properties open the possibility for the construction of very
efficient and compact metamaterial-based bandpass filters with a high
selectivity.
In this paper, we present two types of spectral bandpass filters for the THz
frequency range based on the excitation of trapped modes. The corresponding
resonances of the subwavelength elements have been optimized to obtain a high
transmission in the passband and an efficient suppression of transmitted
radiation in the stopbands. The two implemented metamaterial designs are a
cross-slot structure and a wire-and-plate structure (see Figs. 1(a) and 1(b)).
Such structures have been originally introduced in the microwave regime
munk2000 ; behdad2006 . They operate at normal incidence and are independent
of the polarization of the incident light. The polarization insensitivity is a
direct consequence of the 4-fold rotational symmetry of the structure
mackay1989 . In order to enhance the bandpass effect we employed a multilayer
technique to embed several functional layers in films of BCB. The BCB serves
as a homogeneous background matrix and enables us to fabricate large-area,
free-standing and flexible metamaterial membranes. This is especially
important with regard to a practical integration of such metamaterial
components in THz systems since the beam diameter of THz radiation is usually
in the order of several millimeters. The designed and fabricated bandpass
filters were experimentally characterized by means of THz time-domain
spectroscopy.
## II Filter design und fabrication
The cross-slot structure is set-up by an array of 3 µm wide cross-shaped slots
(Fig. 1(a)). The enclosed crosses are formed by 46 µm long and 9 µm wide cross
bars. The cross bars act as small electric dipoles that can be excited by an
incident THz wave. The lattice constants of the structure are 68 µm in the x-
and y-direction and 40 µm in the z-direction. In contrast, the wire-and-plate
structure (Fig. 1(b)) is composed by two separated layers being 9.5 µm apart
from each other. The front layer consists of a two-dimensional wire grid
formed by 17 µm wide wires whereas the background layer is represented by an
array of square plates with a side length of 50 µm. The lattice constants for
this structure are 60 µm in the x- and y-direction and 35 µm in the
z-direction. Since the adjacent edges of each two plates act as a capacitor
whereas the facing strip of the wire grid acts as an inductor, the composite
structure forms an LC-resonant circuit that can be excited by a normally
incident THz wave.
The fabrication of the metamaterial films was performed in a multilayer
process with alternating layers of BCB 3022-63 and copper on top of a silicon
substrate. The BCB layers were fabricated by a spin coating technique followed
by a thermal curing process in a vacuum oven at 300 ∘C for about 5 h. The
metal layers were patterned by standard UV-lithography using an AZ nLof 2035
photoresist, an EVG 620 mask aligner and an electron beam evaporation of 200
nm copper. For the plate-and-wire design a strict alignment of the plates and
wires layers within a unit cell is necessary to ensure the functionality of
the structure. For this purpose we used alignment marks providing an accuracy
in the order of 1 µm. A microscope image of one layer of unit cells of both
designs is shown in Figs. 1(a) and 1(b), respectively. The films were then
removed from the silicon substrate in a 30 % solution of KOH. The resulting
free-standing membranes are 17$\times$17 mm2 large, mechanically and
chemically stable and quite flexible. A photograph of the resulting membrane
is presented in Fig. 1(c).
We fabricated membranes with one layer of unit cells of the cross-slot
structure and two layers of unit cells of the wire-and-plate structure.
However, as shown in paul2008 , the free-standing membranes can be stacked on
top of another to further increase the number of layers. Similar to the
double-cross structure reported in paul2008 , the structures used for the
filter designs are independent of the polarization and the coupling between
the functional metal layers in neighboring membranes can be neglected due to
the thick BCB spacer. Hence, the alignment of individual membranes is not
crucial to the orientation or the relative position of the membranes and can
be performed under simple visual control.
## III Results and discussion
The transmission characteristics of the metamaterial filters was analyzed by
standard THz time-domain spectroscopy with a detectable frequency range of 0.1
– 2.5 THz and a frequency resolution of 9 GHz. The THz radiation was linearly
polarized and was focused under normal incidence on the sample surface to a
spot size of 1.5 mm. Finally, the measured transmission spectra have been
normalized by a reference spectrum without sample to obtain the amplitude
transmittance of the filters.
We analyzed one and two layers of unit cells of the cross-slot structure by
measuring a single and two stacked membranes, each fabricated with one layer
of unit cells. For the wire-and-plate structure we analyzed two and four
layers of unit cells by using a single and two stacked membranes where each
membrane consisted of two layers of unit cells. The experimentally obtained
spectral transmission data were compared to numerical simulations which have
been carried out by a commercially available time-domain solver (CST Microwave
Studio), where the BCB can be described by a dielectric constant of
$\epsilon=2.67$ and a loss parameter of $\tan\delta=0.01$ paul2008 . However,
we varied the permittivity of BCB in order to fit the numerical data to the
experimental results and obtained reasonable agreement by using
$\epsilon=2.45$ for the cross-slot and $\epsilon=1.85$ for the wire-and-plate
design.
Fig. 2: Experimental (Exp) and numerical (Sim) amplitude transmission and
reflection results for (a) the cross-slot and (b) the wire-and-plate structure
for different numbers of layers of unit cells. Fig. 3: Surface current
distribution at the center frequency of the passband for (a) the cross-slot
structure and (b) the front plane (left) and the backplane (right) of the
wire-and-plate structure. The incident electric field is vertically polarized.
Figs. 2(a) and 2(b) show the spectral amplitude transmission and reflection of
the cross-slot-structure and the wire-and-plate structure, respectively. The
experimental transmission results (colored solid lines) are in good agreement
with the numerical simulations (colored dashed lines). Both filter designs
reveal a pronounced passband around 1.3 THz. As expected, the frequency
selectivity of the bandpass filters increases with increasing number of layers
of unit cells. For two layers of the cross-slot structure and for four layers
of the wire-and-plate structure the FWHM bandwidth of the passband is $\Delta
f=0.3\,\mathrm{THz}$ in each case. Both filters offer a very high amplitude
transmission over 80 %, a fast roll-off and a very efficient blocking in the
lower and upper rejection bands down to the noise level where the incident
radiation is almost completely reflected. Moreover, the transmission response
of the cross-slot structure is ripple-free, whereas the wire-and-plate filter
exhibits a faster roll-off.
The origin of these strong resonances can be attributed to so-called trapped
modes fedotov2007 ; zhang2008 ; Papasimakis2008 ; tassin2009 ; liu2009 , i.e.
modes that are weakly coupled to electromagnetic waves incident from free
space. For such modes the radiation losses are very small in comparison to the
stored field energy which leads to an enhanced transmission at the resonance
frequency. The excitation of trapped modes is evidenced by the simulation of
the surface current distribution in the metamaterial structures presented in
Fig. 3. At the resonance frequency, the induced currents are counter
propagating at distinct sections of the structure with almost similar
magnitude. As a consequence, the resulting dipole moment and therefore the
dipolar coupling to external electromagnetic fields is strongly reduced which
results in a high transmission of electromagnetic radiation through the
metamaterial structure at frequencies near the resonance.
In particular, for the cross-slot structure the surface currents are counter
propagating at the two opposing edges of the slot (Fig. 3(a)). This specific
current distribution is related to the fact that the outer metal frame is just
the complementary of a cross structure which causes the driven currents to
oscillate with opposite phase. For the wire-and-plate structure, it’s the
currents in the front layer (metal wire grid) and the background layer (square
metal patches) that are in opposite phase (Fig. 3(b)). This can be explained
by the different functions of the layers in the equivalent LC-resonant
circuit. As mentioned in Sec. 2, the plates act as capacitors, i.e. the phase
of the driven currents is shifted by $+\pi/2$ with respect to the external
electric field, whereas the wires act as inductors which causes a phase shift
of $-\pi/2$. This implies that the excited currents in the two layers must be
opposite in phase.
Fig. 4: Retrieved values of (a) the effective index of refraction and (b) the
effective permittivity of the bandpass filters where ($\cdot$)’ and ($\cdot$)”
denote the real and imaginary part, respectively. The spectral passband is
shaded.
## IV Effective material parameters
For a more quantitative characterization of the investigated metamaterial
bandpass filters, we further applied a retrieval algorithm chen2004 to
calculate the effective values of the refractive index $n$ and the
permittivity $\epsilon$ from the simulated transmission and reflection data.
The retrieval was supported by additional computation of the phase advance of
a propagating plane wave across the material to ensure that the correct branch
of the refractive index was chosen.
The resulting effective refractive index and the permittivity are plotted in
Fig. 4 for both the cross-slot structure and the wire-and-plate structure.
Thereby, ($\cdot$)’ and ($\cdot$)” denote the real and imaginary part,
respectively and the spectral passband is indicated by a blue box. It can bee
seen from Fig. 4(b) that the permittivity of both structures exhibits a
characteristic narrow resonance, where $\epsilon^{\prime}$ is only positive in
the vicinity of the resonance frequency. As a consequence, only in this region
$n^{\prime\prime}$ is sufficiently small to allow high transmission leading to
the observed passband. It should be noted that both structures exhibit similar
effective material parameters even though their constituent elements widely
differ in shape and geometry. This is due to the fact, that the transmission
response of both structures is related to the same origin: the excitation of
trapped modes in subwavelength elements. Moreover, since the quality factors
of the excited resonances are equal, the media that are composed by the two
structures are equivalent in the framework of effective medium theory. Another
remarkable result is the rapid increase of $n^{\prime}$ within the passband as
displayed in Fig. 4(a). This strong frequency dispersion leads to an increase
of the group index which is given by $n_{g}=\frac{c}{v_{g}}=c\frac{\partial
k^{\prime}}{\partial\omega}=n^{\prime}+\frac{\partial
n^{\prime}}{\partial\omega}\omega$ with the group velocity $v_{g}$ and the
dispersion relation $k^{\prime}=n^{\prime}\frac{\omega}{c}$. From the plotted
curves for $n^{\prime}$, the average group index can be calculated in the
passband to be $n_{g}=7.4$ for the wire-and-plate structure and even
$n_{g}=9.3$ for cross-slot structure. This means that a propagating pulse
whose spectrum covers the passband will be transmitted with a significant time
delay. Although the group refractive index is not as high as can be expected
in the case of electromagnetically induced transparency liu2001 or
accordingly plasmon-induced transparency zhang2008 where a dark mode is
phase-coupled to a broadband dipole resonance, the calculations demonstrate
the highly dispersive character of trapped mode excitation.
## V Conclusion
In summary, we have presented two types of metamaterial bandpass filters in
the THz frequency range. The implemented metamaterials are based on a cross-
slot and a wire-and-plate structure, respectively. The filters are embedded in
membranes of BCB allowing free-standing, flexible films and are designed to
operate at normal incidence and to be independent of the polarization of the
incident light. We have shown that the observed transmission response is
related to the excitation of trapped modes where the reduced coupling to the
electromagnetic field leads to an enhanced transmission at the resonance
frequency.
The special characteristics of the presented filters is an outstanding high
transmission over 80 % in the passband and a fast roll-off down to the noise
level in the stopbands. The spectral bandwidth of the realized band-pass
filters is 0.3 THz. Such highly selective filters can be used to remove
unwanted transmitted signals in pre-defined frequency bands and have potential
applications in the field of THz diagnostics.
We thank Dr. Christian Imhof from the Department of Electrical and Computer
Engineering, University of Kaiserslautern, for supportive comments and
discussions, and the Nano+Bio Center at the University of Kaiserslautern for
their support in the sample fabrication.
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|
arxiv-papers
| 2009-09-18T12:31:16 |
2024-09-04T02:49:05.419199
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Oliver Paul, Rene Beigang, and Marco Rahm",
"submitter": "Marco Rahm",
"url": "https://arxiv.org/abs/0909.3422"
}
|
0909.3570
|
11footnotetext: Weierstrass Institute for Applied Analysis and Stochastics,
Mohrenstr. 39, 10117 Berlin, Germany. belomest@wias-berlin.de.
# On the rates of convergence of simulation based optimization algorithms for
optimal stopping problems
Denis Belomestny${}^{1,\,}$ supported in part by the SFB 649 ‘Economic Risk’.
###### Abstract
In this paper we study simulation based optimization algorithms for solving
discrete time optimal stopping problems. This type of algorithms became
popular among practioneers working in the area of quantitative finance. Using
large deviation theory for the increments of empirical processes, we derive
optimal convergence rates and show that they can not be improved in general.
The rates derived provide a guide to the choice of the number of simulated
paths needed in optimization step, which is crucial for the good performance
of any simulation based optimization algorithm. Finally, we present a
numerical example of solving optimal stopping problem arising in option
pricing that illustrates our theoretical findings.
_Keywords:_ optimal stopping, simulation based algorithms, entropy with
bracketing, increments of empirical processes
## 1 Introduction
The theory of optimal stopping is concerned with the problem of choosing a
time to take a particular action, in order to maximise an expected reward or
minimise an expected cost. Optimal stopping problems can be found in many
areas of statistics, economics, and mathematical finance. They can often be
written in the form of a Bellman equation, and are therefore often solved
using dynamic programming. Results on optimal stopping were first developed in
the discrete case. The formulation of optimal stopping problems for discrete
stochastic processes was in sequential analysis, an area of mathematical
statistics where the number of observations is not fixed in advance but is a
random number determined by the behavior of the data being observed. Snell
(1952) was the first person to come up with results on optimal stopping theory
for stochastic processes in discrete time. We refer to the book of Peskir and
Shiryaev (2006) for a comprehensive review on different aspects of optimal
stopping problems.
A huge impetus to the development of optimal stopping theory was provided by
option pricing theory, developed in the late 1960s and the 1970s. According to
the modern financial theory, pricing an American option in a complete market
is equivalent to solving an optimal stopping problem (with a corresponding
generalization in incomplete markets), the optimal stopping time being the
rational time for the option to be exercised. Due to the enormous importance
of the early exercise feature in finance, this line of research has been
intensively pursued in recent times. Solving the optimal stopping problem and
hence pricing an American option is straightforward in low dimensions.
However, many problems arising in practice have high dimensions, and these
applications have motivated the development of Monte Carlo methods for pricing
American option. Solving a high-dimensional optimal stopping problems or
pricing American style derivatives with Monte Carlo is a challenging task
because the determination of the optimal value function requires a backwards
dynamic programming algorithm that appears to be incompatible with the forward
nature of Monte Carlo simulation. Much research was focused on the development
of fast methods to compute approximations to the optimal value function.
Notable examples include mesh method of Broadie and Glasserman (1997), the
regression-based approaches of Carriere (1996), Longstaff and Schwartz (2001),
Tsitsiklis and Van Roy (1999) and Egloff (2005). All these methods aim at
approximating the so called continuation values that can be used later to
construct suboptimal strategies and to produce lower bounds for the optimal
value function. The convergence analysis for this type of methods was
performed in several papers including Egloff (2005), Egloff, Kohler and
Todorovic (2007) and Belomestny (2009). An alternative to trying to
approximate the continuation values is to find the best value function within
a class of stopping rules. This reduces the optimal stopping problem to a much
more tractable finite dimensional optimization problem. Such optimization
problems appear naturally if one considers finite dimensional or parametric
approximations for the corresponding stopping regions. The latter type of
algorithms became particularly popular among practioneers (see e.g. Andersen
(2000) or Garcia (2001)). However, the practical success of simulation-based
optimization algorithms has not been yet fully explained by existing theory,
and our analysis here represents a further step toward an improved
understanding. The main goal of this work is to provide rigorous convergence
analysis of simulation based optimization algorithms for discrete time optimal
stopping problems.
Let us start with a general stochastic programming problem
(1.1) $\displaystyle
h^{*}:=\min_{\theta\in\Theta}\operatorname{E}_{\operatorname{P}}[h(\theta,\xi)],$
where $\Theta$ is a subset of $\mathbb{R}^{m}$, $\xi$ is a $\mathbb{R}^{d}$
valued random variable on the probability space
$(\Omega,\mathcal{F},\operatorname{P})$ and
$h:\mathbb{R}^{m}\times\mathbb{R}^{d}\to\mathbb{R}.$ Draw an i.i.d. sample
$\xi^{(1)},\ldots,\xi^{(M)}$ from the distribution of $\xi$ and define
$\displaystyle
h_{M}:=\min_{\theta\in\Theta}\left[\frac{1}{M}\sum_{m=1}^{M}h(\theta,\xi^{(m)})\right].$
It is well known (see e.g. Shapiro (1993)) that under very mild conditions it
holds $h_{M}-h^{*}=O_{\operatorname{P}}(M^{-1/2}).$ In their pioneering work
Shapiro and Homem-de-Mello (2000) (see also Kleywegt, Shapiro and Homem-de-
Mello (2001)) showed that in the case of discrete random variable $\xi,$ the
convergence of $h_{N}$ to $h^{*}$ can be much faster than $M^{-1/2},$ making
Monte Carlo method particularly efficient in this situation. Turn now to the
discrete time optimal stopping problem:
(1.2) $\displaystyle V=\sup_{1\leq\tau\leq K}\operatorname{E}[Z_{\tau}],$
where $\tau$ is a stopping time taking values in the set $\\{1,\ldots,K\\}$
and $(Z_{k})_{k\geq 0}$ is a Markov chain. Since the random variable $\tau$
takes only discrete values, one can ask whether the simulation based methods
in the case of discrete time optimal stopping problem (1.2) can be as
efficient as in the case of (1.1) with discrete r.v. $\xi$. In this work we
give an affirmative answer to this question by deriving the optimal rates of
convergence for the corresponding Monte Carlo estimate of $V$ based on $M$
paths and showing that these rates are usually faster than $M^{-1/2}$.
## 2 Main setup
Let us consider a Markov chain $X=(X_{k})_{k\geq 0}$ defined on a filtered
probability space $(\Omega,\mathcal{F},(\mathcal{F}_{k})_{k\geq
0},\operatorname{P}_{x})$ and taking values in a measurable space
$(E,\mathcal{B}),$ where for simplicity we assume that $E=\mathbb{R}^{d}$ for
some $d\geq 1$ and $\mathcal{B}=\mathcal{B}(\mathbb{R}^{d})$ is the Borel
$\sigma$-algebra on $\mathbb{R}^{d}.$ It is assumed that the chain $X$ starts
at $x$ under $\operatorname{P}_{x}$ for some $x\in E$ . We also assume that
the mapping $x\mapsto P_{x}(A)$ is measurable for each $A\in\mathcal{F}$ . Fix
some natural number $K>0.$ Given a set of measurable functions
$G_{k}:E\mapsto\mathbb{R}$, $k=1,\ldots,K,$ satisfying
$\displaystyle\operatorname{E}_{x}\left[\sup_{1\leq k\leq
K}|G_{k}(X_{k})|\right]<\infty$
for all $x\in E$ , we consider the optimal stopping problems
(2.3) $\displaystyle V^{*}_{k}(x):=\sup_{k\leq\tau\leq
K}\operatorname{E}_{k,\,x}\left[G_{\tau}(X_{\tau})\right],\quad k=1,\ldots,K,$
where for any $x\in E$ the expectation in (2.3) is taken w.r.t. the measure
$\operatorname{P}_{k,\,x}$ such that $X_{k}=x$ under
$\operatorname{P}_{k,\,x}$ and the supremum is taken over all stopping times
$\tau$ with respect to $(\mathcal{F}_{n})_{n\geq 0}.$ Introduce the stopping
region
$\boldsymbol{\mathcal{S}}^{*}=\mathcal{S}^{*}_{1}\times\ldots\times\mathcal{S}^{*}_{K}$
with $\mathcal{S}^{*}_{K}=E$ and
$\mathcal{S}^{*}_{k}:=\\{x\in E:V^{*}_{k}(x)=G_{k}(x)\\}\\\ =\left\\{x\in
E:\operatorname{E}\left[\left.V^{*}_{k+1}(X_{k+1})\right|\mathcal{F}_{k}\right]\leq
G_{k}(x)\right\\},\quad k=1,\ldots,K-1.$
Introduce also the first entry times $\tau^{*}_{k}$ into
$\boldsymbol{\mathcal{S}}^{*}$ by setting
$\displaystyle\tau^{*}_{k}:=\tau_{k}(\boldsymbol{\mathcal{S}}^{*}):=\min\\{k\leq
l\leq K:X_{l}\in\mathcal{S}_{l}\\}.$
It is well known that the value functions $V^{*}_{k}(x)$ satisfy the so called
Wald-Bellman equations
$\displaystyle
V^{*}_{k}(x)=\max\\{G_{k}(x),\operatorname{E}_{n,x}[V^{*}_{k+1}(X_{k+1})]\\},\quad
k=1,\ldots,K-1,\quad x\in E$
with $V^{*}_{K}(x)\equiv G_{K}(x)$ by definition. Moreover, the stopping times
$\tau^{*}_{k}$ are optimal in (2.3), i.e.
$\displaystyle
V^{*}_{k}(x)=\operatorname{E}_{k,\,x}\left[G_{\tau^{*}_{k}}(X_{\tau^{*}_{k}})\right],\quad
k=1,\ldots,K.$
Let $(X^{(m)}_{k})_{k=0,\ldots,K},\,m=1,\ldots,M$ be $M$ independent processes
with the same distribution as $X$ all starting from the point $x\in E.$ We can
think of $(X^{(1)}_{k},\ldots,X^{(M)}_{k}),$ $k=0,\ldots,K,$ as a new process
defined on the product probability space equipped with the product measure
$\operatorname{P}_{x}^{\otimes M}.$ Let $\mathfrak{B}$ be a collection of sets
from the product $\sigma$-algebra
$\mathcal{B}^{K}:=\underbrace{\mathcal{B}\otimes\ldots\otimes\mathcal{B}}_{K}$
that contains all sets $\boldsymbol{\mathcal{S}}\in\mathcal{B}^{K}$ of the
form
$\boldsymbol{\mathcal{S}}=\mathcal{S}_{1}\times\ldots\times\mathcal{S}_{K-1}\times
E$ with $\mathcal{S}_{k}\in\mathcal{B},\,k=1,\ldots,K-1.$ Here we take into
account the fact that the stopping set $\mathcal{S}_{K}$ must coincide with
$E.$ Let $\mathfrak{S}$ be a subset of $\mathfrak{B}.$ Define
$\displaystyle\boldsymbol{\mathcal{S}}_{M}:=\arg\max_{\boldsymbol{\mathcal{S}}\in\mathfrak{S}}\left\\{\frac{1}{M}\sum_{m=1}^{M}G_{\tau_{1}(\boldsymbol{\mathcal{S}})}\left(X^{(m)}_{\tau_{1}(\boldsymbol{\mathcal{S}})}\right)\right\\}.$
The stopping rule
$\displaystyle\tau_{M}:=\tau_{1}(\boldsymbol{\mathcal{S}}_{M})=\min\\{1\leq
k\leq K:X_{k}\in\mathcal{S}_{M,k}\\}.$
is generally suboptimal and therefore the corresponding Monte Carlo estimate
(2.4) $\displaystyle
V_{M,N}:=\frac{1}{N}\sum_{n=1}^{N}G_{\tau^{(n)}_{M}}\left(\widetilde{X}^{(n)}_{\tau^{(n)}_{M}}\right)$
with
$\displaystyle\tau^{(n)}_{M}:=\min\\{1\leq k\leq
K:\widetilde{X}^{(n)}_{k}\in\mathcal{S}_{M,k}\\},\quad n=1,\ldots,N$
based on a new, independent of $(X^{(1)},\ldots,X^{(M)})$ set of trajectories
$(\widetilde{X}^{(n)}_{0},\ldots,\widetilde{X}^{(n)}_{K}),\quad n=1,\ldots,N,$
fulfills
(2.5) $\displaystyle
V_{M}:=\operatorname{E}_{x}\left[V_{M,N}|X^{(1)},\ldots,X^{(M)}\right]$
$\displaystyle\leq$
$\displaystyle\sup_{\boldsymbol{\mathcal{S}}\in\mathfrak{S}}\operatorname{E}_{x}\left[G_{\tau_{1}(\boldsymbol{\mathcal{S}})}\left(X_{\tau_{1}(\boldsymbol{\mathcal{S}})}\right)\right].$
If the set $\mathfrak{S}$ is rich enough, then
$\displaystyle\sup_{\boldsymbol{\mathcal{S}}\in\mathfrak{S}}\operatorname{E}_{x}\left[G_{\tau_{1}(\boldsymbol{\mathcal{S}})}\left(X_{\tau_{1}(\boldsymbol{\mathcal{S}})}\right)\right]=:\operatorname{E}_{x}\left[G_{\tau_{1}(\bar{\boldsymbol{\mathcal{S}}})}\left(X_{\tau_{1}(\bar{\boldsymbol{\mathcal{S}}})}\right)\right]\approx\operatorname{E}_{x}\left[G_{\tau_{1}(\boldsymbol{\mathcal{S}}^{*})}\left(X_{\tau_{1}(\boldsymbol{\mathcal{S}}^{*})}\right)\right]$
and $V_{M,N}$ can serve as a good approximation for $V^{*}$ for large enough
$M$ and $N.$ In the next section we are going to study the question: how fast
does $V_{M}$ converge to $V^{*}=V^{*}_{1}$ as $M\to\infty$ ? We will show that
the corresponding rates of convergence are always faster than usual rates
$M^{-1/2}.$ This fact has a practical implication since it indicates that $M$,
the number of simulated paths used in the optimization step, can be taken much
smaller than $N,$ the number of paths used to compute the final estimate
$V_{M,N}$.
## 3 Main results
#### Definition
Let $\delta>0$ be a given number and $d_{X}(\cdot,\cdot)$ be a pseudedistance
between two elements of $\mathfrak{B}$ defined as
(3.6) $\displaystyle d_{X}(G_{1}\times\ldots\times
G_{K},G^{\prime}_{1}\times\ldots\times
G^{\prime}_{K})=\sum_{k=1}^{K}\operatorname{P}_{x}(X(t_{k})\in G_{k}\triangle
G^{\prime}_{k}),$
where $\\{G_{k}\\}$ and $\\{G^{\prime}_{k}\\}$ are subsets of $E.$ Define
$N(\delta,\mathfrak{S},d_{X})$ be the smallest value $n$ for which there exist
pairs of sets
$(G_{j,1}^{L}\times\ldots\times G_{j,K}^{L},G_{j,1}^{U}\times\ldots\times
G_{j,K}^{U}),\quad j=1,\ldots,n,$
such that $d_{X}(G_{j,1}^{L}\times\ldots\times
G_{j,K}^{L},G_{j,1}^{U}\times\ldots\times G_{j,K}^{U})\leq\delta$ for all
$j=1,\ldots,n,$ and for any $G\in\mathfrak{S}$ there exists
$j(G)\in\\{1,\ldots,n\\}$ for which
$G^{L}_{j(G),k}\subseteq G_{k}\subseteq G^{U}_{j(G),k},\quad k=1,\ldots,K.$
Then the value
$\mathcal{H}(\delta,\mathfrak{S},d):=\log[N(\delta,\mathfrak{S},d_{X})]$ is
called the $\delta$-entropy with bracketing of $\mathfrak{S}$ for the
pseudedistance $d_{X}$.
#### Assumption
We assume that the family of stopping regions $\mathfrak{S}$ is such that
(3.7) $\displaystyle\mathcal{H}(\delta,\mathfrak{S},d_{X})\leq
A\delta^{-\rho}$
for some constant $A>0$, any $0<\delta<1$ and some $\rho>0$.
#### Example
Let $\mathfrak{S}=\mathfrak{S}_{\gamma}$, where $\mathfrak{S}_{\gamma}$ is a
class of subsets of
$\overbrace{\mathbb{R}^{d}\times\ldots\times\mathbb{R}^{d}}^{K}$ with
boundaries of Hölder smoothness $\gamma>0$ defined as follows. For given
$\gamma>0$ and $d\geq 2$ consider the functions $b(x_{1},\ldots,x_{d-1}),$
$b:\mathbb{R}^{d-1}\to\mathbb{R}$ having continuous partial derivatives of
order $l$, where $l$ is the maximal integer that is strictly less than
$\gamma$. For such functions $b$, we denote the Taylor polynomial of order $l$
at a point $x\in\mathbb{R}^{d-1}$ by $\pi_{b,x}$. For a given $H>0$, let
$\Sigma(\gamma,H)$ be the class of functions $b$ such that
$\displaystyle|b(y)-\pi_{b,x}(y)|\leq H\|x-y\|^{\gamma},\quad
x,y\in\mathbb{R}^{d-1},$
where $\|y\|$ stands for the Euclidean norm of $y\in\mathbb{R}^{d-1}.$ Any
function $b$ from $\Sigma(\gamma,H)$ determines a set
$\displaystyle S_{b}:=\\{(x_{1},\ldots,x_{d})\in\mathbb{R}^{d}:0\leq x_{d}\leq
b(x_{1},\ldots,x_{d-1})\\}.$
Define the class
(3.8) $\displaystyle\mathfrak{S}_{\gamma}:=\\{S_{b_{1}}\times\ldots\times
S_{b_{K-1}}\times E:\,b_{1},\ldots,b_{K-1}\in\Sigma(\gamma,H)\\}.$
It can be shown (see Dudley, 1999, Section 8.2) that the class
$\mathfrak{S}_{\gamma}$ fulfills
$\mathcal{H}(\delta,\mathfrak{S}_{\gamma},d_{X})\leq
A\delta^{-(K-1)(d-1)/\gamma}$
for some $A>0$ and all $\delta>0$ small enough. Now we are in the position to
formulate the main result of our study.
###### Theorem 3.1.
Let $\mathfrak{S}$ be a subset of $\mathfrak{B}$ such that assumption (3.7) is
fulfilled with some $0<\rho\leq 1$ and
(3.9)
$\displaystyle\operatorname{E}_{x}\left[G_{\tau_{1}(\boldsymbol{\mathcal{S}}^{*})}\left(X_{\tau_{1}(\boldsymbol{\mathcal{S}}^{*})}\right)\right]-\bar{V}\leq
DM^{-1/(1+\rho)}$
with
$\bar{V}:=\sup_{\boldsymbol{\mathcal{S}}\in\mathfrak{S}}\operatorname{E}_{x}\left[G_{\tau_{1}(\boldsymbol{\mathcal{S}})}\left(X_{\tau_{1}(\boldsymbol{\mathcal{S}})}\right)\right]$
and some constant $D>0.$ Assume that all functions $G_{k}$ are uniformly
bounded and the inequalities
(3.10)
$\displaystyle\operatorname{P}_{x}(|G_{k}(X_{k})-\operatorname{E}[V^{*}_{k+1}(X_{k+1})|\mathcal{F}_{k}]|<\delta)\leq
A_{0,k}\delta^{\alpha},\quad\delta<\delta_{0}$
hold for some $\alpha>0$, $A_{0,k}>0,$ $k=1,\ldots,K-1$, and $\delta_{0}>0$.
Then for any $U>U_{0}$ and $M>M_{0}$
(3.11) $\displaystyle\operatorname{P}^{\otimes
M}_{x}\left(V^{*}-V_{M}\geq(U/M)^{\frac{1+\alpha}{2+\alpha(1+\rho)}}\right)\leq
C\exp(-\sqrt{U}/B).$
with some constants $U_{0}>0$, $M_{0}>0$, $B>0$ and $C>0.$
###### Remark 3.2.
Without condition (3.9) the inequality (3.11) continues to hold with $V^{*}$
replaced by $\bar{V}$, the best approximation of $V^{*}$ within the class of
stopping regions $\mathfrak{S}.$
###### Remark 3.3.
The requirement that functions $G_{k}$ are uniformly bounded can be replaced
by the existence of all moments of $G_{k}(X_{k}),\,k=1,\ldots,K-1,$ under
$\operatorname{P}.$ In this case on can reformulate Theorem 6.1 using
generalized entropy with bracketing instead of usual entropy with bracketing
(see Chapter 5.4 in Van de Geer (2000)).
The above convergence rates can not be in general improved as shown in the
next theorem.
###### Proposition 3.4.
Consider the problem (2.3) with $k=1$ and two possible stopping dates, i.e.
$\tau\in\\{1,2\\}$. Fix a pair of non-zero functions $G_{1},G_{2}$ such that
$G_{2}:\mathbb{R}^{d}\to\\{0,1\\}$ and $0<G_{1}(x)<1$ on $[0,1]^{d}.$ Fix some
$\gamma>0$ and $\alpha>0$ and let $\mathcal{P}_{\alpha,\gamma}$ be a class of
pricing measures such that the condition (3.10) is fulfilled and for any
$\operatorname{P}\in\mathcal{P}_{\alpha,\gamma}$ the corresponding stopping
set $\boldsymbol{\mathcal{S}}^{*}_{\operatorname{P}}$ is in
$\mathfrak{S}_{\gamma}.$ Then there exist a subset $\mathcal{P}$ of
$\mathcal{P}_{\alpha,\gamma}$ and a constant $B>0$ such that for any $M\geq
1$, any stopping time $\tau_{M}\in\\{1,2\\}$ measurable w.r.t.
$\mathcal{F}^{\otimes M}$
$\displaystyle\sup_{\operatorname{P}\in\mathcal{P}}\left\\{\sup_{\tau\in\\{1,2\\}}\operatorname{E}_{\operatorname{P}}[G_{\tau}(X_{\tau})]-\operatorname{E}_{\operatorname{P}^{\otimes
M}}[\operatorname{E}_{\operatorname{P}}G_{\tau_{M}}(X_{\tau_{M}})]\right\\}\geq
BM^{-\frac{1+\alpha}{2+\alpha(1+(d-1)/\gamma)}}.$
#### Discussion
It follows from Theorem 3.1 that
$\displaystyle
V^{*}-V_{M}=O_{\operatorname{P}}\left(M^{-\frac{1+\alpha}{2+\alpha(1+\rho)}}\right)=o_{\operatorname{P}}(M^{-1/2})$
as long as $\alpha>0.$ Using the decomposition
$\displaystyle V^{*}-V_{M,N}=V^{*}-V_{M}+V_{M}-V_{M,N}$
and the fact that $V_{M}-V_{M,N}=O_{\operatorname{P}}(1/\sqrt{N})$ for any
$M>0$, we conclude that
$\displaystyle
V^{*}-V_{M,N}=O_{\operatorname{P}}\left(M^{-\frac{1+\alpha}{2+\alpha(1+\rho)}}+N^{-\frac{1}{2}}\right).$
Hence, given $N$, a reasonable choice of $M$, the number of Monte Carlo paths
used in the optimization step, can be defined as $M\asymp
N^{\frac{2+\alpha(1+\rho)}{2(1+\alpha)}}.$ In the case when there exists a
parametric family of stopping regions satisfying (3.9) (see Section 4 for some
examples), one gets
(3.12) $\displaystyle M\asymp N^{\frac{2+\alpha}{2(1+\alpha)}}$
since any parametric family of stopping regions with finite dimensional
parameter set fulfills (3.7) for arbitrary small $\rho>0.$ Let us also make a
few remarks on the condition (3.10) and the parameter $\alpha$. If each
function
$G_{k}(x)-\operatorname{E}_{k,x}[V^{*}_{k+1}(X_{k+1})],\,k=1,\ldots,K-1,$ has
a non-vanishing Jacobian in the vicinity of the stopping boundary
$\partial\mathcal{S}_{k}$ and $X_{k}$ has continuous distribution, then (3.10)
is fulfilled with $\alpha=1.$ In fact, it is not difficult to construct
examples showing that the parameter $\alpha$ can take any value from
$\mathbb{R}_{+}$. If $\alpha=1$ (the most common case) (3.12) simplifies to
$M\asymp N^{3/4}$, the choice supported by our numerical example.
Finally, we would like to mention an interesting methodological connection
between our analysis and the analysis of statistical discrimination problem
performed in Mammen and Tsybakov (1999) (see also Devroye, Györfi and Lugosi
(1996)). In particular, we need similar results form the theory of empirical
processes and the condition (3.10) formally resembles the so called “margin”
condition often encountered in the literature on discrimination analysis.
## 4 Applications
In this section we illustrate our theoretical results by some financial
applications. Namely, we consider the problem of pricing Bermudan options. The
pricing of American-style options is one of the most challenging problems in
computational finance, particularly when more than one factor affects the
option values. Simulation based methods have become increasingly attractive
compared to other numerical methods as the dimension of the problem increases.
The reason for this is that the convergence rates of simulation based methods
are generally independent of the number of state variables. In the context of
our paper we consider the so called parametric approximation algorithms (see
Glasserman, 2003, Section 8.2). In essence, these algorithms represent the
optimal stopping sets $\mathcal{S}^{*}_{k}$ by a finite numbers of parameters
and then find the Bermudan option price by maximizing, over the parameter
space, a Monte Carlo approximation of the corresponding value function. The
important question here is wether on can parametrize the optimal stopping
region $\boldsymbol{\mathcal{S}}^{*}$ by a finite dimensional set of
parameters, i.e.
$\boldsymbol{\mathcal{S}}^{*}=\boldsymbol{\mathcal{S}}(\theta),\,\theta\in\Theta,$
where $\Theta$ is a compact finite dimensional set. It turns out that that
this is possible in many situations (see Garcia (2001)). The assumption (3.7)
and (3.9) are then automatically fulfilled with arbitrary small $\rho>0.$
### 4.1 Numerical example: Bermudan max call
This is a benchmark example studied in Broadie and Glasserman (1997) and
Glasserman (2003) among others. Specifically, the model with $d$ identically
distributed assets is considered, where each underlying has dividend yield
$\delta$. The risk-neutral dynamic of the asset
$X(t)=(X^{1}(t),\ldots,X^{d}(t))$ is given by
$\frac{dX^{l}(t)}{X^{l}(t)}=(r-\delta)dt+\sigma dW^{l}(t),\quad
X^{l}(0)=x_{0},\quad l=1,...,d,$
where $W^{l}(t),\,l=1,...,d$, are independent one-dimensional Brownian motions
and $x_{0},r,\delta,\sigma$ are constants. At any time
$t\in\\{t_{1},...,t_{K}\\}$ the holder of the option may exercise it and
receive the payoff
$G_{k}(X_{k}):=\left(\max\left(X^{1}_{k},...,X^{d}_{k}\right)-\kappa\right)^{+},$
where $X_{k}:=X(t_{k})$ for $k=1,\ldots,K.$ We take $d=2$, $r=5\%$,
$\delta=10\%$, $\sigma=0.2$, $\kappa=100$, $x_{0}=90$ and
$t_{k}=kT/K,\,k=1,\ldots,K$, with $T=3,\,K=9$ as in Glasserman (2003, Chapter
8).
To describe the optimal early exercise region at date $t_{k},\,k=1,\ldots,K,$
one can divide $\mathbb{R}^{2}$ into three different connected sets: one
exercise region and two continuation regions (see Broadie and Detemple (1997)
for more details). All these regions can be parameterized by using two
functions depending on two dimensional parameter
$\theta_{k}\in\mathbb{R}^{2}.$ Making use of this characterization, we define
a parametric family of stopping regions as in Garcia (2001) via
$\displaystyle\mathcal{S}_{k}(\theta_{k}):=\\{(x_{1},x_{2}):\max(\max(x_{1},x_{2})-K,0)>\theta^{1}_{k};\,|x_{1}-x_{2}|>\theta^{2}_{k}\\},$
where $\theta_{k}\in\Theta,\,k=1,\ldots,K$ and $\Theta$ is a compact subset of
$\mathbb{R}^{2}.$ Furthermore, we simplify the corresponding optimization
problem by setting $\theta_{1}=\ldots=\theta_{K}.$ This will introduce an
additional bias and hence may increase the left hand side of (3.9) (see Remark
3.2). However, this bias turns out to be rather small in practice. In order to
implement and analyze the simulation based optimization based algorithm in
this situation, we perform the following steps:
* •
Simulate $L$ independent sets of trajectories of the process $(X_{k})$ each of
the size $M$:
$\displaystyle(X^{(l,m)}_{1},\ldots,X^{(l,m)}_{K}),\quad m=1,\ldots,M,$
where $l=1,\ldots,L.$
* •
Compute estimates $\theta_{M}^{(1)},\ldots,\theta_{M}^{(L)}$ via
$\displaystyle\theta_{M}^{(l)}:=\arg\max_{\theta\in\Theta}\left\\{\frac{1}{M}\sum_{m=1}^{M}G_{\tau_{1}(\boldsymbol{\mathcal{S}}(\theta))}\left(X^{(l,m)}_{\tau_{1}(\boldsymbol{\mathcal{S}}(\theta))}\right)\right\\}.$
* •
Simulate a new set of trajectories of size $N$ independent of
$(X^{(l,m)}_{k}):$
$(\widetilde{X}^{(n)}_{1},\ldots,\widetilde{X}^{(n)}_{K}),\quad n=1,\ldots,N.$
* •
Compute $L$ estimates for the optimal value function $V^{*}_{1}$ as follows
$\displaystyle
V^{(l)}_{M,N}:=\frac{1}{N}\sum_{n=1}^{N}G_{\tau^{(l,n)}_{M}}\left(\widetilde{X}^{(n)}_{\tau^{(l,n)}_{M}}\right),\quad
l=1,\ldots,L,$
with
$\displaystyle\tau^{(l,n)}_{M}:=\min\left\\{1\leq k\leq
K:\widetilde{X}^{(n)}_{k}\in\mathcal{S}_{k}\left(\theta^{(l)}_{M}\right)\right\\},\quad
n=1,\ldots,N.$
Denote by $\sigma_{M,N,l}$ the standard deviation computed from the sample
$(G_{\tau^{(l,n)}_{M}},\,n=1,\ldots,N)$ and set
$\sigma_{M,N}=\min_{l}\sigma_{M,N,l}.$
* •
Compute
$\displaystyle\mu_{M,N,L}:=\frac{1}{L}\sum_{l=1}^{L}V^{(l)}_{M,N},\quad\vartheta_{M,N,L}:=\sqrt{\frac{1}{L-1}\sum_{l=1}^{L}\left(V^{(l)}_{M,N}-\mu_{M,N,L}\right)^{2}}.$
By the law of large numbers
(4.13) $\displaystyle\mu_{M,N,L}$
$\displaystyle\stackrel{{\scriptstyle\operatorname{P}}}{{\to}}$
$\displaystyle\operatorname{E}_{\operatorname{P}^{\otimes
M}}\left[V_{M,N}\right],\quad L\to\infty,$ (4.14)
$\displaystyle\vartheta_{M,N,L}$
$\displaystyle\stackrel{{\scriptstyle\operatorname{P}}}{{\to}}$
$\displaystyle\operatorname{Var}_{\operatorname{P}^{\otimes
M}}\left[V_{M,N}\right],\quad L\to\infty,$
where
$\displaystyle
V_{M,N}:=\frac{1}{N}\sum_{n=1}^{N}G_{\tau^{(n)}_{M}}\left(\widetilde{X}^{(n)}_{\tau^{(n)}_{M}}\right).$
The difference $\bar{V}-V_{M,N}$ with
$\displaystyle\bar{V}:=\max_{\theta\in\Theta}\operatorname{E}[G_{\tau_{1}(\boldsymbol{\mathcal{S}}(\theta))}(X_{\tau_{1}(\boldsymbol{\mathcal{S}}(\theta))})]$
can be decomposed into the sum of three terms
(4.15) $\displaystyle(\bar{V}-\operatorname{E}_{\operatorname{P}^{\otimes
M}}\left[V_{M}\right])+(\operatorname{E}_{\operatorname{P}^{\otimes
M}}\left[V_{M}\right]-V_{M})+V_{M}-V_{M,N}.$
The first term in (4.15) is deterministic and can be approximated by
$Q_{1}(M):=\mu_{M^{*},N^{*},L^{*}}-\mu_{M,N^{*},L^{*}}$ with large enough
$L^{*}$, $M^{*}$ and $N^{*}.$ The variability of the second, zero mean,
stochastic term can be measured by
$\sqrt{\operatorname{Var}_{\operatorname{P}^{\otimes M}}\left[V_{M}\right]}$
which in turn can be estimated by
$Q_{2}(M):=\sqrt{\vartheta_{M,N^{*},L^{*}}}$, due to (4.14). The standard
deviation of $V_{M}-V_{M,N}$ for any $M$ can be approximated by
$Q_{3}(N)=\sigma_{M^{*},N}/\sqrt{N}$. In our simulation study we take
$N^{*}=1000000,\,L^{*}=500,\,M^{*}=10000$ and obtain
$\bar{V}\approx\mu_{M^{*},N^{*},L^{*}}=7.96$ (note that $V^{*}=8.07$ according
to Glasserman (2003)). In the left-hand side of Figure 1 we plot both
quantities $Q_{1}(M)$ and $Q_{2}(M)$ as functions of $M.$ Note that $Q_{2}(M)$
dominates $Q_{1}(M)$, especially for large $M.$ Hence, by comparing $Q_{2}(M)$
with $Q_{3}(N)$ and approximately solving the equation $Q_{2}(M)=Q_{3}(N)$ in
$N$, one can infer on the optimal relation between $M$ and $N$. In Figure 1
(on the right-hand side) the resulting empirical relation is depicted by
crosses. Additionally, we plotted two benchmark curves $N=M^{4/3}$ and
$N=M^{4.5/3}$. As one can see the choice $M=N^{3/4}$ is likely to be
sufficient in this situation since it always leads to the inequality
$Q_{1}(M)+\sigma Q_{2}(M)\leq\sigma Q_{3}(N)$ for any $\sigma>1.$ As a
consequence, for $M=N^{3/4}$ and any $N$, $\bar{V}$ lies with high probability
in the interval $[\mu_{M,N,L^{*}}-\sigma Q_{3}(N),\mu_{M,N,L^{*}}+\sigma
Q_{3}(N)],$ provided that $\sigma$ is large enough.
Figure 1: Left: functions $Q_{1}(M)$ and $Q_{2}(M)$; Right: optimal empirical
relationship between $M$ and $N$ (crosses) together with benchmark curves
$N=M^{4/3}$ (dashed line) and $N=M^{4.5/3}$ (dotted line).
## 5 Proof of main results
### 5.1 Proof of Theorem 3.1
Define
$\displaystyle\Delta_{M}(\boldsymbol{\mathcal{S}})$ $\displaystyle:=$
$\displaystyle\sqrt{M}\sum_{m=1}^{M}\left\\{G_{\tau_{1}(\boldsymbol{\mathcal{S}})}\left(X^{(m)}_{\tau_{1}(\boldsymbol{\mathcal{S}})}\right)-\operatorname{E}\left[G_{\tau_{1}(\boldsymbol{\mathcal{S}})}\left(X_{\tau_{1}(\boldsymbol{\mathcal{S}})}\right)\right]\right\\}$
and
$\Delta_{M}(\boldsymbol{\mathcal{S}}^{\prime},\boldsymbol{\mathcal{S}}):=\Delta_{M}(\boldsymbol{\mathcal{S}}^{\prime})-\Delta_{M}(\boldsymbol{\mathcal{S}})$
for any
$\boldsymbol{\mathcal{S}}^{\prime},\boldsymbol{\mathcal{S}}\in\mathfrak{S}.$
Since
$\frac{1}{M}\sum_{m=1}^{M}G_{\tau_{1}(\bar{\boldsymbol{\mathcal{S}}})}\left(X^{(m)}_{\tau_{1}(\bar{\boldsymbol{\mathcal{S}}})}\right)\leq\frac{1}{M}\sum_{m=1}^{M}G_{\tau_{1}(\boldsymbol{\mathcal{S}}_{M})}\left(X^{(m)}_{\tau_{1}(\boldsymbol{\mathcal{S}}_{M})}\right)$
with probability $1$, it holds
(5.16) $\displaystyle\Delta(\boldsymbol{\mathcal{S}}_{M})$ $\displaystyle\leq$
$\displaystyle\Delta(\bar{\boldsymbol{\mathcal{S}}})+\frac{\left[\Delta_{M}(\boldsymbol{\mathcal{S}}^{*},\bar{\boldsymbol{\mathcal{S}}})+\Delta_{M}(\boldsymbol{\mathcal{S}}_{M},\boldsymbol{\mathcal{S}}^{*})\right]}{\sqrt{M}}$
with
$\Delta(\boldsymbol{\mathcal{S}}):=\operatorname{E}[G_{\tau^{*}_{1}}(X_{\tau^{*}_{1}})]-\operatorname{E}[G_{\tau_{1}(\boldsymbol{\mathcal{S}})}(X_{\tau_{1}(\boldsymbol{\mathcal{S}})})].$
Set $\varepsilon_{M}=M^{-1/2(1+\rho)}$ then
$\displaystyle\Delta(\boldsymbol{\mathcal{S}}_{M})$ $\displaystyle\leq$
$\displaystyle\Delta(\bar{\boldsymbol{\mathcal{S}}})+\frac{2}{\sqrt{M}}\sup_{\boldsymbol{\mathcal{S}}\in\mathfrak{S}:\,\Delta_{G}(\boldsymbol{\mathcal{S}}^{*},\boldsymbol{\mathcal{S}})\leq\varepsilon_{M}}|\Delta_{M}(\boldsymbol{\mathcal{S}}^{*},\boldsymbol{\mathcal{S}})|$
$\displaystyle+2\times\frac{\Delta_{G}^{(1-\rho)}(\boldsymbol{\mathcal{S}}^{*},\boldsymbol{\mathcal{S}}_{M})}{\sqrt{M}}\times\sup_{\boldsymbol{\mathcal{S}}\in\mathfrak{S}:\,\Delta_{G}(\boldsymbol{\mathcal{S}}^{*},\boldsymbol{\mathcal{S}})>\varepsilon_{M}}\left[\frac{|\Delta_{M}(\boldsymbol{\mathcal{S}}^{*},\boldsymbol{\mathcal{S}})|}{\Delta_{G}^{(1-\rho)}(\boldsymbol{\mathcal{S}}^{*},\boldsymbol{\mathcal{S}})}\right].$
Define
$\displaystyle\mathcal{W}_{1,M}:=\sup_{\boldsymbol{\mathcal{S}}\in\mathfrak{S}:\,\Delta_{G}(\boldsymbol{\mathcal{S}}^{*},\boldsymbol{\mathcal{S}})\leq\varepsilon_{M}}|\Delta_{M}(\boldsymbol{\mathcal{S}}^{*},\boldsymbol{\mathcal{S}})|,$
$\displaystyle\mathcal{W}_{2,M}:=\sup_{\boldsymbol{\mathcal{S}}\in\mathfrak{S}:\,\Delta_{G}(\boldsymbol{\mathcal{S}}^{*},\boldsymbol{\mathcal{S}})>\varepsilon_{M}}\frac{|\Delta_{M}(\boldsymbol{\mathcal{S}}^{*},\boldsymbol{\mathcal{S}})|}{\Delta_{G}^{(1-\rho)}(\boldsymbol{\mathcal{S}}^{*},\boldsymbol{\mathcal{S}})}$
and set $\mathcal{A}_{0}:=\\{\mathcal{W}_{1,M}\leq
U\varepsilon_{M}^{1-\rho}\\}$ for $U>U_{0}.$ Note that under assumption (3.7)
the condition (6.17) of Theorem 6.1 is fulfilled with $\nu=2\rho$ due to
Corollary 6.3. Hence Theorem 6.1 yields
$\operatorname{P}(\bar{\mathcal{A}}_{0})\leq
C\exp(-U\varepsilon_{M}^{-2\rho}/C^{2}).$ Denote
$\displaystyle\Delta_{G}(\boldsymbol{\mathcal{S}},\boldsymbol{\mathcal{S}}^{\prime}):=\left\\{\operatorname{E}\left[G_{\tau_{1}(\boldsymbol{\mathcal{S}})}\left(X_{\tau_{1}(\boldsymbol{\mathcal{S}})}\right)-G_{\tau_{1}(\boldsymbol{\mathcal{S}}^{\prime})}\left(X_{\tau_{1}(\boldsymbol{\mathcal{S}}^{\prime})}\right)\right]^{2}\right\\}^{1/2}$
for any
$\boldsymbol{\mathcal{S}},\boldsymbol{\mathcal{S}}^{\prime}\in\mathfrak{B}.$
Since $\Delta(\bar{\boldsymbol{\mathcal{S}}})\leq DM^{-1/(1+\rho)}$ and
$\varepsilon^{1-\rho}_{M}/\sqrt{M}=M^{-1/(1+\rho)}$, we get on
$\mathcal{A}_{0}$
$\displaystyle\Delta(\boldsymbol{\mathcal{S}}_{M})$ $\displaystyle\leq$
$\displaystyle
C_{0}M^{-1/(1+\rho)}+2\times\frac{\Delta_{G}^{(1-\rho)}(\boldsymbol{\mathcal{S}}^{*},\boldsymbol{\mathcal{S}}_{M})}{\sqrt{M}}\mathcal{W}_{2,M}$
with $C_{0}=D+2U$. Combining Corollary 6.3 with Corollary 6.4 leads to the
inequality
$\displaystyle\Delta_{G}(\boldsymbol{\mathcal{S}}^{*},\boldsymbol{\mathcal{S}}_{M})\leq
2\sqrt{2}A_{G}v^{-\alpha/2(1+\alpha)}_{\alpha}\Delta^{\alpha/2(1+\alpha)}(\boldsymbol{\mathcal{S}}_{M})$
which holds on the set
$\mathcal{A}_{1}:=\\{\Delta_{X}(\boldsymbol{\mathcal{S}}^{*},\boldsymbol{\mathcal{S}}_{M})\leq\delta_{\alpha}\\},$
where $\delta_{\alpha}$ and $v_{\alpha}$ are defined in Corollary 6.4. Denote
$\displaystyle\mathcal{A}_{2}:=\left\\{\Delta(\boldsymbol{\mathcal{S}}_{M})>C_{0}(1+\varkappa)M^{-1/(1+\rho)}\right\\}$
with some $\varkappa>0.$ It then holds on
$\mathcal{A}_{0}\cap\mathcal{A}_{1}\cap\mathcal{A}_{2}$
$\displaystyle\Delta(\boldsymbol{\mathcal{S}}_{M})\leq
2\frac{\Delta^{\alpha(1-\rho)/(2(1+\alpha))}(\boldsymbol{\mathcal{S}}_{M})}{\varkappa\sqrt{M}}\mathcal{W}_{2,M}$
and therefore
$\displaystyle\Delta(\boldsymbol{\mathcal{S}}_{M})\leq(\varkappa/2)^{-\nu}M^{-\nu/2}\mathcal{W}^{\nu}_{2,M}$
with $\nu=\frac{2(1+\alpha)}{2+\alpha(1+\rho)}.$ Let us now estimate
$\operatorname{P}(\bar{\mathcal{A}}_{1}).$ Using Corollary 6.4, we get
$\operatorname{P}_{x}^{\otimes
M}(\Delta_{X}(\boldsymbol{\mathcal{S}}^{*},\boldsymbol{\mathcal{S}}_{M})>\delta_{\alpha})\leq\\\
\operatorname{P}_{x}^{\otimes
M}\left(\left(\frac{2^{1/\alpha}}{\delta_{0}}\right)\Delta(\boldsymbol{\mathcal{S}}_{M})+\frac{\delta_{\alpha}}{2(1+\alpha)}>\delta_{\alpha}\right)\\\
=\operatorname{P}_{x}^{\otimes
M}(\Delta(\boldsymbol{\mathcal{S}}_{M})>c_{\alpha})$
with
$c_{\alpha}=\delta_{0}\delta_{\alpha}2^{-1/\alpha}\left(1-\frac{1}{2(1+\alpha)}\right).$
Furthermore, due to (5.16)
$\displaystyle\operatorname{P}_{x}^{\otimes
M}(\Delta(\boldsymbol{\mathcal{S}}_{M})>c_{\alpha})$ $\displaystyle\leq$
$\displaystyle\operatorname{P}_{x}^{\otimes
M}\left(DM^{-1/(1+\rho)}+2M^{-1/2}\sup_{\boldsymbol{\mathcal{S}}\in\mathfrak{S}}|\Delta_{M}(\boldsymbol{\mathcal{S}})|>c_{\alpha}\right)$
$\displaystyle\leq$ $\displaystyle\operatorname{P}^{\otimes
M}_{x}\left(\sup_{\boldsymbol{\mathcal{S}}\in\mathfrak{S}}|\Delta_{M}(\boldsymbol{\mathcal{S}})|>c_{\alpha}\sqrt{M}/4\right)$
for large enough $M.$ Theorem 6.1 implies
$\displaystyle\operatorname{P}^{\otimes
M}_{x}\left(\sup_{\boldsymbol{\mathcal{S}}\in\mathfrak{S}}|\Delta_{M}(\boldsymbol{\mathcal{S}})|>c_{\alpha}\sqrt{M}/4\right)\leq
B_{1}\exp(-MB_{2})$
with some constants $B_{1}>0$ and $B_{2}=B_{2}(\alpha)>0.$ Applying Theorem
6.1 to $\mathcal{W}^{\nu}_{2,M}$ and using the fact that $\nu/2\leq
1/(1+\rho)$ for all $0<\rho\leq 1,$ we finally obtain the inequality
$\displaystyle\operatorname{P}^{\otimes
M}_{x}\left(\Delta(\boldsymbol{\mathcal{S}}_{M})>(V/M)^{\nu/2}\right)$
$\displaystyle\leq$ $\displaystyle C\exp(-\sqrt{V}/B_{3})$
$\displaystyle+C\exp\left(-\frac{U\varepsilon_{M}^{-2\rho}}{C^{2}}\right)+B_{1}\exp(-MB_{2})$
which holds for all $V>V_{0}$ and $M>M_{0}$ with some constant $B_{3}$
depending on $\varkappa.$
### 5.2 Proof of Proposition 3.4
For simplicity, we give the proof only for the case $d=2$ (an extension to
higher dimensions is straightforward). In the case of two exercise dates the
corresponding optimal stopping problem is completely specified by the
distribution of the vector $(X_{1},G_{2}(X_{2})).$ Because of a digital
structure of $G_{2}$ the distribution of $(X_{1},G_{2}(X_{2}))$ would be
completely determined if the marginal distribution of $X_{1}$ and the
probability $\operatorname{P}(G_{2}(X_{2})=1|X_{1}=x)$ are defined. Taking
into account this, we now construct a family of distributions for
$(X_{1},G_{2}(X_{2}))$ indexed by elements of the set $\Omega=\\{0,1\\}^{m}.$
First, the marginal distribution of $X_{1}$ is supposed to be the same for all
$\omega\in\Omega$ and posseses a density $p(x)$ satisfying
$0<p_{\ast}\leq p(x)\leq p^{\ast}<\infty,\quad x\in[0,1]^{2}.$
Let us now construct a family of conditional distributions
$\operatorname{P}_{\omega}(G_{2}(X_{2})=1|X_{1}=x)$, $\omega\in\Omega.$ To
this end let $\phi$ be an infinitely many times differentiable function on
$\mathbb{R}$ with the following properties: $\phi(z)=0$ for $|z|\geq 1,$
$\phi(z)\geq 0$ for all $z$ and $\sup_{z\in\mathbb{R}}[\phi(z)]\leq 1.$ For
$j=1,\ldots,m$ put
$\phi_{j}(z):=\delta
m^{-\gamma}\phi\left(m\left[z-\frac{2j-1}{m}\right]\right),\quad
z\in\mathbb{R}$
with some $0<\delta<1.$ For vectors $\omega=(\omega_{1},\ldots,\omega_{m})$ of
elements $\omega_{j}\in\\{0,1\\}$ and for any $z\in\mathbb{R}$ define
$b(z,\omega):=\sum_{j=1}^{m}\omega_{j}\phi_{j}(z).$
Put for any $\omega\in\Omega$ and any $x\in\mathbb{R}^{2},$
$\displaystyle C_{\omega}(x)$ $\displaystyle:=$
$\displaystyle\operatorname{P}_{\omega}(G_{2}(X_{2})=1|X_{1}=x)=$
$\displaystyle=$ $\displaystyle
G_{1}(x)-Am^{-\gamma/\alpha}\mathbf{1}\left\\{0\leq x_{2}\leq
b(x_{1},\omega)\right\\}$
$\displaystyle+Am^{-\gamma/\alpha}\mathbf{1}\left\\{b(x_{1},\omega)<x_{2}\leq\delta
m^{-\gamma}\right\\},$
where $A$ is a positive constant. Due to our assumptions on $G_{1}(x)$, there
are constants $0<G_{-}<G_{+}<1$ such that
$G_{-}\leq G_{1}(x)\leq G_{+},\quad x\in[0,1]^{2}.$
Hence, the constant $A$ can be chosen in such a way that $C_{\omega}(x)$
remains positive and strictly less than $1$ on $[0,1]^{2}$ for any
$\omega\in\Omega.$ The stopping set
$\mathcal{S}_{\omega}:=\left\\{x:C_{\omega}(x)\leq
G_{1}(x)\right\\}=\left\\{(x_{1},x_{2}):0\leq x_{2}\leq
b(x_{1},\omega)\right\\}$
belongs to $\mathfrak{S}_{\gamma}$ since $b(\cdot,\omega)\in\Sigma(\gamma,L)$
for $\delta$ small enough. Moreover, for any $\eta>0$
$\displaystyle\operatorname{P}_{\omega}\left(|G_{1}(X_{1})-C_{\omega}(X_{1})|\leq\eta\right)$
$\displaystyle=$ $\displaystyle\operatorname{P}_{\omega}(0\leq
X_{1}^{2}\leq\delta m^{-\gamma})\mathbf{1}(Am^{-\gamma/\alpha}\leq\eta)$
$\displaystyle\leq$ $\displaystyle\delta
p^{\ast}m^{-\gamma}\mathbf{1}(Am^{-\gamma/\alpha}\leq\eta)\leq\delta
p^{\ast}A^{-\alpha}\eta^{\alpha}$
and the condition (3.10) is fulfilled. Let $\tau_{M}$ be a stopping time
w.r.t. $\mathcal{F}^{\otimes M}$, then the identity (see Lemma 6.2)
$\displaystyle\operatorname{E}_{\operatorname{P}_{\omega}}[G_{\tau^{\ast}}(X_{\tau^{\ast}})]-\operatorname{E}_{\operatorname{P}_{\omega}}[G_{\tau_{M}}(X_{\tau_{M}})]$
$\displaystyle=$ $\displaystyle\newline
\operatorname{E}_{\operatorname{P}_{\omega}}\left[(G_{1}(X_{1})-G_{2}(X_{2}))\mathbf{1}(\tau^{\ast}=1,\tau_{M}=2)\right]\newline
$
$\displaystyle+\operatorname{E}_{\operatorname{P}_{\omega}}\left[(G_{2}(X_{2})-G_{1}(X_{1}))\mathbf{1}(\tau^{\ast}=2,\tau_{M}=1)\right]\newline
$ $\displaystyle=$
$\displaystyle\operatorname{E}_{\operatorname{P}_{\omega}}\left[|G_{1}(X_{1})-\operatorname{E}(G_{2}(X_{2})|\mathcal{F}_{1})|\mathbf{1}\\{\tau_{M}\neq\tau^{\ast}\\}\right]$
leads to
$\operatorname{E}_{\operatorname{P}_{\omega}}[G_{\tau^{\ast}}(X_{\tau^{\ast}})]-\operatorname{E}_{\operatorname{P}_{\omega}^{\otimes
M}}\left\\{\operatorname{E}_{\operatorname{P}_{\omega}}[G_{\tau_{M}}(X_{\tau_{M}})]\right\\}=\operatorname{E}_{\operatorname{P}_{\omega}^{\otimes
M}}\operatorname{E}_{\operatorname{P}_{\omega}}\left[|\Delta_{\omega}(X_{1})|\mathbf{1}\\{\tau_{M}\neq\tau^{\ast}\\}\right]$
with $\Delta_{\omega}(x):=G_{1}(x)-C_{\omega}(x)$. By conditioning on $X_{1}$
we get
$\displaystyle\operatorname{E}_{\operatorname{P}^{\otimes
M}}\operatorname{E}_{\operatorname{P}_{\omega}}\left[|\Delta_{\omega}(X_{1})|\mathbf{1}\\{\tau_{M}\neq\tau^{\ast}\\}\right]$
$\displaystyle=$ $\displaystyle Am^{-\gamma/\alpha}\operatorname{P}(0\leq
X_{1}^{2}\leq\delta m^{-\gamma})\operatorname{P}_{\omega}^{\otimes
M}\left(\tau_{M}\neq\tau^{\ast}\right)$ $\displaystyle\geq$ $\displaystyle
Am^{-\gamma/\alpha}p_{\ast}\delta
m^{-\gamma}\operatorname{P}_{\omega}^{\otimes
M}\left(\tau_{M}\neq\tau^{\ast}\right).$
Using now a well known Birgé’s or Huber’s lemma, (see, e.g. Devroye, Györfi
and Lugosi, 1996, p. 243), we get
$\sup_{\omega\in\\{0,1\\}^{m}}\operatorname{P}_{\omega}^{\otimes
M}(\widehat{\tau}_{M}\neq\tau^{\ast})\geq\left[0.36\wedge\left(1-\frac{MK_{\mathcal{H}}}{\log(\left|\mathcal{H}\right|)}\right)\right],$
where $K_{\mathcal{H}}:=\sup_{P,Q\in\mathcal{H}}K(P,Q),$
$\mathcal{H}:=\\{\operatorname{P}_{\omega},\,\omega\in\\{0,1\\}^{m}\\}$ and
$K(P,Q)$ is a Kullback-Leibler distance between two measures $P$ and $Q$.
Since for any two measures $P$ and $Q$ from $\mathcal{H}$ with $Q\neq P$
$\displaystyle K(P,Q)$ $\displaystyle\leq$
$\displaystyle\sup_{\begin{subarray}{c}\omega_{1},\omega_{2}\in\\{0,1\\}^{m}\\\
\omega_{1}\neq\omega_{2}\end{subarray}}\operatorname{E}\left[C_{\omega_{1}}(X_{1})\log\left\\{\frac{C_{\omega_{1}}(X_{1})}{C_{\omega_{2}}(X_{1})}\right\\}\right.$
$\displaystyle\left.+(1-C_{\omega_{1}}(X_{1}))\log\left\\{\frac{1-C_{\omega_{1}}(X_{1})}{1-C_{\omega_{2}}(X_{1})}\right\\}\right]$
$\displaystyle\leq$ $\displaystyle(1-G_{+}-A)^{-1}(G_{-}-A)^{-1}$
$\displaystyle\times\operatorname{P}(0\leq X_{1}^{2}\leq\delta
m^{-\gamma})\left[A^{2}m^{-2\gamma/\alpha}\right]$ $\displaystyle\leq$
$\displaystyle CMm^{-\gamma-2\gamma/\alpha-1}$
with some constant $C>0$ for small enough $A$, and
$\log(|\mathcal{H}|)=m\log(2)$, we get
$\sup_{\omega\in\\{0,1\\}^{m}}\operatorname{P}_{\omega}^{\otimes
M}(\widehat{\tau}_{M}\neq\tau^{\ast})\geq\left[0.36\wedge\left(1-CMm^{-\gamma-2\gamma/\alpha-1}\right)\right]\quad$
with some constant $C>0.$ Hence,
$\sup_{\omega\in\\{0,1\\}^{m}}\operatorname{P}_{\omega}^{\otimes
M}(\widehat{\tau}_{M}\neq\tau^{\ast})>0$
provided that $m=qM^{1/(\gamma+2\gamma/\alpha+1)}$ for small enough real
number $q>0$. In this case
$\sup_{\omega\in\\{0,1\\}^{m}}\left\\{\operatorname{E}_{\operatorname{P}_{\omega}}[G_{\tau^{\ast}}(X_{\tau^{\ast}})]-\operatorname{E}_{\operatorname{P}_{\omega}^{\otimes
M}}\left\\{\operatorname{E}_{\operatorname{P}_{\omega}}[G_{\tau_{M}}(X_{\tau_{M}})]\right\\}\right\\}\\\
\geq Ap_{\ast}\delta
q^{-\gamma/\alpha-\gamma}M^{-(\gamma/\alpha+\gamma)/(\gamma+2\gamma/\alpha+1)}=BM^{-\frac{(1+\alpha)}{2+\alpha(1+1/\gamma)}}$
with $B=Ap_{\ast}\delta q^{-\gamma/\alpha-\gamma}.$
## 6 Auxiliary results
We have
$\displaystyle\Delta_{M}(\boldsymbol{\mathcal{S}})$ $\displaystyle:=$
$\displaystyle\sqrt{M}\sum_{m=1}^{M}\left\\{g_{\boldsymbol{\mathcal{S}}}(X^{(m)}_{1},\ldots,X^{(m)}_{K})-\operatorname{E}\left[g_{\boldsymbol{\mathcal{S}}}(X_{1},\ldots,X_{K})\right]\right\\}$
with functions
$g_{\boldsymbol{\mathcal{S}}}:\underbrace{\mathbb{R}^{d}\times\ldots\times\mathbb{R}^{d}}_{K}\to\mathbb{R}$
defined as
$g_{\boldsymbol{\mathcal{S}}}(x_{1},\ldots,x_{K}):=\sum_{k=0}^{K-1}G_{k+1}(x_{k+1})\mathbf{1}_{\\{x_{1}\not\in\mathcal{S}_{1},\ldots,x_{k}\not\in\mathcal{S}_{k},x_{k+1}\in\mathcal{S}_{k+1}\\}}.$
Denote
$\mathcal{G}=\\{g_{\boldsymbol{\mathcal{S}}}:\boldsymbol{\mathcal{S}}\in\mathfrak{S}\\}.$
Obviously $\mathcal{G}$ is a class of uniformly bounded functions provided
that all functions $G_{k}$ are uniformly bounded.
#### Definition
Let $\mathcal{N}_{B}(\delta,\mathcal{G},\operatorname{P})$ be the smallest
value of $n$ for which there exist pairs of functions
$\\{[g_{j}^{L},g_{j}^{U}]\\}_{j=1}^{n}$ such that
$\|g_{j}^{U}-g_{j}^{L}\|_{L_{2}(\operatorname{P})}\leq\delta$ for all
$j=1,\ldots,n,$ and such that for each $g\in\mathcal{G},$ there is
$j=j(g)\in\\{1,\ldots,n\\}$ such that
$\displaystyle g_{j}^{L}\leq g\leq g_{j}^{U}.$
Then
$\mathcal{H}_{B}(\delta,\mathcal{G},\operatorname{P})=\log\left[\mathcal{N}_{B}(\delta,\mathcal{G},\operatorname{P})\right]$
is called the entropy with bracketing of $\mathcal{G}$. The following theorem
follows directly from Theorem 5.11 in Van de Geer (2000).
###### Theorem 6.1.
Assume that there exists a constant $A>0$ such that
(6.17) $\displaystyle\mathcal{H}_{B}(\delta,\mathcal{G},\operatorname{P})\leq
A\delta^{-\nu}$
for any $\delta>0$ and some $\nu>0$, where
$\mathcal{H}_{B}(\delta,\mathcal{G},\operatorname{P})$ is the $\delta$-entropy
with bracketing of $\mathcal{G}.$ Fix some
$\boldsymbol{\mathcal{S}}_{0}\in\mathfrak{S}$ then for any $\varepsilon\geq
M^{-1/(2+\nu)}$
$\displaystyle\operatorname{P}\left(\sup_{\boldsymbol{\mathcal{S}}\in\mathfrak{S},\,\|g_{\boldsymbol{\mathcal{S}}}-g_{\boldsymbol{\mathcal{S}}_{0}}\|_{L_{2}(\operatorname{P})}\leq\varepsilon}|\Delta_{M}(\boldsymbol{\mathcal{S}})-\Delta_{M}(\boldsymbol{\mathcal{S}}_{0})|>U\varepsilon^{1-\frac{\nu}{2}}\right)\leq
C\exp(-U\varepsilon^{-\nu}/C^{2}),$
$\displaystyle\operatorname{P}\left(\sup_{\boldsymbol{\mathcal{S}}\in\mathfrak{S},\,\|g_{\boldsymbol{\mathcal{S}}}-g_{\boldsymbol{\mathcal{S}}_{0}}\|_{L_{2}(\operatorname{P})}\leq\varepsilon}\frac{|\Delta_{M}(\boldsymbol{\mathcal{S}})-\Delta_{M}(\boldsymbol{\mathcal{S}}_{0})|}{\|g_{\boldsymbol{\mathcal{S}}}-g_{\boldsymbol{\mathcal{S}}_{0}}\|_{L_{2}(\operatorname{P})}^{1-\nu/2}}>U\right)\leq
C\exp(-U/C^{2}).$
for all $U>C$ and $M>M_{0},$ where $C$ and $M_{0}$ are two positive constants.
Moreover, for any $z>0$
$\displaystyle\operatorname{P}\left(\sup_{\boldsymbol{\mathcal{S}}\in\mathfrak{S}}|\Delta_{M}(\boldsymbol{\mathcal{S}})-\Delta_{M}(\boldsymbol{\mathcal{S}}_{0})|>z\sqrt{M}\right)\leq
C\exp(-Mz^{2}/C^{2}B)$
with some positive constant $B>0$.
Let us define a pseudedistance $\Delta_{X}$ between any two sets
$\boldsymbol{\mathcal{S}},\boldsymbol{\mathcal{S}}^{\prime}\in\mathfrak{B}$ in
the following way
$\Delta_{X}(\mathcal{S}_{1}\times\ldots\times\mathcal{S}_{K},\mathcal{S}^{\prime}_{1}\times\ldots\times\mathcal{S}^{\prime}_{K}):=\sum_{k=1}^{K}\operatorname{P}\left(X_{k}\in(\mathcal{S}_{k}\triangle\mathcal{S}^{\prime}_{k})\setminus\left(\bigcap_{l=k}^{K-1}\mathcal{S}^{\prime}_{l}\right)\right).$
It obviously holds
$\Delta_{X}(\boldsymbol{\mathcal{S}},\boldsymbol{\mathcal{S}}^{\prime})\leq
d_{X}(\boldsymbol{\mathcal{S}},\boldsymbol{\mathcal{S}}^{\prime})$ for the
pseudodistance $d_{X}$ defined in The following Lemma will be frequently used
in the sequel.
###### Lemma 6.2.
For any
$\boldsymbol{\mathcal{S}},\boldsymbol{\mathcal{S}}^{\prime}\in\mathfrak{B}$ it
holds with probability one
(6.18)
$\left|G_{\tau_{k}(\boldsymbol{\mathcal{S}})}\left(X_{\tau_{k}(\boldsymbol{\mathcal{S}})}\right)-G_{\tau_{k}(\boldsymbol{\mathcal{S}}^{\prime})}\left(X_{\tau_{k}(\boldsymbol{\mathcal{S}}^{\prime})}\right)\right|\\\
\leq\sum_{l=k}^{K-1}|G_{l}(X_{l})-G_{\tau_{l+1}(\boldsymbol{\mathcal{S}})}(X_{\tau_{l+1}(\boldsymbol{\mathcal{S}})})|\mathbf{1}_{\left\\{X_{l}\in(\mathcal{S}_{l}\triangle\mathcal{S}^{\prime}_{l})\setminus\left(\bigcap_{l^{\prime}=l}^{K-1}\mathcal{S}^{\prime}_{l^{\prime}}\right)\right\\}}$
and
(6.19)
$V^{*}_{k}(X_{k})-\operatorname{E}\left[G_{\tau_{k}(\boldsymbol{\mathcal{S}})}(X_{\tau_{k}(\boldsymbol{\mathcal{S}})})|\mathcal{F}_{k}\right]\\\
=\operatorname{E}\left[\left.\sum_{l=k}^{K-1}\left|G_{l}(X_{l})-\operatorname{E}[V^{*}_{l+1}(X_{l+1})|\mathcal{F}_{l}]\right|\mathbf{1}_{\left\\{X_{l}\in(\mathcal{S}^{*}_{l}\triangle\mathcal{S}_{l})\setminus\left(\bigcap_{l^{\prime}=l}^{K-1}\mathcal{S}_{l^{\prime}}\right)\right\\}}\right|\mathcal{F}_{k}\right]$
for $k=1,\ldots,K-1$.
###### Proof.
We prove (6.19) by induction. The inequality (6.18) can be proved in a similar
way. For $k=K-1$ we get
(6.20) $V^{*}_{K-1}(X_{K-1})-V_{K-1}(X_{K-1})=\\\
=\operatorname{E}\left[\left.(G_{K-1}(X_{K-1})-G_{K}(X_{K}))\mathbf{1}_{\\{\tau^{*}_{K-1}=K-1,\,\tau_{K-1}=K\\}}\right|\mathcal{F}_{K-1}\right]\\\
+\operatorname{E}\left[\left.(G_{K}(X_{K})-G_{K-1}(X_{K-1}))\mathbf{1}_{\\{\tau^{*}_{K-1}=K,\,\tau_{K-1}=K-1\\}}\right|\mathcal{F}_{K-1}\right]\\\
=|G_{K-1}(X_{K-1})-\operatorname{E}[G_{K}(X_{K})|\mathcal{F}_{K-1}]|\mathbf{1}_{\\{\tau_{K-1}\neq\tau^{*}_{K-1}\\}}$
since events $\\{\tau^{*}_{K-1}=K\\}$ and $\\{\tau_{K-1}=K\\}$ are measurable
w.r.t. $\mathcal{F}_{K-1}$ and
$G_{K-1}(X_{K-1})\geq\operatorname{E}[G_{K}(X_{K})|\mathcal{F}_{K-1}]$ on the
set $\\{\tau^{*}_{K-1}=K-1\\}.$ Thus, (6.19) holds with $k=K-1$. Suppose that
(6.19) holds with $k=K^{\prime}+1$. Let us prove it for $k=K^{\prime}$.
Consider a decomposition
(6.21) $\displaystyle
G_{\tau^{*}_{K^{\prime}}}(X_{\tau^{*}_{K^{\prime}}})-G_{\tau_{K^{\prime}}}(X_{\tau_{K^{\prime}}})$
$\displaystyle=$ $\displaystyle S_{1}+S_{2}+S_{3}$
with
$\displaystyle S_{1}$ $\displaystyle:=$
$\displaystyle\left(G_{\tau^{*}_{K^{\prime}}}(X_{\tau^{*}_{K^{\prime}}})-G_{\tau_{K^{\prime}}}(X_{\tau_{K^{\prime}}})\right)\mathbf{1}_{\\{\tau^{*}_{K^{\prime}}>K^{\prime},\,\tau_{K^{\prime}}>K^{\prime}\\}},$
$\displaystyle S_{2}$ $\displaystyle:=$
$\displaystyle\left(G_{\tau^{*}_{K^{\prime}}}(X_{\tau^{*}_{K^{\prime}}})-G_{\tau_{K^{\prime}}}(X_{\tau_{K^{\prime}}})\right)\mathbf{1}_{\\{\tau^{*}_{K^{\prime}}>K^{\prime},\,\tau_{K^{\prime}}=K^{\prime}\\}},$
$\displaystyle S_{3}$ $\displaystyle:=$
$\displaystyle\left(G_{\tau^{*}_{K^{\prime}}}(X_{\tau^{*}_{K^{\prime}}})-G_{\tau_{K^{\prime}}}(X_{\tau_{K^{\prime}}})\right)\mathbf{1}_{\\{\tau^{*}_{K^{\prime}}=K^{\prime},\,\tau_{K^{\prime}}>K^{\prime}\\}}.$
Using the fact that $\tau_{k}=\tau_{k+1}$ if $\tau_{k}>k$ for any
$k=1,\ldots,K-1$, we get
$\displaystyle\operatorname{E}\left[S_{1}|\mathcal{F}_{K^{\prime}}\right]$
$\displaystyle=$
$\displaystyle\operatorname{E}\left[\left.\left(V^{*}_{K^{\prime}+1}(X_{K^{\prime}+1})-V_{K^{\prime}+1}(X_{K^{\prime}+1})\right)\mathbf{1}_{\\{\tau^{*}_{K^{\prime}}>K^{\prime},\,\tau_{K^{\prime}}>K^{\prime}\\}}\right|\mathcal{F}_{K^{\prime}}\right],$
$\displaystyle\operatorname{E}\left[S_{2}|\mathcal{F}_{K^{\prime}}\right]$
$\displaystyle=$
$\displaystyle\left(\operatorname{E}\left[\left.G_{\tau^{*}_{K^{\prime}+1}}(X_{\tau^{*}_{K^{\prime}+1}})\right|\mathcal{F}_{K^{\prime}}\right]-G_{K^{\prime}}(X_{K^{\prime}})\right)\mathbf{1}_{\\{\tau^{*}_{K^{\prime}}>K^{\prime},\,\tau_{K^{\prime}}=K^{\prime}\\}}$
$\displaystyle=$
$\displaystyle\left(\operatorname{E}\left[\left.V^{*}_{K^{\prime}+1}(X_{K^{\prime}+1})\right|\mathcal{F}_{K^{\prime}}\right]-G_{K^{\prime}}(X(t_{K^{\prime}}))\right)\mathbf{1}_{\\{\tau^{*}_{K^{\prime}}>K^{\prime},\,\widehat{\tau}_{K^{\prime}}=K^{\prime}\\}}$
and
$\displaystyle\operatorname{E}\left[S_{3}|\mathcal{F}_{K^{\prime}}\right]$
$\displaystyle=$
$\displaystyle\left(G_{K^{\prime}}(X_{K^{\prime}})-\operatorname{E}\left[G_{\tau_{K^{\prime}+1}}(X_{\tau_{K^{\prime}+1}})|\mathcal{F}_{K^{\prime}}\right]\right)\mathbf{1}_{\\{\tau^{*}_{K^{\prime}}=K^{\prime},\,\tau_{K^{\prime}}>K^{\prime}\\}}$
$\displaystyle=$
$\displaystyle\left(G_{K^{\prime}}(X_{K^{\prime}})-\operatorname{E}[V^{*}_{K^{\prime}+1}(X_{K^{\prime}+1})|\mathcal{F}_{K^{\prime}}]\right)\mathbf{1}_{\\{\tau^{*}_{K^{\prime}}=K^{\prime},\,\tau_{K^{\prime}}>K^{\prime}\\}}$
$\displaystyle+\operatorname{E}\left[\left.\left(V^{*}_{K^{\prime}+1}(X_{K^{\prime}+1})-V_{K^{\prime}+1}(X_{K^{\prime}+1})\right)\mathbf{1}_{\\{\tau^{*}_{K^{\prime}}=K^{\prime},\,\tau_{K^{\prime}}>K^{\prime}\\}}\right|\mathcal{F}_{K^{\prime}}\right],$
with probability one. Hence
$\displaystyle
V^{*}_{K^{\prime}}(X_{K^{\prime}})-V_{K^{\prime}}(X_{K^{\prime}})$
$\displaystyle=$
$\displaystyle\left|G_{K^{\prime}}(X_{K^{\prime}})-\operatorname{E}[V^{*}_{K^{\prime}+1}(X_{K^{\prime}+1})|\mathcal{F}_{K^{\prime}}]\right|\mathbf{1}_{\\{\tau_{K^{\prime}}\neq\tau^{*}_{K^{\prime}}\\}}$
$\displaystyle+\operatorname{E}\left[\left.\left(V^{*}_{K^{\prime}+1}(X_{K^{\prime}+1})-V_{K^{\prime}+1}(X_{K^{\prime}+1})\right)\right|\mathcal{F}_{K^{\prime}}\right]\mathbf{1}_{\\{\tau_{K^{\prime}}>K^{\prime}\\}}$
since
$G_{K^{\prime}}(X_{K^{\prime}})-\operatorname{E}[V^{*}_{K^{\prime}+1}(X_{K^{\prime}+1})\geq
0$ on the set $\\{\tau^{*}_{K^{\prime}}=K^{\prime}\\}.$ Our induction
assumption implies now that
$V_{K^{\prime}}^{*}(X_{K^{\prime}})-V_{K^{\prime}}(X_{K^{\prime}})=\\\
\operatorname{E}\left[\sum_{k=K^{\prime}}^{K-1}|G_{l}(X_{l})-\operatorname{E}[V^{*}_{l+1}(X_{l+1})|\mathcal{F}_{l}]|\mathbf{1}_{\\{\tau_{k}\neq\tau^{*}_{k},\tau_{k}>k,\ldots,\tau_{K-1}>K-1\\}}|\mathcal{F}_{K^{\prime}}\right]$
and hence (6.19) holds with $k=K^{\prime}$. ∎
###### Corollary 6.3.
If $\max_{k=1,\ldots,K}\|G_{k}\|_{\infty}<A_{G}$ with some constant $A_{G}>0$,
then
$\displaystyle\Delta_{G}(\boldsymbol{\mathcal{S}},\boldsymbol{\mathcal{S}}^{\prime})\leq
2A_{G}\sqrt{2\Delta_{X}(\boldsymbol{\mathcal{S}},\boldsymbol{\mathcal{S}}^{\prime})}$
for any
$\boldsymbol{\mathcal{S}},\boldsymbol{\mathcal{S}}^{\prime}\in\mathfrak{B}.$
###### Proof.
Follows directly from (6.18) since $G_{\tau}(X_{\tau})\leq A_{G}$ a.s. for any
stopping time $\tau$ taking values in $\\{1,\ldots,K\\}.$ ∎
###### Corollary 6.4.
Assume that (3.10) holds for $\delta<\delta_{0}<1/2$, then there exist
constants $\upsilon_{\alpha}$ and $\delta_{\alpha}$ such that
(6.22)
$\displaystyle\Delta(\boldsymbol{\mathcal{S}})\geq\upsilon_{\alpha}\Delta^{(1+\alpha)/\alpha}_{X}(\boldsymbol{\mathcal{S}}^{*},\boldsymbol{\mathcal{S}})$
for all $\boldsymbol{\mathcal{S}}\in\mathfrak{B}$ satisfying
$\Delta_{X}(\boldsymbol{\mathcal{S}}^{*},\boldsymbol{\mathcal{S}})\leq\delta_{\alpha}$.
Moreover it holds
(6.23)
$\displaystyle\Delta_{X}(\boldsymbol{\mathcal{S}}^{*},\boldsymbol{\mathcal{S}})\leq\left(\frac{2^{1/\alpha}}{\delta_{0}}\right)\Delta(\boldsymbol{\mathcal{S}})+\frac{\delta_{\alpha}}{2(1+\alpha)}.$
for any $\boldsymbol{\mathcal{S}}\in\mathfrak{B}.$
###### Proof.
For any $\delta\leq\delta_{0}$ define the sets
$\mathcal{A}_{k}:=\left\\{x\in\mathbb{R}^{d}:\left|\operatorname{E}[V^{*}_{k+1}(X_{k+1})|X_{k}=x]-G_{k}(x)\right|>\delta\right\\},\quad
k=1,\ldots,K-1.$
Due to (6.19) we have
(6.24) $\displaystyle\Delta(\boldsymbol{\mathcal{S}})$ $\displaystyle\geq$
$\displaystyle\delta\sum_{k=1}^{K-1}\operatorname{P}\left(X_{k}\in(\mathcal{S}^{*}_{k}\triangle\mathcal{S}_{k})\setminus\left(\bigcap_{l=k}^{K-1}\mathcal{S}_{k}\right)\bigcap\mathcal{A}_{k}\right)$
$\displaystyle\geq$
$\displaystyle\delta\sum_{k=1}^{K-1}\left\\{\operatorname{P}\left(X_{k}\in(\mathcal{S}^{*}_{k}\triangle\mathcal{S}_{k})\setminus\left(\bigcap_{l=k}^{K-1}\mathcal{S}_{k}\right)\right)-\operatorname{P}(\bar{\mathcal{A}}_{k})\right\\}$
$\displaystyle\geq$
$\displaystyle\delta[\Delta_{X}(\boldsymbol{\mathcal{S}}^{*},\boldsymbol{\mathcal{S}})-A_{0}\delta^{\alpha}]$
with $A_{0}=\sum_{k=1}^{K-1}A_{k,0},$ where $A_{k,0}$ were defined in (3.10).
The maximum of (6.24) is attained at
$\delta^{*}=[\Delta_{X}(\boldsymbol{\mathcal{S}}^{*},\boldsymbol{\mathcal{S}})/(\alpha+1)A_{0}]^{1/\alpha}$.
Since $\delta^{*}\leq\delta_{0}$ for
$\Delta_{X}(\boldsymbol{\mathcal{S}}^{*},\boldsymbol{\mathcal{S}})\leq
A_{0}(\alpha+1)\delta^{\alpha}_{0}$ the inequality (6.22) holds with
$\upsilon_{\alpha}:=A_{0}^{-1/\alpha}\alpha(1+\alpha)^{-1-1/\alpha}$ and
$\delta_{\alpha}:=A_{0}(\alpha+1)\delta^{\alpha}_{0}$. The inequality (6.23)
follows directly from (6.24) by taking $\delta=\delta_{0}/2^{1/\alpha}.$ ∎
## References
* Andersen (2000) L. Andersen (2000). A simple approach to the pricing of Bermudan swaptions in the multi-factor Libor Market Model. Journal of Computational Finance, 3, 5-32.
* Belomestny (2009) D. Belomestny (2009). Pricing Bermudan options using nonparametric regression: optimal rates of convergence for lower estimates, http://arxiv.org/abs/0907.5599, forthcoming in Finance and Stochastics.
* Broadie and Glasserman (1997) M. Broadie and P. Glasserman (1997). Pricing American-style securities using simulation. J. of Economic Dynamics and Control, 21, 1323-1352.
* Broadie and Detemple (1997) M. Broadie and J. Detemple (1997). The valuation of American options on multiple assets. Mathematical Finance, 7(3), 241-286.
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* Dudley (1999) R.M. Dudley (1999). Uniform central limit theorems. Cambridge University Press.
* Egloff (2005) D. Egloff (2005). Monte Carlo algorithms for optimal stopping and statistical learning. Ann. Appl. Probab., 15, 1396-1432.
* Egloff, Kohler and Todorovic (2007) D. Egloff, M. Kohler and N. Todorovic (2007). A dynamic look-ahead Monte Carlo algorithm for pricing Bermudan options, Ann. Appl. Probab., 17, 1138-1171.
* Garcia (2001) D. Garcia (2001). Convergence and biases of Monte Carlo estimates of American option prices using a parametric exercise rule. Working paper.
* Glasserman (2003) P. Glasserman (2003). Monte Carlo Methods in Financial Engineering. Springer.
* Kleywegt, Shapiro and Homem-de-Mello (2001) A.J. Kleywegt, A. Shapiro and T. Homem-de-Mello (2001). The sample average approximation method for stochastic discrete optimization, SIAM J. Optim., 12, 479-502.
* Longstaff and Schwartz (2001) F. Longstaff and E. Schwartz (2001). Valuing American options by simulation: a simple least-squares approach. Review of Financial Studies, 14, 113-147.
* Mammen and Tsybakov (1999) E. Mammen and A. Tsybakov (1999). Smooth discrimination analysis. Ann. Statist., 27, 1808-1829.
* Peskir and Shiryaev (2006) G. Peskir and A. Shiryaev (2006). Optimal Stopping and Free-Boundary Problems. LM - Lectures in Mathematics ETH Z rich.
* Shapiro (1993) A. Shapiro (1993). Asymptotic behavior of optimal solutions in stochastic programming, Math. Oper. Res., 18, 829-845.
* Shapiro and Homem-de-Mello (2000) A. Shapiro and T. Homem-de-Mello (2000). On the rate of convergence of optimal solutions of Monte Carlo approximations of stochastic programs. SIAM J. Optim., 11(1), 70-86.
* Snell (1952) J. L. Snell (1952). Applications of martingale system theorems. Trans. Amer. Math. Soc. 73, 293–312.
* Tsitsiklis and Van Roy (1999) J. Tsitsiklis and B. Van Roy (1999). Regression methods for pricing complex American style options. IEEE Trans. Neural. Net., 12, 694-703.
* Van de Geer (2000) S. Van de Geer (2000). Applications of Empirical Process Theory. Cambridge Univ. Press.
|
arxiv-papers
| 2009-09-19T08:44:22 |
2024-09-04T02:49:05.426243
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Denis Belomestny",
"submitter": "Denis Belomestny",
"url": "https://arxiv.org/abs/0909.3570"
}
|
0909.3673
|
# The Density Functional via Effective Action
Yi-Kuo Yu National Center for Biotechnology Information, National Library of
Medicine
National Institutes of Health, Bethesda, MD 20894, USA
(August 9th, 2009 )
###### Abstract
A rigorous derivation of the density functional via the effective action in
the Hohenberg-Kohn theory is outlined. Using the auxiliary field method, in
which the electric coupling constant $e^{2}$ need not be small, we show that
the loop expansion of the exchange-correlation functional can be reorganized
so as to be expressed entirely in terms of the Kohn-Sham single-particle
orbitals and energies.
###### pacs:
71.15.Mb
Interactions among electrons largely determine the structure, phases, and
stability of matter. Pragmatic advances in this subject, however, are
nontrivial. When the number of electrons involved becomes large, calculations
based on constructing many-electron wave functions soon lose accuracy and will
be stopped by an “exponential wall”Kohn (1999). Density functional theory
(DFT), using the three-dimensional electronic density as the basic variable,
is free from this wall. DFT originated from the theorem of Hohenberg and Kohn
(HK)Hohenberg and Kohn (1964), which states that there exists a unique
description of a many-body system in its ground state in terms of the
expectation value of the particle-density operator. The HK theorem assures
that the ground state energy $E_{g}$ is obtained by minimizing the energy
functional $E_{\upsilon}$ with respect to the electronic density $n$:
$E_{g}=\min\limits_{n}E_{\upsilon}\left[n\right].\vspace*{-4pt}\phantom{12}$
(1)
Mermin Mermin (1965) extended this theorem to finite-temperature.
To make practical use of the HK theorem, a suitable computational scheme is
necessary. Kohn and Sham Kohn and Sham (1965) proposed a decomposition scheme,
aiming to express $E_{\upsilon}[n]$ via an auxiliary, noninteracting system
that yields a particle density identical to that of the physical ground state.
For a nonrelativistic fermion system described by
$\displaystyle\hat{H}$ $\displaystyle=$ $\displaystyle\int d{\bf
x}{\hat{\psi}}^{{\dagger}}({{\bf
x}})\left(-\frac{1}{2m}\nabla^{2}+\upsilon({{\bf
x}})-\mu\right)\hat{\psi}({{\bf x}})$ (2) $\displaystyle\
+\frac{e^{2}}{2}\int\int\frac{{\hat{\psi}}^{{\dagger}}({{\bf
x}}){\hat{\psi}}^{{\dagger}}({{\bf y}})\hat{\psi}({{\bf y}})\hat{\psi}({{\bf
x}})}{|{{\bf x}}-{{\bf y}}|}d{{\bf x}}d{{\bf y}},$
the energy functional, with $e^{2}$ representing the electric coupling
constant and $T_{0}[n]$ being the kinetic energy of the auxiliary system,
takes the form
$\displaystyle E_{\upsilon}\left[n\right]$ $\displaystyle=$
$\displaystyle\int\upsilon({\bf x})\,n\left({\bf x}\right)d{{\bf x}}-\mu
N_{e}+T_{0}\left[n\right]$ (3)
$\displaystyle+\frac{e^{2}}{2}\int\int\frac{n({{\bf x}})n({{\bf y}})}{|{{\bf
x}}-{{\bf y}}|}d{\bf x}d{\bf y}+E_{xc}\left[n\right],$
where $\mu=$ chemical potential, $N_{e}=$ number of electrons, $\upsilon({\bf
x})=$ external potential, and $E_{xc}\left[n\right]$ is the so-called
exchange-correlation energy functional. This exact decomposition cannot exist
without the quantity $\frac{\delta E_{xc}[n]}{\delta n}$ being well defined.
Being independent of $\upsilon({\bf x})$, the sum of the last three terms in
(3) is universal. All of the many-particle complexity is now completely hidden
in $E_{xc}\left[n\right]$.
Although $T_{0}[n]+E_{xc}[n]$ admits no free parameter and is universal
Hohenberg and Kohn (1964), its explicit construction remains elusive, and
parameter-containing empirical functionals are therefore introduced. Cases of
failure and limitations of these empirical functionals have been discussed
Kümmel and Kronik (2008); Cohen et al. (2008). On the other hand, a number of
groups Fukuda et al. (1994, 1995); Valiev and Fernando (1997); Polonyi and
Sailer (2002) have pursued first-principle derivation of the density
functional via effective action. These efforts either introduce an auxiliary
field Fukuda et al. (1994); Polonyi and Sailer (2002) or expand in powers of
$e^{2}$ Fukuda et al. (1995); Valiev and Fernando (1997). The strengths of the
auxiliary field approach are the simplicity of the effective action expression
and the fact that each term already includes infinitely many Feynman diagrams
Jackiw (1974). However, this approach seems Fukuda et al. (1994) to lack a
direct connection to the Kohn-Sham (KS) scheme. Such a connection can be made
in the expansion in powers of $e^{2}$ Sham (1985); Valiev and Fernando (1997),
but that expansion is good only when $e^{2}$ is small Negele and Orland
(1988). The validity of that assumption depends on the strength and variation
of $\upsilon({\bf x})$.
In this Letter, without assuming $e^{2}$ small, we report our development Yu
(2009) of an auxiliary field method that makes a direct connection to the KS
scheme. To lighten the mathematical expressions in our finite-temperature
formalism, we suppress the spin degree of freedom (as it is easy to include)
and denote by a dot (circle) the three (four) dimensional integral contraction
(with $\tau$ denoting the Euclidean time, $x\equiv(\tau,{\bf x})$) 12
$\displaystyle a{\cdot}b$ $\displaystyle\equiv$ $\displaystyle\int d{\bf
x}\;a({\bf x})\,b({\bf x})\vspace*{-4pt}$ $\displaystyle
a{\scriptstyle\circ}b$ $\displaystyle\equiv$ $\displaystyle\int\\!d\tau d{\bf
x}\,a(\tau,{\bf x})\,b(\tau,{\bf x})\equiv\int
dx\,a(x)\,b(x)\;.\vspace*{-3pt}\phantom{12}$
To probe the electron density, one introduces to $\hat{H}$ a classical source
term $J({\bf x})$ coupled to ${\hat{\psi}}^{{\dagger}}({\bf x})\hat{\psi}({\bf
x})$,
$\hat{H}\to\hat{H}+J{\cdot}({\hat{\psi}}^{{\dagger}}\hat{\psi})\equiv\hat{H}_{J}$.
Let $\beta$ be the temperature inverse, $\beta
J{\cdot}({\hat{\psi}}^{{\dagger}}\hat{\psi})$ is written as
$J{\scriptstyle\circ}({\hat{\psi}}^{{\dagger}}\hat{\psi})=\int
dxJ(x){\hat{\psi}}^{{\dagger}}(x)\hat{\psi}(x)$. The partition function now is
a functional of $J$, that is
$Z[J]\Rightarrow e^{-\beta
W[J]}\\!\\!=\text{Tr}\left[e^{-\beta\left[\hat{H}+J{\cdot}({\hat{\psi}}^{{\dagger}}\hat{\psi})\right]}\right]\\!\equiv\\!\text{Tr}\left[e^{-\beta\hat{H}_{\\!J}}\right].$
(4)
To disentangle the quartic fermionic interaction, we use the standard
procedure of introducing an auxiliary field $\phi$ and express $Z[J]$ as a
path integral over both the Grassmann fields and the auxiliary field
$e^{-\beta W[J]}=\int D\phi
D\psi^{{\dagger}}D\psi\;e^{-S\left[\phi,\psi^{{\dagger}},\psi\right]}\;,\vspace*{-4pt}\phantom{12}$
(5)
where 12
$\displaystyle
S\left[\phi,\psi^{{\dagger}}\\!,\psi\right]=-\frac{1}{2}\text{Tr}\ln(u)+\frac{1}{2}\phi{\scriptstyle\circ}u{\scriptstyle\circ}\phi+\psi^{{\dagger}}{\scriptstyle\circ}G^{-1}{\scriptstyle\circ}\psi$
(6) $\displaystyle G^{-1}(x,x^{\prime})=\left({\partial\tau}+\hat{h}({\bf
x})+i(u{\scriptstyle\circ}\phi)_{x}+J(x)\right)\delta(x-x^{\prime})$ (7)
$\displaystyle\hat{h}({\bf x})=-\frac{\nabla^{2}}{2m}+\upsilon_{\rm ion}({\bf
x})-\mu$ (8) $\displaystyle
u(x,x^{\prime})=\delta(\tau-\tau^{\prime})e^{2}/|{\bf x}-{\bf
x}^{\prime}|\equiv\delta(\tau-\tau^{\prime})u({\bf x},{\bf
x}^{\prime})\;,\vspace*{-2pt}\phantom{12}$ (9)
with $\psi^{(\dagger)}$ denoting the Grassmann fields satisfying
$\psi^{(\dagger)}(\beta,{\bf x})=-\psi^{(\dagger)}(0,{\bf x})$. It is easy to
verify that
$\frac{\delta(\beta W\left[J\right])}{\delta
J(x)}=\langle{\hat{\psi}}^{{\dagger}}(x)\hat{\psi}(x)\rangle_{J}=\langle\hat{n}(x)\rangle_{J}\equiv
n_{J}(x)\;.$ (10)
Eq. (10) expresses $n$ in terms of $J$. The effective action is defined as the
Legendre transformation of $\beta W[J]$
$\Gamma[n_{J}]\equiv\beta W[J]-J{\scriptstyle\circ}n_{J}\;,$ (11)
where the subscript $J$ indicates that the domain of $\Gamma[n]$ is the set of
density profiles reachable by varying $J$. Eq. (11) also leads to
$\frac{\delta\Gamma[n]}{\delta n}=-J\;.\vspace*{-2pt}\phantom{12}$ (12)
We now show that
$E_{\upsilon}[n]=\lim_{\beta\to\infty}\frac{1}{\beta}\Gamma[n]$. Eq. (4)
assures that at the zero temperature limit $W[J]$ is simply the ground state
energy corresponding to $\hat{H}_{J}$. Evidently, when $J=0$,
$\lim_{\beta\to\infty}\frac{1}{\beta}\Gamma[n]|_{n=n_{g}}=W[J]|_{J=0}=E_{g}$
where $E_{g}$ stands for the ground state energy corresponding to $\hat{H}$
and $n_{g}$ represents the electron density at the physical ($J=0$) ground
state. When $J\neq 0$, the corresponding electronic density $n_{J}$ is
different from $n_{g}$ and
$\lim_{\beta\to\infty}\frac{1}{\beta}\Gamma[n]|_{n=n_{J}}$ represents the
expectation value of $\hat{H}$, calculated using the ground state wave
function corresponding to a different Hamiltonian $\hat{H}_{J}$. Thus by the
definition of the ground state,
$\lim_{\beta\to\infty}\frac{1}{\beta}\Gamma[n]|_{n=n_{J}}>\lim_{\beta\to\infty}\frac{1}{\beta}\Gamma[n]|_{n=n_{g}}$.
This means that $\lim_{\beta\to\infty}\frac{1}{\beta}\Gamma[n]$ reaches its
minimum at $n_{g}$. Thus $\lim_{\beta\to\infty}\frac{1}{\beta}\Gamma[n]$ has
all the properties attributed to the energy functional $E_{\upsilon}$ in (1)
and (3). Since the HK theorem states that this functional is unique, it must
in fact be equal to $\lim_{\beta\to\infty}\frac{1}{\beta}\Gamma[n]$.
If we make a change of variable $\phi\to\phi+iu^{-1}{\scriptstyle\circ}J$ in
(5-7) and integrate over the Grassmann fields, we obtain
$\displaystyle e^{-\beta W[J]}$ $\displaystyle\equiv$ $\displaystyle
e^{\frac{1}{2}J{\scriptstyle\circ}u^{-1}{\scriptstyle\circ}J}e^{-\beta
W_{\phi}[J]}$ (13) $\displaystyle=$ $\displaystyle
e^{\frac{1}{2}J{\scriptstyle\circ}u^{-1}{\scriptstyle\circ}J}\int
D\phi\;e^{-I\left[\phi\right]-iJ{\scriptstyle\circ}\phi}\;,\vspace*{-2pt}\phantom{12}$
where
$I[\phi]=-\frac{1}{2}\text{Tr}\ln(u)+\frac{1}{2}\phi{\scriptstyle\circ}u{\scriptstyle\circ}\phi-\text{Tr}\ln(G_{\phi}^{-1})\,,\vspace*{-2pt}\phantom{1}$
(14)
and
${G}_{\phi}^{-1}(x,x^{\prime})=\left(\partial_{\tau}+\hat{h}({\bf
x})+i(u{\scriptstyle\circ}\phi)_{x}\right)\delta(x-x^{\prime})\,.\vspace*{-2pt}\phantom{1}$
(15)
Eq. (13) implies that
$\beta W[J]=\beta
W_{\phi}[J]-\frac{1}{2}J{\scriptstyle\circ}u^{-1}{\scriptstyle\circ}J\,,$ (16)
and thus the left-hand side of (10) can be expressed differently, leading to
$n_{J}=i\varphi-u^{-1}{\scriptstyle\circ}J\;,$ (17)
where $i\varphi\equiv\delta(\beta W_{\phi}[J])/{\delta J}$. To evaluate $\beta
W_{\phi}$, we follow Jackiw Jackiw (1974) and let $\phi\to\phi+\varphi$ in
(13-15). In particular, (15) is rewritten as
$G_{\phi+\varphi}^{-1}(x,x^{\prime})=G_{\varphi}^{-1}(x,x^{\prime})+i\delta(x-x^{\prime})\left(u{\scriptstyle\circ}\phi\right)_{x}\,,$
(18)
and one obtains Jackiw (1974)
$\displaystyle\beta W_{\phi}[J]=\frac{1}{2}\text{Tr}\ln(\tilde{\cal
D}^{-1}{\scriptstyle\circ}u)+\frac{1}{2}\varphi{\scriptstyle\circ}u{\scriptstyle\circ}\varphi-\text{Tr}\ln\left(G_{\varphi}^{-1}\right)$
$\displaystyle+iJ{\scriptstyle\circ}\varphi-\sum_{n=1}^{\infty}\frac{1}{n!}\langle\left[\sum_{k=3}^{\infty}I^{(k)}[\varphi]{\scriptstyle\circ}\,b_{1}\ldots{\scriptstyle\circ}\,b_{k}\right]^{n}\rangle_{\rm
1PI,\leavevmode\nobreak\ conn.},$ (19)
where the subscript “${\rm 1PI,\leavevmode\nobreak\ conn.}$” means to include
only connected, one-particle-irreducible diagrams, $b\equiv
u{\scriptstyle\circ}\phi$,
$\displaystyle\tilde{\mathcal{D}}^{-1}$ $\displaystyle=$ $\displaystyle
u^{-1}-D\;,$ $\displaystyle D(x,y)$ $\displaystyle=$ $\displaystyle
G_{\varphi}(x,y)G_{\varphi}(y,x)\;,\vspace*{-2pt}\phantom{12}$
and
$\displaystyle
I^{(k)}[\varphi]{\scriptstyle\circ}b_{1}\ldots{\scriptstyle\circ}b_{k}\equiv\frac{(-1)^{k-1}}{k}\int\\!dx_{1}\ldots
dx_{k}$ $\displaystyle G_{\varphi}(x_{k},x_{1})\ldots
G_{\varphi}(x_{k-1},x_{k})(ib(x_{1}))\ldots(ib(x_{k}))\;.\vspace*{-2pt}\phantom{12}$
(20)
Fukuda et al. Fukuda et al. (1994) obtained an expression similar to (19) and
used it to derive an effective action as a functional of $\varphi$. They also
noted that this auxiliary field approach does not make a direct connection to
the KS scheme.
Coming to the point of departure from typical auxiliary field approaches, we
show below how an exact correspondence to the KS scheme can be made for the
auxiliary field method by decomposing the source $J$ in a particular way. Let
us define a free fermion propagator ${\mathcal{G}}_{0}$ by
${\mathcal{G}}_{0}^{-1}(x,x^{\prime})=\left[\partial_{\tau}+\hat{h}({{\bf
x}})+J_{0}(x)\right]\delta(x-x^{\prime})\;,$ (21)
where $J_{0}$ is chosen (if $\frac{\delta E_{xc}[n]}{\delta n}|_{n_{J}}$
exists, $J_{0}$ exists and can be written Yu (2009) as
$u{\cdot}n_{J}+\frac{\delta E_{xc}[n]}{\delta n}|_{n_{J}}+J$ ) such that
$-{\mathcal{G}}_{0}(x,x)=n_{J}(x)\;.$ (22)
Eq. (22) demands that this non-interacting (KS) system have electron density,
$-{\mathcal{G}}_{0}(x,x)$, identical to $n_{J}(x)$, the electronic density of
the physical system (where Coulomb interactions exist). In (19), each
occurrence of $iu{\scriptstyle\circ}\varphi$ through $G_{\varphi}$ is to be
replaced by $J+u{\scriptstyle\circ}n_{J}$ (from (17)).
To bring out the KS scheme, we perform the following source decomposition
$J[n]=(J_{0}[n]-u{\scriptstyle\circ}n_{J})+J^{\prime}[n]\equiv\tilde{J}_{0}[n]+J^{\prime}[n]\;.$
(23)
Then from (15) and (17) we have
$G_{\varphi}^{-1}(x,x^{\prime})={\mathcal{G}}_{0}^{-1}(x,x^{\prime})+J^{\prime}(x)\delta(x-x^{\prime})\;.$
(24)
Although the source decomposition (23) is introduced here for the first time
in the auxiliary field approach, a similar method was used in Fukuda et al.
(1995); Valiev and Fernando (1997) to perform perturbative calculations using
$e^{2}$ as the expansion parameter.
Substituting (17) and (19) into (16), one obtains an expression for $\beta
W[J]$, which, upon introducing a parameter $\lambda$ (to be set $=1$ in the
end) to denote the loop order, has the form $\beta
W[J]=\beta\tilde{W}_{0}[J]+\sum_{i=1}^{\infty}\lambda^{i}(\beta
W_{i}[J+u{\scriptstyle\circ}n_{J}])$, where in particular Yu (2009)
$\beta\tilde{W}_{0}[J]=\beta
W_{0}[J+u{\scriptstyle\circ}n_{J}]-\frac{1}{2}n_{J}{\scriptstyle\circ}u{\scriptstyle\circ}n_{J}\;,\vspace*{-2pt}\phantom{12}$
(25)
with $\beta
W_{0}[J+u{\scriptstyle\circ}n_{J}]=-\text{Tr}\ln(G_{\varphi}^{-1})$.
To arrive at an expansion headed by $-\text{Tr}\ln({\mathcal{G}}_{0}^{-1})$
instead of $-\text{Tr}\ln(G_{\varphi}^{-1})$, and containing the expression
$W_{l}[J_{0}]$ instead of $W_{l}[J+u{\scriptstyle\circ}n_{J}]$, we expand
$W_{l}[J+u{\scriptstyle\circ}n_{J}]=W_{l}[J_{0}+J^{\prime}]$ in powers of
$J^{\prime}$ (subscript $l$ omitted in the equation below)
$W=W[J_{0}]+\frac{\delta W[J_{0}]}{\delta
J_{0}}{\scriptstyle\circ}J^{\prime}+\frac{1}{2}\frac{\delta^{2}W[J_{0}]}{\delta
J_{0}\,\delta
J_{0}}{\scriptstyle\circ}J^{\prime}{\scriptstyle\circ}J^{\prime}+\ldots$ (26)
The expression $W_{l}[J_{0}]$ means that $J$ is replaced by $\tilde{J}_{0}$
but $u{\scriptstyle\circ}n_{J}$ is kept unchanged.Yu (2009) With (26), we may
express $\beta W[J]$ as a double series
$\beta
W[J]=\beta\tilde{W}_{00}+\beta\sum_{i,k}W_{ik}\left(1-\delta_{i,0}\delta_{k,0}\right){J^{\prime}}^{k}\lambda^{i}\;,$
(27)
where each $W_{ik}$ involves the $k$’th derivative of $W_{i}$. In particular,
$\tilde{W}_{00}$ is given by (with $n_{J}\to n$ hereafter)
$\beta\tilde{W}_{00}=\beta
W_{00}-\frac{1}{2}n{\scriptstyle\circ}u{\scriptstyle\circ}n=-\text{Tr}\ln({\mathcal{G}}_{0}^{-1})-\frac{1}{2}n{\scriptstyle\circ}u{\scriptstyle\circ}n\;,$
(28)
and in view of (22) $W_{01}$ is given by
$\frac{\delta(\beta W_{0}[J_{0}])}{\delta
J_{0}}=n=\frac{\delta(\beta\tilde{W}_{00}[\tilde{J}_{0}])}{\delta\tilde{J}_{0}}\;.$
(29)
The second half of (29) suggests that we define
$\tilde{\Gamma}_{0}[n]=\beta\tilde{W}_{00}[\tilde{J}_{0}]-\tilde{J}_{0}{\scriptstyle\circ}n\;,$
(30)
the Legendre transformation of the zeroth order contribution from $\beta W[J]$
(in terms of $J^{\prime}$ and $\lambda$), leading to
$\frac{\delta\tilde{\Gamma}_{0}[n]}{\delta n}=-\tilde{J}_{0}\;.$ (31)
Comparing (31) with (12), we find
$\frac{\delta(\Gamma[n]-\tilde{\Gamma}_{0}[n])}{\delta
n}=-J^{\prime}\;.\vspace*{-2pt}\phantom{12}$ (32)
The idea now is to develop a series for $\Gamma[n]$ led by
$\tilde{\Gamma}_{0}[n]$. Subtracting (30) from (11), we have
$\Gamma[n]-\tilde{\Gamma}_{0}[n]=\beta
W[J]-\beta\tilde{W}_{00}[\tilde{J}_{0}]-J^{\prime}{\scriptstyle\circ}n\;,\vspace*{-1pt}\phantom{12}$
(33)
in which the last two terms on the right hand side exactly cancel the terms in
$\tilde{W}_{00}$ and $W_{01}$ contributing to $\beta W[J]$. So the series for
$\Gamma-\tilde{\Gamma}_{0}$ is just (27) with those two terms removed. Next we
convert the double sum in (27) into a single sum by expanding $J^{\prime}$ as
a series in $\lambda$. We write
$J^{\prime}[n]=\sum_{l=1}^{\infty}J_{l}[n]\lambda^{l}\;,\vspace*{-2pt}\phantom{12}$
(34)
where the precise expressions for $J_{1},J_{2},\ldots$ are as yet undetermined
since (34) is not a loop expansion. We substitute (34) formally into (33) and
(27) to obtain a series 12
$\Gamma[n]-\tilde{\Gamma}_{0}[n]=\sum_{l=1}^{\infty}\Gamma_{l}[n]\;\lambda^{l}\;,\vspace*{-2pt}\phantom{12}$
(35)
in which each $\Gamma_{l}$ is defined explicitly in terms of the $J_{k}$,
$\beta W_{k\leq l}[J_{0}]$, and their derivatives. Because $W_{01}$ is missing
from (33), any occurrence of $J_{k}$ is accompanied by at least one other
factor $J_{k^{\prime}}$ or else by an occurrence of some $W_{i>0}$, and hence
by a power of $\lambda$ higher than the $k$’th. In other words, the expression
for $\Gamma_{l\geq 1}$ involves only $J_{k}$ with $k<l$. We finally remove the
indeterminacy in (34) by imposing (32) to hold order by order in $\lambda$,
leading to
$\frac{\delta\Gamma_{l}[n]}{\delta n}=-J_{l}\;.$ (36)
Since $\Gamma_{l\geq 1}$ involves only $J_{k<l}$, all the $J_{l}$ and
$\Gamma_{l}$ can be found explicitly by applying (35) and (36) alternately.
The first few expressions are $\Gamma_{1}=\beta
W_{1}[J_{0}]=-\frac{1}{2}\text{Tr}\ln(\tilde{\mathcal{D}}_{\\!\\!J\to\tilde{J}_{0}}^{-1}{\scriptstyle\circ}u)$,
$J_{1}=-\frac{\delta(\beta W_{1}[J_{0}])}{\delta
J_{0}}{\scriptstyle\circ}\frac{\delta J_{0}}{\delta n}$, $\Gamma_{2}=\beta
W_{2}[J_{0}]+\frac{\delta(\beta W_{1}[J_{0}])}{\delta
J_{0}}{\scriptstyle\circ}J_{1}+\frac{1}{2}J_{1}{\scriptstyle\circ}\frac{\delta^{2}(\beta
W_{0}[J_{0}])}{\delta J_{0}\delta J_{0}}{\scriptstyle\circ}J_{1}$.
For an arbitrary $J_{0}$, one will obtain a corresponding density $\tilde{n}$.
The computation of $\frac{1}{\beta}\Gamma[n]$ using (30), (35) and (36)
evaluates the energy functional at density $\tilde{n}$, which may or may not
be the ground state density. To obtain the ground state density and the
corresponding $J_{0}$, one needs to solve at zero temperature limit the
extremal equation $0=\frac{\delta\Gamma[n]}{\delta n}$, which we turn to
shortly.
To carry out the calculation of $\Gamma[n]$, we need to compute $J_{l}$ (see
(36)) via the functional derivative
$\frac{\delta}{\delta n}=\left(\frac{\delta n}{\delta
J_{0}}\right)^{-1}{\scriptstyle\circ}\frac{\delta}{\delta
J_{0}}\,\equiv\,D_{0}^{-1}{\scriptstyle\circ}\frac{\delta}{\delta J_{0}}\;.$
(37)
Diagrams corresponding to $\beta W_{l}[J_{0}]$ and their derivatives contain
the $u$, ${\mathcal{G}}_{0}$, and
$\tilde{\mathcal{D}}_{0}\equiv\tilde{\mathcal{D}}_{\\!\\!J\to\tilde{J}_{0}}$
propagators. It is easy to show that one may express $\delta n(x)/\delta
J_{0}(y)$ as
$-\frac{\delta{\mathcal{G}}_{0}(x,x)}{\delta
J_{0}(y)}={\mathcal{G}}_{0}(x,y)\,{\mathcal{G}}_{0}(y,x)=D_{J\to\tilde{J}_{0}}(x,y)$
(38)
and thus $D_{0}^{-1}=D_{J\to\tilde{J}_{0}}^{-1}$, which we call the inverse
density correlator. The differentiation rules of ${\mathcal{G}}_{0}$,
$\tilde{\mathcal{D}}_{0}$, and $D_{0}^{-1}$ with respect to $J_{0}$ can be
expressed diagrammatically:
$\displaystyle\frac{\delta{\mathcal{G}}_{0}(x,x^{\prime})}{\delta
J_{0}(y)}=\frac{\delta}{\delta J_{0}(y)}\;\begin{picture}(10.0,40.0)(0.0,-3.0)
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \raise-24.0pt\hbox
to0.0pt{\kern 5.0pt\makebox(0.0,0.0)[tc]{$x^{\prime}$}\hss} \ignorespaces
\raise 24.0pt\hbox to0.0pt{\kern 5.0pt\makebox(0.0,0.0)[bc]{$x$}\hss}
\ignorespaces\end{picture}$ $\displaystyle=$
$\displaystyle-\;\begin{picture}(10.0,40.0)(0.0,-3.0) \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\raise-24.0pt\hbox to0.0pt{\kern 5.0pt\makebox(0.0,0.0)[tc]{$x^{\prime}$}\hss}
\ignorespaces \raise 24.0pt\hbox to0.0pt{\kern
5.0pt\makebox(0.0,0.0)[bc]{$x$}\hss} \ignorespaces \raise-4.0pt\hbox
to0.0pt{\kern 11.0pt\makebox(0.0,0.0)[bc]{$y$}\hss}
\ignorespaces\end{picture}$
$\displaystyle\frac{\delta\tilde{\mathcal{D}}_{0}(x,x^{\prime})}{\delta
J_{0}(y)}=\frac{\delta}{\delta
J_{0}(y)}\;\begin{picture}(10.0,55.0)(0.0,-3.0)\put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \raise-24.0pt\hbox to0.0pt{\kern
5.0pt\makebox(0.0,0.0)[tc]{$x^{\prime}$}\hss} \ignorespaces \raise 24.0pt\hbox
to0.0pt{\kern 5.0pt\makebox(0.0,0.0)[bc]{$x$}\hss} \ignorespaces\end{picture}$
$\displaystyle=$
$\displaystyle-\;\begin{picture}(30.0,40.0)(0.0,-3.0)\put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \raise-24.0pt\hbox to0.0pt{\kern
15.0pt\makebox(0.0,0.0)[tc]{$x^{\prime}$}\hss} \ignorespaces \raise
24.0pt\hbox to0.0pt{\kern 15.0pt\makebox(0.0,0.0)[bc]{$x$}\hss} \ignorespaces
\raise-4.0pt\hbox to0.0pt{\kern 11.0pt\makebox(0.0,0.0)[bc]{$y$}\hss}
\ignorespaces\end{picture}\;-\;\begin{picture}(30.0,40.0)(0.0,-3.0)\put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \raise-24.0pt\hbox to0.0pt{\kern
15.0pt\makebox(0.0,0.0)[tc]{$x^{\prime}$}\hss} \ignorespaces \raise
24.0pt\hbox to0.0pt{\kern 15.0pt\makebox(0.0,0.0)[bc]{$x$}\hss} \ignorespaces
\raise-4.0pt\hbox to0.0pt{\kern 11.0pt\makebox(0.0,0.0)[bc]{$y$}\hss}
\ignorespaces\end{picture}$ $\displaystyle\frac{\delta
D_{0}^{-1}(x,x^{\prime})}{\delta J_{0}(y)}=\frac{\delta}{\delta
J_{0}(y)}\;\begin{picture}(10.0,55.0)(0.0,-3.0) \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \raise-24.0pt\hbox to0.0pt{\kern
5.0pt\makebox(0.0,0.0)[tc]{$x^{\prime}$}\hss} \ignorespaces \raise 24.0pt\hbox
to0.0pt{\kern 5.0pt\makebox(0.0,0.0)[bc]{$x$}\hss} \ignorespaces\end{picture}$
$\displaystyle=$ $\displaystyle+\;\begin{picture}(30.0,40.0)(0.0,-3.0)
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \raise-24.0pt\hbox to0.0pt{\kern
15.0pt\makebox(0.0,0.0)[tc]{$x^{\prime}$}\hss} \ignorespaces \raise
24.0pt\hbox to0.0pt{\kern 15.0pt\makebox(0.0,0.0)[bc]{$x$}\hss} \ignorespaces
\raise-4.0pt\hbox to0.0pt{\kern 11.0pt\makebox(0.0,0.0)[bc]{$y$}\hss}
\ignorespaces\end{picture}\;+\;\begin{picture}(30.0,40.0)(0.0,-3.0)
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \raise-24.0pt\hbox to0.0pt{\kern
15.0pt\makebox(0.0,0.0)[tc]{$x^{\prime}$}\hss} \ignorespaces \raise
24.0pt\hbox to0.0pt{\kern 15.0pt\makebox(0.0,0.0)[bc]{$x$}\hss} \ignorespaces
\raise-4.0pt\hbox to0.0pt{\kern 11.0pt\makebox(0.0,0.0)[bc]{$y$}\hss}
\ignorespaces\end{picture}\;.$
The differentiation rules of ${\mathcal{G}}_{0}$, $\tilde{\mathcal{D}}_{0}$,
and $D_{0}^{-1}$ with respect to $n$ are simply obtained by compounding the
results above with (37). We show only one example:
$\frac{\delta{\mathcal{G}}_{0}(x,x^{\prime})}{\delta
n(z)}=\frac{\delta}{\delta n(z)}\;\begin{picture}(10.0,40.0)(0.0,-3.0)
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \raise-24.0pt\hbox
to0.0pt{\kern 5.0pt\makebox(0.0,0.0)[tc]{$x^{\prime}$}\hss} \ignorespaces
\raise 24.0pt\hbox to0.0pt{\kern 5.0pt\makebox(0.0,0.0)[bc]{$x$}\hss}
\ignorespaces\end{picture}\;=\;-\;\begin{picture}(30.0,40.0)(-20.0,-3.0)
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \raise-24.0pt\hbox
to0.0pt{\kern 5.0pt\makebox(0.0,0.0)[tc]{$x^{\prime}$}\hss} \ignorespaces
\raise 24.0pt\hbox to0.0pt{\kern 5.0pt\makebox(0.0,0.0)[bc]{$x$}\hss}
\ignorespaces \raise-4.0pt\hbox
to0.0pt{\kern-15.0pt\makebox(0.0,0.0)[tc]{$z$}\hss}
\ignorespaces\end{picture}\;.$
Equipped with these differentiation rules, one may use standard diagrammatic
expansion to compute the $W_{l}[J_{0}]$s, their functional derivatives with
respect to $J_{0}$, as well as $J_{l}$s to facilitate the calculations of
$\Gamma_{l}$s. Because
$D_{0}(x,y)={\mathcal{G}}_{0}(x,y){\mathcal{G}}_{0}(y,x)$, both
$\tilde{\mathcal{D}}_{0}=\left(u^{-1}-D_{0}\right)^{-1}$ and $D_{0}^{-1}$ can
be expressed in terms of single-particle orbitals and energies through
${\mathcal{G}}_{0}(x,y)$ –the propagator of the KS system– which can be
expressed as
$\displaystyle{\mathcal{G}}_{0}(x,y)$ $\displaystyle=$
$\displaystyle\sum_{\alpha}\phi_{\alpha}({\bf x})\phi_{\alpha}^{*}({\bf
y})e^{-(\varepsilon_{\alpha}-\mu)(\tau_{x}-\tau_{y})}\times$ (41)
$\displaystyle\times\left\\{\begin{array}[]{l r}(-n_{\alpha})&{\rm
if\leavevmode\nobreak\ }\tau_{x}\leq\tau_{y}\\\ (1-n_{\alpha})&{\rm
if\leavevmode\nobreak\ }\tau_{x}>\tau_{y}\end{array}\right.\;,$
where $n_{\alpha}=1/(e^{\beta(\varepsilon_{\alpha}-\mu)}+1)$,
$\sum_{\alpha}n_{\alpha}=N_{e}$, and the single particle orbital
$\phi_{\alpha}({\bf x})$ satisfies
$\left[\hat{h}({\bf x})+J_{0}({\bf x})\right]\phi_{\alpha}({\bf
x})=(\varepsilon_{\alpha}-\mu)\phi_{\alpha}({\bf x})\;.$
Since $\frac{\delta\tilde{\Gamma}_{0}[n]}{\delta n}=-\tilde{J}_{0}$, the
extremal condition $0=\frac{\delta\Gamma[n]}{\delta n}$ that determines
$n_{g}$ and $J_{0}[n_{g}]$ (as $\beta\to\infty$) becomes
$\frac{\delta\left(\sum_{i=1}^{\infty}\Gamma_{i}[n]\right)}{\delta
n}=D_{0}^{-1}{\scriptstyle\circ}\frac{\delta\left(\sum_{i=1}^{\infty}\Gamma_{i}[J_{0}[n]]\right)}{\delta
J_{0}}=\tilde{J}_{0}\;.$ (42)
Eq. (42) has to be solved self-consistently by keeping $\Gamma_{i}$ terms up
to some order in $\lambda$. Although a truncation is necessary, we note that
each diagram in our expression already corresponds to infinitely many Feynman
diagrams when using $e^{2}$ as the expansion parameter. This is easily seen by
performing the small $e^{2}$ expansion of $\tilde{\mathcal{D}}_{0}$
$\tilde{\mathcal{D}}_{0}=u+u{\scriptstyle\circ}D_{0}{\scriptstyle\circ}u+u{\scriptstyle\circ}D_{0}{\scriptstyle\circ}u{\scriptstyle\circ}D_{0}{\scriptstyle\circ}u+\ldots\;,$
a sum of infinitely many (dressed) propagators. Interestingly, in the strong
coupling limit where one must treat $u^{-1}$ as a small parameter, we may
express $\tilde{\mathcal{D}}_{0}$ as
$\tilde{\mathcal{D}}_{0}=-D_{0}^{-1}-D_{0}^{-1}{\scriptstyle\circ}u^{-1}{\scriptstyle\circ}D_{0}^{-1}-D_{0}^{-1}{\scriptstyle\circ}u^{-1}{\scriptstyle\circ}D_{0}^{-1}{\scriptstyle\circ}u^{-1}{\scriptstyle\circ}D_{0}^{-1}-\ldots$
while the traditional $e^{2}$ expansion fails completely.
Finally we sketch how (3) arises from $\Gamma[n]$. Eq. (30) may be rewritten
as
$\frac{1}{\beta}\tilde{\Gamma}_{0}[n]=\frac{1}{\beta}\left[-\text{Tr}\ln({\mathcal{G}}_{0}^{-1})-J_{0}{\scriptstyle\circ}n\right]+\frac{1}{2\beta}n{\scriptstyle\circ}u{\scriptstyle\circ}n$.
Because
$-\text{Tr}\ln({\mathcal{G}}_{0}^{-1})=\sum_{\alpha}\ln(1-n_{\alpha})$, at
zero temperature limit, the first two terms of
$\frac{1}{\beta}\tilde{\Gamma}_{0}$ above give rise to the $T_{0}[n]-\mu
N_{e}+\int\upsilon({\bf x})n(x)d{\bf x}$ while the last part is exactly the
Hartree term Yu (2009). The exchange-correlation functional $E_{xc}[n]$ equals
$\lim_{\beta\to\infty}\frac{1}{\beta}\sum_{i=1}^{\infty}\Gamma_{i}[n]$. We
also comment that the excitations of the system can be studied Yu (2009) under
this formalism and the energy functional shown in this letter has the correct
single-electron limit Yu (2009).
Providing a scheme beyond perturbative expansion in $e^{2}$, we have proposed
an effective action construction that will contribute to the development of
the parameter-free universal density functional.
This research was supported by the Intramural Research Program of the National
Library of Medicine of the National Institutes of Health.
## References
* Kohn (1999) W. Kohn, Rev. Mod. Phys. 71, 1253 (1999).
* Hohenberg and Kohn (1964) P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964).
* Mermin (1965) N. D. Mermin, Phys. Rev. 137, A1441 (1965).
* Kohn and Sham (1965) W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965).
* Kümmel and Kronik (2008) S. Kümmel and L. Kronik, Reviews of Modern Physics 80, 3 (pages 58) (2008).
* Cohen et al. (2008) A. J. Cohen, P. Mori-Sanchez, and W. Yang, Science 321, 792 (2008).
* Fukuda et al. (1994) R. Fukuda, T. Kotani, Y. Suzuki, and S. Yokojima, Progress of Theoretical Physics 92, 833 (1994).
* Fukuda et al. (1995) R. Fukuda, M. Komachiya, S. Yokojima, Y. Suzuki, K. Okumura, and T. Inagaki, Progress of Theoretical Physics Supplement 121, 1 (1995).
* Valiev and Fernando (1997) M. Valiev and G. W. Fernando (1997), eprint cond-mat/9702247.
* Polonyi and Sailer (2002) J. Polonyi and K. Sailer, Phys. Rev. B 66, 155113 (2002).
* Jackiw (1974) R. Jackiw, Phys. Rev. D 9, 1686 (1974).
* Sham (1985) L. J. Sham, Phys. Rev. B 32, 3876 (1985).
* Negele and Orland (1988) J. W. Negele and H. Orland, _Quantum Many-Particle Systems_ (Addison-Wesley, Redwood city, CA, 1988).
* Yu (2009) Y.-K. Yu (2009), eprint to be submitted to PRB.
|
arxiv-papers
| 2009-09-21T03:07:45 |
2024-09-04T02:49:05.436269
|
{
"license": "Public Domain",
"authors": "Yi-Kuo Yu",
"submitter": "Yi-Kuo Yu",
"url": "https://arxiv.org/abs/0909.3673"
}
|
0909.3713
|
A possible scenario of the Pioneer anomaly in the framework of Finsler
geometry
Xin Li∗,‡ 111lixin@itp.ac.cn and Zhe Chang†,‡ 222changz@mail.ihep.ac.cn
∗Institute of Theoretical Physics, Chinese Academy of Sciences, 100190
Beijing, China
†Institute of High Energy Physics, Chinese Academy of Sciences, 100049
Beijing, China
‡Theoretical Physics Center for Science Facilities, Chinese Academy of
Sciences
###### Abstract
The weak field approximation of geodesics in Randers-Finsler space is
investigated. We show that a Finsler structure of Randers space corresponds to
the constant and sunward anomalous acceleration demonstrated by the Pioneer 10
and 11 data. The additional term in the geodesic equation acts as “electric
force”, which provides the anomalous acceleration.
PACS numbers: 02.40.-k, 04.50.Kd, 95.10.Ce
Newton’s theory of gravitation was proposed almost three hundred and fifty
years ago. Einstein’s general relativity reveals the intrinsic geometric
property of gravity. Newton’s theory of gravitation is the main guideline of
the celestial mechanics, especially for the solar system. General relativity
provides small corrections. It is well-known that the Newton and Einstein’s
theory of gravitation still faces problems. One of them is that the flat
rotation curves of spiral galaxies violate the prediction of Newton and
Einstein’s gravity. Another is related with recent astronomical
observations[1]. Our universe is acceleratedly expanding. This result can not
be obtained directly from Einstein’s gravity and his cosmological principle.
In fact, new puzzle has also arisen in the solar system. That is the Pioneer
anomaly. The Pioneer spacecrafts are excellent tools for dynamical astronomy
studies in the solar system. The radio metric data from the Pioneer 10/11
spacecraft indicate the presence of a small, anomalous, Doppler frequency
drift over the range of 20–70 astronomical units[2]. The drift is blue-shift,
uniformly changing with a rate of $~{}6\times 10^{-9}Hz/s$[3]. It has revealed
an anomalous constant sunward acceleration, $a_{P}=(8.74\pm 1.33)\times
10^{-10}m/s^{2}$.
The Pioneer data have been studied in three different navigational computer
programs. Namely: the JPL’s Orbit Determination Program (ODP), the Aerospace
Corporation’s CHASPM code extended for deep space navigation[2], and a code
written in the Goddard Space Flight Center[4]. These data analysis all confirm
the existence of anomalous acceleration with the following basic properties:
the direction of the anomalous acceleration of the spacecraft towards the Sun,
the anomalous acceleration appears close to $20AU$ and up to $70AU$, the
anomalous acceleration seems to be a constant with 10% order of temporal and
spatial variations of the anomaly’s magnitude. Turyshev et al.[5] summarized
recent results on researches of the anomaly. Several conventional physical
mechanisms have been proposed to explain the anomaly, such as the unknown
systematic–the gas leaks from the propulsion system or a recoil force due to
the on-board thermal power inventory, and the conventional gravitational force
due to a known mass distribution in the outer solar system–the Kuiper Belt
Objects or dust, and the expansion of the universe motivated by the numerical
coincidence $a_{P}\simeq cH_{0}$. However, it was pointed out that these
conventional physical mechanisms can not be the answer of the anomalous
accelerations[2, 6].
The failure of the conventional physical mechanisms imply that the Pioneer
anomaly may correspond to ‘new physics’. One of the most popular ‘new physics’
is the dark matter hypothesis. A specific distribution of dark matter in the
solar system would yield the wanted result[7]. However, this special
distribution of dark matter is not like the consequence of gravity. Thus, to
explain the Pioneer anomaly, dark matter hypothesis still need more work.
Several modified gravitational theories also was suggested to explain the
Pioneer anomaly, such as the scalar-tensor vector gravity (STVG)theory[8],
brane-world models with large extra dimensions[9], and conformal gravity with
dynamic mass generation[2]. These modified gravitational theories seem
appealing, however, most of them either much more complicated or involves too
much hypothesis which does not verified by experiments.
Finsler geometry, which takes Riemann geometry as its special case, is a good
candidate to solve the facing problems of the theory of gravitation. The
gravity in Finsler space has been studied for a long time[10, 11, 12, 13]. In
our previous paper[14], a modified Newton’s gravity was obtained as the weak
field approximation of the Einstein’s equation in Finsler space of Berwald
type. We have shown that the prediction of the modified Newton’s gravity is in
good agreement with the rotation curves of spiral galaxies without invoking
dark matter hypothesis.
Randers space, as a special kind of Finsler space, was first proposed by G.
Randers[15]. Within the framework of Finsler geometry, modified dispersion
relation of free particle in Randers space has been discussed[16]. A modified
Friedmann model in Randers space is proposed. It is showed that the
accelerated expanding universe is guaranteed by a constrained Randers-Finsler
structure without invoking dark energy[17].
In this Letter, in the framework of Finsler geometry we will try to give a
simple and clear description of the Pioneer anomaly. As well-known, the length
in Riemann geometry is a function of positions. However, this is not the case
in Finsler geometry. In Finsler geometry, the length is a function of both
position and velocity. Finsler geometry is base on the so called Finsler
structure $F$ with the property $F(x,\lambda y)=\lambda F(x,y)$, where $x$
represents position and $y$ represents velocity. The Finsler metric is given
as[18]
$\displaystyle g_{\mu\nu}\equiv\frac{\partial}{\partial
y^{\mu}}\frac{\partial}{\partial y^{\nu}}\left(\frac{1}{2}F^{2}\right).$ (1)
The Randers metric is a Finsler structure $F$ on $TM$ of the form
$\displaystyle F(x,y)\equiv\alpha(x,y)+\beta(x,y)~{},$ (2)
where
$\displaystyle\alpha(x,y)$ $\displaystyle\equiv$
$\displaystyle\sqrt{\tilde{a}_{\mu\nu}(x)y^{\mu}y^{\nu}}$
$\displaystyle\beta(x,y)$ $\displaystyle\equiv$
$\displaystyle\tilde{b}_{\mu}(x)y^{\mu}.$ (3)
Here $\tilde{\alpha}$ is a Riemannian metric on the manifold $M$. In this
Letter, the indices decorated with a tilde are lowered and raised by
$\tilde{\alpha}_{\mu\nu}$ and its inverse matrix $\tilde{\alpha}^{\mu\nu}$,
otherwise lower and raise the indices are carried by $g_{\mu\nu}$ and
$g^{\mu\nu}$. We will show that the Finsler structure in Randers space with
$\tilde{b}$ taking the specific form $\tilde{b}_{\mu}=\\{-kr,0,0,0\\}$
corresponds to the Pioneer anomaly. The above form of $\tilde{b}$ is given in
spherical coordinate and $k$ is a constant.
The parallel transport in Finsler space has been studied in terms of Cartan
connection[20, 21, 22]. The notation of parallel transport in Finsler manifold
means that the length $F\left(\frac{d\sigma}{d\tau}\right)$ is constant.
Following the calculus of variations, one gets the autoparallel equation in
Finsler space as[18]
$\displaystyle\frac{d^{2}\sigma^{\lambda}}{d\tau^{2}}+\gamma^{\lambda}_{\mu\nu}\frac{d\sigma^{\mu}}{d\tau}\frac{d\sigma^{\nu}}{d\tau}=0.$
(4)
The autoparallel equation (4) is directly derived from the integral length of
$\sigma$
$\displaystyle L=\int F\left(\frac{d\sigma}{d\tau}\right)d\tau,$ (5)
the inner product
$\left(\sqrt{g_{\mu\nu}\frac{d\sigma^{\mu}}{d\tau}\frac{d\sigma^{\nu}}{d\tau}}=F\left(\frac{d\sigma}{d\tau}\right)\right)$
of two parallel transported vectors is preserved. To get a modified Newton’s
gravity, we consider a particle moving slowly in a weak stationary
gravitational field[19]. Here, we suppose that the Riemannian metric
$\tilde{\alpha}$ is close to Minkowskian metric, and
$|\tilde{b}_{\mu}\tilde{b}^{\mu}|$ is very small
$\displaystyle\tilde{a}_{\mu\nu}(x)=\tilde{\eta}_{\mu\nu}+\tilde{h}_{\mu\nu}(x),$
(6)
where $\tilde{\eta}_{\mu\nu}$ is the Minkowskian metric and
$|\tilde{h}_{\mu\nu}|\ll 1$. Deducing from (4), we obtain the geodesic of
Randers space with constant Riemanian speed (namely,
$\alpha(\frac{d\sigma}{d\tau})$ is constant)
$\displaystyle\frac{d^{2}\sigma^{\lambda}}{d\tau^{2}}+\tilde{\gamma}^{\lambda}_{\mu\nu}\frac{d\sigma^{\mu}}{d\tau}\frac{d\sigma^{\nu}}{d\tau}+\tilde{a}^{\lambda\mu}f_{\mu\nu}\alpha\left(\frac{d\sigma}{d\tau}\right)\frac{d\sigma^{\nu}}{d\tau}=0,$
(7)
where $f_{\mu\nu}\equiv\frac{\partial\tilde{b}_{\mu}}{\partial
x^{\nu}}-\frac{\partial\tilde{b}_{\nu}}{\partial x^{\mu}}$.
Randers[15] has already found that the Randers metric is related to five
dimensional Riemannian geometry. The five dimensional Riemannian metric
$\gamma_{mn}$ ($m,n=1,2,3,4,5;\mu,\nu=1,2,3,4$) is given as
$\gamma_{\mu\nu}=\tilde{a}_{\mu\nu}-\tilde{b}_{\mu}\tilde{b}_{\nu};~{}\gamma_{\mu
5}=\gamma_{5\mu}=\tilde{b}_{\mu};~{}\gamma_{55}=-1.$ (8)
And the geodesic equation (7) of Randers metric is a solution of the five
dimensional Einstein’s field equation. The five dimensional Einstein tensor is
expressed as
$\displaystyle G^{\mu\nu}$ $\displaystyle=$
$\displaystyle\left(\tilde{R}^{\mu\nu}-\frac{1}{2}\tilde{a}^{\mu\nu}\tilde{R}\right)+\frac{1}{2}\tilde{E}^{\mu\nu},$
(9) $\displaystyle G_{5}^{~{}\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{2}f^{\nu\mu}_{~{}~{}~{};\mu},$ (10) $\displaystyle
G_{55}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\tilde{R}-\frac{3}{4}f^{\mu\nu}f_{\mu\nu}\right),$
(11)
where
$\tilde{E}^{\mu\nu}=-f^{\mu\lambda}f_{\lambda}^{~{}\nu}+\frac{1}{4}\tilde{a}^{\mu\nu}f^{\lambda\theta}f_{\lambda\theta}$,
$\tilde{R}^{\mu\nu}$ is the Ricci tensor of the four dimensional Riemannian
metric $\alpha$, and the covariant derivative of four dimensional Riemannian
metric $\alpha$ is denoted by “;”. In a geometrical viewpoint, the Randers
metric arised from the Zermelo navigation problem [23]. It aims to find the
paths of shortest travel time in a Riemannian manifold under the influence of
a drift (“wind”). Shen [24] has shown that these minimum time trajectories are
exactly the geodesics of a particular Finsler geometry-Randers metric. The map
between the Randers metric to a Riemannian space in the viewpoint of Zermelo
navigation problem is investigated in the paper [25].
In weak field approximation, the second term of the left side of the equation
(7) represents the Newtonian gravitational acceleration. And the third term
may induce the anomalous acceleration. One should notice that the trajectory
of the Pioneer 10 spacecraft is different from that of the Pioneer 11
spacecraft. The basic property of the Pioneer anomaly that the direction of
the anomalous acceleration of the spacecraft towards the Sun tells us that the
non vanish components of $f_{\mu\nu}$ is $f_{0i}$. The term $f_{\mu\nu}$ acts
as electromagnetic force. In dealing with the Pioneer anomaly, one need take
only the “electric force” into account. Also, due to physical consideration,
the “electric force” should be static. Thus, in the approximation of moving
slowly and weak field, the geodesic equation (7) reduces to
$\displaystyle\frac{d^{2}t}{d\tau^{2}}$ $\displaystyle=$ $\displaystyle 0,$
$\displaystyle\frac{d^{2}\sigma^{i}}{d\tau^{2}}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\frac{\partial
h_{00}}{\partial\sigma^{i}}\left(\frac{dt}{d\tau}\right)^{2}-\frac{\partial
b_{0}}{\partial\sigma^{i}}\alpha\left(\frac{d\sigma}{d\tau}\right)\left(\frac{dt}{d\tau}\right).$
(12)
The solution of the first equation in (S0.Ex2) is $dt/d\tau=const.$. Dividing
the second equation in (S0.Ex2) by $(dt/d\tau)^{2}$, we obtain
$\displaystyle\frac{d\sigma^{i}}{dt}=-\frac{1}{2}\frac{\partial
h_{00}}{\partial\sigma^{i}}-\frac{\partial b_{0}}{\partial\sigma^{i}}.$ (13)
In spherical coordinate, the above equation changes as
$\displaystyle a=\nabla\varphi+k,$ (14)
where $\varphi\equiv-\frac{GM_{\odot}}{r}$ is the Newtonian gravitational
potential. Then, from the equation (14) one can see clearly that the anomalous
acceleration
$\displaystyle a_{p}=k.$ (15)
Taking the average value of $a_{p}$, we can set the constant $k$ as
$9.71\times 10^{-25}m^{-1}$.
At a position of 20AU far from the Sun, the Newtonian gravitational potential
$\varphi=-4.43\times 10^{7}m^{2}/s^{2}$ and the perturbation of Minkowskian
metric $h_{00}=9.85\times 10^{-10}$, and $b_{0}=2.91\times 10^{-14}$. Thus,
the Finsler structure of Randers space is a good description for metric
fluctuation around the Minkowskian one. At a position of 1AU far from the Sun,
the Newtonian gravitational potential $\varphi=-8.87\times 10^{8}m^{2}/s^{2}$
and the perturbation of Minkowskian metric $h_{00}=1.97\times 10^{-8}$, and
$b_{0}=1.46\times 10^{-15}$. Einstein’s relativity offers high order
correction for Newtonian mechanics (post-Newtonian approximation)[19], the
corresponding metric correction approximately equals $h_{00}^{2}$. Here, we
can see that $h_{00}^{2}$ is very close to $b_{0}$. While the Pioneer is not
far from the earth, it is hard to distinguish the effect of general
relativity(or Riemann geometry) and Finsler geometry. This is a reason for why
the anomaly appears in the position of 20AU far from the Sun.
The equation (14) implies that the modified gravitational potential is
$\varphi_{P}=krc^{2},$ (16)
where $c$ is the speed of light. Since the parameter $k$ is set as $9.71\times
10^{-25}m^{-1}$, the ration $|\frac{\varphi_{P}}{\varphi}|$ is less than
$10^{-8}$ for the solar system. The classical tests of general relativity are
carried in solar system. Thus, the geodesic equation (7) also predict the same
astrophysical phenomena that Einstein’s general relativity are able to
predict. One also could directly obtain this fact from the field equation (9),
for the tensor $\tilde{E}^{\mu\nu}$ in it is the second order in $f^{\mu\nu}$.
The existence of the Pioneer anomaly suggests the Newton’s theory of
gravitation and general relativity need to be modified even in the solar
system. Here, we have suggested that Finsler geometry could give a clear and
simply description of the Pioneer anomaly. The specific Finsler structure of
the Randers space corresponds to the Pioneer anomaly. We hope that the gravity
anomalies mentioned in the beginning of the Letter can be solved
systematically in the framework of Finsler geometry.
Acknowledgements
We would like to thank Prof. C. J. Zhu, H. Y. Guo and C. G. Huang for useful
discussions. The work was supported by the NSF of China under Grant No.
10525522 and 10875129.
## References
* [1] A. G. Riess, et al., Astrophys J. 117 (1999) 707; S. Perlmutter, et al., Astrophys J. 517 (1999) 565; C. L. Bennett, et al., Astrophys J. 148 (Suppl.) (2003) 1.
* [2] J. D. Anderson, et al., Phys. Rev. Lett. 81 (1998) 2858, J. D. Anderson, et al., Phys. Rev. D 65 (2002) 082004, J. D. Anderson, et al., Mod. Phys. Lett. A 17 (2002) 875.
* [3] S. G. Turyshev, et al., Stanford e-Conf #C041213, #0310, arXiv:gr-qc/0503021.
* [4] C. Markwardt, arXiv:gr-qc/0208046.
* [5] S. G. Turyshev, et al., EAS Publ. Ser. 20 (2006) 243.
* [6] M. M. Nieto, et al., Phys. Lett. B 613 (2005) 11.
* [7] R. Foot and R. R. Volkas, Phys. Lett. B 517 (2001) 13.
* [8] J. W. Moffat, arXiv:gr-qc/0405076; J. Cosmol. Astropart. Phys. JCAP 03 (2006) 004.
* [9] O. Bertolami and J. Páramos, Phys. Rev. D 71 (2005) 023521.
* [10] Y. Takano, Lett. Nuovo Cimento 10 (1974) 747.
* [11] S. Ikeda, Ann. der Phys. 44 (1987) 558.
* [12] R. Tavakol and N. van den Bergh, Phys. Lett. A 112 (1985) 23.
* [13] G. Yu. Bogoslovsky, Phys. Part. Nucl. 24 (1993) 354.
* [14] Z. Chang and X. Li, Phys. Lett. B 668 (2008) 453.
* [15] G. Randers, Phys. Rev. 59 (1941) 195.
* [16] Z. Chang and X. Li, Phys. Lett. B 663 (2008) 103.
* [17] Z. Chang and X. Li, Phys. Lett. B 676 (2009) 173.
* [18] D. Bao, S. S. Chern and Z. Shen, An Introduction to Riemann–Finsler Geometry, Graduate Texts in Mathmatics 200, Springer, New York, 2000.
* [19] S. Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, Wiley, New York, 1972.
* [20] M. Matsumoto, Foundations of Finsler Geometry and Special Finsler Spaces, Kaiseisha Press, Saikawa Shigaken, Japan 1986.
* [21] P. L. Antonelli and S. F. Rutz, in: Finsler Geometry, Sapporo 2005 C In Memory of Makoto Matsumoto, in: S.V. Sabau, H. Shimada (Eds.), Advanced Studies in Pure Mathematics, vol. 48, World Scientific, 2007, p. 210.
* [22] Z. Szabo, Ann. Glob. Anal. Geom 34 (2008) 381.
* [23] E. Zermelo, Z. Angew. Math. Mech. 11(2) (1931) 114.
* [24] Z. Shen, Canadian J. Math. 55 (2003) 112, arXiv:math/0109060.
* [25] G. W. Gibbons, C. A. R. Herdeiro, C. M. Warnick, M. C. Werner, Phys. Rev. D 79 (2009) 044022.
|
arxiv-papers
| 2009-09-21T08:54:59 |
2024-09-04T02:49:05.442512
|
{
"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/0909.3713"
}
|
0909.3747
|
# Discrete Algebraic Equations and Discrete Operator Equations(Presentations
for ICM 2010)
Wu Zi qian Fangda group company,Shenzhen city,Guangdong province,China
$runton_{-}runton$$@$ruc.edu.cn,woodschain$@$sohu.com
###### Abstract.
We give constructive results of Hilbert’s 13th problem for discrete functions.
By them we give formula solution expressed by a superposition of functions of
one variable to equations constructed by discrete functions and equations with
parameterized discrete functions. Further more we give formula solution
expressed by a superposition of operators of one variable to equations
constructed by discrete operators and equations with parameterized discrete
operators. This is a Short communication, Section 9,Functional Analysis and
Application, Saturday, August 21, 2010,18:00-18:15, Room No. T3.
###### Key words and phrases:
Discrete function,commutation operator,tension-compression
operator,superposition operator,decomposition operator,discrete operator,high
operator
## 1\. Introduction
Problems about equations are very important and difficult. Solving quadratic
equation and cubic equation and quartic equation had cost the mathematicians
in history a great deal of time.
Babylonians solve quadratics in radicals in 2000 BC. Cubic equation and
quartic equation were solved by Italian mathematicians
Girolamo.Cardano(1501-1576) and Ludovico.Ferrari(1522-1565)in 16th
century,respectively.
But mathematicians met big troubles when they tried to solve quintic equation.
Leonhard.Euler(1707-783) believed quintic equation can be changed to a quartic
equation by transformation of variable. Niels.Henrik.Abel (1802-1829) got a
conclusion that there is no solution by radicals for a general polynomial
algebraic equation if n$\geq$5\. Evariste.Galois (1811-1832) built group
theory and got the same conclusion. His method come down to now and can be
found in any textbook about Galois group theory.
There is no solution by radicals. Are there any solutions of other forms such
as numerical solution and solution expressed in function of two variables or
of many variables or solution expressed in series or in integral?
We do not discuss numerical solutions because they belong to applied
mathematics. We prefer formula solution expressed in binary function to other
ones. What is a formula solution expressed in binary function? It contains
only function of two variables. We can give a expression of a alone binary
function at the beginning. We can replace any one of variables by a binary
function then we get a new expression. We can replace any one of variables of
this new expression by a binary function again and get a more complex
expression. We can repeat the procedure for any finite times. But it is not
easy to get solution expressed in binary function. It’s easier to get
solutions of other forms.History developed just like this.
Camille.Jordan (1838-1922) shows that algebraic equations of any degree can be
solved in terms of modular functions in 1870. Ferdinand.von.Lindemann (1852-
1939) expresses the roots of an arbitrary polynomial in terms of theta
functions in 1892. In 1895 Emory.McClintock (1840-1916) gives series solutions
for all the roots of a polynomial. Robert Hjalmal.Mellin (1854-1933) solves an
arbitrary polynomial equation with Mellin integrals in 1915. In 1925
R.Birkeland shows that the roots of an algebraic equation can be expressed
using hypergeometric functions in several variables. Hiroshi.Umemura expresses
the roots of an arbitrary polynomial through elliptic Siegel functions
in1984[1].
All of solutions mentioned above are not ones expressed in binary function.
By Tschirnhausen transformation a quintic equation or a sextic equation can be
changed to ones containing only two parameters so there are solutions
expressed in binary function for them. David.Hilbert presumed that there is no
solution expressed in binary function for polynomial equations of n when
n$\geq$7 and wrote his doubt into his famous 23 problems as the 13th one[2].
Hilbert published his last mathematical paper [3] in 1927 where he reported on
the status of his problems, he devoted 5 pages to the 13th problem and only 3
pages to the remaining 22 problems. We can see that so much attention Hilbert
paid to 13th problem. In 1957 V.I.Arnol’d proved that every continuing
function of many variables can be represented as a superposition of functions
of two variables and refuted Hilbert conjecture[4][5]. Furthermore,
A.N.Kolmogorov proved that every continuous function of several variables can
be represented as a superposition of continuous functions of one variable and
the operation of addition [6].
Result for Hilbert’s 13th problem is very important for us and it points us a
quite right direction to solve polynomial equations and general algebraic
equations. But method used in it is topological and the result is not a
constructive one. In this paper we will give a constructive result in discrete
situation. This result is very important. We can construct profuse discrete
algebraic equations and discrete operator equations and for this result we can
give any of them a formula solution. There is never such a mathematical
structure in the history of mathematics. This is the first time! A.G.Vitushkin
dissatisfies the current results about 13th problem and points out that the
algebraic core of the problem remains untouched[7]. We believe we have gotten
the algebraic core Vitushkin wanted.
## 2\. Constructive results for Hilbert’s 13th problem
A.N.Kolmogorov expresses function of several variables as a superposition of
functions of one variable like this:
(2.1) $\displaystyle
W(x_{1},x_{2}\cdots,x_{n})=\sum_{i=1}^{2n+1}f_{i}\Big{[}g_{i1}(x_{1})+g_{i2}(x_{2})+\cdots+g_{in}(x_{n})\Big{]}$
This is a existence result but it’s easy to give a constructive result for
discrete functions.
Definition 2.1 Let A={-1,0,1}, a three numbers function of M variables is
defined as:
g:$A^{M}$$\longrightarrow$A
There are 9 discrete points for a binary three numbers function. A binary
three numbers function can be indicated by a table with 4x4 elements. Its
first column indicates the first variable and the first row indicates the
second variable. To give a table with 4x4 elements is to define a binary three
numbers function and vice versa. For example:
| -1 | 0 | 1
---|---|---|---
-1 | 1 | -1 | 0
0 | -1 | 0 | 1
1 | 0 | 1 | -1
| -1 | 0 | 1
---|---|---|---
-1 | 0 | 0 | 0
0 | 0 | 0 | 0
1 | 0 | 0 | 0
| -1 | 0 | 1
---|---|---|---
-1 | 1 | 1 | 1
0 | 0 | 0 | 0
1 | -1 | -1 | -1
There are three functions in above tables. The first one is linear binary
three numbers function and the second one is identity function with o value
and value of the third one is not change with the second variable. There is
only one value for each discrete point in these three functions and they are
called single-valued binary three numbers function. It’s easy to know there
are $3^{9}$=19683 single-valued binary three numbers function. Can it be two-
valued or three-valued in a discrete point?certainly! There are three
combinations -1,0 and -1,1 and 0,1 for two-valued and only one
combination-1,0,1 for three-valued and numbers will be partitioned by symbol
’*’ if it’s a multi-valued. Can it be no-valued in a discrete point? Yes! We
will indicate it in ’N’. A binary three numbers function can be no-valued in
all 9 discrete points in uttermost like:
| -1 | 0 | 1
---|---|---|---
-1 | N | N | N
0 | N | N | N
1 | N | N | N
There is single-valued,two-valued,three-valued and no valued point in below
three numbers function.
| -1 | 0 | 1
---|---|---|---
-1 | -1 | 0 | 1
0 | -1*0 | -1*1 | 0*1
1 | N | -1*0*1 | N
It’s easy to know there are $8^{9}$ binary three numbers functions. A unary
three numbers function can be indicated by three value numbers partitioned by
symbol ’,’ in bracket and numbers will be partitioned by the symbol ’*’if it’s
many-valued, for example:(-1*0,N,-1*0*1).
The expression $H(x_{1},x_{2})=f[g_{11}(x_{1})+g_{12}(x_{2})]$ contains
$x_{1}$,$x_{2}$, but we intend to take H as a independent object not
containing $x_{1}$,$x_{2}$. We can’t express H in $f[g_{11}+g_{12}]$ because
we will get a unary function $[g_{11}+g_{12}]$ by adding $g_{11}$ and
$g_{12}$. $f[g_{11}+g_{12}]$ is also a unary function and is never equal to
binary the function H.
We express H by only f, $g_{11}$, $g_{12}$ without $x_{1},x_{2}$like this:
$H=f[g_{11}\widetilde{\alpha_{1}}+g_{12}\widetilde{\alpha_{2}}]$
To define a function is to give a rule to get it’s values. For such an
expression we are very clear the rule about getting values of the function if
we replace $\widetilde{\alpha_{1}}$ or $\widetilde{\alpha_{2}}$by
$x_{1}orx_{2}$ ,respectively. That is enough.
Binary three numbers function is called single term binary three numbers
function if it can be represented as
$H=f[g_{11}\widetilde{\alpha_{1}}+g_{12}\widetilde{\alpha_{2}}]$ in which f
,$g_{ij}$ is unary three numbers function and it will be called L term binary
three numbers function if it can be expressed as $\sum
f_{i}[g_{i1}\widetilde{\alpha_{1}}+g_{i2}\widetilde{\alpha_{2}}](i=1,L)$.
Expressing a function of many variables as this form is also called
representing it as a superposition of functions of one variable or decomposing
it to functions of one variable.
For example F is a single term binary three numbers function:
$F=(0,0,1)\Big{[}(0,0,1)\widetilde{\alpha_{1}}+(-1,-1,0)\widetilde{\alpha_{2}}\Big{]}$
Theorem 2.1 Every binary three numbers function can be represented as a
superposition of three numbers functions of one variable.
A binary three numbers function is called singular binary three numbers
function if it’s zero in all discrete points but except one. It’s called
standard singular three numbers binary function if non-zero point is in lower-
right location. Definitions for singular three numbers function of three
variables and for standard singular three numbers function of three variables
are similar.
First we prove that the standard singular three numbers binary function is a
single term one. It’s clear the standard singular binary three numbers
function is F above. (0,0,1) in(0,0,1)$\widetilde{\alpha_{1}}$ in the
expression of F is called raw function. Raw of none-zero point will change if
we adjust the location of ‘1’ in (0,0,1). (-1,-1,0) of(-1,-1,0)
$\widetilde{\alpha_{2}}$ in it is called column fuction.Column of none-zero
point will change if we adjust the location of ‘0’in (-1,-1,0). The first
(0,0,1) in it is called value function. Value which may be single-valued or
multi-valued or no-valued of non-zero point will change if we modify ‘1’ in
(0,0,1). Thus we know that every singular binary three numbers function can be
represented as a superposition of three numbers functions of one variable.
Because every binary three numbers function can be transformed to sum of 9
singular binary three numbers functions then we get our theorem.
So every binary three numbers function can be represented as:
(2.2)
$\displaystyle\Psi_{2}=\sum_{i=1}^{L}f_{i}[g_{i1}\widetilde{\alpha_{1}}+g_{i2}\widetilde{\alpha_{2}}]$
Here L is not greater than 9. Thus we can express and can construct a binary
three numbers function by unary three numbers functions.
We can extend all these result to N numbers function of several variables.In
the decomposition of standard singular binary three numbers function if we
replace raw function (0,0,1) by (0,0,$\cdots$0,1),column function (-1,-1,0) by
(-1,-1,$\cdots$-1,0) and value function (0,0,1)by(0,0,$\cdots$0,1)
,respectively.Then we can extend this expression to N numbers functions of two
variables. Situation for N numbers functions of M variables is similar. So we
get conclusion below.
If N$\geq$M+1 a general N numbers function of M variables can be decomposed
as:
(2.3)
$\displaystyle\psi=\sum_{i=1}^{L}fi\sum_{j=1}^{M}g_{ij}\widetilde{\alpha_{i}}$
If N$<$M+1, the number of unary function in expression of singular discrete
function will be bigger than M+1. For example a standard singular three
numbers function of three variables 3 can be represented as:
(2.4)
$\displaystyle\Psi_{3}=(0,0,1)\\{(0,0,1)[(0,0,1)\widetilde{\alpha_{1}}+(-1,-1,0)\widetilde{\alpha_{2}}]+(-1,-1,0)\widetilde{\alpha_{3}}\\}$
Here are more location functions (-1,-1,0) and (0,0,1) than one of the
standard singular binary three numbers function. Expressions for singular
three numbers function of three variables and for general three numbers
function of three variables are similar to ones of binary three numbers
functions.
All conclusions here are not suit to two numbers function. So we have:
Theorem 2.2 Every N numbers (N$\geq$3) function of M variables can be
represented as a superposition of N numbers functions of one variable.
Note we not only prove the existence of representation by superposition of
functions of one variable and give a constructive procedure. We just only gave
the method to decomposing a function but expression is not the shortest one.
Decomposition with terms being equal to its discrete points is called a
trivial decomposition. Actual terms are more less. Decomposition with less
terms than trivial decomposition is called non-trivial decomposition. It’s an
important topic to study non-trivial decompositions and will not be stated
here.
## 3\. Equations constructed by three numbers functions
There are $3^{9}$ single-valued binary three numbers function and $8^{9}$ ones
if they contain many-valued or no-valued ones. How many equations can we
construct with these functions? So many! How many things need to study about
group of the order 3 ? Too poor! So we know there are ample mineral resources
in this task.
Theorem 3.1 Every algebraic equation constructed by three numbers functions of
two variables can be represented as a superposition of three numbers functions
of one variable.
It’s simple to improve it. Solution of any equation is always function of
several variables. By substituting -1,0,1 to the equation respectively we can
get this function easily because field of definition of it is only three
numbers -1,0,1. We can get the solution expressed by function of one variable
by decomposing this function. That is wonderful that we can construct
equations and solve them freely in a mathematics system! In this paper we just
only solve the equation though there are multitudinous equations:
$(x\psi_{1}a)\psi_{3}(x\psi_{2}b)=c$
Here and in this paper we do’nt write functions of two variables in prefix
form like $\psi_{3}[\psi_{1}(x,a),\psi_{2}(x,b)]]=c$ for clearness.This
equation is called two branches equation. $\psi_{i}$ is parameterized function
and can be any one of $8^{9}$ three numbers functions of two variables. So
actually we solve not one equation but a kind of equation and the method
possesses universality.
Assume function $\psi_{1}$,$\psi_{2}$ and $\psi_{3}$ in the two branches
equation is $\Omega_{1}$,$\Omega_{2}$ and $\Omega_{3}$ ,respectively:
| -1 | 0 | 1
---|---|---|---
-1 | -1 | 1 | 0
0 | 0 | -1 | 1
1 | 1 | 0 | -1
| -1 | 0 | 1
---|---|---|---
-1 | 0 | -1 | 1
0 | -1 | 0 | -1
1 | 1 | 1 | 0
| -1 | 0 | 1
---|---|---|---
-1 | 1 | -1 | 0
0 | 0 | 1 | -1
1 | -1 | 0 | 1
When a=b=c=-1 we get numerical equation
$[x\Omega_{1}(-1)]\Omega_{3}[x\Omega_{2}(-1)]=-1$. We know only -1 is the
solution of this equation by substituting -1,0,1 to it. So we can know that
W(-1,-1,-1)=-1. By the same way we can get other values of W(a,b,c). W(a,b,c)
can be expressed by table below.
c=-1 | -1 | 0 | 1 | c=0 | -1 | 0 | 1 | c=1 | -1 | 0 | 1
---|---|---|---|---|---|---|---|---|---|---|---
-1 | -1 | N | N | | 0 | N | -1*0*1 | | 1 | -1*0*1 | N
0 | 1 | -1*0*1 | N | | -1 | N | N | | 0 | N | -1*0*1
1 | 0 | N | -1*0*1 | | 1 | -1*0*1 | N | | -1 | N | N
In this table the first column indicates the first function number a and the
first row indicates the second function number b and c indicates the third
function number. Decomposing this function of three variables we get the
solution expressed by a superposition of functions of one variable.
$x=(0,0,-1)\Bigg{\\{}(0,0,1)\Big{[}(1,0,0)a+(0,-1,-1)b\Big{]}+(0,-1,-1)c\Bigg{\\}}$
+$(0,0,1)\Bigg{\\{}(0,0,1)\Big{[}(1,0,0)a+(0,-1,-1)b\Big{]}+(-1,-1,0)c\Bigg{\\}}$
+$(0,0,N)\Bigg{\\{}(0,0,1)\Big{[}(1,0,0)a+(-1,0,-1)b\Big{]}+(0,-1,-1)c\Bigg{\\}}$
+$(0,0,N)\Bigg{\\{}(0,0,1)\Big{[}(1,0,0)a+(-1,0,-1)b\Big{]}+(-1,0,-1)c\Bigg{\\}}$
+$(0,0,$-1*0*1$)\Bigg{\\{}(0,0,1)\Big{[}(1,0,0)a+(-1,0,-1)b\Big{]}+(-1,-1,0)c\Bigg{\\}}$
+$(0,0,N)\Bigg{\\{}(0,0,1)\Big{[}(1,0,0)a+(-1,-1,0)b\Big{]}+(0,-1,-1)c\Bigg{\\}}$
+$(0,0,$-1*0*1$)\Bigg{\\{}(0,0,1)\Big{[}(1,0,0)a+(-1,-1,0)b\Big{]}+(-1,0,-1)c\Bigg{\\}}$
+$(0,0,N)\Bigg{\\{}(0,0,1)\Big{[}(1,0,0)a+(-1,-1,0)b\Big{]}+(-1,-1,0)c\Bigg{\\}}$
+$(0,0,1)\Bigg{\\{}(0,0,1)\Big{[}(0,1,0)a+(0,-1,-1)b\Big{]}+(0,-1,-1)c\Bigg{\\}}$
+$(0,0,-1)\Big{]}\Bigg{\\{}(0,0,1)\Big{[}(0,1,0)a+(0,-1,-1)b\Big{]}+(-1,0,-1)c\Bigg{\\}}$
+$(0,0,$-1*0*1$)\Bigg{\\{}(0,0,1)\Big{[}(0,1,0)a+(-1,0,-1)b\Big{]}+(0,-1,-1)c\Bigg{\\}}$
+$(0,0,N)\Bigg{\\{}(0,0,1)\Big{[}(0,1,0)a+(-1,0,-1)b\Big{]}+(-1,0,-1)c\Bigg{\\}}$
+$(0,0,N)\Big{]}\Bigg{\\{}(0,0,1)\Big{[}(0,1,0)a+(-1,0,-1)b\Big{]}+(-1,-1,0)c\Bigg{\\}}$
+$(0,0,N)\Bigg{\\{}(0,0,1)\Big{[}(0,1,0)a+(-1,-1,0)b\Big{]}+(0,-1,-1)c\Bigg{\\}}$
+$(0,0,N)\Bigg{\\{}(0,0,1)\Big{[}(0,1,0)a+(-1,-1,0)b\Big{]}+(-1,0,-1)c\Bigg{\\}}$
+$(0,0,$-1*0*1$)\Bigg{\\{}(0,0,1)\Big{[}(0,1,0)a+(-1,-1,0)b\Big{]}+(-1,-1,0)c\Bigg{\\}}$
+$(0,0,1)\Bigg{\\{}(0,0,1)\Big{[}(0,0,1)a+(0,-1,-1)b\Big{]}+(-1,0,-1)c\Bigg{\\}}$
+$(0,0,-1)\Bigg{\\{}(0,0,1)\Big{[}(0,0,1)a+(0,-1,-1)b\Big{]}+(-1,-1,0)c\Bigg{\\}}$
+$(0,0,N)\Bigg{\\{}(0,0,1)\Big{[}(0,0,1)a+(-1,0,-1)b\Big{]}+(0,-1,-1)c\Bigg{\\}}$
+$(0,0,$-1*0*1$)\Bigg{\\{}(0,0,1)\Big{[}(0,0,1)a+(-1,0,-1)b\Big{]}+(-1,0,-1)c\Bigg{\\}}$
+$(0,0,N)\Bigg{\\{}(0,0,1)\Big{[}(0,0,1)a+(-1,0,-1)b\Big{]}+(-1,-1,0)c\Bigg{\\}}$
+$(0,0,$-1*0*1$)\Bigg{\\{}(0,0,1)\Big{[}(0,0,1)a+(-1,-1,0)b\Big{]}+(0,-1,-1)c\Bigg{\\}}$
+$(0,0,N)\Bigg{\\{}(0,0,1)\Big{[}(0,0,1)a+(-1,-1,0)b\Big{]}+(-1,0,-1)c\Bigg{\\}}$
+$(0,0,N)\Big{]}\Bigg{\\{}(0,0,1)\Big{[}(0,0,1)a+(-1,-1,0)b\Big{]}+(-1,-1,0)c\Bigg{\\}}$
There are 24 but not 27 terms because there are three discrete point with 0
value.
## 4\. Four special operators
How to get a new function from a known one? Operator is correspondence between
functions. To give a correspondence between known functions and new functions
is to give an operator. Four special operators mentioned here are easy to be
understood intuitively and are important to solve equations with parameterized
functions however so we must pay attention to them.
Definition 4.1Commutation operators. Assume there is an function of two
variables $\psi$, $a_{1}\psi a_{2}=a_{0}$,its commutation functions
$\psi$(1,2,0),$\psi$(1,0,2),$\psi$(0,2,1),$\psi$(
2,1,0),$\psi$(2,0,1),$\psi$(0,1,2) will be defined by following formulas and
we introduce commutation operators of one variable C[1,2,0], C[1,0,2],
C[0,2,1], C[2,1,0], C[2,0,1], C[0,1,2] then new functions can be expressed by
$\psi$ and commutation operators.
(4.1a) $\displaystyle
a_{1}\psi[1,2,0]a_{2}=a_{0}\qquad\qquad\psi(1,2,0)=C(1,2,0)(\psi)$ (4.1b)
$\displaystyle
a_{1}\psi[1,0,2]a_{0}=a_{2}\qquad\qquad\psi(1,0,2)=C(1,0,2)(\psi)$ (4.1c)
$\displaystyle
a_{0}\psi[0,2,1]a_{2}=a_{1}\qquad\qquad\psi(0,2,1)=C(0,2,1)(\psi)$ (4.1d)
$\displaystyle
a_{2}\psi[2,1,0]a_{1}=a_{0}\qquad\qquad\psi(2,1,0)=C(2,1,0)(\psi)$ (4.1e)
$\displaystyle
a_{2}\psi[2,0,1]a_{0}=a_{1}\qquad\qquad\psi(2,0,1)=C(2,0,1)(\psi)$ (4.1f)
$\displaystyle
a_{0}\psi[0,1,2]a_{1}=a_{2}\qquad\qquad\psi(0,1,2)=C(0,1,2)(\psi)$
Note $\psi$(1,2,0)is $\psi$ itself.
Numbers in brackets indicates new locations of function numbers and of
function result after commutating. That is say original function doesn’t
satisfy the new relation gotten by commuting location of function numbers and
of function result but new one satisfies it. New relation with new function
and new location is equivalent to original one in despite of their forms are
different. For example: if $\Omega$ is the first table below then
$C(1,0,2)(\Omega)$, $C(0,2,1)(\Omega)$ , $C(2,1,0)(\Omega)$,
$C(2,0,1)(\Omega)$, $C(0,1,2)(\Omega)$will be other tables ,respectively.
| -1 | 0 | 1
---|---|---|---
-1 | -1 | 1 | 0
0 | 0 | -1 | 1
1 | 1 | 0 | -1
| -1 | 0 | 1
---|---|---|---
-1 | -1 | 1 | 0
0 | 0 | -1 | 1
1 | 1 | 0 | -1
| -1 | 0 | 1
---|---|---|---
-1 | -1 | 0 | 1
0 | 0 | 1 | -1
1 | 1 | -1 | 0
| -1 | 0 | 1
---|---|---|---
-1 | -1 | 0 | 1
0 | 1 | -1 | 0
1 | 0 | 1 | -1
| -1 | 0 | 1
---|---|---|---
-1 | -1 | 0 | 1
0 | 0 | 1 | -1
1 | 1 | -1 | 0
| -1 | 0 | 1
---|---|---|---
-1 | -1 | 0 | 1
0 | 1 | -1 | 0
1 | 0 | 1 | -1
We can get any combination of function numbers and of function result for
$C(1,0,2)(\Omega)$ by commuting the second function number and function result
for $C(1,2,0)(\Omega)$. Situations for other commutation functions are similar
to it. We don’t limit function at all when we do commutation operator. May be
we get a many-valued function by a not monotonic function or get an function
with no values in some discrete points by a not surjective function. The same
situation may be exists in other three special operators. We have shown our
opinion above. An mathematics system is extensive and open if it involves
solving equation so it’s impossible to limit functions in it. I have ever
tried to limit function in ones of single-valued but failed because function
of many-valued or of no-valued can be introduced from function of single-
valued by special operators. This problem had troubled me for a long time
until I read materials about extension of group. I known functions of many-
valued or of no-valued are not difficult to be accepted by mathematicians.
Commutation operator for binary functions can be extended to function of many
variables. Showing all commutation functions is integrity in logical and not
all of them will be used in solving equations. There are only two commutation
functions for a unary function:
(4.2a) $\beta_{e}(a)=a_{0}\qquad\qquad\qquad\beta_{e}=\beta$ (4.2b)
$\beta_{t}(a_{0})=a\qquad\qquad\qquad\beta_{t}=C(\beta)$
Definition 4.2Tension-compression operator.Assume there is a binary function
$\psi$ and an unary function $\beta$, $\beta(a_{1})\psi a_{2}=a_{0}$, we can
introduce a new binary function $\psi_{1}$ by $\psi$ and $\beta$, $\psi_{1}$
will meet the relation: $a_{1}\psi_{1}a_{2}=a_{0}$, that is say,
$a_{1}\psi_{1}a_{2}=\beta(a_{1})\psi a_{2}$. Introduce a special operator
$T_{1}$ to express the relation between $\psi_{1}$and $\psi$,$\beta$ .
(4.3a) $\psi_{1}=\psi T_{1}\beta$ In the same way if
$a_{1}\psi\beta(a_{2})=a_{0}$, we can introduce a new binary function
$\psi_{2}$ by $\psi$ and $\beta$, $\psi_{2}$ will meet the relation:
$a_{1}\psi_{2}a_{2}=a_{0}$, that is say,
$a_{1}\psi_{2}a_{2}=a_{1}\psi\beta(a_{2})$. Introduce a special operator
$T_{2}$ to express the relation between $\psi_{2}$and $\psi$,$\beta$. (4.3b)
$\psi_{2}=\psi T_{2}\beta$ If $a_{1}\psi a_{2}=\beta(a_{0})$,that is say
$\beta^{-1}[a_{1}\psi a_{2}]=a_{0}$,we can introduce a new binary function
$\psi_{0}$ by $\psi$ and $\beta$, $\psi_{0}$ will meet the relation:
$a_{1}\psi_{0}a_{2}=a_{0}$,that is say
$a_{1}\psi_{0}a_{2}=\beta^{-1}[a_{1}\psi a_{2}]$, and there is $T_{0}$: (4.3c)
$\psi_{0}=\psi T_{0}\beta$
For example, (1,-1,0) is an function of one variable and written in $\gamma$
then $\Omega$$T_{1}$$\gamma$ and $\Omega$$T_{2}$$\gamma$ and
$\Omega$$T_{0}$$\gamma$ will be
| -1 | 0 | 1
---|---|---|---
-1 | 1 | 0 | -1
0 | -1 | 1 | 0
1 | 0 | -1 | 1
| -1 | 0 | 1
---|---|---|---
-1 | 0 | -1 | 1
0 | 1 | 0 | -1
1 | -1 | 1 | 0
| -1 | 0 | 1
---|---|---|---
-1 | 0 | -1 | 1
0 | 1 | 0 | -1
1 | -1 | 1 | 0
respectively.
It’s occasional that $\Omega$$T_{2}$$\gamma$ is equal to
$\Omega$$T_{0}$$\gamma$. Only $T_{0}$ will be used in solving equation.
For an unary function we have only T and $T_{0}$:
(4.4a) $\beta_{1}T\beta_{2}=\beta_{1}\beta_{2}$ (4.4b)
$\beta_{1}T_{0}\beta_{2}=\beta_{2}^{-1}\beta_{1}$
Note,$\beta_{1}$$\beta_{2}$ means applying$\beta_{2}$ first and then
applying$\beta_{1}$.That is say
(4.5) $\\\ \beta_{1}\beta_{2}(x)=\beta_{1}\Big{[}\beta_{2}(x)\Big{]}$
A discrete point for $\beta_{1}$$\beta_{2}$ will be no-valued if it for any of
$\beta_{1}$ or $\beta_{2}$ is no-valued. $\beta_{1}$ and $\beta_{2}$ will be
each other inverse function if$\beta_{1}$$\beta_{2}$ =e . There are $8^{3}$
three numbers functions of one variable in which there is always inverse
function for any three numbers function of one variable.
This rule is right for many-valued functions of two variables because tension-
compression operators for functions of two variables involves actually only
composition of two functions of one variable.
Definition 4.3Superposition operator. Assume there are P functions of many
variables$\psi_{k}$(k=1,p),their superposition function $\psi$ will be:
(4.6) $\psi=\sum_{k=1}^{P}\psi_{k}$
Value of $\psi$ will be the sum of value of $\psi_{k}$(k=1,p). This is a kind
of operator by it we can get a new function by several known functions with
same variables. $\psi$ will be no-valued in a point if any of $\psi_{k}$ is
no-valued in this point. $\psi_{1}+\psi_{2}$ will be many-valued in a point if
$\psi_{1}$ is single-valued and $\psi_{2}$ is many-valued in this point.
Definition 4.4Decomposition operator.
(4.7)
$\psi_{3}=\sum_{i=1}^{27}f_{i}\Bigg{\\{}g_{i4}\Big{[}g_{i1}(\widetilde{\alpha_{1}})+g_{i2}(\widetilde{\alpha_{2}})\Big{]}+g_{i3}(\widetilde{\alpha_{3}})\Bigg{\\}}$
We can express the relations between$f_{i}$ or $g_{ij}$ and $\psi_{3}$with
special operators $V_{3}$ and $P_{ij}$ and actually $g_{ij}$ is not change
with $\psi_{3}$.
(4.8a) $f_{i}=V_{i}(\psi_{3})\qquad\qquad\qquad\qquad(i=1,27)$ (4.8b)
$g_{ij}=P_{ij}(\psi_{3})\qquad\qquad(i=1,27,j=1,4)$
Otherwise there are more than one decomposition for any function of 3
variables but we select only one of them. Correspondence between $\psi_{3}$
and$f_{i}$ , $g_{ij}$ is clear and easy to be gotten. So decomposition
operator is not occult at all.
Please note commutation operator or tension-compression operator or
decomposition operator or superposition operator will be close within all
three numbers functions if they contain ones being many-valued and no-valued.
This is very important and is the sufficient reason for existing of many-
valued functions and no-valued functions.
So four special operators are very clear and not perplexed at all.
Definition 4.5False function of M+K variables. We can change an function of M
variables to a false one of (M+K ) variables by adding
$o\widetilde{\alpha_{k}}$ in which o is a zero function and function $\psi$
will not change with K variables.
(4.9)
$\psi=\sum_{i=1}^{L}fi\sum_{j=1}^{M}g_{ij}\widetilde{\alpha_{i}}=\sum_{i=1}^{L}fi\Bigg{\\{}\sum_{j=1}^{M}g_{ij}\widetilde{\alpha_{i}}+\sum_{k=M+1}^{M+K}o\widetilde{\alpha_{k}}\Bigg{\\}}$
We can also get false function of (M+K ) variables from one of M variables by
$T_{k}o$ (k=M,M+K). For example:
c=-1 | -1 | 0 | 1 | c=0 | -1 | 0 | 1 | c=1 | -1 | 0 | 1
---|---|---|---|---|---|---|---|---|---|---|---
-1 | 0 | 1 | -1 | | 0,1 | N | -1*0*1 | | 1 | 0 | N
0 | 0 | 1 | -1 | | 0,1 | N | -1*0*1 | | 1 | 0 | N
1 | 0 | 1 | -1 | | 0,1 | N | -1*0*1 | | 1 | 0 | N
This is a false function of three variables and value of it will not change
with the first variable. Below table is a false function of two variables.
| -1 | 0 | 1
---|---|---|---
-1 | 1 | 1 | 1
0 | -1 | -1 | -1
1 | 0 | 0 | 0
False function of many variables will be used in solving equations with
parameterized functions.
## 5\. Formula solution for equations with parameterized functions
What’s an analytic solution or formula solution for an equation?Formula
solution can only contain known parameters or constants and known
parameterized functions or known numerical functions and four kinds of special
operators and we call them valid symbols and all others invalid ones. This is
the standard to verify a formula solution of an equation. Commutation
operators and tension-compression operators and superposition operators and
decomposition operators are the sufficient condition but not the necessary
condition to give formula solutions of equations. There may be another
equivalence set of operators that can express formula solutions of equations.
We will solve two branches equation as a example below. At the same time we
will solve an equation with digital functions below then we can understand the
procedure more clearly. We must believe that it’s not complex to solve this
equation because we have known already the solution exists surely and only
four special operators will be deal with to get it. We will take any new
function met in procedure of solving the equation as a normal one and will
never be puzzled by its appearance.
Step 1: Decomposing function $\psi_{3}$ as:
$\psi_{3}=\sum_{i=1}^{9}f_{i}(g_{i1}\widetilde{\alpha_{1}}+g_{i2}\widetilde{\alpha_{2}})$
$\sum_{i=1}^{9}f_{i}\Big{[}g_{i1}(x\psi_{1}a)+g_{i2}(x\psi_{2}b)\Big{]}=c$
$\Omega_{3}=~{}(0,0,1)\Big{[}(1,0,0)\widetilde{\alpha_{1}}+(0,-1,-1)\widetilde{\alpha_{2}}\Big{]}+(0,0,-1)\Big{[}(1,0,0)\widetilde{\alpha_{1}}+(-1,0,-1)\widetilde{\alpha_{2}}\Big{]}$
$+(0,01)\Big{[}(0,1,0)\widetilde{\alpha_{1}}+(-1,0,-1)\widetilde{\alpha_{2}}\Big{]}+(0,0,-1)\Big{[}(0,1,0)\widetilde{\alpha_{1}}+(-1,-1,0)\widetilde{\alpha_{2}}\Big{]}$
$+(0,0,-1)\Big{[}(0,0,1)\widetilde{\alpha_{1}}+(0,-1,-1)\widetilde{\alpha_{2}}\Big{]}+(0,0,1)\Big{[}(0,0,1)\widetilde{\alpha_{1}}+(-1,-1,0)\widetilde{\alpha_{2}}\Big{]}$
$(x\Omega_{1}a)\Omega_{3}(x\Omega_{2}b)=$
$~{}(0,0,1)\Big{[}(1,0,0)(x\Omega_{1}a)+(0,-1,-1)(x\Omega_{2}b)\Big{]}+(0,0,-1)\Big{[}(1,0,0)(x\Omega_{1}a)+(-1,0,-1)(x\Omega_{2}b)\Big{]}$
$+(0,01)\Big{[}(0,1,0)(x\Omega_{1}a)+(-1,0,-1)(x\Omega_{2}b)\Big{]}+(0,0,-1)\Big{[}(0,1,0)(x\Omega_{1}a)+(-1,-1,0)(x\Omega_{2}b)\Big{]}$
$+(0,0,-1)\Big{[}(0,0,1)(x\Omega_{1}a)+(0,-1,-1)(x\Omega_{2}b)\Big{]}+(0,0,1)\Big{[}(0,0,1)(x\Omega_{1}a)+(-1,-1,0)(x\Omega_{2}b)\Big{]}$
=c
Step 2: By tension-compression of$g_{i1},g_{i2}$ we have:
$\sum_{i=1}^{9}f_{i}\Big{[}x(\psi_{1}T_{0}g_{i1}^{-1})a+x(\psi_{2}T_{0}g_{i2}^{-1})b\Big{]}=c$
Note,$(\psi_{1}T_{0}g_{i1}^{-1})$ in $x(\psi_{1}T_{0}g_{i1}^{-1})a$ and
$(\psi_{2}T_{0}g_{i2}^{-1})$ in $x(\psi_{2}T_{0}g_{i2}^{-1})b$ are two
functions of two variables.
$(x\Omega_{1}a)\Omega_{3}(x\Omega_{2}b)=~{}$
$(0,0,1)\Bigg{\\{}x\Big{[}\Omega_{1}T_{0}(1,0,0)^{-1}\Big{]}a+x\Big{[}\Omega_{2}T_{0}(0,-1,-1)^{-1}\Big{]}b\Bigg{\\}}+$
$(0,0,-1)\Bigg{\\{}x\Big{[}\Omega_{1}T_{0}(1,0,0)^{-1}\Big{]}a+x\Big{[}\Omega_{2}T_{0}(-1,0,-1)^{-1}\Big{]}b\Bigg{\\}}+$
$(0,01)\Bigg{\\{}x\Big{[}\Omega_{1}T_{0}(0,1,0)^{-1}\Big{]}a$
$+x\Big{[}\Omega_{2}T_{0}(-1,0,-1)^{-1}\Big{]}b\Bigg{\\}}+$
$(0,0,-1)\Bigg{\\{}x\Big{[}\Omega_{1}T_{0}(0,1,0)^{-1}\Big{]}a+x\Big{[}\Omega_{2}T_{0}(-1,-1,0)^{-1}\Big{]}b\Bigg{\\}}+$
$(0,0,-1)\Bigg{\\{}x\Big{[}\Omega_{1}T_{0}(0,0,1)^{-1}\Big{]}a+x\Big{[}\Omega_{2}T_{0}(0,-1,-1)^{-1}\Big{]}b\Bigg{\\}}+$
$(0,0,1)\Bigg{\\{}x\Big{[}\Omega_{1}T_{0}(0,0,1)^{-1}\Big{]}a+$
$x\Big{[}\Omega_{2}T_{0}(-1,-1,0)^{-1}\Big{]}b\Bigg{\\}}=c$
$\Omega_{1}T_{0}(1,0,0)^{-1}$, $\Omega_{1}T_{0}(0,1,0)^{-1}$,
$\Omega_{1}T_{0}(0,0,1)^{-1}$ is
| -1 | 0 | 1
---|---|---|---
-1 | 1 | 0 | 0
0 | 0 | 1 | 0
1 | 0 | 0 | 1
| -1 | 0 | 1
---|---|---|---
-1 | 0 | 0 | 1
0 | 1 | 0 | 0
1 | 0 | 1 | 0
| -1 | 0 | 1
---|---|---|---
-1 | 0 | 1 | 0
0 | 0 | 0 | 1
1 | 1 | 0 | 0
$\Omega_{2}T_{0}(0,-1,-1)^{-1}$, $\Omega_{2}T_{0}(-1,0,-1)^{-1}$,
$\Omega_{2}T_{0}(-1,-1,0)^{-1}$ is
| -1 | 0 | 1
---|---|---|---
-1 | -1 | 0 | -1
0 | 0 | -1 | 0
1 | -1 | -1 | -1
| -1 | 0 | 1
---|---|---|---
-1 | 0 | -1 | -1
0 | -1 | 0 | -1
1 | -1 | -1 | 0
| -1 | 0 | 1
---|---|---|---
-1 | -1 | -1 | 0
0 | -1 | -1 | -1
1 | 0 | 0 | -1
respectively.
Step 3: Changing $\psi_{1}T_{0}g_{i1}^{-1}$ by $T_{3}o$ to get a false
function of three variables $\psi_{1}T_{3}oT_{0}g_{i1}^{-1}$ in which variable
c is a false one and Changing $\psi_{2}T_{0}g_{i2}^{-1}$ by $T_{2}$o to get a
false function of three variables $\psi_{2}T_{2}oT_{0}g_{i2}^{-1}$ in which
variable b is a false one ,respectively. Adding them to get a real function of
three variables $\psi_{i3}$. This is the application of tension-compression
operator in solving equation.
$\psi_{i3}=\psi_{1}T_{3}oT_{0}g_{i1}^{-1}+\psi_{2}T_{2}oT_{0}g_{i2}^{-1}\qquad(i=1,9)$
$\theta_{1}=\Omega_{1}T_{3}oT_{0}(1,0,0)^{-1}+\Omega_{2}T_{2}oT_{0}(0,-1,-1)^{-1}$
is:
c=-1 | -1 | 0 | 1 | c=0 | -1 | 0 | 1 | c=1 | -1 | 0 | 1
---|---|---|---|---|---|---|---|---|---|---|---
-1 | 0 | -1 | -1 | | 1 | 0 | 0 | | 0 | -1 | -1
0 | 0 | 1 | 0 | | -1 | 0 | -1 | | 0 | 1 | 0
1 | -1 | -1 | 0 | | -1 | -1 | 0 | | -1 | -1 | 0
$\theta_{2}=\Omega_{1}T_{3}oT_{0}(1,0,0)^{-1}+\Omega_{2}T_{2}oT_{0}(-1,0,-1)^{-1}$
is:
c=-1 | -1 | 0 | 1 | c=0 | -1 | 0 | 1 | c=1 | -1 | 0 | 1
---|---|---|---|---|---|---|---|---|---|---|---
-1 | 1 | 0 | 0 | | 0 | -1 | -1 | | 0 | -1 | -1
0 | -1 | 0 | -1 | | 0 | 1 | 0 | | -1 | 0 | -1
1 | -1 | -1 | 0 | | -1 | -1 | 0 | | 0 | 0 | 1
$\theta_{3}=\Omega_{1}T_{3}oT_{0}(0,1,0)^{-1}+\Omega_{2}T_{2}oT_{0}(-1,0,-1)^{-1}$
is:
c=-1 | -1 | 0 | 1 | c=0 | -1 | 0 | 1 | c=1 | -1 | 0 | 1
---|---|---|---|---|---|---|---|---|---|---|---
-1 | 0 | 0 | 1 | | -1 | -1 | 0 | | -1 | -1 | 0
0 | 0 | -1 | -1 | | 1 | 0 | 0 | | 0 | -1 | -1
1 | -1 | 0 | -1 | | -1 | 0 | -1 | | 0 | 1 | 0
$\theta_{4}=\Omega_{1}T_{3}oT_{0}(0,1,0)^{-1}+\Omega_{2}T_{2}oT_{0}(-1,-1,0)^{-1}$
is:
c=-1 | -1 | 0 | 1 | c=0 | -1 | 0 | 1 | c=1 | -1 | 0 | 1
---|---|---|---|---|---|---|---|---|---|---|---
-1 | -1 | -1 | 0 | | -1 | -1 | 0 | | 0 | 0 | 1
0 | 0 | -1 | -1 | | 0 | -1 | -1 | | 0 | -1 | -1
1 | 0 | 1 | 0 | | 0 | 1 | 0 | | -1 | 0 | -1
$\theta_{5}=\Omega_{1}T_{3}oT_{0}(0,0,1)^{-1}+\Omega_{2}T_{2}oT_{0}(0,-1,-1)^{-1}$
is:
c=-1 | -1 | 0 | 1 | c=0 | -1 | 0 | 1 | c=1 | -1 | 0 | 1
---|---|---|---|---|---|---|---|---|---|---|---
-1 | -1 | 0 | -1 | | 0 | 1 | 0 | | -1 | 0 | -1
0 | 0 | 0 | 1 | | -1 | -1 | 0 | | 0 | 0 | 1
1 | 0 | -1 | -1 | | 0 | -1 | -1 | | 0 | -1 | -1
$\theta_{6}=\Omega_{1}T_{3}oT_{0}(0,0,1)^{-1}+\Omega_{2}T_{2}oT_{0}(-1,-1,0)^{-1}$
is:
c=-1 | -1 | 0 | 1 | c=0 | -1 | 0 | 1 | c=1 | -1 | 0 | 1
---|---|---|---|---|---|---|---|---|---|---|---
-1 | -1 | 0 | -1 | | -1 | 0 | -1 | | 0 | 1 | 0
0 | -1 | -1 | 0 | | -1 | -1 | 0 | | -1 | -1 | 0
1 | 1 | 0 | 0 | | 1 | 0 | 0 | | 0 | -1 | -1
Step 4: Changing $\psi_{i3}$ by $T_{0}f_{i}^{-1}$we get:
$\psi_{i4}=\psi_{i3}T_{0}f_{i}^{-1}\qquad(i=1,9)$
$\theta_{1}T_{0}(0,0,1)^{-1}$ is:
c=-1 | -1 | 0 | 1 | c=0 | -1 | 0 | 1 | c=1 | -1 | 0 | 1
---|---|---|---|---|---|---|---|---|---|---|---
-1 | 0 | 0 | 0 | | 1 | 0 | 0 | | 0 | 0 | 0
0 | 0 | 1 | 0 | | 0 | 0 | 0 | | 0 | 1 | 0
1 | 0 | 0 | 0 | | 0 | 0 | 0 | | 0 | 0 | 0
$\theta_{2}T_{0}(0,0,-1)^{-1}$ is:
c=-1 | -1 | 0 | 1 | c=0 | -1 | 0 | 1 | c=1 | -1 | 0 | 1
---|---|---|---|---|---|---|---|---|---|---|---
-1 | -1 | 0 | 0 | | 0 | 0 | 0 | | 0 | 0 | 0
0 | 0 | 0 | 0 | | 0 | -1 | 0 | | 0 | 0 | 0
1 | 0 | 0 | 0 | | 0 | 0 | 0 | | 0 | 0 | -1
$\theta_{3}T_{0}(0,0,1)^{-1}$ is:
c=-1 | -1 | 0 | 1 | c=0 | -1 | 0 | 1 | c=1 | -1 | 0 | 1
---|---|---|---|---|---|---|---|---|---|---|---
-1 | 0 | 0 | 1 | | 0 | 0 | 0 | | 0 | 0 | 0
0 | 0 | 0 | 0 | | 1 | 0 | 0 | | 0 | 0 | 0
1 | 0 | 0 | 0 | | 0 | 0 | 0 | | 0 | 1 | 0
$\theta_{4}T_{0}(0,0,-1)^{-1}$ is:
c=-1 | -1 | 0 | 1 | c=0 | -1 | 0 | 1 | c=1 | -1 | 0 | 1
---|---|---|---|---|---|---|---|---|---|---|---
-1 | 0 | 0 | 0 | | 0 | 0 | 0 | | 0 | 0 | -1
0 | 0 | 0 | 0 | | 0 | 0 | 0 | | 0 | 0 | 0
1 | 0 | -1 | 0 | | 0 | -1 | 0 | | 0 | 0 | 0
$\theta_{5}T_{0}(0,0,-1)^{-1}$ is:
c=-1 | -1 | 0 | 1 | c=0 | -1 | 0 | 1 | c=1 | -1 | 0 | 1
---|---|---|---|---|---|---|---|---|---|---|---
-1 | 0 | 0 | 0 | | 0 | -1 | 0 | | 0 | 0 | 0
0 | 0 | 0 | -1 | | 0 | 0 | 0 | | 0 | 0 | -1
1 | 0 | 0 | 0 | | 0 | 0 | 0 | | 0 | 0 | 0
$\theta_{6}T_{0}(0,0,1)^{-1}$ is:
c=-1 | -1 | 0 | 1 | c=0 | -1 | 0 | 1 | c=1 | -1 | 0 | 1
---|---|---|---|---|---|---|---|---|---|---|---
-1 | 0 | 0 | 0 | | 0 | 0 | 0 | | 0 | 1 | 0
0 | 0 | 0 | 0 | | 0 | 0 | 0 | | 0 | 0 | 0
1 | 1 | 0 | 0 | | 1 | 0 | 0 | | 0 | 0 | 0
Step 5: To sum $\psi_{i4}$ we get:
$\psi_{5}=\sum_{i=1}^{9}\psi_{i4}$
Original equation will be:
$\psi_{5}(x,a,b)=c$
$\theta_{7}=\theta_{1}T_{0}(0,0,1)^{-1}+\theta_{2}T_{0}(0,0,-1)^{-1}+\theta_{3}T_{0}(0,0,1)^{-1}+\theta_{4}T_{0}(0,0,-1)^{-1}$
+$\theta_{5}T_{0}(0,0,-1)^{-1}+\theta_{6}T_{0}(0,0,1)^{-1}$ is:
c=-1 | -1 | 0 | 1 | c=0 | -1 | 0 | 1 | c=1 | -1 | 0 | 1
---|---|---|---|---|---|---|---|---|---|---|---
-1 | -1 | 0 | 1 | | 1 | -1 | 0 | | 0 | 1 | -1
0 | 0 | 1 | -1 | | 1 | -1 | 0 | | 0 | 1 | -1
1 | 1 | -1 | 0 | | 1 | -1 | 0 | | 0 | 1 | -1
Step6: By commutation operator we get:
$x=\Big{[}C(2,3,0,1)\psi_{5}\Big{]}(a,b,c)=W(a,b,c)$
$C(2,3,0,1)\theta_{7}$is:
c=-1 | -1 | 0 | 1 | c=0 | -1 | 0 | 1 | c=1 | -1 | 0 | 1
---|---|---|---|---|---|---|---|---|---|---|---
-1 | -1 | N | N | | 0 | N | -1*0*1 | | 1 | -1*0*1 | N
0 | 1 | -1*0*1 | N | | -1 | N | N | | 0 | N | -1*0*1
1 | 0 | N | -1*0*1 | | 1 | -1*0*1 | N | | -1 | N | N
It’s not oddball there are many-valued discrete points or no-valued discrete
points. Not all commutation operators are used in solving equation.
Step 7: by decomposition operator we get:
$x=\sum_{k=1}^{27}u_{k}\Bigg{\\{}v_{k4}\Big{[}v_{k1}(a)+v_{k2}(b)\Big{]}+v_{k3}(c)\Bigg{\\}}$
$=\sum_{k=1}^{27}(V_{k}W)\Bigg{\\{}(P_{k4}W)\Big{[}(P_{k1}W)(a)+(P_{k2}W)(b)\Big{]}+(P_{k3}W)(c)\Bigg{\\}}$
we replace logogram symbols by complete ones.
$x=\sum_{k=1}^{27}V_{k}\Big{[}C(2,3,0,1)\Big{(}\sum_{i=1}^{9}\big{\\{}\big{[}\psi_{1}T_{3}oT_{0}(P_{i1}\psi_{3})^{-1}+\psi_{2}T_{2}oT_{0}(P_{i2}\psi_{3})^{-1}\big{]}T_{0}(V_{i}\psi_{3})^{-1}\big{\\}}\Big{)}\Big{]}$
$\Bigg{[}P_{k4}\Big{[}C(2,3,0,1)\Big{(}\sum_{i=1}^{9}\big{\\{}\big{[}\psi_{1}T_{3}oT_{0}(P_{i1}\psi_{3})^{-1}+\psi_{2}T_{2}oT_{0}(P_{i2}\psi_{3})^{-1}\big{]}T_{0}(V_{i}\psi_{3})^{-1}\big{\\}}\Big{)}\Big{]}$
$\Bigg{(}P_{k1}\Big{[}C(2,3,0,1)\Big{(}\sum_{i=1}^{9}\big{\\{}\big{[}\psi_{1}T_{3}oT_{0}(P_{i1}\psi_{3})^{-1}+\psi_{2}T_{2}oT_{0}(P_{i2}\psi_{3})^{-1}\big{]}T_{0}(V_{i}\psi_{3})^{-1}\big{\\}}\Big{)}\Big{]}(a)$
$+P_{k2}\Big{[}C(2,3,0,1)\Big{(}\sum_{i=1}^{9}\big{\\{}\big{[}\psi_{1}T_{3}oT_{0}(P_{i1}\psi_{3})^{-1}+\psi_{2}T_{2}oT_{0}(P_{i2}\psi_{3})^{-1}\big{]}T_{0}(V_{i}\psi_{3})^{-1}\big{\\}}\Big{)}\Big{]}(b)\Bigg{)}$
$+P_{k3}\Big{[}C(2,3,0,1)\Big{(}\sum_{i=1}^{9}\big{\\{}\big{[}\psi_{1}T_{3}oT_{0}(P_{i1}\psi_{3})^{-1}+\psi_{2}T_{2}oT_{0}(P_{i2}\psi_{3})^{-1}\big{]}T_{0}(V_{i}\psi_{3})^{-1}\big{\\}}\Big{)}\Big{]}(c)\Bigg{]}$
Actually location functions $P_{ij}\psi_{k}$ do not change with $\psi_{k}$ and
can be called constant functions. Solution of equation with function
$\Omega_{1}$,$\Omega_{2}$ and $\Omega_{3}$ has been given already above by
getting a function of many variables . Giving the procedure of it is just only
make the method clearer. We deal with the function of three variables in
solving this equation. Can we avoid to use it in the procedure? Never! In
history one reason to introduce complex number is that we have to deal with
complex number even if three roots of a cubic equation are all real number.
It’s the most important that we have gotten the solution expressed by function
of one variable however.
## 6\. Composition of special operators
There are 10 compositions for commutation operators and tension-compression
operators and superposition operators and decomposition operators as bellow
tables:
table1
commutation tension-compression superposition decomposition commutation 1
tension-compression 2 3 superposition 4 5 6 decomposition 7 8 9 10
We will mention them below. Here we give only results of binary function and
they can be extended to functions of many variables easily.
Composition 1 commutation and commutation:
see table 2
Composition 2 tension-compression and commutation:
see table 3
Composition 3 tension-compression and tension-compression:
see table4
Composition 4 superposition and commutation:
is equal to commutation and superposition for commutation C(2,1,0):
(6.1)
$C(2,1,0)(\sum_{k=1}^{H}\psi_{k})=\sum_{k=1}^{H}\Big{[}C(2,1,0)(\psi_{k})\Big{]}\qquad$
It will be complex for commutation C(0,2,1) and commutation C(1,0,2).
Composition 5 superposition and tension-compression:
is equal to tension-compression and superposition for tension-compression
$T_{1}$ and $T_{2}$:
(6.2a)
$(\sum_{k=1}^{H}\psi_{k})T_{1}\beta=\sum_{k=1}^{H}(\psi_{k}T_{1}\beta)\qquad\qquad\qquad$
(6.2b)
$(\sum_{k=1}^{H}\psi_{k})T_{2}\beta=\sum_{k=1}^{H}(\psi_{k}T_{2}\beta)\qquad\qquad\qquad$
is complex for tension-compression $T_{0}$.
Composition 6 superposition and superposition:
It is very simple.
Composition 7 decomposition - commutation:
Value functions will hold the line and location functions will exchange for
commutation C(2,1,0)
(6.3a) $V_{i}\Big{[}C(2,1,0)(\psi)\Big{]}=V_{i}(\psi)\qquad(i=1,L)$ (6.3b)
$P_{i1}\Big{[}C(2,1,0)(\psi)\Big{]}=P_{i2}(\psi)\qquad(i=1,L)$ (6.3c)
$P_{i2}\Big{[}C(2,1,0)(\psi)\Big{]}=P_{i1}(\psi)\qquad(i=1,L)$
is complex for commutation C(0,2,1) and commutation C(1,0,2).
Composition 8 decomposition and tension-compression:
Value functions will hold the line and location functions will be acted by
$T_{1}\beta$ or $T_{2}\beta$ for tension-compression $T_{1}$ and $T_{2}$.
(6.4a) $V_{i}(\psi T_{j}\beta)=V_{i}\psi\qquad(i=1,L\qquad j=1,2)$ (6.4b)
$P_{ij}(\psi T_{j}\beta)=(P_{ij}\psi)T_{j}\beta\qquad(i=1,L\qquad j=1,2)$
is complex for $T_{0}$. But there is relation between value functions of
$\psi$ and of $\psi$ acted by $T_{0}$ if it’s a trivial decomposition.
(6.5) $V_{i}(\psi T_{0}\beta)=(V_{i}\psi)T_{0}\beta\qquad(i=1,L)$
This relation is very important.
Composition 9 decomposition and superposition:
Value functions will be composition of value functions and location functions
will be any location functions.
(6.6a)
$V_{i}(\sum_{k=1}^{H}\psi_{k})=\sum_{k=1}^{H}V_{i}(\psi_{k})\qquad(i=1,L)$
(6.6b) $P_{ij}(\sum_{k=1}^{H}\psi_{i})=P_{ij}(\psi_{k})\qquad(i=1,L\qquad
j=1,2)$
Composition 10 decomposition and decomposition:
None.
Law and composition of special operators can be extended to high degree
operators in form.
Table2 commutation and commutation
| C(1,2,0) | C(1,0,2) | C(0,2,1) | C(2,1,0) | C(2,0,1) | C(0,1,2)
---|---|---|---|---|---|---
C(1,2,0) | C(1,2,0) | C(1,0,2) | C(0,2,1) | C(2,1,0) | C(2,0,1) | C(0,1,2)
C(1,0,2) | C(1,0,2) | C(1,2,0) | C(2,0,1) | C(0,1,2) | C(0,2,1) | C(2,1,0)
C(0,2,1) | C(0,2,1) | C(0,1,2) | C(1,2,0) | C(2,0,1) | C(2,1,0) | C(1,0,2)
C(2,1,0) | C(2,1,0) | C(2,0,1) | C(0,1,2) | C(1,2,0) | C(1,0,2) | C(0,2,1)
C(2,0,1) | C(2,0,1) | C(2,1,0) | C(1,0,2) | C(0,2,1) | C(0,1,2) | C(1,2,0)
C(0,1,2) | C(0,1,2) | C(0,2,1) | C(2,1,0) | C(1,0,2) | C(1,2,0) | C(2,0,1)
Table3 tension-compression and commutation
| C(1,2,0) | C(1,0,2) | C(0,2,1) | C(2,1,0) | C(2,0,1) | C(0,1,2)
---|---|---|---|---|---|---
$T_{1}\beta$ | $T_{1}\beta$ | $C(1,0,2)T_{1}\beta$ | $C(0,2,1)T_{0}\beta$ | $C(2,1,0)T_{2}\beta$ | $C(2,0,1)T_{0}\beta$ | $C(0,1,2)T_{2}\beta$
$T_{2}\beta$ | $T_{2}\beta$ | $C(1,0,2)T_{0}\beta$ | $C(0,2,1)T_{2}\beta$ | $C(2,1,0)T_{1}\beta$ | $C(2,0,1)T_{1}\beta$ | $C(0,1,2)T_{0}\beta$
$T_{0}\beta$ | $T_{0}\beta$ | $C(1,0,2)T_{2}\beta$ | $C(0,2,1)T_{1}\beta$ | $C(2,1,0)T_{0}\beta$ | $C(2,0,1)T_{2}\beta$ | $C(0,1,2)T_{1}\beta$
table4 tension-compression and tension-compression
| $T_{1}\beta_{2}$ | $T_{2}\beta_{2}$ | $T_{0}\beta_{2}$
---|---|---|---
$T_{1}\beta_{1}$ | $T_{1}(\beta_{1}\beta_{2})$ | $(T_{2}\beta_{2})T_{1}\beta_{1}$ | ($T_{0}\beta_{2})T_{1}\beta_{1}$
$T_{2}\beta_{1}$ | $(T_{1}\beta_{2})T_{2}\beta_{1}$ | $T_{2}(\beta_{1}\beta_{2})$ | $(T_{0}\beta_{2})T_{2}\beta_{1}$
$T_{0}\beta_{1}$ | $(T_{1}\beta_{2})T_{0}\beta_{1}$ | $(T_{2}\beta_{2})T_{0}\beta_{1}$ | $T_{0}(\beta_{1}\beta_{2})$
All of them are easy to be validated by readers.
## 7\. Extend results to discrete operators
Now we extend results about discrete functions to discrete operators. We limit
the field of definition and range of operators within three discrete functions
-e=(1,0,-1),o=(0,0,0),e=(-1,0,1) for simplicity.
Definition 7.1Assume there are three numbers functions -e=(1,0,-1),o=(0,0,0)
and e=(-1,0,1)we let A=$\\{$-e,0,e$\\}$ and define three functions operator of
one variable $S_{1}$ as
$S_{1}$:A$\longrightarrow$A
define three functions operator of two variables $S_{2}$ as
$S_{2}$:$A^{2}$$\longrightarrow$A
define three functions operator of three variables $S_{3}$ as
$S_{3}$:$A^{3}$$\longrightarrow$A
There are $3^{3}$ single-valued three functions operators of one variable and
$8^{3}$ ones if they contain many-valued or no-valued.There are $3^{9}$
single-valued three functions operators of two variables and $8^{9}$ ones if
they contain many-valued or no-valued.There are $3^{27}$ single-valued three
functions operators of three variable and $8^{27}$ ones if they contain many-
valued or no-valued.
Functions will be partitioned by the symbol ’*’for many-valued point and no-
valued point will be indicated by ’N’.
’+’operation will be expressed as:
| -1 | 0 | 1
---|---|---|---
-1 | 1 | -1 | 0
0 | -1 | 0 | 1
1 | 0 | 1 | -1
’+’operator will be expressed as:
| -e | o | e
---|---|---|---
-e | e | -e | o
o | -e | o | e
e | o | e | -e
Compare two tables we know -e,o,e in discrete operators system is like -1,0,1
in discrete functions system ,respectively. We can also introduce concepts of
singular three functions operator and standard singular three functions
operator.
A standard singular three functions operator of two variables can be expressed
by table:
| -e | o | e
---|---|---|---
-e | o | o | o
o | o | o | o
e | o | o | e
It can be represented as a superposition of three functions operators of one
variable:
$G=(o,o,e)\Big{[}(o,o,e)\widetilde{\beta_{1}}+(-e,-e,o)\widetilde{\beta_{2}}\Big{]}$
By the same reason for three numbers function we know a standard singular
binary three functions operator can be represented as a superposition of unary
three functions operators and so does a general singular binary three
functions operator. A general binary three functions operator can be expressed
to sum of 9 singular binary three functions operators so we have
Theorem 7.1 Every binary three functions operator can be represented as a
superposition of three functions operators of one variable.
A standard singular three functions operator of three variables $\phi_{3}$ can
be represented as:
$\phi_{3}=(o,o,e)\Bigg{\\{}(o,o,e)\Big{[}(o,o,e)(\widetilde{\beta_{1}})+(-e,-e,o)(\widetilde{\beta_{2}})\Big{]}+(-e,-e,o)(\widetilde{\beta_{3}})\Bigg{\\}}$
Theorem 7.2 Every three functions operator of two or of three variables can be
represented as a superposition of three functions operators of one variable.
All conclusions here are not suit to discrete 2 operator.
There are great number of operator equations constructed by $8^{9}$ operators
of two variables.
Theorem 7.3 Every operator equation constructed by three functions operators
of two variables can be give formula solution represented as a superposition
of three functions operators of one variable.
Although there are many operator equations we give formula solution for only
double branches operator equation with digital operators and with
parameterized operators.
$(y\phi_{1}f)\phi_{3}(y\phi_{2}g)=h$
Assume $\phi_{1}$,$\phi_{2}$,$\phi_{3}$ is $\Theta_{1},\Theta_{2},\Theta_{3}$
as below,respectively:
| -e | o | e
---|---|---|---
-e | -e | e | o
o | o | -e | e
e | e | o | -e
| -e | o | e
---|---|---|---
-e | o | -e | e
o | -e | o | -e
e | e | e | o
| -e | o | e
---|---|---|---
-e | e | -e | o
o | o | e | -e
e | -e | o | e
Solution expressed by superposition of operators of one variable will be:
$y=(o,o,-e)\Bigg{\\{}(o,o,e)\Big{[}(e,o,o)f+(o,-e,-e)g\Big{]}+(o,-e,-e)h\Bigg{\\}}$
+$(o,o,e)\Bigg{\\{}(o,o,e)\Big{[}(e,o,o)f+(o,-e,-e)g\Big{]}+(-e,-e,o)h\Bigg{\\}}$
+$(o,o,N)\Bigg{\\{}(o,o,e)\Big{[}(e,o,o)f+(-e,o,-e)g\Big{]}+(o,-e,-e)h\Bigg{\\}}$
+$(o,o,N)\Bigg{\\{}(o,o,e)\Big{[}(e,o,o)f+(-e,o,-e)g\Big{]}+(-e,o,-e)h\Bigg{\\}}$
+$(o,o,$-e*o*e$)\Bigg{\\{}(o,o,e)\Big{[}(e,o,o)f+(-e,o,-e)g\Big{]}+(-e,-e,o)h\Bigg{\\}}$
+$(o,o,N)\Bigg{\\{}(o,o,e)\Big{[}(e,o,o)f+(-e,-e,o)g\Big{]}+(o,-e,-e)h\Bigg{\\}}$
+$(o,o,$-e*o*e$)\Bigg{\\{}(o,o,e)\Big{[}(e,o,o)f+(-e,-e,o)g\Big{]}+(-e,o,-e)h\Bigg{\\}}$
+$(o,o,N)\Bigg{\\{}(o,o,e)\Big{[}(e,o,o)f+(-e,-e,o)g\Big{]}+(-e,-e,o)h\Bigg{\\}}$
+$(o,o,e)\Bigg{\\{}(o,o,e)\Big{[}(o,e,o)f+(o,-e,-e)g\Big{]}+(o,-e,-e)h\Bigg{\\}}$
+$(o,o,-e)\Big{]}\Bigg{\\{}(o,o,e)\Big{[}(o,e,o)f+(o,-e,-e)g\Big{]}+(-e,o,-e)h\Bigg{\\}}$
+$(o,o,$-e*o*e$)\Bigg{\\{}(o,o,e)\Big{[}(o,e,o)f+(-e,o,-e)g\Big{]}+(o,-e,-e)h\Bigg{\\}}$
+$(o,o,N)\Bigg{\\{}(o,o,e)\Big{[}(o,e,o)f+(-e,o,-e)g\Big{]}+(-e,o,-e)h\Bigg{\\}}$
+$(o,o,N)\Big{]}\Bigg{\\{}(o,o,e)\Big{[}(o,e,o)f+(-e,o,-e)g\Big{]}+(-e,-e,o)h\Bigg{\\}}$
+$(o,o,N)\Bigg{\\{}(o,o,e)\Big{[}(o,e,o)f+(-e,-e,o)g\Big{]}+(o,-e,-e)h\Bigg{\\}}$
+$(o,o,N)\Bigg{\\{}(o,o,e)\Big{[}(o,e,o)f+(-e,-e,o)g\Big{]}+(-e,o,-e)h\Bigg{\\}}$
+$(o,o,$-e*o*e$)\Bigg{\\{}(o,o,e)\Big{[}(o,e,o)f+(-e,-e,o)g\Big{]}+(-e,-e,o)h\Bigg{\\}}$
+$(o,o,e)\Bigg{\\{}(o,o,e)\Big{[}(o,o,e)f+(o,-e,-e)g\Big{]}+(-e,o,-e)h\Bigg{\\}}$
+$(o,o,-e)\Bigg{\\{}(o,o,e)\Big{[}(o,o,e)f+(o,-e,-e)g\Big{]}+(-e,-e,o)h\Bigg{\\}}$
+$(o,o,N)\Bigg{\\{}(o,o,e)\Big{[}(o,o,e)f+(-e,o,-e)g\Big{]}+(o,-e,-e)h\Bigg{\\}}$
+$(o,o,$-e*o*e$)\Bigg{\\{}(o,o,e)\Big{[}(o,o,e)f+(-e,o,-e)g\Big{]}+(-e,o,-e)h\Bigg{\\}}$
+$(o,o,N)\Bigg{\\{}(o,o,e)\Big{[}(o,o,e)f+(-e,o,-e)g\Big{]}+(-e,-e,o)h\Bigg{\\}}$
+$(o,o,$-e*o*e$)\Bigg{\\{}(o,o,e)\Big{[}(o,o,e)f+(-e,-e,o)g\Big{]}+(o,-e,-e)h\Bigg{\\}}$
+$(o,o,N)\Bigg{\\{}(o,o,e)\Big{[}(o,o,e)f+(-e,-e,o)g\Big{]}+(-e,o,-e)h\Bigg{\\}}$
+$(o,o,N)\Big{]}\Bigg{\\{}(o,o,e)\Big{[}(o,o,e)f+(-e,-e,o)g\Big{]}+(-e,-e,o)h\Bigg{\\}}$
Definition 7.2 High Commutation Operators. Assume there is an operator of two
variables $\phi$, $y_{1}\phi y_{2}=y_{0}$, its commutation operators
$\phi$(1,2,0),$\phi$(1,0,2),$\phi$(0,
2,1),$\phi$(2,1,0),$\phi$(2,0,1),$\phi$(0,1,2) will be defined by following
formulas and we introduce high commutation operators $\overline{C}$[1,2,0],
$\overline{C}$[1,0,2], $\overline{C}$[0,2,1], $\overline{C}$[2,1,0],
$\overline{C}$[2,0,1],$\overline{C}$[0,1,2] then new operators can be
expressed by $\phi$ and high commutation operators.
(7.1a)
$y_{1}\phi[1,2,0]y_{2}=y_{0}\qquad\qquad\phi[1,2,0]=\overline{C}[1,2,0](\phi)$
(7.1b)
$y_{1}\phi[1,0,2]y_{0}=y_{2}\qquad\qquad\phi[1,0,2]=\overline{C}[1,0,2](\phi)$
(7.1c)
$y_{0}\phi[0,2,1]y_{2}=y_{1}\qquad\qquad\phi[0,2,1]=\overline{C}[0,2,1](\phi)$
(7.1d)
$y_{2}\phi[2,1,0]y_{1}=y_{0}\qquad\qquad\phi[2,1,0]=\overline{C}[2,1,0](\phi)$
(7.1e)
$y_{2}\phi[2,0,1]y_{0}=y_{1}\qquad\qquad\phi[2,0,1]=\overline{C}[2,0,1](\phi)$
(7.1f)
$y_{0}\phi[0,1,2]y_{1}=y_{2}\qquad\qquad\phi[0,1,2]=\overline{C}[0,1,2](\phi)$
There are only two high commutation functions for a unary operator:
(7.2a) $\zeta_{e}(y)=y_{0}\qquad\qquad\qquad\zeta_{e}=\zeta$
(7.2b) $\zeta_{t}(y_{0})=y\qquad\qquad\qquad\zeta_{t}=\overline{C}(\zeta)$
Definition 7.3 High Tension-compression Operator.Assume there is a binary
operator $\phi$ and an unary operator $\zeta$, $\zeta(y_{1})\phi y_{2}=y_{0}$,
we can introduce a new binary operator $\phi_{1}$ by $\phi$ and $\zeta$,
$\phi_{1}$ will meet the relation: $y_{1}\phi_{1}y_{2}=y_{0}$, that is say,
$y_{1}\phi_{1}y_{2}=\zeta(y_{1})\phi y_{2}$. Introduce a special high operator
$\overline{T}_{1}$ to express the relation between $\phi_{1}$and
$\phi$,$\zeta$ .
(7.3a) $\phi_{1}=\phi\overline{T}_{1}\zeta$ In the same way if
$y_{1}\phi\zeta(y_{2})=y_{0}$, we can introduce a new binary operator
$\phi_{2}$ by $\phi$ and $\zeta$, $\phi_{2}$ will meet the relation:
$y_{1}\phi_{2}y_{2}=y_{0}$, that is say,
$y_{1}\phi_{2}y_{2}=y_{1}\phi\zeta(y_{2})$. Introduce a special high operator
$\overline{T}_{2}$ to express the relation between $\phi_{2}$and
$\phi$,$\zeta$. (7.3b) $\phi_{2}=\phi\overline{T}_{2}\zeta$ If $y_{1}\phi
y_{2}=\zeta(y_{0})$,that is say $\zeta^{-1}[y_{1}\phi y_{2}]=y_{0}$, we can
introduce a new binary operator $\phi_{0}$ by $\phi$ and $\zeta$, $\phi_{0}$
will meet the relation: $y_{1}\phi_{0}y_{2}=y_{0}$, that is say
$y_{1}\phi_{0}y_{2}=\zeta^{-1}[y_{1}\phi y_{2}]$, and there is
$\overline{T}_{0}$: (7.3c) $\phi_{0}=\phi\overline{T}_{0}\zeta$
For an unary operator we have only $\overline{T}$ and $\overline{T}_{0}$:
(7.4a) $\zeta_{1}\overline{T}\zeta_{2}=\zeta_{1}\zeta_{2}$ (7.4b)
$\zeta_{1}\overline{T}_{0}\zeta_{2}=\zeta_{2}^{-1}\zeta_{1}$
Definition 7.4High Superposition Operator. Assume there are P operators of
many variables $\phi_{k}$ (k=1,p), its superposition operator $\phi$ will be:
(7.5) $\phi=\sum_{k=1}^{P}\phi_{k}$
Function of $\phi$ will be the sum of function of $\phi_{k}$(k=1,p). $\phi$
will be no-valued in a point if any of $\phi_{k}$ is no-valued in this point.
$\phi_{1}+\phi_{2}$ will be many-valued in a point if $\phi_{1}$ is single-
valued and $\phi_{2}$ is many-valued in this point.
Definition 7.5High Decomposition Operator.
(7.6)
$\phi_{3}=\sum_{i=1}^{27}\zeta_{i}\Bigg{\\{}\eta_{i4}\Big{[}\eta_{i1}(\widetilde{\beta_{1}})+\eta_{i2}(\widetilde{\beta_{2}})\Big{]}+\eta_{i3}(\widetilde{\beta_{3}})\Bigg{\\}}$
We can express the relations between$\zeta_{i}$ or $\eta_{ij}$ and
$\phi_{3}$with special operators $\overline{V}_{i}$ and $\overline{P}_{ij}$
and actually $\eta_{ij}$ is not change with $\phi_{3}$.
(7.7a) $\zeta_{i}=\overline{V}_{i}(\phi_{3})\qquad\qquad\qquad\qquad(i=1,27)$
(7.7b) $\eta_{ij}=\overline{P}_{ij}(\phi_{3})\qquad\qquad(i=1,27,j=1,4)$
Please note high commutation operator or high tension-compression operator or
high decomposition operator or high superposition operator will be close
within all three numbers operators if they contain ones being many-valued and
no-valued.
Definition 7.6False operator of M+K variables. We can change an operator of M
variables to a false one of (M+K ) variables by adding
$\sigma\widetilde{\zeta_{k}}$ in which $\sigma$ is a zero operator and
operator $\phi$ will not change with K variables.
(7.8)
$\phi=\sum_{i=1}^{L}fi\sum_{j=1}^{M}g_{ij}\widetilde{\zeta_{i}}=\sum_{i=1}^{L}fi\Bigg{\\{}\sum_{j=1}^{M}g_{ij}\widetilde{\zeta_{i}}+\sum_{k=M+1}^{M+K}\sigma\widetilde{\zeta_{k}}\Bigg{\\}}$
We can also get false operator of (M+K ) variables from one of M variables by
$\overline{T}_{k}\sigma$ (k=M,M+K).
Formula solution of double branches operator equation with parameterized
operators will be:
$y=\sum_{k=1}^{27}\overline{V}_{k}\Big{[}\overline{C}(2,3,0,1)\Big{(}\sum_{i=1}^{9}\big{\\{}\big{[}\phi_{1}T_{3}o\overline{T}_{0}(\overline{P}_{i1}\phi_{3})^{-1}+\phi_{2}\overline{T}_{2}\sigma\overline{T}_{0}(\overline{P}_{i2}\phi_{3})^{-1}\big{]}\overline{T}_{0}(\overline{V}_{i}\phi_{3})^{-1}\big{\\}}\Big{)}\Big{]}$
$\Bigg{[}\overline{P}_{k4}\Big{[}\overline{C}(2,3,0,1)\Big{(}\sum_{i=1}^{9}\big{\\{}\big{[}\phi_{1}T_{3}o\overline{T}_{0}(\overline{P}_{i1}\phi_{3})^{-1}+\phi_{2}\overline{T}_{2}\sigma\overline{T}_{0}(\overline{P}_{i2}\phi_{3})^{-1}\big{]}\overline{T}_{0}(\overline{V}_{i}\phi_{3})^{-1}\big{\\}}\Big{)}\Big{]}$
$\Bigg{(}\overline{P}_{k1}\Big{[}\overline{C}(2,3,0,1)\Big{(}\sum_{i=1}^{9}\big{\\{}\big{[}\phi_{1}T_{3}o\overline{T}_{0}(\overline{P}_{i1}\phi_{3})^{-1}+\phi_{2}\overline{T}_{2}\sigma\overline{T}_{0}(\overline{P}_{i2}\phi_{3})^{-1}\big{]}\overline{T}_{0}(\overline{V}_{i}\phi_{3})^{-1}\big{\\}}\Big{)}\Big{]}(f)$
$+\overline{P}_{k2}\Big{[}\overline{C}(2,3,0,1)\Big{(}\sum_{i=1}^{9}\big{\\{}\big{[}\phi_{1}T_{3}o\overline{T}_{0}(\overline{P}_{i1}\phi_{3})^{-1}+\phi_{2}\overline{T}_{2}\sigma\overline{T}_{0}(\overline{P}_{i2}\phi_{3})^{-1}\big{]}\overline{T}_{0}(\overline{V}_{i}\phi_{3})^{-1}\big{\\}}\Big{)}\Big{]}(g)\Bigg{)}$
$+\overline{P}_{k3}\Big{[}\overline{C}(2,3,0,1)\Big{(}\sum_{i=1}^{9}\big{\\{}\big{[}\phi_{1}T_{3}o\overline{T}_{0}(\overline{P}_{i1}\phi_{3})^{-1}+\phi_{2}\overline{T}_{2}\sigma\overline{T}_{0}(\overline{P}_{i2}\phi_{3})^{-1}\big{]}\overline{T}_{0}(\overline{V}_{i}\phi_{3})^{-1}\big{\\}}\Big{)}\Big{]}(h)\Bigg{]}$
Please note solution for double branches operator equation has the same form
with one for double branches algebraic equation. Is it appropriate to class
algebraic equation and operator equation to different fields? But we have done
it! Mathematics has been parted to many alone islands. This situation is not
good and will be changed in future.
These results mean that there is a new accurate analytical route beside
approximate numerical method and topological way in study of operator
equations certainly including functional equations and function equations and
differential equations.We can extend results to N numbers operators of M
variables but there are many works to be done.
## 8\. Try to extend to continuous situation
We can extend results about discrete functions to continue functions if we
accept results about Hilbert’s 13th problem. We can express formula solution
of equation constructed by continue functions in the same form of equation
constructed by discrete functions even though we can’t give a procedure to
decompose a continue function of many variables to a superposition of
functions of one variable. But there are many tasks to be done if we want to
make results to be strict in logic.
We must prove that every continue operator of many variables can be
represented as a superposition of continue operators of one variable if we
want to extend results in this paper to continue operators and equations
constructed by them. I don’t know if there is such a result in current
literature. Please give it if there isn’t.
There are enough space for us to write our results so we are luckier than
Pierre de Fermat (1601-1665) who could not write the proof of his last
theorem. Now we have only poor results shown here but mathematicians will find
more and more good results because there is huge mineral deposit in this
direction. Please believe this point!
## References
* [1] H.Umemura, _Solution of algebraic equations in terms of theta constants_ , In D.Mumford, Tata.Lectures on Theta II, Progress in Mathematics. 43, Birkh user, Boston, 1984.
* [2] D.Hilbert, _Mathematical Problemsüller space,_ Bull.Amer.Math Soc8 (1902), 461–462.
* [3] D.Hilbert, _ber die Gleichung neunten Gradesüller space,_ Mathematische Annalen 97 (1927), 243–250.
* [4] V.I.Arnol d, _On functions of three variablesüller space,Dokl.Akad. Nauk SSSR 114 (1957), 679–681._ Amer.Math.Soc.Transl.(2) 28 (1963), 51–54.
* [5] V.I.Arnol d, _On the representation of continuous functions of three variables by superpositions of continuous functions of two variablesüller space,Mat.Sb 48 (1959), 3–74._ Amer.Math. Soc.Transl.(2) 28 (1963), 61–147.
* [6] A.N.Kolmogorov, _On the representation of continuous functions of several variables by superpositions of continuous functions of one variable and additionüller space,Dokl.Akad.Nauk SSSR 114 (1957), 953–956._ Amer.Math. Soc.Transl.(2) 28 (1963), 55–59.
* [7] A. G.Vitushkin, _On Hilbert’s thirteenth problem and related questionsüller space,_ Russian Math. Surveys 59:1 (2004), 11–25.
|
arxiv-papers
| 2009-09-21T12:14:13 |
2024-09-04T02:49:05.448075
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "ZiQian Wu",
"submitter": "Ziqian Wu sir",
"url": "https://arxiv.org/abs/0909.3747"
}
|
0909.3939
|
# The exchange coupling between the valence electrons of the fullerene cage
and the electrons of the N atoms in N@C${}_{60}^{-1,3}$
L. Udvardi Budapest University of Technology and Economics, Department of
Theoretical Physics, Budafoki út 8, H-1111 Budapest, Hungary
###### Abstract
MCSCF calculations are performed in order to determine the exchange coupling
between the 2p electrons of the N atom and the LUMOs of the fullerene cage in
the case of mono- and tri-anions of N@C60. The exchange coupling resulted by
our calculations is large compared to the hyperfine interaction. The strong
coupling can explain the missing EPR signal of the nitrogen in paramagnetic
anions.
###### pacs:
31.15.Ar
## I Introduction
Since the discovery of the first endohedral fullerene a lot of interests have
been attracted by this area of the nanotechnology. Many metal atom can be
encapsulated by using discharge techniques or ion implantation. In all the
cases the metal atom interacts strongly with the fullerene and acts as an
electron donor occupying an ’off-centered’ position inside the cage. In
contrast the nitrogen in N@C60 is situated at the center of the molecule and
retains its S=3/2 spin quartet atomic state waib . This amazing property of
the encapsulated N atom triggered several research on its possible application
in quantum computing and spin labeling. Several publications qc1 ; qc2 ; qc3 ;
qc4 studied the promise and limitations of using endohedral fullerenes as
quantum information carriers. Mehring et al. qc5 recently pointed out
experimentally the entanglement of the nuclear spin and the electronic spin of
the encaged N atom.
The changes of the characteristic EPR signal of the quartet electronic spin of
the N atom makes it an ideal probe for monitoring chemical reactions of C60
spinl . During the last decade a great deal of excitement has been brought by
the discovery of the superconductivity of the alkali-doped fullerenes. In this
type of fullerene compounds the valence electrons of the ionized alkali atoms
partially occupy the bands formed by the LUMOs of the C60 molecules. The
applicability of the quartet atomic state of the N atom as a spin label
depends on the strength of the interaction between the 2p electrons of the N
atom and the valence electron of the fullerene cage. An interaction which is
small compared to the hyper-fine interaction, results in a line width effect
of the EPR signal and the N@C60 is a good candidate for a spin labeling agent.
In the case of strong coupling the EPR signal of the system is completely
changed and the lines corresponding to the valence electrons of the N atom are
hard to identify in the signal of the paramagnetic system.
The interaction between the 2p electrons of the N atom and the valence
electrons of the C60 can be described by a Heisenberg like effective
Hamiltonian $H_{int}=J\mathbf{S}_{N}\mathbf{S}_{C_{60}}$ where
$\mathbf{S}_{N}$ and $\mathbf{S}_{C_{60}}$ denote the spin operator for the
valence electrons of the N atom and the the C60, respectively, and J is the
exchange coupling characterizing the strength of the interaction. The aim of
the present paper is to determine theoretically the exchange coupling of the
effective Hamiltonian. The exchange coupling has importance not only for EPR
measurements but it plays essential role in the description of the transport
through magnetic molecules transport which is particularly interesting from
the point of view of spintronics.
## II Computational details
The calculations have been performed using the Gamess quantum chemical program
package gamess . The proper description of the open-shell N@C${}_{60}^{-1}$
and N@C${}_{60}^{-3}$ anions requires multi-determinant wave functions. The
restricted open-shell (ROHF) calculations in the Gamess package are accessible
via the generalized valence bond (GVB) or the multi-configurational self-
consistent field (MCSCF) methods using an appropriate active space. The energy
of the anions of N@C60 with different multiplicity has been determined by
means of CAS SCF calculations where the active space is confined to the 2p
orbitals of the nitrogen atom and the three fold degenerate LUMOs of the
fullerene molecule. For the clear interpretation of the results the
excitations from the orbitals of the nitrogen to the LUMOs of the cage, and
vice versa, were excluded from the active space applying the occupation
restricted multiple active space ormas technique. The MCSCF treatment of the
open-shell systems using such a small active space is practically equivalent
to the ROHF level of calculations. The calculation has been performed using
split valence 631g basis on the carbon atoms. For the better description of
the week interaction between the encapsulated atom and the fullerene molecule
jcp the basis on the N atom is extended by additional diffuse p orbitals and
two d polarization functions (631+g(dd)). It is well known that in order to
describe the electronic structure of negatively charged species application of
diffuse basis functions is necessary. In our case the excess charge is
distributed uniformly among the 60 carbon atoms and the lack of the diffuse
basis on the carbon atoms does not affect dramatically our results. However,
in order to check the sensitivity of the exchange coupling to the applied
basis the calculations have also been performed with the Dunning’s double zeta
dh and the split valence 631+g basis sets on the carbon atoms. The geometry
of the N@C${}_{60}^{-1}$ molecule in the S=1 state and the N@C${}_{60}^{-3}$
in the high spin S=3 state have been optimized at ROHF level and it is
retained during the calculations of the energy of the systems with different
multiplicity.
## III Results and discussions
It has been shown experimentally that in the highly reduced states of the
N@C60 the excess electrons occupy the LUMOs of the fullerene and the N atom
inside the cage remains in spin quartet state anion . In order to check the
consistency of our calculations to the experimental findings we performed a
set of ROHF calculations on the mono- and tri-anions populating at first the
2p orbitals of the nitrogen and then populating the LUMOs of the C60. The
results are summarized in Table 1. The valence electrons of the nitrogen
referred as $N2p$ in Table 1 occupy the $7t_{1u}$ orbitals of the endohedral
complex between the $6h_{u}$ HOMO and $8t_{1u}$ LUMO of the C60 in agreement
with the result of ref Lu . Rather different value for the one-electron energy
of the $N2p$ orbitals is reported by Greer Greer . This discrepancy is
originated from the different treatment of the open-shell problem as it is
discussed in ref. Plakhutin . In the case of the mono-anion the energy of the
two triplet states were compared while in the case of the triply ionized
molecule the energy of the singlet state with fully occupied valence orbitals
of N was compared to the high spin state of the N@C${}_{60}^{-3}$. For both
ions the system with intact N atom were energetically more favorable in
agreement with the EPR measurements anion .
The interaction between the electrons of the nitrogen atom and the valence
electrons on the C60 anion is described by a Heisenberg-like effective
Hamiltonian:
$H_{int}=J\mathbf{S}_{N}\mathbf{S}_{c_{60}}$ (1)
where $J$ is the coupling constant, $\mathbf{S}_{N}$ and $\mathbf{S}_{c_{60}}$
are the spin of the nitrogen atom and the C60 anion, respectively. The square
of the total spin operator
$\mathbf{S}^{2}=(\mathbf{S}_{N}+\mathbf{S}_{c_{60}})^{2}$ commutes with the
Hamiltonian of the full system $H=H_{N}+H_{C_{60}}+H_{int}$ and ,consequently,
its eigenvalue is a good quantum number. Expressing the interaction in terms
of the spin of the subsystems and the spin of the whole molecule:
$H_{int}=\frac{1}{2}J\left(\mathbf{S}^{2}-\mathbf{S}_{N}^{2}-\mathbf{S}_{c_{60}}^{2}\right)$
(2)
the energy can be simply given as:
$E_{S}=E_{0}+\frac{1}{2}JS(S+1)\;\;,$ (3)
where $E_{0}$ denotes the energy of the separated systems and the subscript
$S$ indicates the explicit dependence of the energy on the multiplicity. Since
our interaction Hamiltonian can describe only such processes in which
$\mathbf{S}_{N}$ and $\mathbf{S}_{c_{60}}$ are unchanged the excitations
altering the spin of the subsystems, namely the hole and the particle are on
different species, has to be excluded from the configuration space. In the
case of N@C${}_{60}^{-1}$ $S_{N}=3/2$ and $S_{C_{60}}=1/2$ spanning an 8
dimensional direct product space. The total spin can have the values of $S=1$
or $S=2$ with the corresponding energies
$E_{S=1}=E_{0}+J\;,\;\;\;\;\;E_{S=2}=E_{0}+3J\;.$ (4)
Comparing the energy of the triplet and quintet state one can easily extract
the exchange coupling as:
$J=\frac{1}{2}\left(E_{S=2}-E_{S=1}\right)$ (5)
The results of the MCSCF calculations using 631g and DH basis are summarized
in Table 2. Although the application of the double zeta basis resulted in
considerably deeper total energy the deviation of the exchange couplings is
small.
In the case of the triply ionized N@C60 the valence electrons form a
$S_{C_{60}}=3/2$ state on the LUMOs of the fullerene molecule according to the
Hund’s rule. From the two quartet states, $\mathbf{S}_{N}$,
$\mathbf{S}_{C_{60}}$, four eigenstate of the $\mathbf{S}^{2}$ operator can be
constructed with the spin of $S=0,1,2,3$, respectively. The corresponding
energies as a function of $S$ must be on a parabola according to Eq. 3. The
results provided by the MCSCF calculations using three different basis sets
are shown by Fig. 1. The energies can nicely be fitted by the parabola given
by Eq. 3. The exchange couplings obtained by using the split valence basis
with and without diffuse p orbitals are practically the same. The inclusion of
the diffuse basis functions on the carbon atoms resulted in negligible change.
Although the magnitude of the exchange coupling corresponding to the double
zeta basis is somewhat smaller than those provided by the split valence basis
the agreement between them is satisfactory.
Ferromagnetic exchange couplings between the $2p$ orbitals of the N atom and
the valence electrons of the fullerene molecule have been found in both
anions. The exchange coupling of approximately 1 meV provided by our
calculations for both systems is within the range of those found in organic
ferromagnets metal . This relatively strong coupling between the valence
electrons of the nitrogen and the valence electrons of the fullerene cage
could be responsible for the disappearance of the nitrogen lines in the EPR
spectrum of N@C60 anions with partially filled LUMOs anion .
## IV Conclusions
In conclusion, ROHF and MCSF calculations have been performed on singly and
triply ionized anions of N@C60 in order to determine the effective exchange
coupling between the valence electrons of the encapsulated N atom and the
fullerene cage. In agreement with experiments we found that the excess
electrons occupy the LUMOs of the fullerene molecule and the entrapped atom
keeps its atomic character. The interaction between the valence electrons of
the N atom and the LUMOs of the C60 can be well described by a Heisenberg like
Hamiltonian. The size of the exchange couplings obtained by our calculations
are much larger then the hyperfine interaction and can explain the results of
EPR measurements on radical anions of N@C60.
## V Acknowlegments
This work is supported by the Hungarian National Science Foundation (contracts
OTKA T038191 and T037856).
## References
* (1) T.Almeida Murphy, T. Pawlik, A. Weidinger, M. Höhne, R. Alcala, J.-M. Spaeth, Phys.Rev. Letter. 77 (1996) 1075 B. Pietzak, M. Waiblinger, T.Almeida Murphy, A. Weidinger, M. Höhne, E. Dietel, A. Hirsch, Chem. Phys. Letters 279 (1997) 259
* (2) W. Harneit, Phys. Rev. A 65 (2002) 032322
* (3) D. Suter and K. Lim, Phys. Rev. A 65 (2002) 052309
* (4) J. Twamley, Phys. Rev. A 67 (2003) 052318
* (5) M. Feng and J. Twamley, Phys. Rev. A 70 (2004) 032318
* (6) M. Mehring, W. Scherer, and A. Weidinger, Phys. Rev. Lett. 93 (2004) 206603
* (7) E. Dietel, A. Hirsch, B. Pietzak, M. Waiblinger, K. Lips, A. Weidinger, A. Gruss, K.P. Dinse, J. Am. Chem. Soc. 121 (1999) 2432
* (8) F. Elste and C. Timm, Phys. Rev. B 71 (2005) 155403
* (9) M.W.Schmidt, K.K.Baldridge, J.A.Boatz, S.T.Elbert, M.S.Gordon, J.H.Jensen, S.Koseki, N.Matsunaga, K.A.Nguyen, S.J.Su, T.L.Windus, M.Dupuis,
J.A.Montgomery J.Comput.Chem. 14, 1347-1363(1993)
* (10) J.Ivanic J.Chem.Phys. 119 ((2003) 9364, 9377
* (11) J.M. Park, P. Tarakeshwar, and K.S. Kim J. Chem. Phys.116 (2002) 10684
* (12) T.H.Dunning, Jr., P.J.Hay Chapter 1 in ”Methods of Electronic Structure Theory”, H.F.Shaefer III, Ed. Plenum Press, N.Y. 1977, pp 1-27.
* (13) KP. Dinse, B. Godde, P. Jakes, M. Waiblinger, A. Weidinger, A. Hirsch, Abstr. Pap. - Am. Chem. Soc. (2001) 221st IEC-199. CODEN: ACSRAL ISSN: 0065-7727., P. Jakes, B. Godde, M. Waiblinger, N. Weiden, K.P. Dinse, A. Weidinger, AIP Conference Proceedings 544 (2000) 174
* (14) D.A. Shultz,K.E. Vostrikova, S.H. Bodnar, Hyun-Joo Koo, Myung-Hwan Whangbo, M.L. Kirk, E.C. Depperman, and J.W. Kampf J. Am. Chem. Soc. 125 (2003) 1607
* (15) J. Lu, X. Zhang, X. Zhao, Chem. Phys. Lett. 312 (1999) 85
* (16) J.C. Greer, Chem. Phys. Lett. 326 (2000) 567
* (17) B.N. Plakhutin, N.N. Breslavskaya, E.V. Gorelik, A.V. Arbuznikov, Journal of Molecular Structure: THEOCHEM 727 (2005) 149
| Configuration | Etotal(Hartree) | $\Delta$ E (eV) |
---|---|---|---|---
N@C${}_{60}^{-1}$ | N$2p^{4}$C${}_{60}8t_{1u}^{0}$ S = 1 | -2325.30365 | | (a)
| N$2p^{3}$C${}_{60}8t_{1u}^{1}$ S = 1 | -2325.37155 | -1.84 | (b)
N@C${}_{60}^{-3}$ | N$2p^{6}$C${}_{60}8t_{1u}^{0}$ S = 0 | -2324.74514 | | (a)
| N$2p^{3}$C${}_{60}8t_{1u}^{3}$ S = 3 | -2325.10707 | -9.84 | (b)
Table 1: Energies of N@C${}_{60}^{-1}$ and N@C${}_{60}^{-3}$ with excess electron(s) occupying the $2p$ orbitals of the N atom (a) and the LUMOs of the C60 molecule (b). basis | ES=1 (Hartree) | ES=2 (Hartree) | J (meV)
---|---|---|---
631g | -2325.371558 | -2325.371673 | -1.56
DH | -2325.515025 | -2325.515134 | -1.49
Table 2: Energy of the N@C${}_{60}^{-1}$ resulted by MCSCF calculations using
split valence (631g) and double zeta (DH) basis on the carbon atoms and the
exchange coupling extracted from the energies.
Figure 1: Energies of N@C${}_{60}^{-3}$ corresponding to different
multiplicity and the parabola fitted to the data points. The energy $E_{0}$
independent of spin is subtracted.
|
arxiv-papers
| 2009-09-22T09:25:38 |
2024-09-04T02:49:05.457659
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "L. Udvardi",
"submitter": "L\\'aszl\\'o Udvardi",
"url": "https://arxiv.org/abs/0909.3939"
}
|
0909.3955
|
# Deciphering solar turbulence from sunspots records
F. Plunian1, G. R. Sarson2, R. Stepanov3
1 LGIT, UJF, CNRS, B.P. 53, 38041 Grenoble Cedex 9, France
2 School of Mathematics and Statistics, Newcastle University, Newcastle NE1
7RU, UK
3 Institute of Continuous Media Mechanics, Korolyov 1, 614013 Perm, Russia
(Accepted 2009 …. Received 2009 ….; in original form 2009)
###### Abstract
It is generally believed that sunspots are the emergent part of magnetic flux
tubes in the solar interior. These tubes are created at the base of the
convection zone and rise to the surface due to their magnetic buoyancy. The
motion of plasma in the convection zone being highly turbulent, the surface
manifestation of sunspots may retain the signature of this turbulence,
including its intermittency. From direct observations of sunspots, and
indirect observations of the concentration of cosmogenic isotopes 14C in tree
rings or 10Be in polar ice, power spectral densities in frequency are plotted.
Two different frequency scalings emerge, depending on whether the Sun is
quiescent or active. From direct observations we can also calculate scaling
exponents. These testify to a strong intermittency, comparable with that
observed in the solar wind.
###### keywords:
MHD, turbulence, statistics, sunspots, magnetic fields, plasmas
††pagerange: Deciphering solar turbulence from sunspots
records–References††pubyear: 2009
## 1 Introduction
Sunspots observed at the surface of the convection zone of the Sun are usually
understood as the manifestation of solar magnetic activity. Naked-eye and
telescope observations of sunspots are available from AD 1610, providing
reliable records of sunspot numbers (SSN). Several sets of data exist, varying
in how they have been sampled, averaged (daily or monthly), whether they
concern sunspots or sunspots groups, and on the scientific societies who have
compiled the records. Here we consider the American daily SSN (D)
111ftp://ftp.ngdc.noaa.gov/STP/SOLAR_DATA/SUNSPOT_NUMBERS/
AMERICAN_NUMBERS/RADAILY.PLT, the American daily group SSN (G)
222ftp://ftp.ngdc.noaa.gov/STP/SOLAR_DATA/SUNSPOT_NUMBERS/
GROUP_SUNSPOT_NUMBERS/dailyrg.dat, and the International monthly averaged SSN
(M) 333http://solarscience.msfc.nasa.gov/greenwch/spot_num.txt (figure 1). In
addition SSN at earlier times have been reconstructed from proxies, based on
the concentration of cosmogenic isotopes 14C in tree rings
444ftp://ftp.ncdc.noaa.gov/pub/data/paleo/climate_forcing/solar_variability/
solanki2004-ssn.txt or 10Be in ice core bubbles. The production rate of such
isotopes increases with the cosmic-ray flux, which is higher when the solar
magnetic activity is low. Plotting the SSN versus time reveals a cycle of
about 11 years known as the Schwabe cycle. This cyclical solar magnetic
activity is sufficiently robust to be detected in 10Be concentration records,
even during some long periods with almost no visible sunspots, like the
Maunder minimum (1645-1715) (Beer et al., 1998). Analyzing long time series of
14C and 10Be, it has been shown that the solar activity of the last 70 years
has been exceptionally high (Usoskin et al., 2003; Solanki et al., 2004), and
that a decline is expected within the next two or three cycles (Abreu et al.,
2008).
Figure 1: Data sets of 10 year-averaged SSN from 14C (top blue) and 10Be (top
red), International SSN monthly averaged M (second), American daily group SSN
G (third), and American daily SSN D (bottom).
The occurrence of sunspots is of course an important diagnostic that must be
reproduced by any solar dynamo model. The fact that it is irregular (in spite
of the Schwabe cycle) reflects the complexity inherent in the nonlinear
coupling between the turbulent flow and the magnetic field in the solar
convection zone (Browning et al., 2006). With the long time series of direct
SSN observations and 14C and 10Be data available, it is tempting to calculate
the corresponding frequency spectra, to infer some signature of the underlying
turbulence in the convection zone. Similar attempts have been made for the
Earth (Courtillot & Le Mouel, 1988; Constable & Johnson, 2005; Sakuraba &
Hamano, 2007); the frequency spectrum of the geomagnetic dipole moment
obtained from paleomagnetic data is consistent with an inertial range of
$f^{-5/3}$ scaling, and a dissipation range of $f^{-11/3}$ as expected in
magnetohydrodynamic turbulence (Alemany et al., 2000). For the Sun, analysis
of International daily SSN have led to a $f^{-2/3}$ scaling (Morfill et al.,
1991; Lawrence et al., 1995), and this has been attributed to some sequential
sampling of the field upon arrival at the photosphere on top of a Kolmogorov
spatial scaling due to the underlying turbulence. Here we extend this analysis
to the other SSN records mentioned above and test how robust the $f^{-2/3}$
scaling is, in particular during minima and maxima of sunspot activity. In
addition, for time scales smaller than 2 years, the stochastic character of
the SSN records suggests strong intermittency (Lawrence et al., 1995), as
opposed to a low-dimensional chaotic interpretation. We shall characterize
this intermittency by calculating the corresponding scaling exponents.
## 2 Wavelet spectra
Figure 2: Wavelet coefficients (logarithm of absolute value) of 14C (top), M
(second), G (third), and D (bottom).
In order to filter out the noise, we use a wavelet decomposition of the
signal. In figure 2, the wavelet coefficients are plotted versus time (in
years) and frequency (in month-1). The light (resp. dark) colors correspond to
low (resp. high) values of these coefficients. For a given year, the curve
giving the wavelet coefficient versus frequency corresponds to a power
spectral density of the signal. The dark horizontal stripe for $f\sim 0.01$
month-1 corresponds to the Schwabe cycle. It is not visible in the 14C data
set due to the coarse sampling of this data (averaged over 10 years). On the
other hand a clear dark stripe is visible for $f\sim 3.5\times 10^{-5}$
month-1, corresponding to a $\sim$2400 years cycle. In the figure for M data
the dark stripe of the Schwabe cycle almost disappears during the Dalton
minimum (1790-1820) (Frick et al., 1997). In the G and D data figures we
identify another horizontal stripe at $f\sim 1$ month-1, corresponding to the
solar mean rotation rate. This indicates that the latitudinal repartition of
sunspots is not homogeneous. Finally the vertical light stripes correspond to
minima of magnetic activity.
Figure 3: Wavelet spectral density versus frequency, on average (top), and for
magnetic activity minima (bottom, dashed curves) and maxima (bottom, solid
curves). The dashed straight lines correspond to $-2/3$ and $-8/3$ slopes
(top), and $-2/3$ and $-1$ slopes (bottom).
The time average of the wavelet coefficients are shown in figure 3 (top) for
the five data sets. As mentioned earlier the two peaks around $0.01$ month-1
and $1$ month-1 correspond to the Schwabe cycle and solar mean rotation rate.
Between them the three spectra of the directly observed SSN (M, G, D) present
a common scaling in $f^{-2/3}$. The other data sets 14C and 10Be are
compatible with a $f^{-2/3}$ scaling as well, although the 14C PSD is
overestimated by roughly a factor 10, probably due to the proxy used in the
reconstruction of the SSN from the 14C data.
Figure 4: Same data as in figure 1 (the last three panels) but subdivided into
sets of maximum (dark) and minimum (light) magnetic activity.
For the three data sets (M), (G) and (D), instead of averaging on all times,
we now average on periods corresponding to either maximum or minimum magnetic
activity as shown in figure 4. For the maximum (resp. minimum) activity
subset, the excluded data is that centred around the times of the Schwabe
minima (resp. maxima) The corresponding spectral densities are plotted in
figure 3 (bottom). The slopes for minima are systematically steeper than those
for maxima, indicating two different regimes. To estimate these slopes we vary
both the range of frequency $\left[f_{\min},f_{\max}\right]$ on which they are
calculated, and the way the data sets are split into subsets of maximum and
minimum activity. For the former we take $f_{\max}=0.7$ month-1 for the data
sets (G) and (D) in order to escape from the influence of the peak $f=1$
month-1, and $f_{\max}=0.5$ month-1 for the data set (M). When changing
$f_{\min}$ the slopes change. We vary $f_{\min}$ such that the ratio
$f_{\max}/f_{\min}$ is about 10, and the standard deviation of the slope
remains 10 % or less of its average value. This leads to
$f_{\min}\in\left[0.05,0.07\right]$ for (G),
$f_{\min}\in\left[0.06,0.08\right]$ for (D), and
$f_{\min}\in\left[0.03,0.05\right]$ for (M). In addition we consider at least
three different degrees of splitting for each data set, this splitting degree
being related to the time length of the subsets of maximum and minimum
activity. The choices of this splitting degree are made such that both subsets
are long enough to provide good statistics, but remain separated by sufficient
time lags that the SSN values falling into each subset do not overlap too
much. Then for both the maximum and minimum subsets, we calculate the mean
slope and the standard deviation obtained when varying both the frequency
range and the degree of splitting. The corresponding slope estimates are given
in table 1. They are consistent with power spectra in $f^{-2/3}$ and $f^{-1}$.
The corresponding dashed lines are plotted in figure 3 to guide the eye. The
standard deviations are small, showing that these slopes are robust with
respect to the details of our analysis. The formal standard errors from each
of the individual regressions (for specific frequency ranges and degrees of
splitting) are of comparable magnitude.
Activity | (M) | (G) | (D)
---|---|---|---
Max. | $-0.61\pm 0.05$ | $-0.69\pm 0.05$ | $-0.63\pm 0.05$
Min. | $-1.03\pm 0.03$ | $-0.94\pm 0.07$ | $-0.98\pm 0.02$
Table 1: Slope estimates for the PSD curves plotted in figure 3 (bottom). They
correspond to average values plus standard deviation errors when varying both
the frequency range and the degree of splitting of each data set (into the two
subsets of maximum and minimum magnetic activity).
As noted by Lawrence et al. (1995), the question of causality complicates the
interpretation of such temporal data. The difference of spectral slopes
between minima and maxima can be attributed to two different effects: a change
of the underlying turbulence, affecting the spatial structure the magnetic
field; or a change in the frequency of the sequential sampling of the magnetic
field, as suggested by Lawrence et al. (1995). Although the latter effect
cannot be excluded, there is a simple argument in favour of the former. It is
generally accepted that the occurrence of sunspots at the photosphere is due
to the magnetic buoyancy force $\nabla B^{2}$, where $B$ is some magnetic
induction intensity in the convection zone (Tobias et al., 2001). It is then
tempting to interpret the two spectral slopes as the signatures of this
buoyancy, assuming that the frequency of sunspot occurrence at the photosphere
is proportional to this force. Then the $f^{-2/3}$ and $f^{-1}$ SSN spectra
would correspond to buoyancy spectra of $k^{-2/3}$ and $k^{-1}$, where $k$ is
the spatial wave number. During maxima this implies a Kolmogorov magnetic
energy spectrum of $k^{-5/3}$, compatible with inertia-driven turbulence in
the convection zone. During minima it implies a magnetic energy spectrum of
$k^{-2}$, compatible with turbulence dominated by the solar rotation (Zhou,
1995). In the transport scenario proposed by Tobias et al. (2001), the field
which arises at the surface is the strongest part of a poloidal field
generated by cyclonic turbulence in the convection zone. Our interpretation
then suggests two different regimes for such cyclonic turbulence, controlled
by either inertia or rotation.
## 3 Intermittency
The stochastic nature of SSN occurrence for times scales smaller than 2 years
has been shown by Lawrence et al. (1995), suggesting an intermittent
turbulence. Here our goal is to quantify this intermittency for the three sets
(M, G, D), calculating the corresponding scaling exponents. For that we first
calculate the associated generalized structure function (GSF) and look for its
scaling exponents, as usually done in turbulence. We define the SSN increment
by
$\delta y(t,\tau)=S(t+\tau)-S(t)\;,$ (1)
where $S(t)$ denotes the SSN at time $t$. Assuming statistical stationarity in
the frequency range of interest, the $t$ dependence in $\delta y(t,\tau)$ can
be dropped and the GSF is then given by (Nicol et al., 2008)
$S_{m}(\tau)=\left\langle\left|\delta
y\right|^{m}\right\rangle=\int^{\infty}_{-\infty}\left|\delta
y\right|^{m}P(\delta y,\tau)d(\delta y)\;,$ (2)
where $P$ is the probability density function (PDF) of $\delta y$, the angle
brackets $\left\langle\cdot\right\rangle$ denote time averaging, and $m$ is a
positive integer.
Figure 5: Probability density functions of $S(t+\tau)-S(t)$ for M (top), G
(middle) and D (bottom). The labels indicate the value of $\tau$ in years.
In figure 5 PDFs for the three data sets (M, G, D) are plotted for selected
values of $\tau$. The PDFs of the 14C data are poorly defined, and so we drop
this data set for the rest of the study. The PDFs of the other data sets show
peaks at $\tau=11/2$ years, corresponding to the Schwabe cycle. For other
values of $\tau$ they exhibit tails containing a higher number of rare events
than for a gaussian distribution, suggesting intermittency. Similar results
were shown in Lawrence et al. (1995).
To quantify this intermittency we first check whether the GSF obey a scaling
law in the form
$S_{m}(\tau)\sim\tau^{\zeta(m)}.$ (3)
We find (not shown) that this is clearly the case. In homogeneous and
isotropic fully developed turbulence, intermittency corresponds to
$\zeta(m)<m/3$. The ratio $\zeta(m)/\zeta(3)$ is calculated, estimating the
scaling power of $S_{m}(\tau)/S_{m}(3)$. Plotting the ratio
$\zeta(m)/\zeta(3)$ versus $m$ for the three data sets (figure 6) we see a
clear departure from the Kolmogorov straight line $\zeta(m)/\zeta(3)=m/3$, and
clear indications of intermittency. It is remarkable that the (D) set, which
has the best sampling, leads to the largest intermittency. It is also
remarkable that the (M) and (G) sets lead to similar scaling exponents,
supporting the equivalence between averaging over space and time.
Figure 6: Scaling exponents $\zeta(m)/\zeta(3)$ plotted versus $m$ for the
three sets of data M, G, D, and for 2 months $<\tau<$ 14 months. The dashed
line corresponds to a Kolmogorov scaling $\zeta(m)/\zeta(3)=m/3$.
The exponents can be fitted to the standard $p$-model derived for hydrodynamic
(Meneveau & Sreenivasan, 1987) and magnetohydrodynamic turbulence (Carbone,
1993). This model is defined by
$\zeta(m)=1-\log_{2}\left[p^{m/3}+(1-p)^{m/3}\right].$ (4)
We find $p_{\rm(G)}=0.68$, $p_{\rm(M)}=0.68$, and $p_{\rm(D)}=0.83$. The last
value compares surprising well with those for the solar wind measured by the
Ulysses spacecraft (Pagel & Balogh, 2002; Nicol et al., 2008), and for the
magnetospheric cusp measured by the Polar satellite (Yordanova et al., 2004),
even though the frequencies differ by several orders of magnitude.
## 4 Summary
In conclusion, the wavelet spectral analysis of sunspot records has revealed
two different behaviors, depending on whether the Sun is quiescent or active.
This suggests two different kinds of turbulence in the convection zone,
controlled either by inertia or by rotation. The signature of such fully
developed turbulence is confirmed by the calculation of the GSF scaling
exponents, which indicate strong intermittency.
## Acknowledgments
We are grateful for support from the Dynamo Program at KITP (supported in part
by the National Science Foundation under Grant No. PHY05-51164), during which
this work was started. We thank Prof. Steve Tobias for helpful comments, and
Prof. Ilya Usoskin for providing the 10Be data. Finally F.P. and R.S. are
grateful for support from a RFBR/CNRS 07-01-92160 PICS grant.
## References
* Abreu et al. (2008) Abreu J. A., Beer J., Steinhilber F., Tobias S. M., Weiss N. O., 2008, Geophys. Research Lett., 35, 20109
* Alemany et al. (2000) Alemany A., Marty P., Plunian F., Soto J., 2000, Journal of Fluid Mechanics, 403, 263
* Beer et al. (1998) Beer J., Tobias S., Weiss N., 1998, Solar Physics, 181, 237
* Browning et al. (2006) Browning M. K., Miesch M. S., Brun A. S., Toomre J., 2006, ApJ Lett., 648, L157
* Carbone (1993) Carbone V., 1993, Physical Review Letters, 71, 1546
* Constable & Johnson (2005) Constable C., Johnson C., 2005, Physics of the Earth and Planetary Interiors, 153, 61
* Courtillot & Le Mouel (1988) Courtillot V., Le Mouel J. L., 1988, Annual Review of Earth and Planetary Sciences, 16, 389
* Frick et al. (1997) Frick P., Galyagin D., Hoyt D. V., Nesme-Ribes E., Schatten K. H., Sokoloff D., Zakharov V., 1997, A&A, 328, 670
* Lawrence et al. (1995) Lawrence J. K., Cadavid A. C., Ruzmaikin A. A., 1995, ApJ, 455, 366
* Meneveau & Sreenivasan (1987) Meneveau C., Sreenivasan K. R., 1987, Physical Review Letters, 59, 1424
* Morfill et al. (1991) Morfill G. E., Scheingraber H., Voges W., Sonett C. P., 1991, in Sonett C. P., Giampapa M. S., Matthews M. S., eds, The Sun in Time Sunspot number variations - Stochastic or chaotic. pp 30–58
* Nicol et al. (2008) Nicol R. M., Chapman S. C., Dendy R. O., 2008, ApJ, 679, 862
* Pagel & Balogh (2002) Pagel C., Balogh A., 2002, Journal of Geophysical Research (Space Physics), 107, 1178
* Sakuraba & Hamano (2007) Sakuraba A., Hamano Y., 2007, Geophys. Research Lett., 34, 15308
* Solanki et al. (2004) Solanki S. K., Usoskin I. G., Kromer B., Schüssler M., Beer J., 2004, Nat, 431, 1084
* Tobias et al. (2001) Tobias S. M., Brummell N. H., Clune T. L., Toomre J., 2001, ApJ, 549, 1183
* Usoskin et al. (2003) Usoskin I. G., Solanki S. K., Schüssler M., Mursula K., Alanko K., 2003, Physical Review Letters, 91, 211101
* Yordanova et al. (2004) Yordanova E., Grzesiak M., Wernik A., Popielawska B., Stasiewicz K., 2004, Annales Geophysicae, 22, 2431
* Zhou (1995) Zhou Y., 1995, Physics of Fluids, 7, 2092
|
arxiv-papers
| 2009-09-22T18:51:26 |
2024-09-04T02:49:05.462761
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Franck Plunian, Graeme Sarson, Rodion Stepanov",
"submitter": "Rodion Stepanov",
"url": "https://arxiv.org/abs/0909.3955"
}
|
0909.4127
|
# Static Spherically Symmetric Solutions to modified Hořava-Lifshitz Gravity
with Projectability Condition
Jin-Zhang Tang111Electronic address:JinzhangTang@pku.edu.cn, Bin
Chen222Electronic address: bchen01@pku.edu.cn Department of Physics, and State
Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing
100871, China
(August 27, 2024
)
###### Abstract
In this paper we seek static spherically symmetric solutions of Hořava-
Lifshitz-like gravity with projectability condition. We consider the most
general form of gravity action without detailed balance, and require the
spacetime metric to respect the projectability condition. We find that for any
value of $\lambda$, it may exists the solutions of topology
$\mathbb{R}\times\mathbb{M}_{3}$, where $\mathbb{R}$ is the time direction and
$\mathbb{M}_{3}$ is a three-dimensional maximally symmetric space depending on
the value of cosmological constant and the potential of the action. Besides,
in the UV region where $\lambda\neq 1$, we find Minkowski or de-Sitter space-
time as the solution, while in the IR region where $\lambda=1$, we prove that
(dS-)Schwarzschild solution is the only nontrivial solution. We also notice
that the other static spherically symmetric solutions found in the literature
do not satisfy the projectability condition and are not the solutions we get.
Our study shows that in Hořava-Lifshitz gravity with projectability condition,
there is no novel correction to Einstein’s general relativity in solar system
tests.
###### pacs:
98.80.Cq
## I introduction
Diffeomorphism is an essential symmetry of Einstein’s relativity theory of
gravity. It has been widely believed to be exact in any theory of gravity.
However, in the recent proposal by HořavaHorava:2008ih ; Horava:2009uw on
gravity theory, it is no longer an exact symmetry. The basic idea behind
Hořava’s theory is that time and space may have different dynamical scaling in
UV limit. This was inspired by the development in quantum critical phenomena
in condensed matter physics, with the typical model being Lifshitz scalar
field theoryLifshitz ; Chen:2009ka . In this Hořava-Lifshitz theory, time and
space will take different scaling behavior as
$\mathbf{x}\rightarrow b\mathbf{x},\;\;\;\;t\rightarrow b^{z}t,$ (1)
where $z$ is the dynamical critical exponent characterizing the anisotropy
between space and time. Due to the anisotropy, instead of diffeomorphism, we
have the so-called foliation-preserving diffeomorphism. The transformation is
now just
$\displaystyle t$ $\displaystyle\rightarrow$ $\displaystyle\tilde{t}(t),$
$\displaystyle x^{i}$ $\displaystyle\rightarrow$
$\displaystyle\tilde{x^{i}}(x^{j},t).$ (2)
As a result, there is one more dynamical degree of freedom in Hořava-Lifshitz-
like gravity than in the usual general relativity. Such a degree of freedom
could play important role in UV physics, especially in early
cosmologyCai:2009dx ; Chen:2009jr . At IR, due to the emergence of new gauge
symmetry, this degree of freedom is not dynamical any more such that the
kinetic part of the theory recovers the one of the general relativity.
Since time direction plays a privileged role in the whole construction, it is
more convenient to work with ADM metric
$ds^{2}=-N^{2}dt^{2}+g_{ij}(dx^{i}+N^{i}dt)(dx^{j}+N^{j}dt),$ (3)
in which $N$ and $N_{i}$ are called “lapse” and “shift” variables
respectively. Then we have the following transformations on the metric
components:
$\displaystyle\delta g_{ij}$ $\displaystyle=$
$\displaystyle\partial_{i}\xi^{k}g_{jk}+\partial_{j}\xi^{k}g_{ik}+\xi^{k}\partial_{k}g_{ij}+\xi^{0}\dot{g}_{ij}$
$\displaystyle\delta N_{i}$ $\displaystyle=$
$\displaystyle\partial_{i}\xi^{j}N_{j}+\xi^{j}\partial_{j}N_{i}+\dot{\xi}^{j}g_{ij}+\dot{\xi}^{0}N_{i}+\xi^{0}\dot{N}_{i}$
$\displaystyle\delta N$ $\displaystyle=$
$\displaystyle\xi^{j}\partial_{j}N+\dot{\xi}^{0}N+\xi^{0}\dot{N}$ (4)
It seems natural to choose the lapse function $N$ to be projectable function
on the spacetime foliation, i.e. only a function of $t$. Such a choice makes
the above gauge transformations simpler and more transparent. More
importantly, with the projectable condition, in the Hamiltonian formulation
the constraints could form a closed algebra Horava:2008ih since the momentum
conjugate to $N$ does not lead to a local constraint. On the contrary, if the
projectable condition on $N$ is abandoned, then the theory would not be well-
defined, as shown in Horava:2008ih ; Li:2009bg . Therefore in this letter, we
will focus on the case with the projectable condition.
Taken Hořava-Lifshitz gravity as a new gravitational theory, it is an
important issue to study its static spherically symmetric solutions. This
issue has been widely studied in the literature, see Lu2009 ; Nastase2009 ;
Kehagias:2009is ; AhmadGhodsi2009 ; Colgain:2009fe ; park2009 . In these
papers, for example Lu2009 ; park2009 , it was assumed that the metric of the
black solutions had the following form
$ds^{2}=-N(r)^{2}dt_{S}^{2}+\frac{dr^{2}}{g(r)}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}).$ (5)
From this metric ansatz, it was found that there were new spherically
symmetric solutions, even at IR. For example, in Kehagias:2009is , based on a
modified Hořava-Lifshitz -type action, an asymptotically flat solution with
$g=N^{2}=1+\omega r^{2}-\sqrt{r(\omega^{2}r^{3}+4\omega M)}$ (6)
was found. This raised the issue that if there is any observational effect in
solar system testsSolar .
However, in the above ansatz (5) the “lapse function” $N(r)$ obviously breaks
the “projectability condition”. As the Hořava gravity is only well defined
when the “projectability condition” is preserved, this naturally leads one to
ask whether the above new solutions still are the solutions of Hořava-Lifshitz
gravity with the projectability condition after proper coordinates
transformation? The answer to this question is not obvious, considering the
freedom in doing coordinate transformation. For instance, a static spherically
symmetric solution in the flat spacetime could be represented in Schwarzschild
coordinates as
$ds^{2}=-(1-\frac{2GM}{r})dt_{S}^{2}+(1-\frac{2GM}{r})^{-1}dr^{2}+r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right),$ (7)
which looks against the projectability condition. By a transformation into the
Painlevé-Gullstrand coordinatesPainleve ; Gullstrand ; Lematre ; Hawking
$dt_{S}=dt_{PG}\mp\frac{\sqrt{2GM/r}}{1-2GM/r}dr,$ (8)
the solution (7) becomes
$ds^{2}=-dt_{PG}^{2}+(dr\pm\sqrt{\frac{2GM}{r}}dt_{PG})^{2}+r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right).$ (9)
Comparing with the ADM metric (3), we find that the “lapse function” $N=1$,
which is in accord with the “projectability condition”.
Furthermore, we would like to know if there are any other new solutions,
especially at IR, which may have significant physical implication in IR
physics. Therefore, in this letter, we study the static spherically symmetric
solutions to modified Hořava-Lifshitz gravity with the projectability
condition. We consider the most general form of the action without the
detailed balance condition. We find that for any value of $\lambda$, if the
potential term is properly chosen, there may exists the solutions of topology
$\mathbb{R}\times\mathbb{M}_{3}$, where $\mathbb{R}$ is the time direction and
$\mathbb{M}_{3}$ is a three-dimensional maximally symmetric space. In the case
without the cosmological constant in the action, $\mathbb{M}_{3}$ is just the
flat spacetime. In the case with the cosmological constant, $\mathbb{M}_{3}$
could be a three-dimensional sphere $\mathbb{S}^{3}$ or hyperboloid
$\mathbb{H}^{3}$, depending on the potential. Moreover, apart from these
solutions, in the UV region where $\lambda\neq 1$, we find either de-Sitter
space-time or Minkowski spacetime, up to the cosmological constant, while in
the IR region where $\lambda=1$, we prove that (dS)-Schwarzschild solution is
the only nontrivial solution. This result seems in accordence with
A.A.Kocharyan . We also notice that the other static spherically symmetric
solutions found in the literature do not satisfy the projectability condition
and are not the solutions we want. Our study shows that in Hořava-Lifshitz-
like Gravity with the projectability condition, there is no novel correction
to Einstein’s general relativity in solar system tests.
We study the topological static spherically symmetric solutions in the Hořava-
Lifshitz-like gravity as well. We choose the metric ansatz in which
$d\Omega_{k}^{2}$ denotes the line element for an 2-dimensional Einstein space
with constant scalar curvature $2k$. Without loss of generality, one may take
$k=0,\pm 1$ respectively. The $k=1$ case has been discussed above. To $k=-1$
case, we find that it may also exists the solutions of topology
$\mathbb{R}\times\mathbb{M}_{3}$ for all $\lambda$. In the UV region where
$\lambda\neq 1$, the only possible solution is either Minkowski or de-Sitter
space-time with topological twist. In the IR region where $\lambda=1$, the
Schwarzschild topological black hole is the only nontrivial solution. For the
case $k=0$, there is not a Schwarzschild solution at IR or de-sitter space-
time in the UV region because $f$ can’t be zero.
## II The modified Hořava-Lifshitz gravity
In this section, we give a brief review of Hořava-Lifshitz gravity and its
modifications. Using the ADM formalism, the action of this Hořava-Lifshitz
gravitational theory is given byHorava:2008ih ; Horava:2009uw
$\displaystyle S$ $\displaystyle=$ $\displaystyle\int
dtd^{3}\mathbf{x}(\mathcal{L}_{K}+\mathcal{L}_{V}),$
$\displaystyle\mathcal{L}_{K}$ $\displaystyle=$
$\displaystyle\sqrt{g}N\left\\{\frac{2}{\kappa^{2}}(K_{ij}K^{ij}-\lambda
K^{2})\right\\},$ $\displaystyle\mathcal{L}_{V}$ $\displaystyle=$
$\displaystyle\sqrt{g}N\left\\{\frac{\kappa^{2}\mu^{2}(\Lambda_{W}R-3\Lambda^{2}_{W})}{8(1-3\lambda)}+\frac{\kappa^{2}\mu^{2}(1-4\lambda)}{32(1-3\lambda)}R^{2}\right.$
(10)
$\displaystyle\left.-\frac{\kappa^{2}}{2\omega^{4}}\left(C_{ij}-\frac{\mu\omega^{2}}{2}R_{ij}\right)\left(C^{ij}-\frac{\mu\omega^{2}}{2}R^{ij}\right)\right\\},$
where $\mathcal{L}_{K}$ is the kinetic term and $\mathcal{L}_{V}$ is the
potential term. In the action, $\lambda,\kappa,\mu,\omega$ and $\Lambda_{W}$
are the coupling parameters, and $C_{ij}$ is the Cotton tensor defined by
$C^{ij}=\epsilon^{ikl}\nabla_{k}\left(R^{j}_{l}-\frac{1}{4}R\delta^{j}_{l}\right).$
(11)
The study of the perturbations around the Minkowski vacuum shows that there is
ghost excitation when $\frac{1}{3}<\lambda<1$. This indicates that the theory
is only well-defined in the region $\lambda\leq\frac{1}{3}$ and $\lambda\geq
1$. Since the theory should be RG flow to IR with $\lambda=1$, we expect that
at UV, $\lambda>1$ to have a well-defined RG flow. At IR, $\lambda=1$, the
kinetic term recovers the one of standard general relativity. Comparing to the
action of the general relativity in the ADM formalism, the speed of light, the
Newton’s constant and the cosmological constant emerge as
$\displaystyle
c=\frac{\kappa^{2}\mu}{4}\sqrt{\frac{\Lambda_{W}}{1-3\lambda}},\hskip
12.91663ptG=\frac{\kappa^{2}}{32\pi c},\hskip
12.91663pt\Lambda=\frac{3}{2}\Lambda_{W}.$ (12)
It follows from (12) that for $\lambda>1/3$ ,the cosmological constant
$\Lambda_{W}$ has to be negative. It was noticed in Lu2009 that if we make an
analytic continuation of the parameters
$\mu\to i\mu,\hskip 17.22217pt\omega^{2}\to-i\omega^{2},$ (13)
the four-dimensional action remains real. In this case, the emergent speed of
light becomes
$c=\frac{\kappa^{2}\mu}{4}\sqrt{\frac{\Lambda_{W}}{3\lambda-1}}.$ (14)
The requirement that this speed be real implies that $\Lambda_{W}$ must be
positive for $\lambda>\frac{1}{3}$.
One important feature of original Hořava-Lifshitz gravity is that it respects
the so-called “detailed balance” conditionHorava:2008ih ; Horava:2009uw .
However, it turns out that the detailed balance condition is not essential to
the theory. It could be just a nice way to organize the action. If abandoning
‘detailed balance” and just requiring the model to be power-counting
renormalizable, we find that the most general form of the action is of the
form Visser_2009
$\displaystyle S$ $\displaystyle=$ $\displaystyle\int
dtd^{3}\mathbf{x}(\mathcal{L}_{K}+\mathcal{L}_{V}),$
$\displaystyle\mathcal{L}_{K}$ $\displaystyle=$
$\displaystyle\sqrt{g}N\left\\{g_{K}(K_{ij}K^{ij}-\lambda K^{2})\right\\},$
$\displaystyle\mathcal{L}_{V}$ $\displaystyle=$
$\displaystyle\sqrt{g}N\left\\{-g_{0}\zeta^{6}+g_{1}\zeta^{4}R+g_{2}\zeta^{2}R^{2}+g_{3}\zeta^{2}R_{ij}R^{ij}\right.$
(15)
$\displaystyle\left.+g_{4}R^{3}+g_{5}R(R_{ij}R^{ij})+g_{6}R^{i}_{j}R^{j}_{k}R^{k}_{i}\right.$
$\displaystyle\left.+g_{7}R\nabla^{2}R+g_{8}\nabla_{i}R_{jk}\nabla^{i}R^{jk}\right\\}.$
where $\zeta$ is a suitable factor to ensure the couplings $g_{a}$ are all
dimensionless. From anisotropic scaling counting, five of these operators are
marginal(renormalizable) and four are relevant(super-renormalizable). And we
can rescale the time and space coordinates to set both $g_{K}\to 1$ and
$g_{1}\to 1$ without loss of generality. In the following, we will study the
static spherically symmetric solution to the action (II).
## III Static spherically symmetric solutions
The static spherically symmetric solutions of Hořava-Lifshitz gravity have
been discussed by Lu2009 ; Nastase2009 ; Kehagias:2009is ; AhmadGhodsi2009 ;
park2009 . In these paper, it was assumed that the metric of the solutions
took the form (5). Consequently, some new kinds of solutions have been found.
For the Horava’s original model, three types of solutions were found in Lu2009
. The first one is given by
$g=1+x^{2},\;\;\;x=\sqrt{-\Lambda_{W}}r,$ (16)
without any restriction on the function $N(r)$. This is valid for all
$\lambda$. And the other two solutions are given by
$g=1+x^{2}-\alpha
x^{\frac{2\lambda\pm\sqrt{6\lambda-2}}{\lambda-1}},\;\;\;\;N=x^{-\frac{1+3\lambda\pm
2\sqrt{6\lambda-2}}{\lambda-1}}g,$ (17)
where $\alpha$ is an integration constant. For the solution to be real, it is
necessary that $\lambda>1/3$.
In paper park2009 , Park got a more general solution in the IR region when
$\lambda=1$, basing on an action softly breaking the detailed balance
condition
$N^{2}=g=1+(\omega-\Lambda_{W})r^{2}-\sqrt{r\left[\omega\left(\omega-2\Lambda_{W}\right)r^{3}+\beta\right]}.$
(18)
Certainly, for a general form of the action like (II), it may exists other
kinds of solution with the metric ansatz (5).
For the metric of the form (5), we can work in the Painlevé-Gullstrand
coordinates by making a transformation
$dt_{S}=dt_{PG}-\frac{\sqrt{1-N^{2}}}{N^{2}}dr.$ (19)
Then the ansatz (5) becomes
$ds^{2}=-dt_{PG}^{2}+(dr+\sqrt{1-N^{2}}dt_{PG})^{2}+(\frac{1}{g}-\frac{1}{N^{2}})dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}).$ (20)
Comparing with the ADM metric, we find that $N(t_{PG})=1$ and if
$g=N^{2},$ (21)
we reach (3). So the solutions (17) of paper Lu2009 can not preserve the
“projectability condition” after the coordinate transformation. And it seems
that the solution (18) could preserve the “projectability condition” after the
coordinate transformation. However note that (21) is only a necessary
condition but not a sufficient condition. Actually from the study below, we
will see that (18) could not satisfy the “projectability condition” neither.
We now seek the static, spherically symmetric solutions with the metric ansatz
$ds^{2}=-N(t)^{2}dt^{2}+\frac{1}{f(r)}(dr+N^{r}dt)(dr+N^{r}dt)+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}).$ (22)
By the coordinate transformation $dt=dt_{s}+\frac{N_{r}}{N^{2}-fN_{r}^{2}}dr$,
we can transform the metric ansatz to the Schwarzschild coordinates type,
$ds^{2}=-(N^{2}-fN_{r}^{2})dt_{S}^{2}+\frac{N^{2}}{f(N^{2}-fN_{r}^{2})}dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}).$ (23)
Substituting the metric ansatz (22) into the Lagrangian (II), up to an overall
scaling constant, we get
$\displaystyle\mathcal{L}_{K}=$
$\displaystyle\frac{1}{\sqrt{f}}\frac{1}{N(t)}\left\\{(1-\lambda)r^{2}f^{2}\left(N^{{}^{\prime}}_{r}+N_{r}\frac{f^{{}^{\prime}}}{2f}\right)^{2}+2(1-2\lambda)f^{2}N_{r}^{2}\right.$
$\displaystyle\left.-4\lambda
rf^{2}N_{r}\left(N^{{}^{\prime}}_{r}+N_{r}\frac{f^{{}^{\prime}}}{2f}\right)\right\\},$
$\displaystyle\mathcal{L}_{V}=$
$\displaystyle\frac{1}{\sqrt{f}}N(t)r^{2}\left\\{-g_{0}\zeta^{6}+\zeta^{4}\left[\frac{2(1-f)}{r^{2}}-\frac{2f^{{}^{\prime}}}{r}\right]+g_{2}\zeta^{2}\left[\frac{2(1-f)}{r^{2}}-\frac{2f^{{}^{\prime}}}{r}\right]^{2}\right.$
(24)
$\displaystyle\left.+g_{3}\zeta^{2}\left[\frac{f^{{}^{\prime}2}}{r^{2}}+\frac{2}{r^{4}}(1-f-\frac{r}{2}f^{{}^{\prime}})^{2}\right]+g_{4}\left[\frac{2(1-f)}{r^{2}}-\frac{2f^{{}^{\prime}}}{r}\right]^{3}\right.$
$\displaystyle\left.+g_{5}\left[\frac{2(1-f)}{r^{2}}-\frac{2f^{{}^{\prime}}}{r}\right]\left[\frac{f^{{}^{\prime}2}}{r^{2}}+\frac{2}{r^{4}}(1-f-\frac{r}{2}f^{{}^{\prime}})^{2}\right]\right.$
$\displaystyle\left.+g_{6}\left[-\frac{f^{{}^{\prime}3}}{r^{3}}+\frac{2}{r^{6}}(1-f-\frac{rf^{{}^{\prime}}}{2})^{3}\right]+g_{7}\left[\frac{2(1-f)}{r^{2}}-\frac{2f^{{}^{\prime}}}{r}\right]\frac{\sqrt{f}}{r^{2}}\partial_{r}\left\\{\frac{1}{\sqrt{f}}r^{2}f\partial_{r}\left[\frac{2(1-f)}{r^{2}}-\frac{2f^{{}^{\prime}}}{r}\right]\right\\}\right.$
$\displaystyle\left.+g_{8}\left[f^{3}\left(\frac{f^{{}^{\prime}}}{r^{2}f}-\frac{f^{{}^{\prime\prime}}}{rf}\right)^{2}+\frac{2f}{r^{4}}\left(\frac{f^{{}^{\prime}}}{2}+\frac{rf^{{}^{\prime\prime}}}{2}+\frac{2(1-f)}{r}\right)^{2}\right]\right\\}.$
Here $N_{r}=N^{r}/f$ and ′ means the derivative with respect to $r$. The full
Lagrangian is $\mathcal{L}=\mathcal{L}_{K}+\mathcal{L}_{V}$. By varying the
action with respect to the functions $N_{r}$ , $f$ and $N(t)$, we obtain three
equations of motions,
$\displaystyle 0$ $\displaystyle=$
$\displaystyle\sqrt{f}\left\\{\partial_{r}\frac{\partial\mathcal{L}}{\partial
N_{r}^{{}^{\prime}}}-\frac{\partial\mathcal{L}}{\partial N_{r}}\right\\}$ (25)
$\displaystyle=$ $\displaystyle
2(1-\lambda)r^{2}f^{2}\frac{1}{N(t)}\left\\{N_{r}^{{}^{\prime\prime}}+\frac{f^{{}^{\prime\prime}}}{2f}N_{r}+\frac{3}{2}\frac{f^{{}^{\prime}}}{f}N_{r}^{{}^{\prime}}+2\frac{N_{r}^{{}^{\prime}}}{r}+\frac{1-2\lambda}{1-\lambda}\frac{f^{{}^{\prime}}}{f}\frac{N_{r}}{r}-2\frac{N_{r}}{r^{2}}\right\\},$
$\displaystyle 0$ $\displaystyle=$
$\displaystyle\sqrt{f}\left\\{\partial_{r}\frac{\partial\mathcal{L}}{\partial
f^{{}^{\prime}}}-\frac{\partial\mathcal{L}}{\partial
f}-\partial_{r}\partial_{r}\frac{\partial\mathcal{L}}{\partial
f^{{}^{\prime\prime}}}\right\\}$ (26) $\displaystyle=$
$\displaystyle\sqrt{f}\left\\{\partial_{r}\frac{\partial\mathcal{L}_{V}}{\partial
f^{{}^{\prime}}}-\frac{\partial\mathcal{L}_{V}}{\partial
f}-\partial_{r}\partial_{r}\frac{\partial\mathcal{L}_{V}}{\partial
f^{{}^{\prime\prime}}}\right\\}-\frac{f^{{}^{\prime}}}{2f}\frac{1}{N(t)}\left\\{(1-\lambda)r^{2}fN_{r}\left(N_{r}^{{}^{\prime}}+N_{r}\frac{f^{{}^{\prime}}}{2f}\right)-2\lambda
rfN_{r}^{2}\right\\}$
$\displaystyle\;+\frac{1}{N(t)}\left\\{(1-\lambda)r^{2}fN_{r}N_{r}^{{}^{\prime\prime}}+\frac{1}{2}(1-\lambda)r^{2}f^{{}^{\prime\prime}}N_{r}^{2}-(1-\lambda)r^{2}fN_{r}^{{}^{\prime}2}+(1-\lambda)r^{2}f^{{}^{\prime}}N_{r}N_{r}^{{}^{\prime}}\right.$
$\displaystyle\;+\left.2(1+\lambda)rfN_{r}N_{r}^{{}^{\prime}}+(1-\lambda)rf^{{}^{\prime}}N_{r}^{2}+(6\lambda-4)fN_{r}^{2}\right\\}+\frac{1}{2\sqrt{f}}\mathcal{L}_{K},$
$\displaystyle 0$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}drr^{2}\frac{1}{N(t)}\left(-\mathcal{L}_{K}+\mathcal{L}_{V}\right).$
(27)
The third equation (27) is a spatially integrated Hamiltonian constraint
because of the “projectability condition” on the lapse function $N(t)$. We
find that for all $\lambda$, $N_{r}=0$ is the solution of the equation (25).
In this case, the equations (26),(27) are the equations depending on the form
of the potential. We can make ansatz $f(r)=1+yr^{2}$, where $y$ is a constant
to be determined. Then we have two cubic equations of $y$
$\displaystyle
g_{0}\zeta^{6}+2\zeta^{4}y+4(3g_{2}+g_{3})\zeta^{2}y^{2}-24(9g_{4}+3g_{5}+g_{6})y^{3}$
$\displaystyle=$ $\displaystyle 0,$ (28) $\displaystyle
g_{0}\zeta^{6}+6\zeta^{4}y-12(3g_{2}+g_{3})\zeta^{2}y^{2}+24(9g_{4}+3g_{5}+g_{6})y^{3}$
$\displaystyle=$ $\displaystyle 0.$ (29)
Here the equation (29) is from the non-local Hamiltonian constraint.
For the solution $f=1+yr^{2}$, the metric now has the form
$ds^{2}=-dt^{2}+\frac{dr^{2}}{1+yr^{2}}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}).$ (30)
Such a metric describes a spacetime of topology
$\mathbb{R}\times\mathbb{M}_{3}$, where $\mathbb{M}_{3}$ is a three-
dimensional maximally symmetric space, could be a flat space, a sphere or a
hyperboloid. If $y=0$, this is just the flat spacetime. If $y<0$, the
spacetime is $\mathbb{R}\times\mathbb{S}^{3}$, where $\mathbb{R}$ is the time
direction, $\mathbb{S}^{3}$ is the three-sphere. If $y>0$, the spacetime is
$\mathbb{R}\times\mathbb{H}^{3}$, where $\mathbb{H}^{3}$ is the three-
dimensional hyperboloid with negative constant curvature. In fact, if one
considers the time-dependent solution, then the latter two solutions are very
similar to closed and open universe with a constant scale factor.
For a general potential, there is no solution to (28) and (29). When
$\zeta=1,\;g_{0}=2\Lambda,\;g_{2}=g_{3}=g_{4}=g_{5}=g_{6}=g_{7}=g_{8}=0$, it
recovers Einstein’s general relativity. The only possible solution requires
$g_{0}=0$ and $y=0$, which corresponds to a flat spacetime. Actually, when the
cosmological constant is vanishing, the flat Minkowski spacetime corresponding
to $y=0$ is always a solution.
For the original Hořava-Lifshitz gravity with the action (II), the equations
(28),(29) become
$\displaystyle y^{2}-2\Lambda_{W}y-3\Lambda_{W}^{2}=0,$ (31) $\displaystyle
y^{2}+2\Lambda_{W}y+\Lambda_{W}^{2}=0.$ (32)
The solution is $y=-\Lambda_{W}$. In this case, the curvature of maximally
symmetric space is determined by the cosmological constant of the theory.
For the general action of modified Hořava-Lifshitz gravity, the existence of
the solution depends on the form of the potential. It is easy to see that the
equations (28),(29) could be reduced to two equations both quadratic in $y$.
It is straightforward to find the condition under which there exist a
solution.
In the IR region, the modified Hořava-Lifshitz gravity recovers the Einstein’s
general relativity except the higher derivative terms on the spatial metric.
When $\lambda=1$, the equation (25) becomes
$\frac{f^{{}^{\prime}}}{f}\frac{N_{r}}{r}=0.$ (33)
Its solutions are $N_{r}=0$ or $f=\mbox{constant}$. The solution $N_{r}=0$ has
been discussed above. When $f$ is a constant, the equations (26),(27) become
$\displaystyle
0=(N_{r}^{2})^{\prime}+\frac{N_{r}^{2}}{r}+\frac{N(t)^{2}}{2f^{2}}$
$\displaystyle\left\\{-g_{0}\zeta^{6}r+\frac{2\zeta^{4}(1-f)}{r}+\frac{2\zeta^{2}(1-f)}{r^{3}}\left[2g_{2}(1+7f)+g_{3}(1+5f)\right]\right.$
(34)
$\displaystyle\left.+\frac{2(1-f)^{2}}{r^{5}}\left[4g_{4}(1+23f)+2g_{5}(1+17f)+g_{6}(1+14f)\right]\right.$
$\displaystyle\left.+\frac{8f(1-f)}{r^{5}}\left[2g_{7}(1+7f)+g_{8}(1-4f)\right]\right\\},$
$\displaystyle 0$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}drr^{3}\left\\{(N_{r}^{2})^{\prime}+\frac{N_{r}^{2}}{r}+\frac{N(t)^{2}}{2f^{2}}\left[-g_{0}\zeta^{6}r+\frac{2\zeta^{4}(1-f)}{r}+\frac{2\zeta^{2}(1-f)^{2}}{r^{3}}\left(2g_{2}+4g_{3}\right)\right.\right.$
(35)
$\displaystyle\left.\left.+\frac{2(1-f)^{3}}{r^{5}}\left(4g_{4}+2g_{5}+g_{6}\right)+\frac{8f(1-f)^{2}}{r^{5}}\left(g_{7}+g_{8}\right)\right]\right\\}.$
It is not hard to find that just when $f=1$ the two equations have the same
solutions of $N_{r}$. In other words, $f$ is constrainted to be $1$. In this
case, the solutions are just
$N_{r}=\pm\;N(t)\sqrt{\frac{g_{0}\zeta^{6}}{6}r^{2}+\frac{M}{r}},$ (36)
where $M$ is an integration constant. For $N_{r}$ is just the function of $r$,
$N(t)$ must be a constant. We could use the freedom of gauge transformation to
set $N(t)=1$. If let $g_{0}\zeta^{6}=3\Lambda_{W}$, the solution (36)
corresponds to a dS-Schwarzschild spacetime written in Painlevé-Gullstrand
type coordinates. The solution is just determined by the kinetic term and the
cosmological constant in the potential. In other words, at IR, the static
spherically symmetric solutions of the modified Hořava-Lifshitz gravity are
the same as the ones in the Einstein’s general relativity. If the theory has a
nonvanishing cosmological constant, the solution is the Schwarzschild solution
in dS spacetime. If the theory has no cosmological constant, the solution is
just the Schwarzschild solution.
In the UV region when $\lambda\neq 1$, similar to the discussion in the IR
region, the equations (25), (26) and (27) have solutions just when $f=1$. In
this case, they become
$\displaystyle 0$ $\displaystyle=$ $\displaystyle
N_{r}^{{}^{\prime\prime}}+2\frac{N_{r}^{{}^{\prime}}}{r}-2\frac{N_{r}}{r^{2}},$
(37) $\displaystyle 0$ $\displaystyle=$
$\displaystyle(1-\lambda)r^{2}N_{r}^{{}^{\prime}2}-4\lambda
rN_{r}N_{r}^{{}^{\prime}}+2(1-2\lambda)N_{r}^{2}+g_{0}N(t)^{2}\zeta^{6}r^{2},$
(38) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}drr^{2}\left\\{(1-\lambda)r^{2}N_{r}^{{}^{\prime}2}-4\lambda
rN_{r}N_{r}^{{}^{\prime}}+2(1-2\lambda)N_{r}^{2}+g_{0}N(t)^{2}\zeta^{6}r^{2}\right\\}.$
(39)
They have solutions as
$N_{r}=\pm\;N(t)\sqrt{\frac{g_{0}\zeta^{6}}{3(3\lambda-1)}}r.$ (40)
We could also use the freedom of gauge transformation to set $N(t)=1$. These
solutions actually describe the same de-Sitter space-time. One easy way to see
this point is to change inversely into the Schwarzschild coordinates.
One subtle issue happens when the cosmological constant $\Lambda_{W}$ is
negative. In this case, $N_{r}$ becomes imaginary in (40). This is not
physical anymore. However, after being transformed into Schwartzschild
coordinates, the metric describes the anti-de-Sitter spacetime. Similarly the
solution (36) becomes imaginary at asymptotic region if $\Lambda_{W}$ is
negative, but it may describe a AdS-Sch. spacetime in the Schwarzschild
coordinates. Since in Hořava-Lifshitz-like gravity, to respect the
projectability condition, the static spherically symmetric solution should
take the form of (22), the solutions with negative $\Lambda_{W}$ are not
acceptable. It would be interesting to see if the AdS and AdS-Sch. spacetime
could be rewritten into a form respecting projectability condition333In Lu2009
, it has been pointed out that the dS-Sch. solution could be rewritten in
terms of the Painlevé-Gullstrand coordinates to respect the projectability
condition. We are also grateful to H.Lu for the discussion on the pathology of
negative $\Lambda_{W}$..
After some tedious calculation, it is straightforward to check that the
solutions (30),(36), and (40) satisfy all the equations of $\delta S/\delta
N(t)=0$, $\delta S/\delta N_{i}=0$ and $\delta S/\delta g_{ij}=0$. Obviously
they are all the solutions of Hořava gravity in the IR region($\lambda=1$). So
the new solutions found in Lu2009 ; park2009 could not satisfy the
“projectability condition”, even though they satisfy the necessary condition
(21). Our result also indicates that in Hořava-Lifshitz-like gravity theory
with the projectability condition, there is no novel correction in solar
system test.
It is also interesting to study the topological black hole in Hořava-Lifshitz
like gravity. It has been discussed in RongCai-2009 without taking into
account of the “projectability condition”. The static spherically symmetric
metric ansatz of a topological spacetime may be written as
$ds^{2}=-dt^{2}+\frac{1}{f(r)}(dr+N^{r}dt)(dr+N^{r}dt)+r^{2}d\Omega_{k}^{2}$
(41)
Here we have set $N(t)=1$ and $d\Omega_{k}^{2}$ denotes the line element for
an 2-dimensional Einstein space with constant scalar curvature $2k$. Without
loss of generality, one may take $k=0,\pm 1$ respectively. Substituting the
metric ansatz (41) into the Lagrangian (II), up to an overall scaling
constant, we get
$\displaystyle\mathcal{L}_{K}=$
$\displaystyle\frac{1}{\sqrt{f}}\left\\{(1-\lambda)r^{2}f^{2}\left(N^{{}^{\prime}}_{r}+N_{r}\frac{f^{{}^{\prime}}}{2f}\right)^{2}+2(1-2\lambda)f^{2}N_{r}^{2}\right.$
$\displaystyle\left.-4\lambda
rf^{2}N_{r}\left(N^{{}^{\prime}}_{r}+N_{r}\frac{f^{{}^{\prime}}}{2f}\right)\right\\},$
$\displaystyle\mathcal{L}_{V}=$
$\displaystyle\frac{1}{\sqrt{f}}r^{2}\left\\{-g_{0}\zeta^{6}+\zeta^{4}\left[\frac{2(k-f)}{r^{2}}-\frac{2f^{{}^{\prime}}}{r}\right]+g_{2}\zeta^{2}\left[\frac{2(k-f)}{r^{2}}-\frac{2f^{{}^{\prime}}}{r}\right]^{2}\right.$
(42)
$\displaystyle\left.+g_{3}\zeta^{2}\left[\frac{f^{{}^{\prime}2}}{r^{2}}+\frac{2}{r^{4}}(k-f-\frac{r}{2}f^{{}^{\prime}})^{2}\right]+g_{4}\left[\frac{2(k-f)}{r^{2}}-\frac{2f^{{}^{\prime}}}{r}\right]^{3}\right.$
$\displaystyle\left.+g_{5}\left[\frac{2(k-f)}{r^{2}}-\frac{2f^{{}^{\prime}}}{r}\right]\left[\frac{f^{{}^{\prime}2}}{r^{2}}+\frac{2}{r^{4}}(k-f-\frac{r}{2}f^{{}^{\prime}})^{2}\right]\right.$
$\displaystyle\left.+g_{6}\left[\frac{f^{{}^{\prime}3}}{r^{3}}+\frac{2}{r^{6}}(k-f-\frac{rf^{{}^{\prime}}}{2})^{3}\right]+g_{7}\left[\frac{2(k-f)}{r^{2}}-\frac{2f^{{}^{\prime}}}{r}\right]\frac{\sqrt{f}}{r^{2}}\partial_{r}\left\\{\frac{1}{\sqrt{f}}r^{2}f\partial_{r}\left[\frac{2(k-f)}{r^{2}}-\frac{2f^{{}^{\prime}}}{r}\right]\right\\}\right.$
$\displaystyle\left.+g_{8}\left[f^{3}\left(\frac{f^{{}^{\prime}}}{r^{2}f}-\frac{f^{{}^{\prime\prime}}}{rf}\right)^{2}+\frac{2f}{r^{4}}\left(\frac{f^{{}^{\prime}}}{2}+\frac{rf^{{}^{\prime\prime}}}{2}+\frac{2(k-f)}{r}\right)^{2}\right]\right\\}.$
Here $N_{r}=N^{r}/f$ and ′ means the derivative with respect to $r$. The full
Lagrangian is $\mathcal{L}=\mathcal{L}_{K}+\mathcal{L}_{V}$. The $k=1$ case
has been discussed above. Comparing with (III), we find that the kinetic term
is exactly the same, and the difference in the potential term coming from the
factor $(k-f)$ in (III) and $(1-f)$ in (III). By varying the action with
respect to the functions $N_{r}$, $f$ and $N(t)$, we could get three equations
of motions which are quite similar to (25),(26) and (27), with $(1-f)$ being
replaced with $(k-f)$. Therefore the solutions are quite similar to the ones
when $k=1$.
The case $k=1$ has been discussed above. In the case $k=-1$, for the solution
with $f$ being a constant, $f$ must be set to $-1$. At IR, $\lambda=1$,
$N_{r}=\pm\;\sqrt{\frac{g_{0}\zeta^{6}}{6}r^{2}+\frac{M^{\star}}{r}}$, where
$M^{\star}$ is an integration constant. They correspond to an
(dS-)Schwarzschild type’s topological black hole written in Painlevé-
Gullstrand type coordinates. When $\lambda\neq 1$,
$N_{r}=\pm\sqrt{\frac{g_{0}\zeta^{6}}{3(3\lambda-1)}}r$. These solutions
actually describe the de-Sitter space-time or Minkowski spacetime with
topological twist. In the case $k=0$, because $f$ can’t be zero, we only have
the solution “$N_{r}=0,\,f=yr^{2}$” in which $y$ satisfy the equation
(28),(29). In any case, these solutions are different from the ones studied in
RongCai-2009 .
## Acknowledgments
The work was partially supported by NSFC Grant No.10535060, 10775002, 10975005
and RFDP.
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* (15) Thomas P. Sotiriou, Matt Visser, Silke Weinfurtner, “Phenomenologically viable Lorentz-violating quantum gravity”, Phys.Rev.Lett.102:251601,2009, arXiv:0904.4464 [hep-th];“Quantum gravity without Lorentz invariance,” arXiv:0905.2798 [hep-th].
* (16) Painlevé P. La mécanique classique el la theorie de la relativité(Classical mechanics of the theory of relativity).C. R. Acad. Sci. (Paris), 173 (1921), 677 C680.
* (17) Gullstrand A. Allegemeine l$\ddot{o}$sung des statischen eink$\ddot{o}$rper-problems in der einsteinshen gravitations theorie (General solution for static onebody problems in Einstein s theory of gravity)._Arkiv. Mat. Astron. Fys._ ,16(8) (1922), 1-15.
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* (19) Hawking S W and Israel S W (editors)._Three hundred years of gravitation_. Cambridge University Press, England (1987). See especially the discussion on page 234.
* (20) Rong-Gen Cai, Li-Ming Cao, Nobuyoshi Ohta, “Topological Black Holes in Horava-Lifshitz Gravity,” Phys. Rev. D 80, 024003 (2009) [arXiv:0904.3670 [hep-th]]
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|
arxiv-papers
| 2009-09-23T03:50:40 |
2024-09-04T02:49:05.470174
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jin-Zhang Tang, Bin Chen",
"submitter": "Jinzhang Tang",
"url": "https://arxiv.org/abs/0909.4127"
}
|
0909.4529
|
# New approach to numerical computation of the eigenfunctions of the
continuous spectrum of three-particle Schrödinger operator.
I. One-dimensional particles, short-range pair potentials
V. S. Buslaev1, S. B. Levin1, P. Neittaanmäki2, T. Ojala2
###### Abstract
Basing on analogy between the three-body scattering problem and the
diffraction problem of the plane wave (for the case of the short range pair
potentials) by the system of six half transparent screens, we presented a new
approach to the few-body scattering problem. The numerical results have been
obtained for the case of the short range nonnegative pair potentials. The
presented method allows a natural generalization to the case of the long range
pair potentials.
1Department of Mathematical and Computational Physics,
St-Petersburg State University, Russia
2Department of Mathematical Information Technology,
University of Jyvaskyla, Finland
## 1 Introduction
### 1.1
The quantum system of two particles interacting via the Coulomb potential is
probably the most known model of the Quantum mechanics. The model allows an
explicit solution. Oppositely, the mathematical status of the system of three
quantum particles with the pair Coulomb interaction is relatively poor. The
system of three particles with short range pair interactions was successfully
studied by L. Faddeev [1], but the direct generalization to the Coulomb type
potentials was found impossible. Something, however, is known: the quantative
nature of the spectrum and the asymptotic behavior of the solutions of the
non-stationary Schrödinger equation. These results were obtained in frameworks
of a non-stationary approach, see [2, 3]. Nevertheless a mathematically
consistent stationary approach similar to the Lippmann-Schwinger integral
equation, or something analogous, was not developed though. Such an approach
is needed if we are interested in numerical parameters of many important
physical processes like dissociative recombination in atomic and molecular
physics with applications to astrophysics, formation and break up processes of
large molecules in bioengineering and medicine, formation of the molecular
resonance states in chemical physics, dynamics of the few electron systems in
wave-conductor nano-technology.
There are specific difficulties that are characteristic for the systems with
Coulomb type interactions.
They are naturally explained by the fact that the long range interactions
crucially affects the asymptotic behavior at infinity in the configuration
space of the eigenfunctions, Green’s functions and other similar objects. The
consequences of that affection on the structure of asymptotics up to now have
not been taken in account in correct mathematical manner. As a result, such
approaches to many particle scattering as Faddeev’s equations [1], AGS
equations [5], successfully applicable to the systems with short range
potentials, do not work for the systems with the long range potentials.
The asymptotic behavior of the wave functions for the systems of few charged
particles has been studied only in some domains of configuration space but not
for all asymptotic directions. Let us shortly list some known results. In [6,
7] there was studied the asymptotic behavior of three charged particles wave
function for the case of large distances between all three particles. Another
limiting case, considered in [8], corresponds to configurations with one
Jacobi coordinates been much larger than another one.
In the list of the literature reflecting the theoretical aspects of the
problem we mention also [10], [11], [12], [13] , [14, 15, 16]. Application to
computational aspects of the problem were treated in [17, 18, 19], [20, 21,
22], [23].
One of the typical computational approaches to such systems is to replace the
Coulomb potentials by the Yukava potentials (or some other cut-off
potentials), to compute the parameters of the scattering for this modified
system, and to consider the results for small screening parameter.
Mathematically, it is not a completely satisfactory procedure. Some other
approximate approaches also exist.
### 1.2
We started a description of a new approach to the mathematically consistent
stationary treatment of the scattering in the systems of three quantum
particles with long range pair interactions in [27]. We are going to consider
in turn the case of three one-dimensional particles with short range
interaction (it is already completed theoretically and published), the case of
three one-dimensional particles with long range (Coulomb type) interactions
(it is also completed at the moment but is not published yet) and the case of
three three-dimensional particles. Each next case will be based on the results
for the preceding stage. We hope that we will be able to illustrate each
theoretical stage by numerical computation of the field. This paper contains
the numerical results illustrating the formulas of paper [27].
We assume here that the pair potentials are non-negative. In this case the
spectrum is purely continuous, covers the positive semi-axis and is in the
natural sense homogeneous. In fact, this case is the most interesting at the
present stage since the lower spectral branches for negative total energy in
case of charged particles was already treated in [28].
It is worth mentioning that the scattering in the system of one-dimensional
particles is not just a first step on the way to the case of three-dimensional
particles. It is interesting by itself, the systems of three one-dimensional
particles (neutral or charged) were intensively studied during many years
(see, for example, [29, 30, 31, 32, 33]). In recent years there appeared a new
interest to such systems since they were realized experimentally (see [34, 35,
36, 37]).
The main idea is to suggest a priori explicit formulas for the asymptotic
behavior of the eigenfunctions of the continuous spectrum (for example, the
scattered plane waves).
The formulas describe the eigenfunctions at infinity up to the simple
diverging waves with smooth amplitudes. If we are able to find such asymptotic
behavior (satisfying certain criteria that will be discussed later on) even
heuristically, we obtain a way for regular numerical computations of the
eigenfunctions. We obtain simultaneously also a method to construct an
appropriate integral equation of the same nature as the Lippmann-Schwinger
equation for the scattering of the plane wave by a quickly decreasing
potential that can be used to justify the asymptotic behavior rigorously
following the ideas of [4].
For one-dimensional particles with quickly decreasing at infinity pair
potentials we can use, for the description of the mentioned asymptotic
behavior, the analogy between the stated problem and the classical problem of
the diffraction of the plane waves by the set of semi-transparent infinite
screens. This analogy was already used in [24, 25, 26, 27]. In case of long
range potentials we are able to treat the diffraction problem analogously with
the replacement of the classical plane waves by plane waves that are
appropriately deformed by the long range tails of the Coulomb potentials. It
is important to mention that the diffraction itself and the corresponding
scattering problems cannot be completely reduced to the scattering of the
plane waves by the screens; we have to add to these processes some genuine
diffraction components that have more complicated analytical structure but
still explicit description. This more complicated structure is also dictated
by the analogy with the classical diffraction theory.
Here we consider a system of three identical one-dimensional quantum particles
interacting via short-range pair potentials. These strict limitations allow to
simplify the narration and the view of the formulas, but the essence of the
main questions which we are interested in and their treatments is not
affected. In the following parts we consequentially will get rid of these
limitations. As we have mentioned above, the theoretical part of this work is
already published, but we decided for the completeness to repeat shortly the
main theoretical ideas of [27]. The main goal of the work is to confirm that
the approach works for the numerical computation of the eigenfunctions of the
continuous spectrum. The approach is new even for the short range pair
potentials.
The structure of the work is as following: it consists of two parts. The first
part is devoted to the known theoretical constructions. The second one is
original and represents the results of numerical computer computations.
## 2 Main formulas
### 2.1 Configuration plane
The configuration space of the system after the separation of motion of the
center of mass is the hyperplane
$\Gamma=\\{\mathbf{x}=(x_{1},x_{2},x_{3}):x_{1}+x_{2}+x_{3}=0\\}$ in
$\mathbf{R}^{3}$. The Schroedinger equation has the form:
$-\triangle\psi+(v(x_{1})+v(x_{2})+v(x_{3}))\psi=E\psi,$ (1)
where $\psi=\psi(\mathbf{x})\in\mathbf{C}$, $\triangle$ is the Laplace
operator on $\Gamma$ that will be described more specifically later on. The
real-valued function $v(x),\ x\in\mathbf{R},$ is the potential of the pair
interaction. In the present text it is supposed to be an even function with a
compact support, $v(x)=0,\ |x|>b/2.$ We suppose $E>0$.
The scalar product on $\Gamma$ is given by the formula
$<\mathbf{x},\mathbf{x}^{\prime}>=\frac{2}{3}(x_{1}x^{\prime}_{1}+x_{2}x^{\prime}_{2}+x_{3}x^{\prime}_{3}).$
(2)
As usual, the norm of the vector is defined by the formula
$|\mathbf{x}|^{2}=<\mathbf{x},\mathbf{x}>.$ The Laplacian is also generated by
this scalar product.
Let us consider on $\Gamma$ three straight lines
$l_{j}=\\{\mathbf{x}:x_{j}=0\\},\ j=1,2,3,$ and three unit vectors
$\mathbf{l}_{j}$ that belong to these lines and oriented such that $x_{j+1}$
increases along $\mathbf{l}_{j}$. Consider also the unit vectors
$\mathbf{k}_{j}$ that are orthogonal to $\mathbf{l}_{j}$ and oriented along
the direction of increasing of $x_{j}$. Consider, at last, three pairs of the
cartesian coordinates $(x_{j},y_{j})$ with respect to the bases
$(\mathbf{k}_{j},\mathbf{l}_{j})$. These are, so called, Jacobian coordinates
on $\Gamma$. With these coordinates
$<\mathbf{x},\mathbf{x}^{\prime}>=x_{j}x^{\prime}_{j}+y_{j}y^{\prime}_{j},\quad|\mathbf{x}|^{2}=x_{j}^{2}+y_{j}^{2},\quad
j=1,2,3,$ (3)
and
$\triangle=\frac{\partial^{2}}{\partial
x_{j}^{2}}+\frac{\partial^{2}}{\partial y_{j}^{2}},\quad j=1,2,3.$ (4)
The lines $l_{j}$ define on the plane $\Gamma$ six sectors. The internal part
of a certain one consists of the vectors $(x_{1},x_{2},x_{3})$ whose
coordinates satisfy the condition $x_{j_{1}}>x_{j_{2}}>x_{j_{3}}$ where
$\sigma=(j_{1},j_{2},j_{3})$ is a permutation of the numbers $(123)$. We will
denote any sector by the corresponding permutation $\sigma$ and will write
$\lambda=\lambda_{\sigma}$ (see Figure 1).
Figure 1
Let the group $S_{3}$ of permutation acts on $\Gamma$ so that
$(\sigma,\mathbf{x})\rightarrow\sigma\mathbf{x}=(x_{j_{1}},x_{j_{2}},x_{j_{3}}),\
\ \ \ \sigma=(j_{1},j_{2},j_{3}),\ \ \ \ \mathbf{x}=(x_{1},x_{2},x_{3}).$
The group contains 6 elements. The permutation can be identical, or a
transposition of two elements, or a composition of two transpositions, some of
the compositions coincide. Introduce the notations for the transpositions:
$\tau_{1}=(132)$, $\tau_{2}=(321)$, $\tau_{3}=(213)$, and notice that
$\tau_{i}^{2}=I,\ \ \ i=1,2,3$. The action of the transposition on $\Gamma$
will be denoted by the same symbol $\tau_{j},\ \ j=1,2,3$. It corresponds to
the reflection with respect to the line $l_{j},\ \ j=1,2,3$. It is clear that
$\tau_{1}(x_{1},x_{2},x_{3})=(-x_{1},-x_{3},-x_{2}),\quad\tau_{1}(y_{1},y_{2},y_{3})=(y_{1},y_{3},y_{2}),$
(5)
the analogous formulas are also satisfied for $\tau_{2},\tau_{3}.$ The
composition of two transpositions generates a rotation, and the following
equalities, in particular, hold:
$\tau_{1}\tau_{2}=\tau_{3}\tau_{1}=\tau_{2}\tau_{3}$,
$\tau_{2}\tau_{1}=\tau_{1}\tau_{3}=\tau_{3}\tau_{2}$.
Six elements $\tau$ of the group $S_{3}$ generate six vectors
$\tau\mathbf{q}$. If $\mathbf{q}\in\lambda_{\sigma}$ then
$\tau\mathbf{q}\in\lambda_{\tau\sigma}$.
### 2.2 Separation of variables
Consider now the eigenfunction that describes the scattering in the system
where just one of three potentials is not equal to zero. Now we deal with the
Schrödinger equation
$-\triangle\chi_{j}+v(x_{j})\chi_{j}=E\chi_{j}.$ (6)
It allows the separation of variables:
$\chi_{j}(\mathbf{x},\mathbf{q})=\chi(x_{j},k_{j})e^{ip_{j}y_{j}}.$ (7)
The sense of the variables $(x_{j},y_{j})$ is clear, $(k_{j},p_{j})$ are the
Jacobian coordinates of a given vector $\mathbf{q}$.
The function $\chi(x,k),x,k\in\mathbf{R},$ is a solution of the ordinary
differential equation
$-\chi_{xx}+v(x)\chi=k^{2}\chi,$ (8)
it has to be described separately. For $k>0$ there exists and is unique the
solution that is characterized by the following asymptotic behavior:
$\chi(x,k)\sim s(k)e^{ikx},\ x\to+\infty;\quad\chi(x,k)\sim
e^{ikx}+r(k)e^{-ikx},\ x\to-\infty.$ (9)
On the whole axis $k$ this solution, due to the evenness of the potential, has
to be extended by the formula $\chi(x,k)=\chi(-x,-k)$. Here $s$ and $r$ are
some complex-valued functions of $k$ that are called the transition and the
reflection coefficients.
We will suppose here that $v(x)\geq 0$ therefore the equation (8) does not
have the bound states.
### 2.3 Formal setting of the problem
Our final goal is to construct the solution $\psi(\mathbf{x},\mathbf{q})$ of
the Schrödinger equation that is characterized by the following behavior at
infinity:
$\psi=n({\hat{\mathbf{x}}},\mathbf{q})\frac{e^{-i|\mathbf{x}||\mathbf{q}}|}{|\mathbf{x}|^{1/2}}+f({\hat{\mathbf{x}}},\mathbf{q})\frac{e^{i|\mathbf{x}||\mathbf{q}|}}{|\mathbf{x}|^{1/2}}+o\left(\frac{1}{|\mathbf{x}|^{1/2}}\right),\quad{\hat{\mathbf{x}}}=\frac{\mathbf{x}}{|\mathbf{x}|}.$
(10)
Here
$n({\hat{\mathbf{x}}},\mathbf{q})=\sqrt{\frac{2\pi}{i|\mathbf{q}|}}\delta({\hat{\mathbf{x}}},{\hat{\mathbf{q}}}),$
(11)
and the $\delta$ function has to be considered with respect to the angle
measure on the unit circle.
The asymptotic behavior has to be treated in a weak sense (in sense of
distributions) with respect to ${\hat{\mathbf{x}}}$. The coefficient $n$
before the converging circle wave coincides with the analogous coefficient
before the converging wave in the weak asymptotic representation of the plane
wave $e^{i<\mathbf{x},\mathbf{q}>}$. Therefore the solution
$\psi(\mathbf{x},\mathbf{q})$ can be naturally called the scattered plane
wave.
Due to the symmetries of the potential
$\psi(\mathbf{x},\mathbf{q})=\psi(\sigma\mathbf{x},\sigma\mathbf{q}),\
\sigma\in S,$ we always can assume that $\mathbf{q}$ belongs to a certain
sector, say $\lambda_{I}\equiv\lambda_{123}$. We restrict ourselves here by
the assumption that $\mathbf{q}$ does not belong to neighborhoods of the
boundaries of the sector. It would be not hard to consider also the case when
$\mathbf{q}$ belongs to the lines $l_{j}$ and their neighborhoods.
The function $f$ is a singular distribution. We will see that it has
singularities on all six directions $\sigma\mathbf{q},\ \sigma\in S$. Four of
them are of $\delta$ \- function type, two (for
$\sigma\mathbf{q}=\tau_{2}\tau_{3}\mathbf{q},\ \tau_{2}\tau_{1}\mathbf{q}$)
are of type of Cauchy’s limiting kernel. It is worth to notice that although
the asymptotic behavior is singular the solution itself is, naturally, a
smooth function.
In the case of the scattering by a quickly decreasing at infinity potential
the asymptotic behavior is given by the formula
$\psi(\mathbf{x},\mathbf{q})=e^{i<\mathbf{x},\mathbf{q}>}+f({\hat{\mathbf{x}}},\mathbf{q})\frac{e^{i|\mathbf{x}||\mathbf{q}|}}{|\mathbf{x}|^{1/2}}+o\left(\frac{1}{|\mathbf{x}|^{1/2}}\right),$
(12)
where that time the scattering amplitude $f$ is not a singular distribution,
but a smooth function, and the asymptotic behavior can be treated in uniform
sense.
Under our assumptions over the potential the scattered plane waves create for
$E>0$ a complete system of the eigenfunctions of the uniform in multiplicity
continuous spectrum of the three particle Schrödinger operator, $E\geq 0$.
Our further plan is following: we construct in explicit form a function
$\psi_{1}(\mathbf{x},\mathbf{q})$ and hope that the difference $\psi-\psi_{1}$
has the diverging asymptotic behavior
$\psi(\mathbf{x},\mathbf{q})-\psi_{1}(\mathbf{x},\mathbf{q})=g({\hat{\mathbf{x}}},\mathbf{q})\frac{e^{i|\mathbf{x}||\mathbf{q}|}}{|\mathbf{x}|^{1/2}}+o\left(\frac{1}{|\mathbf{x}|^{1/2}}\right),$
(13)
where $g$ is a continuous function of the arguments.
Constructing $\psi_{1}$ we use two criteria: (1) The discrepancy
$Q[\psi_{1}](\mathbf{x},\mathbf{q})=-\triangle\psi_{1}+(v(x_{1})+v(x_{2})+v(x_{3}))\psi_{1}-E\psi_{1},\quad
E=|\mathbf{q}|^{2}.$ (14)
sufficiently quickly vanishes at infinity, (2) The asymptotic representation
for $\psi_{1}-e^{i<\mathbf{x},\mathbf{q}>}$ contains asymptotically only the
diverging wave.
Consider the difference
$\xi=\psi-\psi_{1}.$ (15)
It satisfies the equation
$H\xi-E\xi=-Q,\quad H=-\triangle+(v(x_{1})+v(x_{2})+v(x_{3})).$ (16)
Since $Q$ is quickly vanishing one can hope that $\xi$ asymptotically behaves
as the diverging wave
$\xi(\mathbf{x},\mathbf{q})=g(\widehat{\mathbf{x}},\mathbf{q})\frac{e^{i|\mathbf{q}||\mathbf{x}|}}{|\mathbf{x}|^{1/2}}+o\left(\frac{1}{|\mathbf{x}|^{1/2}}\right),$
(17)
with a continuous amplitude $g$. In other words, $\xi$ satisfies the classical
radiation conditions at infinity.
Further, it is naturally to hope that for $\xi$ we can construct an integral
equation with the same properties as the properties of classical Lippmann-
Schwinger equation. We can do it developing the ideas of work [4]. However,
preliminary, we can try to use (16)-(17) for the numerical computation of
$\xi$ and, consequently, of $\psi$. For the numerical computations we can
replace (17) by approximate boundary condition
$\left(\frac{\partial}{\partial|x|}-i\sqrt{E}\right)\xi=0\,,\ \ \text{for}\ \
|x|=R,$ (18)
where $R$ is sufficiently large.
The following construction of $\psi_{1}$ will consist of two steps. At first,
we construct for $\psi_{1}$ so called ray approximation $\psi_{R}$. Its
discrepancy has some singularities. After a natural modification motivated by
some classical diffraction problems the discrepancy will become a smooth
function.
### 2.4 Ray approximation
Consider six vectors $\sigma\mathbf{q}$. These vectors, more precisely,
spanned by them rays, separate six sectors that we denote $K_{j}^{\pm}$. The
indices of the notation coincide with the indices of the vector
$\pm\mathbf{l}_{j}$ that belongs to the sector $K_{j}^{\pm}$.
Now we can give explicit expressions for the ray approximation in different
sectors $K_{j}^{\pm}$.
Sector $K_{1}^{+}$: $\psi_{R}=\psi_{1}^{+}$,
$\psi_{1}^{+}(\mathbf{x},\mathbf{q})=\chi_{1}(\mathbf{x},\mathbf{q})s_{2}s_{3}.$
We use here the following notations: $s_{j}=s(k_{j}),\ \ r_{j}=r(k_{j})$.
Sector $K_{3}^{-}$: $\psi_{R}=\psi_{3}^{-}$,
$\psi_{3}^{-}(\mathbf{x},\mathbf{q})=\chi_{3}(\mathbf{x},\mathbf{q})s_{1}s_{2}.$
Sector $K_{2}^{+}$: $\psi_{R}=\psi_{2}^{+}$,
$\psi_{2}^{+}(\mathbf{x},\mathbf{q})=\chi_{2}(\mathbf{x},\mathbf{q})s_{1}+\chi_{2}(\mathbf{x},\tau_{3}\mathbf{q})s_{2}r_{3}.$
Sector $K_{2}^{-}$: $\psi_{R}=\psi_{2}^{-}$,
$\psi_{2}^{-}(\mathbf{x},\mathbf{q})=\chi_{2}(\mathbf{x},\mathbf{q})s_{3}+\chi_{2}(\mathbf{x},\tau_{1}\mathbf{q})s_{2}r_{1}.$
Sector $K_{1}^{-}$: $\psi_{R}=\psi_{1}^{-}$,
$\psi_{1}^{-}(\mathbf{x},\mathbf{q})=\chi_{1}(\mathbf{x},\mathbf{q})+\chi_{1}(\mathbf{x},\tau_{2}\mathbf{q})r_{2}s_{1}+\chi_{1}(\mathbf{x},\tau_{3}\tau_{1}\mathbf{q})r_{2}r_{1}+\chi_{1}(\mathbf{x},\tau_{3}\mathbf{q})r_{3}.$
Sector $K_{3}^{+}$: $\psi_{R}=\psi_{3}^{+}$,
$\psi_{3}^{+}(\mathbf{x},\mathbf{q})=\chi_{3}(\mathbf{x},\mathbf{q})+\chi_{3}(\mathbf{x},\tau_{2}\mathbf{q})r_{2}s_{3}+\chi_{3}(\mathbf{x},\tau_{1}\tau_{3}\mathbf{q})r_{2}r_{3}+\chi_{3}(\mathbf{x},\tau_{1}\mathbf{q})r_{1}.$
The total field $\psi_{R}$ is defined by the formula
$\psi_{R}=\theta_{1}^{+}\psi_{1}^{+}+\theta_{3}^{-}\psi_{3}^{-}+\theta_{2}^{+}\psi_{2}^{+}+\theta_{2}^{-}\psi_{2}^{-}+\theta_{1}^{-}\psi_{1}^{-}+\theta_{3}^{+}\psi_{3}^{+}.$
The notation $\theta_{j}^{(\pm)}$ is used here for the characteristic function
of the corresponding sector $K_{j}^{\pm}$,
$\theta_{1}^{+}+\theta_{3}^{-}+\theta_{2}^{+}+\theta_{2}^{-}+\theta_{1}^{-}+\theta_{3}^{+}=1.$
In this formula the value of the field $\psi_{R}$ on the boundaries of the
sectors is not defined. In [27] it was shown that on all boundary rays except
two, directed along the vectors
$\mathbf{q}_{23}\equiv\tau_{2}\tau_{3}\mathbf{q},\ \ \
\mathbf{q}_{21}\equiv\tau_{2}\tau_{1}\mathbf{q},$
the field is smooth, and its discrepancy everywhere except the two vectors is
equal to zero.
### 2.5 Diffraction corrections
The diffraction corrections on rays directed along the vectors
$\mathbf{q}_{23}$ and $\mathbf{q}_{21}$, can be constructed quite easily.
Consider the sector $\lambda_{231}$ containing $\mathbf{q}_{23}$. Introduce
the polar coordinates $(r=|\mathbf{x}|,\omega)$. Let us orient the angle from
$\mathbf{l}_{2}^{+}$ to $\mathbf{l}_{1}^{-}$. Let $\omega_{23}$ correspond to
$\mathbf{q}_{23}$. Introduce four angles
$0<\omega_{1}<\omega_{2}<\omega_{23}<\omega_{3}<\omega_{4}<\pi/3$. Consider
the open covering of the interval $(0,\pi/3)$ by the subintervals
$(0,\omega_{2})$, $(\omega_{1},\omega_{4}$), $(\omega_{3},\pi/3)$ and
introduce a subordinated partition of unit:
$1=\zeta_{1}+\zeta_{2}+\zeta_{3}.$ (19)
Further consider the function
$\Phi(\alpha)=\frac{e^{-i\frac{\pi}{4}}}{\sqrt{\pi}}\int_{\infty}^{\alpha}e^{it^{2}}dt.$
(20)
Notice that
$\Phi(\alpha)\to 1,{\text{as}}\,\,\alpha\to+\infty,\quad\Phi(\alpha)\to 0,\
{\text{as}}\,\,\alpha\to-\infty.$ (21)
In more detail:
$\Phi(\alpha)=1+\frac{e^{-i\frac{\pi}{4}}}{\sqrt{\pi}}\frac{e^{i\alpha^{2}}}{2i\alpha}+\Delta\Phi(\alpha),\quad\Delta\Phi(\alpha)=-\frac{e^{-i\frac{\pi}{4}}}{\sqrt{\pi}}\int_{\alpha}^{\infty}\frac{e^{it^{2}}}{2it^{2}}dt=O(\alpha^{-3}),\
\ {\text{when}}\,\,\alpha\to+\infty.$
Introduce the function
$\Phi^{(23)}_{1}=\Phi(sign(\omega_{23}-\omega)||\mathbf{q}_{23}||\mathbf{x}|-<\mathbf{q}_{23},\mathbf{x}>|^{1/2}),$
(22)
$\Phi^{(23)}_{2}=\Phi(sign(\omega-\omega_{23})||\mathbf{q}_{23}||\mathbf{x}|-<\mathbf{q}_{23},\mathbf{x}>|^{1/2}).$
(23)
It is known that
$\phi=e^{i<\mathbf{x},\mathbf{q}_{23}>}\Phi^{(23)}_{j}$ (24)
satisfies the Helmholtz equation $-\triangle\phi-E\phi=0.$
Now we can describe the diffraction corrections to the ray approximation on
$\lambda_{231}$. For that the ray field
$\psi_{R}=\theta_{2}^{+}\psi_{2}^{+}+\theta_{1}^{-}\psi_{1}^{-}$ in the sector
$\lambda_{231}$ is replaced by
$\psi_{D}^{(23)}=\psi_{R}+\zeta_{2}e^{i<{\mathbf{q}}_{23},\mathbf{x}>}[R_{1}(\Phi^{(23)}_{1}-\theta_{2}^{+})+R_{2}(\Phi^{(23)}_{2}-\theta_{1}^{-})].$
(25)
$R_{1}=r_{1}s_{2}r_{3},R_{2}=r_{3}r_{2}s_{1}+s_{3}r_{2}r_{1}$.
Notice that the field $\psi_{R}$ on the interval $(\omega_{1},\omega_{4})$
contains the discontinuous component
$\psi_{J}=e^{i<{\mathbf{q}}_{23},\mathbf{x}>}[\theta_{2}^{+}R_{1}+\theta_{1}^{-}R_{2}],$
(26)
so the sense of the modification in nothing else but a simple replacement of
this discontinuous on $\mathbf{q}_{23}$ component by a smooth solution of the
Helmholtz equation that outside of $(\omega_{2},\omega_{3})$ gradually
transfers to the original discontinuous component up to a diverging circle
wave with a smooth amplitude. Outside of the interval
$(\omega_{1},\omega_{4})$ the function $\psi_{D}$ coincides with the original
ray approximation $\psi_{R}$.
Analogous constructions can be also considered in the sector $\lambda_{312}$.
It is also worth to introduce here the polar coordinates, and again to suppose
that the angle $\omega$ varies in the same limits with the same orientation,
from $\mathbf{l}_{3}$ to $-\mathbf{l}_{2}$. We again can introduce the angles
$\omega_{21},\ \omega_{j},\ j=1,2,3,4$ and a cutoff function $\zeta_{2}.$
After that the modified field on $\lambda_{312}$ can be described by the
formula
$\psi_{D}^{(21)}=\psi_{R}+\zeta_{2}e^{i<{\mathbf{q}}_{21},\mathbf{x}>}[R_{2}(\Phi^{(21)}_{1}-\theta_{3}^{+})+R_{1}(\Phi^{(21)}_{2}-\theta_{2}^{-})].$
(27)
Here
$\Phi^{(21)}_{1}=\Phi(sign(\omega_{21}-\omega)||\mathbf{q}_{21}||\mathbf{x}|-<\mathbf{q}_{21},\mathbf{x}>|^{1/2}),$
(28)
$\Phi^{(21)}_{2}=\Phi(sign(\omega-\omega_{21})||\mathbf{q}_{21}||\mathbf{x}|-<\mathbf{q}_{21},\mathbf{x}>|^{1/2}).$
(29)
As a result everywhere on $\Gamma$ outside of some circle $C_{r_{1}}$ with the
center at $0$ and the radius $r_{1}$ there appears a smooth approximate wave
field $\psi_{0}$:
$\psi_{0}=\psi_{R}\theta_{I}+\psi_{D}^{(23)}\theta_{231}+\psi_{D}^{(21)}\theta_{312}.$
(30)
Here $\theta_{231}$ and $\theta_{312}$ are the characteristic functions of the
corresponding $\lambda$-sectors, and $\theta_{I}$ is the characteristic
function of their complement. Again there are no jumps on the boundaries of
the $\lambda$-sectors.
Consider a circle with the center at the origin. The radius $r_{1}$ of this
circle is defined by the condition that outside of the circle on the rays
directed along the vectors $\sigma\mathbf{q}$ the sum of the pair potentials
is equal to zero. Under this condition the field $\psi_{0}$ can be
additionally modified with the help of the cutoff function
$\zeta(|\mathbf{x}|)$ that is equal to $0$ for $|\mathbf{x}|<r_{1}$ and to $1$
for $|\mathbf{x}|>r_{2}$ where $r_{1}<r_{2}$.
The final expression for the approximate field is now
$\psi_{1}=\psi_{0}\zeta.$ (31)
### 2.6 Discrepancy
We remember that there were proposed two criteria that have to be taken into
account when constructing the function $\psi_{1}$. It is sufficiently clear
that the second one : (2) The difference
$\psi_{1}-e^{i<\mathbf{x},\mathbf{q}>}$ contains asymptotically (in the weak
sense) only the diverging circle wave, is fulfilled. It is remained to check
the first one: (1) The discrepancy
$Q[\psi_{1}](\mathbf{x},\mathbf{q})=-\triangle\psi_{1}+(v(x_{1})+v(x_{2})+v(x_{3}))\psi_{1}-E\psi_{1},\quad
E=|\mathbf{q}|^{2}.$ (32)
sufficiently quickly vanishes at infinity.
From the previous formulas it follows that outside a certain circle of the
radius $r_{1}$ the discrepancy is not equal to zero only on some neighborhoods
the rays generated by the vectors $\mathbf{q}_{23}$ and $\mathbf{q}_{21}$. On
these neighborhoods the discrepancy vanishes as $|\mathbf{x}|^{-5/2}$. It
follows from this that the relative scattering amplitude
$g({\hat{\mathbf{x}}},\mathbf{q})$, see (13), must be continuous. Here we give
for the discrepancy a formula that can be used for the numerical computations
of $\psi$.
Consider now the field $\psi_{1}$ on the neighborhoods of $\mathbf{q}_{23}$
and $\mathbf{q}_{21}$. It is not hard to see that the discrepancy of this
expression is equal to zero on the sectors where there is equal to zero the
derivative of the function $\zeta_{2}$. It means that the discrepancy
$Q[\psi_{0}]$ can differ from zero only on the subintervals
$(\omega_{1},\omega_{2})$ and $(\omega_{3},\omega_{4})$. That implies that the
discrepancy $Q^{(23)}$ on the sector $\lambda_{231}$ can be naturally
represented as the sum:
$Q^{(23)}=Q_{1}^{(23)}+Q_{2}^{(23)}.$ (33)
Similarly, on the sector $\lambda_{312}$
$Q^{(21)}=Q_{1}^{(21)}+Q_{2}^{(21)}.$ (34)
All four terms here can be easily computed. The answers are completely
analogous. In particular,
$Q_{1}^{(23)}=R_{1}(-\Delta-E)e^{i<\mathbf{q}_{23},\mathbf{x}>}(\Phi^{(23)}_{1}-1)\zeta_{2}^{\prime}=$
(35)
$=R_{1}[e^{i<\mathbf{q}_{23},\mathbf{x}>}(\Phi^{(23)}_{1}-1)\frac{-1}{r^{2}}\zeta_{2}^{\prime\prime}-2i\frac{1}{r}\zeta_{2}^{\prime}<\mathbf{q}_{23},w>e^{i<\mathbf{q}_{23},\mathbf{x}>}\Delta\Phi^{(23)}_{1}].$
(36)
where $w$ is a unit vector orthogonal to ${\hat{\mathbf{x}}}$ and oriented
along the direction of increasing $\omega$.
Finally,
$Q[\psi_{0}]=Q_{1}^{(23)}+Q_{2}^{(23)}+Q_{1}^{(21)}+Q_{2}^{(21)}.$ (37)
It is easy to see that all four components of the discrepancy vanish at
infinity like $|\mathbf{x}|^{-5/2}.$
The previous computations of the discrepancy were given for not small
$|\mathbf{x}|$ where the supports of three potentials are separated. Let us
modify now the field $\psi_{1}$ by introducing in it the factor
$\zeta=\zeta(|\mathbf{x}|)$ that is equal to $0$ for $|\mathbf{x}|<r_{1}$, and
is equal to $1$ for $|\mathbf{x}|>r_{2}$, $0<r_{1}<r_{2}.$ It is supposed that
for $|\mathbf{x}|>r_{1}$ three supports do not intersect. The definition of
$\psi_{1}$ is given by the formula
$\psi_{1}=\psi_{0}\zeta.$ (38)
The final expression for the discrepancy is given by the formula
$Q[\psi_{1}]=Q[\psi_{0}]\zeta-2\left[\frac{\partial}{\partial|\mathbf{x}|}\psi_{0}(\mathbf{x},\mathbf{q})\right]\zeta^{{}^{\prime}}-\psi_{0}\frac{1}{|\mathbf{x}|}\frac{\partial}{\partial|\mathbf{x}|}|\mathbf{x}|\frac{\partial}{\partial|\mathbf{x}|}\zeta.$
(39)
There is no problem in explicit computation of the derivative
$\frac{\partial}{\partial|\mathbf{x}|}\psi_{0}(\mathbf{x},\mathbf{q})$.
## 3 Numerical computations
The goal of the computations was to show that the suggested plan is realistic
and can be practically used for the computations of the scattered plane wave
and the corresponding amplitude of scattering. The pair-particle potential
$v(x)$ and the vector $\mathbf{q}$ are two parameters of the problem.
As for $v(x)$ we choose the potential function
$v(x)=\left\\{\begin{array}[]{lc}2e^{\frac{1}{(4x)^{2}-1}+1},&|x|<\frac{1}{4}\\\
0,&\text{otherwise}.\end{array}\right.$ (40)
Any specific choice is not crucial, we could take arbitrary even potential
(even non-necessary continuous) with the compact support. With this potential
we computed the solution $\chi(x,k)$ of one-dimensional Schrödinger equation
(6) and found the corresponding transition $s(k)$ and reflection $r(k)$
coefficients.
Then the solutions $\chi(x,k)$ were interpolated to the actual computational
domain to construct the functions $\chi_{j}$. This interpolation was necessary
only on the support of the potentials. Outside of the supports the analytic
expressions of the functions $\chi(x,k)$ were known after the coefficients
$s(k)$ and $r(k)$ were found numerically.
We took $E=4$. For the vector $\mathbf{q}$ we used two choices: 1) $k_{1}=1,\
\ p_{1}=\sqrt{3},\ \ $ 2) $k_{1}=p_{1}=\sqrt{2}.$ In the first case the field
as a function of $\mathbf{x}$ is symmetric with respect to the straight line
generated by $\mathbf{q}$. It was taken for the control.
The function $\psi_{R}$ was computed directly with the knowledge of
$\chi_{j}$, the Fresnel integral was taken from GSL (Gnu scientific library).
The functions $\psi_{0}(\mathbf{x},\mathbf{q})$ and
$\psi_{1}(\mathbf{x},\mathbf{q})$ were computed with the help of the explicit
formulas for them. The discrepancy $Q$ was also computed with the help of the
explicit formulas. The radii $r_{1}<r_{2}$ were taken as $r_{1}=4,\ \
r_{2}=14.5.$ We think that this choice reasonably corresponds to the selected
value of $|\mathbf{q}|$.
For the diffraction corrections (and near the origin) the chosen partition of
unity corresponds to function $\zeta(z)=z^{3}(10-15z+6z^{2}),\;0<z<1$, where
$z$ is the variable relative to angle $\omega$.
Then we finally considered the boundary problem (16 \- 18). Of course, it was
the main part of the numerical program of the work. The problem on the disc is
not on the spectrum.
For the computations we used mainly FreeFem++, which is a user friendly
language dedicated for solving partial differential equations with the finite
element method. All the necessary steps from mesh creation to solving the
linear system can be done within the same program in a manner that is not of a
black box type. Since we used the finite element method, we introduced the
corresponding weak formulation of the problem: find $\xi\in H^{1}(\Omega)$
such that
$\int_{\Omega}\nabla\xi\cdot\nabla w+(v(x_{1})+v(x_{2})+v(x_{3})-E)\xi
w\;dx-\int_{\partial\Omega}i\sqrt{E}\xi w\;dS\\\
=-\int_{\Omega}Qw\;dx\quad\forall w\in H^{1}(\Omega)$ (41)
The finite element discretization of (41) was then done in a standard fashion
using quadratic Lagrange elements on a triangular mesh. The computational
domain was divided into sub-domains to have the finite element mesh fit better
with the support of the potential $V$ and the constructed function $\chi_{0}$
and the discrepancy $Q$. A relatively uniform mesh was introduced with lengths
of triangle edges between 0.15 and 0.48. With a circular domain of radius 190,
the total number of degrees of freedom was 3 million. We used Matlab’s solver
for large linear systems.
The results are represented by the Fig.2-3, and we think that they are
reasonable.
Figure 2
$real(Q)$
Figure 3
$real(\xi)$
A certain problem was the choice of the radius $R$. To get more precise
results it would be better to take bigger $R$, but the bigger $R$ means the
harder computations. The criterium of the compromise was connected with the
integral form of the radiation conditions. These conditions are:
1)the integral
$\int_{S_{R}}ds|\xi(\mathbf{x},\mathbf{q})|^{2}$ (42)
over the circle $|\mathbf{x}|=R$ must be bounded for large $R$;
2) the integral
$\int_{S_{R}}ds|(\frac{\partial}{\partial|\mathbf{x}|}-i\sqrt{E})\xi(\mathbf{x},\mathbf{q})|^{2}$
(43)
must decrease as $R^{-2}$.
Notice that Fig. 4 shows that the first integral here is asymptotically
approaching a constant at sufficiently large $|\mathbf{x}|$, and the second
integral is quite small for such $|\mathbf{x}|$, but does not decrease for the
present computations with $R=190$.
Figure 4
It, probably, means that such radius is not completely sufficient for the
final computations. However, the bigger radius would mean the harder
computations, so we decided at the moment to restrict the radius of the circle
by $190$.
We considered also the corrected boundary condition where the next term of
asymptotic behavior of $\xi$ was also taken into account:
$\left(\frac{\partial}{\partial
r}-i|\mathbf{q}|+\frac{1}{2r}\right)\xi|_{r=R}=0.$
Nevertheless, the correction did not help to stabilize the calculation in
smaller domain, as it could be expected. The reason is that the term
$\frac{1}{2R}$ appeared to be very small comparatively with other terms.
To clarify further the situation with the stabilization of $L_{2}$ \- norm on
the boundary of the disk we should come back to behavior of $\xi$ as a
function of the angle at fixed radius $r=R$.
Figure 5 $|\xi(\theta)|$
One can easily see from the Figure that the main contribution to the integral
comes from two special directions on the boundaries of shadow and light in
sectors $\lambda_{231}$ and $\lambda_{312}$, see Fig.5. Therefore for the
stabilization of the whole integrals first of all the contributions to them of
two indicated sectors must be stabilized. Namely their stabilization was not
completely reached for considered size of configuration domain and requires
bigger scale of radii. There are also other computational indications that the
behavior of the field in these two sectors is responsible for the
(non)stabilization of the $L_{2}$ norms.
On the other hand, we found the right tendency of the behavior of the solution
for large $r$ what was the aim of the present calculations.
## 4 Acknowledgment
The authors would like to thank Prof. V.B.Belyaev for the fruitful
discussions. The work was partially supported by RFBR grant 08-01-00209.
## References
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|
arxiv-papers
| 2009-09-24T19:52:12 |
2024-09-04T02:49:05.483333
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "V. S. Buslaev, S. B. Levin, P. Neittaanm\\\"aki, T. Ojala",
"submitter": "Sergey Levin",
"url": "https://arxiv.org/abs/0909.4529"
}
|
0909.4589
|
arxiv-papers
| 2009-09-25T02:57:51 |
2024-09-04T02:49:05.489963
|
{
"license": "Public Domain",
"authors": "Kai Cai, Rongquan Feng, and Zhiming Zheng",
"submitter": "Kai Cai",
"url": "https://arxiv.org/abs/0909.4589"
}
|
|
0909.4611
|
Vol.0 (200x) No.0, 000–000
11institutetext: 1Purple Mountain Observatory, Chinese Academy of Sciences,
Nanjing 210008, China
2Shanghai Observatory, Chinese Academy of Sciences, Shanghai 200030, China
11email: ywwu@pmo.ac.cn
# A multiwavelength study of massive star-forming region IRAS 22506+5944
Yuan-Wei Wu 11 Ye Xu 11 Ji Yang 11 Jing-Jing Li 22
(Received 2001 month day; accepted 2001 month day)
###### Abstract
We present a multi-line study of the massive star-forming region IRAS
22506+5944. A new 6.7 GHz methanol maser was detected. 12CO, 13CO, C18O and
HCO+ J = 1-0 transition observations reveal a star formation complex
consisting mainly of two cores. The dominant core has a mass of more than 200
M⊙, while another one only about 35 M⊙. Both cores are obviously at different
evolutionary stages. A 12CO energetic bipolar outflow was detected with an
outflow mass of about 15 M⊙.
###### keywords:
infrared: ISM — ISM: individual (IRAS 22506+5944) — ISM: jets and outflows —
masers — stars: formation
## 1 Introduction
Massive stars play an important role in the evolution of the interstellar
medium (ISM) and galaxies; nevertheless their formation process is still
poorly understood because of large distances, high extinction, and short
timescales of critical evolutionary phases. In addition, massive stars do not
form in isolation but often in clusters and associations, which make the
environment of massive star formation regions more complex.
The 6.7 GHz transition of methanol has been found to be a particularly useful
signpost to trace massive star formation (Minier et al. [2003], Xu et al.
[2003]). On the other hand, the maser phase encompasses the outflow phase (Xu
et al. [2006]), which give us another powerful tool to study the dynamics of
massive star formations.
IRAS 22506+5944, with an infrared luminosity of 1.5$\times$104L⊙, belongs to
the Cepheus molecular cloud complex. Harju et al. ( [1993]) made a NH3 map of
this region, and found a NH3 core is coincident with the peak of the IRAS
source. Both H2O maser (Wouterloot & Walmsley [1986]) and SiO (Harju et al.
[1998]) have been detected. Although searches for 6.7 GHz methanol maser
(Szymczak et al. [2000]) show negative results, recently, we found a weak 6.7
GHz methanol maser in this region. Despite its high luminosity and FIR color
characteristics of the ultra-compact HII region, no radio emission was
detected (Molinari et al. [1998]). The distances used in literatures for this
source range from 5.0 kpc to 5.7 kpc. Here we use the value of 5.0 kpc.
In this paper, we present a multi-line study of this star-forming region. In
Sect. 2, we describe our observations. The results are given in Sect.3. We
give analysis and discussion in Sect.4, and summarize in Sect.5.
## 2 Observations
### 2.1 The Effelsberg 100 m Telescope
Observations of the methanol (CH3OH) maser were made using the Effelsberg 100
m telescope in February 2006. The rest frequency adopted for the 51-60 A+
transition was 6668.519 MHz (Breckenridge & Kukolich [1995]). The spectrometer
was configured to have a 10 MHz bandwidth with 4096 channels yielding a
spectral resolution of 0.11 km s-1 and a velocity coverage of 450 km s-1. The
half-power beam width was $\sim$ 2′ and the telescope has an rms pointing
error of 10′′. The observations were made in position switched mode. The
system temperature was typically around 35 K during our observations. The flux
density scale was determined by observations of NGC7027 (Ott et al. [1994]).
The absolute calibration for flux density is estimated to be accurate to
$\sim$ 10%. The integration time on source was 10 minutes, with a rms noise
level of $\sim$ 0.05 Jy in the spectra. The pointing position was R.A.(J2000)
22h52m36.9s, DEC.(J2000) = +60∘00′48′′.
### 2.2 The PMO 13.7 m Telescope at Delingha
The 12CO, 13CO, C18O and HCO+ J = 1-0 maps were observed with the PMO 13.7 m
millimeter-wave telescope at Delingha, China, during 2008 November. A cooled
SIS receiver was employed, and system temperatures was $\sim 250$ K during the
observations. Three AOS (acousto-optical spectrometer) were used to measure
the J = 1-0 transitions of 12CO, 13CO, C18O and the FFTS (Fast Fourier
Transform Spectrometer) were used to measure the HCO+ J = 1-0 lines. All the
observations were performed in position switch mode. The pointing and tracking
accuracy was better than 10′′. The obtained spectra were calibrated in the
scale of antenna temperature T${}^{*}_{A}$ during the observation, corrected
for atmospheric and ohmic loss by the standard chopper wheel method. The grid
spacings of the mapping observations were 30′′. Table1 summarizes the basic
information about our observations, including: the transitions, the center
rest frequencies $\nu_{rest}$, the half-power beam widths (HPBWs), the
bandwidths, the equivalent velocity resolutions ($\Delta\nu_{res}$), and the
typical rms levels of measured spectra. All of the spectral data were
transformed from the T${}^{*}_{A}$ to the main beam brightness temperature
T${}^{*}_{MB}$ scale. The absolute calibration for intensity was about 10%.
The GILDAS software package (CLASS & GREG) was used for the data reduction.
Table 1: Observation Parameters
Translation | $\nu_{rest}$ | HPBW | Bandwidth | $\Delta\nu_{res}$ | 1$\sigma$ rsma
---|---|---|---|---|---
| (GHz) | (′′) | (MHz) | (km s-1) | (K)
12CO J = 1-0 | 115.271204 | 58 | 145 | 0.37 | 0.10
13CO J = 1-0 | 110.201353 | 61 | 43 | 0.11 | 0.10
C18O J = 1-0 | 109.782182 | 62 | 43 | 0.12 | 0.09
HCO+ J = 1-0 | 89.188521 | 75 | 43 | 0.16 | 0.10
* 1
_a_ typical value in the scale of $T_{R}^{*}$.
## 3 Results and Discussion
### 3.1 spectra
#### 3.1.1 6.7 GHz CH3OH maser spectrum
The spectrum of the CH3OH maser detected in this region is shown in Fig. 1.
There are two features that are separated by about 2.2 km s-1. The stronger
feature is at the LSR (local standard of rest) velocity of -53.7 km s-1, with
a flux density of 0.52 Jy, while the other is only about 0.2 Jy. In order to
get high signal to noise spectra, we did not attempt to refine the position
and just integrated the time at the same position. Hence, the actual position
could be off by 1 arcminute.
Figure 1: Spectrum of the 6.7-GHz CH3OH maser. The spectral resolution is 0.11
km s-1.
#### 3.1.2 12CO, 13CO, C18O and HCO+ spectra
Spectra of 12CO, 13CO, C18O and HCO+ are presented in Fig. 2. The spectra in
left panel come from the peak of core A (dominant core in Fig. 3). Both 12CO
and HCO+ show remarkable broad line wings, with a FW (full width) of 24 km s-1
and 6 km s-1 at 1$\sigma$ level, respectively. Spectra in the right panel are
correspondent to the peak of core B, and spectra at the conjunctive point of
the two cores are given in middle panel. Details of the line in positions of
the peak, including the line central velocities, the fitted line widths, the
bright temperatures and integrated intensity were listed in Table 2.
Table 2: Result of molecular line measurements.
Translation | VLSR | $\Delta\nu_{res}$ | T${}^{*}_{MB}$ | $\int$ T${}^{*}_{MB}$ _d_ $\upsilon$
---|---|---|---|---
(GHz) | (km s-1) | (km s-1) | (K) | (K km s-1)
12CO J = 1-0 _a_ | -51.4 | 4.3 | 22.3 | 98.3
12CO J = 1-0 _b_ | -51.5 | 3.3 | 14.1 | 48.9
13CO J = 1-0 _a_ | -51.4 | 2.3 | 9.5 | 22.1
13CO J = 1-0 _b_ | -51.5 | 1.8 | 6.1 | 11.4
C18O J = 1-0 _a_ | -51.6 | 1.8 | 1.2 | 2.3
C18O J = 1-0 _b_ | -51.5 | 1.1 | 0.9 | 1.0
HCO+ J = 1-0 _a_ | -51.1 | 3.8 | 2.1 | 6.9
HCO+ J = 1-0 _b_ | -51.6 | 2.0 | 0.4 | 0.9
* 1
_a_ and _b_ indicate Core A and Core B.
Figure 2: _Left panel:_ spectra at the C18O east peak. _Middle panel:_ spectra
at the conjunctive point of the two C18O cores. _Right panel:_ spectra at the
C18O west peak. The horizontal dot line is 1$\sigma$ level of each line (Table
1).
### 3.2 Mapping
#### 3.2.1 13CO, C18O and HCO+ maps
Contour maps of the total integrated 13CO J = 1-0, C18O J = 1-0 and HCO+ J =
1-0 line emissions were presented in Figure 3. We used MSX E band (21$\mu$m)
image as background images of the integrated contours to compare the
distributions between gas and dust. The filled triangle denotes 3 millimeter
continuum peak (Su et al. [2004]). H2O (Wooterloot & Walmsley [1986]), SiO
(Harju et al. [1998]) and CH3OH masers were indicated with the open triangle,
star and square, respectively. The IRAS error ellipse is also marked. Contour
levels are 20% to 90% by steps of 10% of the peak emission with the exception
of HCO+ J = 1-0 line, whose contour levels are 10%, 15% and 20% to 90% by
steps of 10%.
From Figure 3, we see that both molecular line and dust emission peak are
roughly coincident with the IRAS source. 13CO and HCO+ are dominated with a
single core. 13CO shows a little elongation. C18O map clearly shows two cores
(Core A and Core B), indicating that such optical thin line traces the inner
part of a molecular cloud than other two lines. The center of the Core A with
an angular extent of (70′′, 60′′), coincides with IRAS 22506+5944 and MSX
peak, indicates that they may be the same source. The Core B has an offset of
(100′′, 60′′) at the north-west of the Core A. The size of Core A is slightly
larger than the Core B (70′′, 60′′). In order to show the kinematic relation
of the two cores we also give the channel maps of the 13CO J = 1-0 lines in
Figure 4.
Figure 3: _upper left_ : contour map of the total integrated 13CO J = 1-0 line
emission in the velocity range from -53.4 to -49.5 km s-1 overlaid on MSX E
band 21 $\mu$m image. _upper right_ : contour map of the total integrated C18O
J = 1-0 line emission in the velocity range from -52.1 to -50.8 km s-1. _lower
left_ : contour map of the total integrated HCO+ J = 1-0 line emission in the
velocity range from -52.6 to -49.5 km s-1. _lower right_ : Contour map for the
12CO J = 1-0 outflow. The blue wing (solid line) emission was integrated over
-60 to -54 km s-1 and -49 to -42 km s-1 for red wing (dashed
line),respectively. Contour levels of the plots are all from 20% to 90% by
steps of 10% of each peak emission with the exception of HCO+ J = 1-0 lines,
whose contour levels are 10%, 15% and 20% to 90% by steps of 10%. 50% contour
levels used to determine core size are plotted with thicker lines. The small
crosses in the contour plots show the measured positions and the ellipses mark
IRAS error ellipse. The filled triangles denote 3 mm continuum peak (Su et al
2004). H2O (Wooterloot & Walmsley 1986), SiO (Harju et al 1998) and CH3OH
masers are indicated with open symbols of triangle, star and square,
respectively. Figure 4: Channel maps of 13CO J = 1-0 lines with contour levels
starting at 0.4 K km s-1 and separated by 0.3 K km s-1.
We derive the physical parameters of the cores, assuming LTE (local
thermodynamic equilibrium) and with an abundance ratio $[H_{2}]/[{}^{12}CO]$ =
104. Given a distance of the source to galactic center, DGC $\sim$ 11.4 kpc
,we adopt an abundance ratio $[{}^{12}CO]/[C^{18}O]\simeq 707$ and
$[{}^{12}CO]/[{}^{13}CO]\simeq 93$ estimated from the relationship
$[{}^{16}O]/[^{18}O]=(58.8\pm 11.8)D_{GC}+(37.1\pm 82.6)$ and
$[{}^{12}C]/[{}^{13}C]=(7.5\pm 1.9)D_{GC}+(7.6\pm 12.9)$ (Wilson & Rood
[1994]). Excitation temperature is calculated Using equation 1, assuming 12CO
J = 1-0 lines are optical thick:
$T_{ex}^{*}=5.532\left\\{\ln\left[1+\frac{5.532}{\left(T_{R}^{*}\left({}^{12}CO\right)+0.819\right)}\right]\right\\}^{-1},$
(1)
13CO and C18O J = 1-0 line optical depths, $\tau$, are estimated with formulas
below:
$\tau(^{13}CO)\thickapprox-\ln[1-\frac{T_{R}^{*}(^{13}CO)}{T_{R}^{*}({}^{12}CO)}]$
(2)
$\tau(C^{18}O)\thickapprox-\ln[1-\frac{T_{R}^{*}(C^{18}O)}{T_{R}^{*}({}^{12}CO)}]$
(3)
13CO and C18O column densities are derived using equation 4 (Kawamura et al.
[1998]) and equation 5 (Sato et al. [1994]). $\tau$ and $\Delta\nu$ are
optical depth and intrinsic line width:
$N\left({}^{13}CO\right)=2.42\times
10^{14}\frac{T_{ex}\tau(^{13}CO)\Delta\nu(^{13}CO)}{1-\exp\left(-5.29/T_{ex}\right)}cm^{-2},$
(4)
$N\left(C^{18}O\right)=2.24\times
10^{14}\frac{T_{ex}\tau(C^{18}O)\Delta\nu(C^{18}O)}{1-\exp\left(-5.27/T_{ex}\right)}cm^{-2}.$
(5)
The nominal core size, _l_ , is determined by de-convolving the telescope
beam, using equation 6:
$l=D\left(A_{1/2}-\theta_{MB}^{2}\right)^{1/2},$ (6)
where _D_ is the distance (5.0 kpc), _A_ 1/2 is the area within the contour at
the half- integrated intensity of the peak and $\theta_{MB}$ is the main beam
size (see Table 1).
Core masses are computed with equation 7, where _m_ is the mass of the
hydrogen molecule, $\mu$ the ratio of total gas mass to hydrogen mass,
$\mu\approx$ 1.36 (Hildebrand [1983]), $N_{H_{2}}$ the column density of H2
and _l_ the de-convolved half power size defined above.
$M_{LTE}=\mu mN_{H_{2}}l^{2}/4$ (7)
The physical parameters derived are tabulated in Table 3.
Table 3: Physical parameters of Core A and Core B
Name | $\Delta\alphaup$ | $\Delta\delta$ | _l_ | Tex | $\Delta\nuup$ _a_ | N(13CO) | N(C18O) | N(H2) _b_ | M(LTE)
---|---|---|---|---|---|---|---|---|---
| (arcsec) | (arcsec) | (pc) | (K) | km s-1 | (cm${}^{-2})$ | (cm-2) | (cm-2) | (M⊙)
core A | 90 | 70 | 0.9 | 26 | 1.5 | 2.7E+16 | 1.7E+15 | 2.7E+22 | 228
core B | 85 | 60 | 0.5 | 18 | 1.0 | 10.7E+15 | 7.5E+14 | 1.1E+22 | 35
* 1
_a_ : $\Delta\nuup$ has been corrected using $\frac{\Delta V_{line}}{\Delta
V_{true}}=\sqrt{\frac{\ln\left[\tau/\ln\left(2/\left(1+e^{-\tau}\right)\right)\right]}{\ln
2}}$ ,considering line broadening due to optical depth.
* 2
_b_ : H2 column densities were derived using 13CO column densities, assuming
$[H_{2}]/[{}^{13}CO]$ = 9.3$\times$105.
#### 3.2.2 Outflows
Molecular outflows are an important signature of the earlier stage in star
formation. An outflow has been detected (Wu et al. [2005]) using the 12CO J =
2-1 line. A comparison of two different transitions will be helpful to our
better understanding of the physical properties of outflows. In Fig. 3, we
present a similar work with the 12CO (1-0) line. The red and blue lobes are
largely overlapped, while the IRAS source is located at the center of the
outflow, probably the driving source of the outflow. Morphology of the 12CO J
= 1-0 outflow is similar to that of the 12CO J = 2-1 outflow (Wu et al.
[2005]), but the former extends a larger area than the latter, spreading from
Core A to Core B.
The outflow parameters, except for 12CO column density which is derived from
Snell et al. ( [1988]), are estimated with the method of Beuther et al. (
[2002]). We assume that the gas is in LTE and the line wings are optically-
thin. Excitation temperature and $[H_{2}]/[{}^{12}CO]$ abundance ratio adopted
are the same as Sect. 3.2.1. In order to better define the kinematics of the
high gas, we divided the wings into low velocity and high velocity segments.
The physical properties, including velocity range, size, column density, mass,
momentum and kinetic energy are summarized in Table 4.
Following the method of Beuther et al. ([2002]), we obtain the characteristic
time scale, _t_ $\approx 8.1\times$ 104 yr, the mass loss rate,
$\dot{M}_{out}\approx 1.8\times 10^{-4}$M⊙ yr-1 , the mechanical force,
F${}_{m}\approx 2.2\times 10^{-3}$ M⊙ km s-1 yr-1, and the mechanical
luminosity, L${}_{m}\approx 2.2$ L⊙. The mass and kinetic energy of the
outflow are significantly larger than typical values from low-mass star
forming regions (Bontemps et al. 1996).
Table 4: Outflow properties
Compoent | Vrange | Size _a_ | N(H2) | Mass | P | Ek
---|---|---|---|---|---|---
| (km s-1) | (pc) | (cm-2) | M⊙ | (M⊙ km s-1) | (erg)
red lobe(L) | (-48.2 -44.0) | 0.8 | 1.3E+20 | 5.3 | 72 | 9.6E+45
red lobe(H) | (-44.0 -38.0) | 0.8 | 4.2E+19 | 2.1 | 29 | 3.8E+45
blue lobe(L) | (-54.7 -59.0) | 1.0 | 1.0E+20 | 6.6 | 68 | 7.1E+45
blue lobe(H) | (-59.0 -62.0) | 1.0 | 1.7E+19 | 1.0 | 11 | 1.1E+45
total | — | —- | — | 15.0 | 180 | 2.2E+46
* 1
_a_ : size of lobes are computed using formula 6.
### 3.3 evolutionary scenario
HCO+ usually traces the geometrically thick envelope of a core, while C18O is
expected to trace the inner part of the core. The C18O map clearly shows two
cores, Core A and Core B, which likely consist of two star forming regions.
Core A has obvious star forming evidences, such as strong middle and far
infrared emission, masers and outflows, while Core B is only associated with
some cold molecular lines, indicating that core A and core B are in different
evolutionary stages, Core A at the phase of protostar core, while Core B
probably at the phase of pre-stellar core. The mass of Core A is more than 200
$M_{\odot}$, while the IRAS source has a luminosity of 1.5 $\times 10^{4}$
$L_{\odot}$. According to the relation between mass and luminosity, $L\sim
M^{3.5}$, the core mass is around one order of magnitude larger than that of
the IRAS source. This indicates there are other sources within the core, which
are not detected due to the resolution limit of the employed telescope. With a
rising steep spectrum, the IRAS colors imply that the source is deeply
embedded in a dense molecular cloud. This IRAS source could be the exciting
source of the H2O, SiO and CH3OH masers. Core B has a mass of about 35
$M_{\odot}$, which might form several low or/and medium mass stars in the
future. The energetic outflow driven by IRAS 22506+5944 covers the whole
region, including both Core A and Core B, and could greatly affect its
surrounding and accelerate Core B to form stars. In summary, the whole region
is a star formation complex, in which stars at different evolutionary stages
live in the same cluster and interact with each other.
## 4 Summary
Our multi-line study reveals a star formation complex around IRAS 22506+5944,
in which a weak 6.7 GHz CH3OH maser was detected. Multi-line Maps reveal a
two-core structure: core A with mass of $\sim$230 M⊙ contains the IRAS source
which is driving an energetic bipolar outflow, while Core B, significantly
smaller than core A, has a large offset from the IRAS source. The two cores
are at different evolutionary stages. The energy released by the more evolved
core (Core A) is influencing the relatively less evolved core (Core B) to
accelerate its step to form stars.
###### Acknowledgements.
We wish to thank all the staff at Qinghai Station of Purple Mountain
Observatory for their assistance with our observations. This work was
supported by the National Natural Science Foundation of China (Grant Nos.
10673024, 10733030, 10703010 and 10621303) and National Basic Research Program
of China-973 Program 2007CB815403.
## References
* [2002] Beuther H., Schilke P., Sridharan T.K., Menten, K.M., et al., 2002, A&A, 383, 892
* [1995] Breckenridge S.M., Kukolich S.G., 1995, ApJ, 438, 504
* [1] Bontemps, S., André, P., Terebey, S., & Cabrit, S. 1996, A&A, 311, 858
* [1993] Harju J., Walmsley C.M., Wouterloot J.G.A., 1993, A&AS, 98, 51
* [1998] Harju J., Lehtinen K., Booth R.S., Zinchenko I., 1998, A&AS, 132, 211
* [1983] Hildebrand R.H., 1983, QJRAS, 24, 267
* [1998] Kawamura A., Onishi T., Yonekura Y., et al., 1998, ApJS, 117, 387
* [2003] Minier V., Ellingsen S. P., Norris R. P., Booth R. s., 2003, A&A, 403, 1095
* [1998] Molinari S., Brand J., Cesaroni R., Palla, F., et al., 1998, A&A, 336, 339
* [1994] Ott M., Witzel A., Quirrenbach A., et al. 1994, A&A, 284, 331
* [1994] Sato F., Mizuno A., Nagahama T., et al. 1994, ApJ, 435, 279
* [1988] Snell R.L., Huang Y.-L., Dickman R.L., Claussen M.J., 1988, ApJ, 325, 853
* [2004] Su Y.-N., Zhang Q.-Z., Lim J., 2004, ApJ, 604, 258
* [2000] Szymczak M., Hrynek G., Kus A.J., 2000, A&AS, 143, 269
* [1994] Wilson T.L., Rood R., 1994, ARA&A, 32, 191
* [1986] Wouterloot J.G.A., Walmsley C.M., 1986, A&A, 168, 237
* [2005] Wu Y., Zhang Q.-Z., Chen H., Yang C., et al., 2005, AJ, 129, 330
* [2003] Xu Y., Zheng X.-W., Jiang D-R., 2003, , 3, 49
* [2006] Xu, Y., Shen, Z.-Q., Yang, J., Zheng, X. W., et al. 2006, AJ, 132, 20
|
arxiv-papers
| 2009-09-25T06:15:29 |
2024-09-04T02:49:05.493353
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yuan-Wei Wu, Ye Xu, Ji Yang and Jing-Jing Li",
"submitter": "Ye Xu",
"url": "https://arxiv.org/abs/0909.4611"
}
|
0909.5008
|
# Simulation of Wave Equation on Manifold using DEC
Zheng Xie1 Yujie Ma2
$1.$ Center of Mathematical Sciences, Zhejiang University (310027),China
$2.$ Key Laboratory of Mathematics Mechanization,
Chinese Academy of Sciences, (100090), China E-mail: lenozhengxie@yahoo.com.cn
Tel./fax: +86 0739 5316081 E-mail: yjma@mmrc.iss.ac.cn This work is partially
supported by CPSFFP (No. 20090460102), NKBRPC (No. 2004CB318000) and NNSFC
(No. 10871170)
###### Abstract
The classical numerical methods play important roles in solving wave equation,
e.g. finite difference time domain method. However, their computational domain
are limited to flat space and the time. This paper deals with the description
of discrete exterior calculus method for numerical simulation of wave
equation. The advantage of this method is that it can be used to compute
equation on the space manifold and the time. The analysis of its stable
condition and error is also accomplished.
Keywords: Discrete exterior calculus, Manifold, Wave equation, Laplace
operator, Numerical simulation.
PASC(2010): 43.20.+g, 02.30.Jr, 02.30.Mv, 02.40.Ky.
## 1 Introduction
The wave equation is the prototypical example of a hyperbolic partial
differential equation of waves, such as sound waves, light waves and water
waves. It arises in fields such as acoustics, electromagnetism, and fluid
dynamics[1, 2]. To investigate the predictions of wave equation of such
phenomena it is often necessary to approximate its solution numerically. A
technique suitable for providing numerical solutions to the wave propagation
problem is the finite difference time domain (FDTD) method. This is normally
defined by looking for an approximate solution on a uniform mesh of points and
by replacing the derivatives in the differential equation by difference
quotients at points of mesh. The computational domain of this algorithm is
limited to flat space and the time [3, 4, 5, 6, 7].
Discrete exterior calculus (DEC) constitutes a discrete realization of the
exterior differential forms, and therefore, the right framework in which to
develop a discretization for differential equations not just on flat space but
on manifold [8, 9, 12, 15, 14, 13, 11, 10, 16]. The differential operators
such as gradient, divergence, and Laplace operator on manifold can also be
naturally discretized using DEC. The numerical solution of wave equation on
space manifold and the time by the methods of DEC is obtained in this paper.
For this equation, an explicit scheme is derived. The analysis of this
scheme’s stability shows that the numerical solution becomes unstable unless
the time step is restricted.
## 2 DEC method for wave equation
### Wave equation
The wave equation is the prototypical example of a hyperbolic partial
differential equation. In its simplest form, the wave equation refers to a
scalar function $u$ that satisfies:
$\frac{\partial^{2}u}{\partial t^{2}}=c^{2}\Delta u,$ $None$
where $\Delta$ is the Laplace operator and $c$ is the propagation speed of the
wave. More realistic differential equations for waves allow the speed of wave
propagation to vary with the frequency of the wave, a phenomenon known as
dispersion. In this case, $c$ is replaced by the phase velocity:
$\frac{\partial^{2}u}{\partial t^{2}}=\left(\frac{\omega}{k}\right)^{2}\Delta
u.$
Another common correction in realistic systems is that the speed is depend on
the amplitude of the wave, leading to a nonlinear wave equation:
$\frac{\partial^{2}u}{\partial t^{2}}=c(u)^{2}\Delta u.$
### DEC scheme for wave equation
A discrete differential $k$-form, $k\in\mathbb{Z}$, is the evaluation of the
differential $k$-form on all $k$-simplices. A dual form is evaluated on the
dual cell. Suppose each simplex contains its circumcenter. The circumcentric
dual cell $D(\sigma_{0})$ of simplex $\sigma_{0}$ is
$D(\sigma_{0}):=\bigcup_{\sigma_{0}\in\sigma_{1}\in\cdots\in\sigma_{r}}\mathrm{Int}(c(\sigma_{0})c(\sigma_{1})\cdots
c(\sigma_{r})),$
where $\sigma_{i}$ is all the simplices which contains $\sigma_{0}$,…,
$\sigma_{i-1}$, $c(\sigma_{i})$ is the circumcenter of $\sigma_{i}$. In DEC,
the basic operators in differential geometry are approximated as follows:
* 1.
Discrete exterior derivative $d$, this operator is the transpose of the
incidence matrix of $k$-cells on $k+1$-cells.
* 2.
Discrete Hodge Star $\ast$, the operator scales the cells by the volumes of
the corresponding dual and primal cells.
* 3.
Discrete Laplace operator is $\ast^{-1}d^{T}\ast+d^{T}\ast d.$
For some situations, a source having azimuthal symmetry about its axis is
considered. In this case, the 2D triangular discrete manifold as the space is
only need to be considered. Now, we show how to derive an explicit DEC scheme
for Eq.(1) in 2D space manifold and the time. The wave equation in 3D space
and the time can also be computed by a similar approach.
Take Fig.1 as an example for a part of 2D mesh, in which $0$,…, $F$ are
vertices, $1$,…,$6$ are the circumcenters of triangles, $a$,…,$f$ are the
circumcenters of edges. Denote $l_{ij}$ as the length of line segment $(i,j)$
and $A_{ijkl}$ as the area of quadrangle $(i,j,k,l)$.
Fig.1. A part of 2D mesh
Define
$l_{12}:=l_{1f}+l_{2f},~{}l_{23}:=l_{2a}+l_{3a},...,l_{61}:=l_{6e}+l_{1e},$
and
$P_{123456}:=A_{01fe}+A_{02fa}+\cdots+A_{06de}.$
The diffusion term $\Delta u$ at vertice $0$ is approximated using discrete
Laplace operator as follows:
$\begin{array}[]{lll}\Delta
u_{0}&\approx&\dfrac{1}{P_{123456}}\left(\dfrac{l_{23}}{l_{A0}}(u_{A}-u_{0})+\dfrac{l_{34}}{l_{B0}}(u_{B}-u_{0})+\dfrac{l_{45}}{l_{C0}}(u_{C}-u_{0})\right.\\\
&&\left.+\dfrac{l_{56}}{l_{D0}}(u_{D}-u_{0})+\dfrac{l_{16}}{l_{E0}}(u_{E}-u_{0})+\dfrac{l_{12}}{l_{F0}}(u_{F}-u_{0})\right).\end{array}$
$None$
The temporal derivative is approximated by middle time differences as follows:
$\frac{\partial^{2}u^{n}}{\partial t^{2}}\approx\frac{1}{(\Delta
t)^{2}}(u^{n+1}-2u^{n}+u^{n-1}),$ $None$
where $\Delta t$ is uniform spacing, and $n\Delta t$ is the coordinate of
time. The approximation of Eq.(1) generated by substituting the left-hand
sides of (2) and (3) into (1), thus satisfies
$\begin{array}[]{lll}\mathrm{{Right}}(2)^{n-1}&=&\dfrac{1}{(c\Delta
t)^{2}}\left(u^{n}_{0}-2u^{n-1}_{0}+u^{n-2}_{0}\right).\end{array}$ $None$
## 3 Stability, convergence and accuracy
### Stability
The Courant-Friedrichs-Lewy condition is a necessary condition for stability
while solving certain partial differential equations numerically. Now, this
condition is derived for scheme (4). First, this DEC scheme is decomposed into
temporal and spacial eigenvalue problems.
The temporal eigenvalue problem:
$\dfrac{\partial^{2}u^{n}}{\partial t^{2}}=\Lambda u^{n}$
It can be approximated by difference equation
$\dfrac{u^{n+1}_{0}-2u^{n}_{0}+u^{n-1}_{0}}{(\Delta t)^{2}}=\Lambda
u^{n}_{0}.$ $None$
Supposing
$u^{n+1}_{0}=u^{n}_{0}\cos(\Delta
t)~{}~{}~{}~{}u^{n-1}_{0}=u^{n}_{0}\cos(-\Delta t)$
and substituting those into Eq.(5), we obtain
$\dfrac{\cos(\Delta t)+\cos(-\Delta t)-2}{(\Delta t)^{2}}=\Lambda,$
therefore
$-\dfrac{4}{(\Delta t)^{2}}\leq\Lambda\leq 0.$
This is the stable condition for the temporal eigenvalue problem.
The spacial eigenvalue problem:
$c^{2}\Delta u=\Lambda u$
It can be approximated by difference equation (6) based on Fig.1.
$\begin{array}[]{lll}\dfrac{P_{123456}}{c^{2}}\Lambda
u_{0}&=&\dfrac{l_{23}}{l_{A0}}(u_{A}-u_{0})+\dfrac{l_{34}}{l_{B0}}(u_{B}-u_{0})+\dfrac{l_{45}}{l_{C0}}(u_{C}-u_{0})\\\
&&+\dfrac{l_{56}}{l_{D0}}(u_{D}-u_{0})+\dfrac{l_{16}}{l_{E0}}(u_{E}-u_{0})+\dfrac{l_{12}}{l_{F0}}(u_{F}-u_{0})\end{array}$
$None$
Let $u_{i}=u_{0}\cos(cl_{0i})$ and substitute into Eq.(6) to obtain
$\begin{array}[]{lll}\dfrac{P_{123456}}{c^{2}}\Lambda&=&\dfrac{l_{23}}{l_{A0}}(\cos(cl_{0A})-1)+\dfrac{l_{34}}{l_{B0}}(\cos(cl_{0B})-1)+\dfrac{l_{45}}{l_{C0}}(\cos(cl_{0C})-1)\\\
&&+\dfrac{l_{56}}{l_{D0}}(\cos(cl_{0D})-1)+\dfrac{l_{16}}{l_{E0}}(\cos(cl_{0E})-1)+\dfrac{l_{12}}{l_{F0}}(\cos(cl_{0F})-1)\end{array}$
So we have
$-\dfrac{2c^{2}}{P_{123456}}\left(\dfrac{l_{23}}{l_{A0}}+\dfrac{l_{34}}{l_{B0}}+\dfrac{l_{45}}{l_{C0}}+\dfrac{l_{56}}{l_{D0}}+\dfrac{l_{16}}{l_{E0}}+\dfrac{l_{12}}{l_{F0}}\right)\leq\Lambda\leq
0.$
In order to keep the stability of scheme (4), we need
$-\dfrac{4}{(\Delta
t)^{2}}\leq-\dfrac{2c^{2}}{P_{123456}}\left(\dfrac{l_{23}}{l_{A0}}+\dfrac{l_{34}}{l_{B0}}+\dfrac{l_{45}}{l_{C0}}+\dfrac{l_{56}}{l_{D0}}+\dfrac{l_{16}}{l_{E0}}+\dfrac{l_{12}}{l_{F0}}\right)$
$None$
for all vertices, namely
$c\Delta t\leq{\mathrm{Min}}_{0\in
V}\sqrt{\dfrac{2P_{123456}}{{\dfrac{l_{23}}{l_{A0}}+\dfrac{l_{34}}{l_{B0}}+\dfrac{l_{45}}{l_{C0}}+\dfrac{l_{56}}{l_{D0}}+\dfrac{l_{16}}{l_{E0}}+\dfrac{l_{12}}{l_{F0}}}}}.$
### Convergence
By the definition of truncation error, the solution $\tilde{u}$ of the Eq.(1)
satisfies the same relation as scheme (4) except for an additional term
$O(\Delta t)^{2}$ on the right hand side. Thus the error
$X^{n}_{i}=\tilde{u}^{n}_{i}-u^{n}_{i}$ is determined from the relation
$\begin{array}[]{lll}X^{n}_{0}&=&2X^{n-1}_{0}-X^{n-2}_{0}+\dfrac{(c\Delta
t)^{2}}{P_{123456}}\left(\dfrac{l_{23}}{l_{A0}}(X^{n-1}_{A}-X^{n-1}_{0})\right.\\\
&&+\left.\dfrac{l_{34}}{l_{B0}}(X^{n-1}_{B}-X^{n-1}_{0})+\dfrac{l_{45}}{l_{C0}}(X^{n-1}_{C}-X^{n-1}_{0})+\dfrac{l_{56}}{l_{D0}}(X^{n-1}_{D}-X^{n-1}_{0})\right.\\\
&&+\left.\dfrac{l_{16}}{l_{E0}}(X^{n-1}_{E}-X^{n-1}_{0})+\dfrac{l_{12}}{l_{F0}}(X^{n-1}_{F}-X^{n-1}_{0})\right)+O(\Delta
t)^{2}.\end{array}$ $None$
Define
$|X^{n}|=\mathrm{Max}_{i\in V}|X^{n}_{i}|.$
From condition (7), we have
$\dfrac{(c\Delta
t)^{2}}{P_{123456}}\left({\dfrac{l_{23}}{l_{A0}}+\dfrac{l_{34}}{l_{B0}}+\dfrac{l_{45}}{l_{C0}}+\dfrac{l_{56}}{l_{D0}}+\dfrac{l_{16}}{l_{E0}}+\dfrac{l_{12}}{l_{F0}}}\right)<2.$
Hence the coefficient of $X^{n-1}_{0}$ in Eq.(8) is nonnegative. It follows
that
$\begin{array}[]{lll}|X^{n}_{0}|&\leq&2|X^{n-1}|+|X^{n-2}|+O(\Delta
t)^{2},\end{array}$
and hence that
$|X^{n}|\leq 2|X^{n-1}|+|X^{n-2}|+O(\Delta t)^{2}.$
Iterating $n$, we obtain
$|X^{n}|<M_{1}|X^{1}|+M_{0}|X^{0}|+O(\Delta t)^{2},$
where $M_{1}$, $M_{0}$ are finite value defined on $n$. Since the initial
conditions ensure $X^{0}=0$ and $X^{1}=0$, we have
$\lim_{\Delta t\rightarrow 0}|X^{n}|=0.$
That is to say the numerical solution approaches the exact solution as the
step size goes to $0$, and scheme (4) is convergent.
### Accuracy
In scheme (4), the space derivative of is approximated by first order
difference. Equivalently, $u$ is approximated by linear interpolation
functions. Consulting the definition about accuracy of finite volume method,
we can say that scheme (4) has first order spacial accuracy. Scheme (4) has
second order temporal accuracy, and second order spacial accuracy on
rectangular grid with uniform spacing.
## 4 Algorithm Implementation
The implementation of DEC scheme (4) for wave equation consists of the
following steps:
* 1.
Set the simulation parameters. These are the dimensions of the computational
mesh and the size of the time step, etc.;
* 2.
Initialize the mesh indexes.
* 3.
Assign current transmitted signal.
* 4.
Compute the value of all spatial nodes and temporarily store the result in the
circular buffer for further computation.
* 5.
Visualize the currently computed grid of spatial nodes.
* 6.
Repeat the process from the step 3, until reach the desired total number of
iterations.
The flowchart of the scheme (4) can be seen in Fig.2.
Fig.2. The flowchart of scheme (4)
The Fig.3 and Fig.4 show the numerical simulation of Gaussian pulse
propagating on the sphere and rabbit by scheme (4) on C# platform.
Fig.3. The propagation of Gaussian pulse on a rabbit Fig.4. The propagation of
Gaussian pulse on a sphere
## 5 Discussion
The DEC scheme for Laplace operator here can also be used to simulate the heat
equation, Laplace equation and Poisson equation on manifold.
### Discrete Laplace equation
The discrete Laplace equation on surface of regular tetrahedron (Fig.(5)) is
$\left(\begin{array}[]{cccc}1&1&1&-3\\\ 1&1&-3&1\\\ 1&-3&1&1\\\ -3&1&1&1\\\
\end{array}\right)\left(\begin{array}[]{c}u_{A}\\\ u_{B}\\\ u_{C}\\\ u_{D}\\\
\end{array}\right)=\left(\begin{array}[]{c}0\\\ 0\\\ 0\\\ 0\\\
\end{array}\right)$ $None$
The solution of Eq.(9) is
$u_{A}=u_{B}=u_{C}=u_{D}=C,$
where $C$ is arbitrary constant. Eq.(9) is an imprecise approximation of
Laplace’s equations on a sphere. Obviously, this equation has constant
solution.
Fig.5. The surface of regular tetrahedron
### Discrete Poisson equation
Consider a discrete Poisson equation on surface of regular tetrahedron.
Suppose the boundary condition is
$u_{A}=H,$
then discrete Poisson equation on Fig.(5) is
$\left(\begin{array}[]{cccc}3&-1&-1\\\ -1&3&-1\\\ -1&-1&3\\\
\end{array}\right)\left(\begin{array}[]{c}u_{B}\\\ u_{C}\\\ u_{D}\\\
\end{array}\right)=\left(\begin{array}[]{c}H\\\ H\\\ H\\\ \end{array}\right)$
$None$
The solution of Eqs.(10) is
$u_{B}=u_{C}=u_{D}=H.$
### Discrete heat equation
The heat equation of temperature $u$ is
$\frac{\partial u}{\partial t}=c\Delta u,$
which can be approximated as
$u^{n}_{0}=u^{n-1}_{0}+c\Delta t~{}\mathrm{{Right}}(2)^{n-1}.$ $None$
The Fig.6 shows the heat diffusion of a constant heat source on the sphere
simulated by scheme (11).
Fig.6. The heat diffusion on a sphere
## References
* [1] K.W. Morton, D.F. Mayers, Numerical Solution of Partial Differential Equations: An Introduction 2nd Edition, Cambridge University Press, (2005).
* [2] S. Larsson, V. Thomée, Parial differential equations with numercial methods, Springer, (2009).
* [3] A. Bondeson, T. Rylander, P. Ingelstrom, Computational electromagnetics, Texts in Applied Mathematics, vol. 51. Springer, New York (2005)
* [4] K.S. Yee, Numerical solution of inital boundary value problems involving Maxwell’s equations in isotropic media. IEEE Trans. Ant. Prop. 14(3), 302-307 (1966)
* [5] A. Bossavit, Computational electromagnetism. Electromagnetism. Academic Press Inc., San Diego, CA (1998). Variational formulations, com- plementarity, edge elements
* [6] A. Bossavit, L. Kettunen, : Yee-like schemes on a tetrahedral mesh, with diagonal lumping. Int. J. Numer. Modell. 12(1-2), 129-142 (1999)
* [7] A. Stern, Computational Electromagnetism with Variational Integrators and Discrete Differential Forms. arXiv:0707.4470v2
* [8] H. Whitney, Geometric integration theory. Princeton University Press, Princeton, (1957).
* [9] D.N. Arnold, R.S. Falk, R. Winther, Finite element exterior calculus, homological techniques, and applications. Acta Numer. 15, 1-155 (2006).
* [10] F. Luo, Variational Principles on Triangulated Surfaces. http://arxiv.org/abs/0803.4232.
* [11] M. Meyer, M. Desbrun, P. Schröder, A.H. Barr, Discrete differential geometry operators for triangulated 2-manifolds. In InternationalWorkshop on Visualization and Mathematics, VisMath, (2002).
* [12] M. Desbrun, A.N. Hirani, M. Leok, J. E. Marsden, Discrete exterior calculus arXiv: math.DG/0508341.
* [13] J. M. Hyman, M. Shashkov, Natural discretizations for the divergence, gradient, and curl on logically rectangular grids. Comput. Math. Appl., 33(4):81-104, (1997).
* [14] R. Hiptmair, Discrete Hodge operators, Numer. Math., 90(2):265-289, (2001).
* [15] M. Leok, Foundations of computational geometric mechanics. Ph.D. thesis, California Institute of Technology (2004).
* [16] Z. Xie, Y.J. Ma, Computation of Maxwell’s equations on manifold using DEC, arXiv:0908.4448
|
arxiv-papers
| 2009-09-28T06:16:29 |
2024-09-04T02:49:05.504255
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zheng Xie, Yujie Ma",
"submitter": "Yujie Ma",
"url": "https://arxiv.org/abs/0909.5008"
}
|
0909.5110
|
# Plasma-based Control of Supersonic Nozzle Flow
Datta V. Gaitonde
Computational Sciences Branch, AFRL/RBAC
Air Vehicles Directorate
Air Force Research Laboratory, WPAFB, OH 45433, USA
###### Abstract
The flow structure obtained when Localized Arc Filament Plasma Actuators
(LAFPA) are employed to control the flow issuing from a perfectly expanded
Mach 1.3 nozzle is elucidated by visualizing coherent structures obtained from
Implicit Large-Eddy Simulations. The computations reproduce recent
experimental observations at the Ohio State University to influence the
acoustic and mixing properties of the jet. Eight actuators were placed on a
collar around the periphery of the nozzle exit and selectively excited to
generate various modes, including first and second mixed ($m=\pm 1$ and $m=\pm
2$) and axisymmetric ($m=0$). In this fluid dynamics video (mpeg-1, mpeg-2),
unsteady and phase-averaged quantities are displayed to aid understanding of
the vortex dynamics associated with the $m=\pm 1$ and $m=0$ modes excited at
the preferred column-mode frequency (Strouhal number $0.3$). The unsteady flow
in both contains a broad spectrum of coherent features. For $m=\pm 1$, the
phase-averaged flow reveals the generation of successive distorted elliptic
vortex rings with axes in the flapping plane, but alternating on either side
of the jet axis. This generates a chain of structures where each interacts
with its predecessor on one side and its successor on the other. Through self
and mutual interaction, the leading segment of each loop is pinched and passes
through the previous ring before rapidly breaking up, and the mean jet flow
takes on an elliptic shape. The $m=0$ mode exhibits relatively stable roll-up
events, with vortex ribs in the braid regions connecting successive large
coherent structures. Results with other modes are described in Ref. 1.
## 1 Introduction
Jet flow behavior and control has significant consequences for many
applications, including mitigation of aircraft noise, and mixing for various
industrial concerns such as combustion and pollution. Numerous types of
control methods have been employed, including passive (chevrons, tabs) and
active (fluidic and plasma) techniques. The present effort uses numerical
simulations to explore the dynamics of plasma-based open-loop flow control of
a Mach 1.3 jet, described by Samimy et al [2]. The plasma devices, denoted
Localized Arc Filament Plasma Actuators (LAFPA), strike an electric arc at a
specified frequency between two closely placed pin electrodes. Their
advantages include rapid on-off capability, low inertia and superior high
frequency performance. The full 3-D Navier-Stokes equations are solved with an
implicit LES approach. The effect of the actuators is simulated with a surface
heating model that successfully reproduces the principal effects observed in
experimental flow visualizations. Since the actuators can be either on or off,
the excitation is applied in a discrete fashion and can be described by a
collective frequency and duty cycle. The fluid dynamics video may be found in
mpeg-1 and mpeg-2 formats. The visualizations explore first mixed (flapping)
and axisymmetric modes at a Strouhal number of $0.3$, corresponding to
$4618Hz$ and a duty cycle of $20\%$. The spreading rate of the jet with the
$m=\pm 1$ mode is significantly increased in the flapping plane, but is
reduced in the non flapping-plane relative to the no-control case. Phase-
averaged observations visualized with isolevels of the Q-criterion colored by
vorticity magnitude, and vorticity magnitude colored by velocity magnitude,
indicate staggered and unstaggered structures on the two planes respectively.
The video shows these to be consistent with rings whose axes lie in the
flapping plane, but alternate about the jet centerline. The resulting
interaction between successive such events results in elongation and partial
pushing through of the leading segment of each ring through the trailing
segment of the previous ring. The jet displays an elliptic cross-section in
the mean. Axisymmetric excitation yields successive roll-up events subjected
to azimuthal disturbances, whose rapid evolution leads to breakdown. Although
not shown in the video, the $m=\pm 2$ mode (see Ref. [1]) also generates
elliptic vortex structures but with major axes being successively aligned
along the two symmetry planes. The minor axes regions move downstream faster
because of the higher velocity near the centerline, yielding rings which
stretch in the streamwise direction and ultimately breakdown.
## References
[1] Gaitonde, D.V., “Simulation of Supersonic Nozzle Flows with Plasma-based
Control,” AIAA-2009-4187, 39th AIAA Fluid Dynamics Conference, San Antonio,
Texas, June 22-25, 2009.
[2] Samimy, M., Kim, J.-H., Kastner, J., Adamovich, I., and Utkin, Y., “Active
Control of High-speed and High-Reynolds-number Jets Using Plasma Actuators,”
J. Fluid Mech., Vol. 578, 2007, pp. 305-330
|
arxiv-papers
| 2009-09-28T15:38:53 |
2024-09-04T02:49:05.511202
|
{
"license": "Public Domain",
"authors": "Datta V. Gaitonde",
"submitter": "Datta Gaitonde",
"url": "https://arxiv.org/abs/0909.5110"
}
|
0909.5174
|
# Sr2VO3FeAs as Compared with Other Fe-based Superconductors
I.I. Mazin Code 6393, Naval Research Laboratory, Washington, D.C. 20375
(Printed on )
###### Abstract
One of the most popular scenarios for the superconductivity in Fe-based
superconductors (FeBSC) posits that the bosons responsible for electronic
pairing are spin-fluctuations with a wave vector spanning the hole Fermi
surfaces (FSs) near $\Gamma$ and the electron FSs near M points. So far all
FeBSC for which neutron data are available do demonstrate such excitations,
and the band structure calculations so far were finding quasi-nested FSs in
all FeBSC, providing for a peak in the spin susceptibility at the desired wave
vectors. However, the newest addition to the family, Sr2VO3FeAs, has been
calculated to have a very complex FS with no visible quasi-nesting features.
It was argued therefore that this material does not fall under the existing
paradigm and calls for revisiting our current ideas about what is the likely
cause of superconductivity in FeBSC. In this paper, I show that the visible
complexity of the FS is entirely due to the V-derived electronic states.
Assuming that superconductivity in Sr2VO3FeAs, as in the other FeBSC,
originates in the FeAs layers, and the superconducting electrons are sensitive
to the susceptibility of the FeAs electronic subsystem, I recalculate the bare
susceptibility, weighting the electronic states with their Fe character, and
obtain a susceptibility that fully supports the existing quasi-nesting model.
Besides, I find that the mean-filed magnetic ground state is the checkerboard
in V sublattuce and stripes in the Fe sublattice.
The recently discovereddisc Fe-based high-temperature superconductors (FeBSC)
represent a challenging case for the theory of superconductivity. They appear
to be rather different from cuprates in terms of their electronic structure,
magnetic order, correlation effects, and superconducting symmetryreview . So
far the most popular suggestion for the pairing mechanism has been one that
assigns the role of an intermediate boson to spin fluctuations with wave
vectors close to Q=($\pi,\pi)$ (in the two-Fe Brillouin zone). There are two
ways to generate such spin fluctuations: one assumes superexchange between the
second neighbors in the Fe lattice and the other exploits the fact that the
non-interacting spin susceptibility calculated using the one-electron band
structure has a peak, or better to say a broad maximum close to ($\pi,\pi)$
(see review Ref. review, ). A strong argument in favor of the latter scenario
was the case of FeSe, where the parent magnetic compound FeTe shows an
antiferromagnetic order at a different wave vector. both in the experiment and
in the calculations, but the calculated spin susceptibility is still peaked
Q=($\pi,\pi),$ and the experiment also observes spin fluctuations with the
same wave vector. Also, the fact that FeBSC lack strong Coulomb
correlationsAnis ; Tom speaks against the former alternative.
Recently, however, a new FeBSC, Sr2VO3FeAs, has been discovered which
seemingly violates this so far meticulously observed rule. The calculated
Fermi surface (FS)WEP appears to be much more complex than in the other
investigated FeBSC, and there is no visual indication of any quasinesting
topology. Lee and PickettWEP argued that Sr2VO3FeAs represents “a new
paradigm for Fe-pnictide superconductors”, and inferred that “there is no
reason to expect an s± symmetry of superconducting order parameter ($i.e.$ a
different sign on the two FSs) in Sr2VO3FeAs.
Figure 1: The Fermi surfaces of Sr2VO3FeAs. The $\Gamma$ points are in the
corners, the M point in the center of the shown Brillouin zone. The colored
(dark) portion are the parts with the predominantly Fe character. The rest is
predominantly V. (color online)
I have repeated the calculations of Lee and Pickett and have obtained the FS
that was similar to theirsnote (Fig. 1). I have also verified that the bare
susceptibility without any account for the matrix elements
$\chi_{0}(\mathbf{q)}=-\sum_{\mathbf{k\alpha\beta}}\frac{f(\varepsilon_{\mathbf{k\alpha}})-f(\varepsilon_{\mathbf{k+q,\beta}})}{\varepsilon_{\mathbf{k\alpha}}-\varepsilon_{\mathbf{k+q,\beta}}+i\delta}$
(1)
indeed does not have any peak at Q=($\pi,\pi)$ (Fig. 2). In fact, it has a
peak at an entirely different wave vector, $(\pi,0.4\pi),$ as anticipated by
Lee and Pickett. However, this does not take into account the fact that the
calculated Fermi surface is really a superposition of two FS systems, one
originating from the FeAs planes, and the other from VO ones. While there is
some hybridization between the two systems of bands (at least along the XM
direction; see Ref. WEP, for details), as well as a magnetic coupling and a
magnetic moment on V, and maybe even Coulomb correlation effects on V site,
electrons derived from the Fe $d$-orbitals couple mostly with the spin
fluctuations on the Fe sites. This is a simple consequence of the Hund’s rule.
With that in mind, I colored the parts of the Fermi surface in Fig. 1 that
have predominantly Fe character.
Figure 2: The bare susceptibility (the real part) calculated with a constant
matrix element independently of the wave function character. The band
structure had been averaged over $k_{z}$ before the integration. The corners
of the plot correspond to $\mathbf{q}=(0,0)$, $(\pi,0)$, $(0,\pi)$, and
$(\pi,\pi)$. The vertical scale is in arbitrary units. (color online)
Figure 3: The bare susceptibility calculated as in Fig.2, but with matrix
elements taken as the product of the Fe weights for the corresponding wave
functions. The top panel shows the real part, the bottom one the imaginary
part. (color online)
Imagine now that the unpainted parts of the FS disappear. What remains after
this mental tour de force closely resembles the familiar FSs of other FeBSC.
Taking into account the above argument regarding the special role of the Fe
spin fluctuations, we can rewrite Eq. 1 as
$\tilde{\chi}_{0}(\mathbf{q)}=-\sum_{\mathbf{k\alpha\beta}}\frac{f(\varepsilon_{\mathbf{k\alpha}})-f(\varepsilon_{\mathbf{k+q,\beta}})}{\varepsilon_{\mathbf{k\alpha}}-\varepsilon_{\mathbf{k+q,\beta}}+i\delta}A_{\mathbf{k\alpha}}A_{\mathbf{k+q,\beta}},$
(2)
where $A_{\mathbf{k\alpha}}$ is the relative weight of the Fe orbitals in the
$|\mathbf{k}\alpha\mathbf{>}$ wave function. The result (Fig. 3), as expected,
shows the same structure as for the other pnictides, especially for the real
part of susceptibility, which is the one relevant for superconductivity.
I conclude that, unfortunately, Sr2VO3FeAs, despite being an interesting and
in many aspects unusual FeBSC, does not represent a new paradigm, but rather
falls into the same class as other pnictides. It is also worth noting that
while it has been established both experimentallyAnis ; Tom and
computationallyAnis ; Antoin that the FeAs subsystem is only weakly
correlated, this had not been obvious a priori, and it is not obvious for the
V-O subsystem in Sr2VO3FeAs. Being essentially in a vanadium oxide layer (and
vanadium oxide is strongly correlated in the bulk form), V in Sr2VO3FeAs may
be subject to strong Hubbard correlations that would remove V states from the
Fermi levelLDAU . Thus, strictly speaking, the conclusion above should be
formulated as follows: even if Sr2VO3FeAs is a weakly correlated metal and the
FS calculated within the density functional theory is realistic, the fact that
the overall topology seems on the first glance to be different from other
pnictides is misleading and the spin fluctuation spectrum is likely to be
rather similar.
At the end, let me briefly touched upon a separate, but equally (if not more)
interesting issue of the magnetic ground state and magnetic properties of
Sr2VO3FeAs within the density functional theory (DFT). It is well knowreview
that DFT seriously overestimates the tendency to magnetism in FeBSCs, so that
the calculated ground state appears strongly antiferromagnetic even in the
materials that sho no long range magnetic order (phosphates, selenide). This
is routinely ascribed to the mean-filead character of DFT. However, it is of
course interesting to see what is the (magnetic) ground state in the mean
filed, even when in real life the ground state is paramagnetic. For all FeBSCs
studied so far the antiferromagnetic stripe magnetic structure is by far the
lowest in energy (energy gain of the order of 200 meV per Fe compared to a
nonmagnetic solution), while the ferromagnetic structure is barely stable if
at all.
Most likely, the DFT ground state of FeBSCs is also antiferromagnetic in-
plane. However, even the nonmagetic unit cell contains 16 atoms, which makes
it extremely difficult to investigate the energy landscape for possible
antiferromagnetic pattern. Thus, it makes sense to study possible
ferro(ferri)magnetic solutions, in hope to extract at least some useful
information. This approach was adapted in Ref. Shein (although these authors
do not present any nonmagnetic calculations, actually relevant for
superconductivity). They found a solution with a moment on V ($\sim 1.5$
$\mu_{B}),$ but not on Fe. Lee and Pickett found another, ferrimagnetic
solution, with opposite moments on V and Fe, the former being largerLP . Using
different starting configurations, I was able to converge to three different
ground states within the same symmetry, as shown in the Table, as well as to
two lower-symmetry states, as illustrated in Fig. 4(b,c,d): interlayer
antiferromagnetic V sublattice, where the V layers are ferromagnetic, and
antiferromagnetically stacked, while Fe is nonmagnetic, and Fe-checkerboard,
where Fe forms a Neel plane and V in nonmagnetic. After that, I have
calculated two configurations in the double (four formula units) cell, which I
feel are the most relevant because of the superexchange interaction in the V
layers: V-checkerboard with nonmagnetic Fe, and V-checkerboard combined with
the stripe order in the Fe layers (Fig. 4)
Figure 4: Magnetic configurations used in the Table 1. Hollow symbols indicate
nonmagnetic atoms, blue (dark) spin-up moments, red (light) spin-down moments.
Circles are Fe atoms, upward and downward pointing triangles are two V layers
in the unit cell. The configurations are: (a) NM:nonmagnetic, (b)
FM:ferromagnetic, (c) half-FM, (d) FiM: ferrimagnetic (Fe and V spins are
antiparallel), (e) V-AF: antiferromagnetically stacked FM V layers,
nonmagnetic Fe (f) Fe-cb: checkerboard Fe planes, weakly ferromagnetic V
planes, (g) V-cb: checkerboard V planes, ferromagnetic Fe planes (h) V-cb
combined with Fe stripes. Minimal crystallographic unit cell is shown in each
case, and in the last panel dashed lines connect V atoms in the same layer
(color online)
A few observations are in place: (1) the state found in Ref. Shein is not the
ground state even within that symmetry; (2) unlike all other FeBSCs, FeAs
planes can support a very stable ferromagnetic state; (3) the interaction
between V and Fe is ferromagnetic, that is, not of superexchange character,
(4) the magnetic coupling between V and Fe is so weak that V does not induce
any magnetization on Fe, unless one already starts with a magnetic Fe; (5) It
is more important, from the total energy point of view, to have magnetic
moment on V that on Fe — a bit surprising, given that V has a weaker Hund’s
rule coupling; (6) V sublattice itself has a net antiferromagnetic
interaction: if Fe is not magnetic, V orders antiferromagnetically; (7) Unless
some more exotic ground state will be discovered, the total energy is
minimized when V layers order in the Neel (checkerboard) fashion, while Fe
orders the same way as in other pnictides, forming stripes; (8) most
importantly, a number of very different magnetic states are nearly degenerate
in energy. This last fact may be the key to the experimental fact that the
actual material is paramagnetic despite the fact that on the mean field level
it is more magnetic than other pnictides. This is an extremely intriguing
situation and the magnetism Sr2VO3FeAs deserves a more elaborated experimental
and theoretical study that is beyond the scope of this paper.
Table 1: Properties of some stable magnetic solutions in the Generalized Gradient Approximation of the DFT. All energies are given with respect to the nonmagnetic state. The magnetic states are described in Fig. 4. For the V-cb configuration I was able to converge to two different solution, high-spin and low-spin, with essentially the same energies | $M_{Fe},$ $\mu_{B}$ | $M_{V},$ $\mu_{B}$ | $\Delta E.$ meV/Fe
---|---|---|---
FM | 2.0 | 1.4 | $-396$
half-FM | 0.0 | 1.5 | $-381$
FiM | 2.1 | -1.4 | $-387$
AFM-V | 0.1 | $\pm 1.4$ | $-385$
Fe-cb | $\pm$2.0 | 0.2 | $-219$
V-cb | 2.0 | $\pm$1.2 | -237
V-cb | 0.1 | $\pm$1.2 | -232
V-cb + Fe-stripes | $\pm$2.2 | $\pm 1.2$ | $-409$
I thank W.E. Pickett and D. Parker for stimulating discussions related to this
work. I also acknowledge funding from the Office of Naval Research.
After this paper was accepted for publication, I became aware of another band
structure calculationChina . These author have considered the “Shein-
Ivanovskii” half-FM states and two antiferromagnetic states, with the
checkerboard (Neel) and stripe ordering in the Fe subslattice, and
unspecified, presumably ferromagnetic, ordering in the V subsystem. As clear
from the above, neither in this states represents an energy minimum even
within the corresponding symmetry group, therefore these authors arrived to an
incorrect conclusion that the lowest-energy magnetic state is characterized by
Neel order in the Fe subsystem.
## References
* (1) X. Zhu, F. Han, G. Mu, P. Cheng, B. Shen, B. Zeng, and H.-H. Wen, Phys. Rev. B 79, 220512(R) (2009)
* (2) I.I. Mazin and J. Schmalian, Physica C, 469, 614 (2009)
* (3) V.I. Anisimov, E.Z. Kurmaev, A. Moewes, I.A. Izyumov, Physica C, 469, 442-447 (2009).
* (4) W. L. Yang, A. P. Sorini, C-C. Chen, B. Moritz, W.-S. Lee, F. Vernay, P. Olalde-Velasco, J. D. Denlinger, B. Delley, J.-H. Chu, J. G. Analytis, I. R. Fisher, Z. A. Ren, J. Yang, W. Lu, Z. X. Zhao, J. van den Brink, Z. Hussain, Z.-X. Shen, and T. P. Devereaux. Phys. Rev. B80, 014508 (2009).
* (5) K.-W. Lee and W. E. Pickett, arXiv:0908.2698 (unpublished)
* (6) I used the standard LAPW code as implemented in the WIEN2k package with generalized gradient corrections. The orbitals weight used for the rest of the calculations are the LAPW orbital projection inside the corresponding muffin-tin spheres.
* (7) M. Aichhorn, L. Pourovskii, V. Vildosola, M. Ferrero, O. Parcollet, T. Miyake, A. Georges, and S. Biermann, Phys. Rev. B 80, 085101 (2009)
* (8) Unfortunately, it is not clear at all whether the LDA+U treatment would be appropriate for V in this fluctuating system even if V itself is subject to strong Hubbard correlations.
* (9) I. R. Shein and A. L. Ivanovskii, arXiv:0904.2671 (unpublished)
* (10) K.-W. Lee and W. E. Pickett, private communication.
* (11) G. Wang, M. Zhang, L. Zheng, and Z. Yang, Phys. Rev. B80, 184501 (2009).
|
arxiv-papers
| 2009-09-28T19:50:13 |
2024-09-04T02:49:05.515632
|
{
"license": "Public Domain",
"authors": "I.I. Mazin",
"submitter": "Igor Mazin",
"url": "https://arxiv.org/abs/0909.5174"
}
|
0909.5184
|
# The central surface density of “dark halos” predicted by MOND
Mordehai Milgrom The Weizmann Institute Center for Astrophysics, Rehovot,
76100, Israel
###### Abstract
Prompted by the recent claim, by Donato & al., of a quasi-universal central
surface density of galaxy dark matter halos, I look at what MOND has to say on
the subject. MOND, indeed, predicts a quasi-universal value of this quantity
for objects of all masses and of any internal structure, provided they are
mostly in the Newtonian regime; i.e., that their mean acceleration is at or
above $a_{\scriptscriptstyle 0}$. The predicted value is
$\gamma\Sigma_{\scriptscriptstyle M}$, with $\Sigma_{\scriptscriptstyle
M}\equiv a_{\scriptscriptstyle 0}/2\pi G=138(a_{\scriptscriptstyle
0}/1.2\times 10^{-8}{\rm cm~{}s^{-2}})M_{\scriptscriptstyle\odot}{\rm
pc}^{-2}$, and $\gamma$ a constant of order 1 that depends only on the form of
the MOND interpolating function. For the nominal value of
$a_{\scriptscriptstyle 0}$, $\log(\Sigma_{\scriptscriptstyle
M}/M_{\scriptscriptstyle\odot}{\rm pc}^{-2})=2.14$, which is consistent with
that found by Doanato & al. of $2.15\pm 0.2$.
MOND predicts, on the other hand, that this quasi-universal value is not
shared by objects with much lower mean accelerations. It permits halo central
surface densities that are arbitrarily small, if the mean acceleration inside
the object is small enough. However, for such low-surface-density objects,
MOND predicts a halo surface density that scales as the square root of the
baryonic one, and so the range of the former is much compressed relative to
the latter. This explains, in part, the finding of Donato & al. that the
universal value applies to low acceleration systems as well. Looking at
literature results for a number of the lowest surface-density disk galaxies
with rotation-curve analysis, I find that, indeed, their halo surface
densities are systematically lower then the above “universal” value.
The prediction of $\Sigma_{\scriptscriptstyle M}$ as an upper limit, and
accumulation value, of halo central surface densities, pertains, unlike most
other MOND predictions, to a pure “halo” property, not to a relation between
baryonic and “dark matter” properties.
galaxies: kinematics and dynamics; cosmology: dark matter, theory.
## 1 introduction
Donato & al. (2009) have recently looked at the distribution of the central
surface densities, $\Sigma_{c}$, of the dark matter halos (hereafter CHSD) of
galaxies of different types. They find that the distribution is rather narrow,
with a central value $\Sigma_{c}=10^{2.15\pm
0.2}M_{\scriptscriptstyle\odot}{\rm pc}^{-2}$. This finding agrees with
previous studies, in particular with that of Milgrom & Sanders 2005, who dealt
with the relevance to MOND, and with others (see references in Donato & al.
2009). $\Sigma_{c}$ is defined by Donato & al. as the product of the central
halo density, $\rho_{0}$, and the core radius, $r_{0}$, both derived by
fitting halo-plus-baryons models to various observations, such as rotation
curves, weak lensing results, or velocity dispersion data. In deducing
$\rho_{0}$ and $r_{0}$, the halo is sometimes assumed to have a density
distribution of the cored isothermal form; Donato & al. assumed a spherical
Burkert profile.
A surface density of special role, $\Sigma_{c}$, translates into an
acceleration of a special role $\Sigma_{c}G$, and this immediately evokes
MOND. One is thus naturally led to consider whether such a special value for
the CHSD is predicted by MOND.
Brada & Milgrom (1999) showed that MOND predicts an absolute acceleration
maximum, of order $a_{\scriptscriptstyle 0}$, that any phantom halo can
produce, anywhere in an object. Milgrom & Sanders (2005), in a precursor to
Donato & al. (2009), tested this MOND prediction by plotting, for a sample of
17 Ursa Major galaxies, the deduced $\rho_{0}$ and $r_{0}$ against each other.
(These were deduced for a cored isothermal sphere model, not a Burkert one,
with a variety of assumptions on the stellar $M/L$ values: maximum disc,
population synthesis values, best MOND fits to rotation curves, etc.) They
found, for their sample, that these parameters lie near a line of constant
$\Sigma_{c}=10^{2}M_{\scriptscriptstyle\odot}{\rm pc}^{-2}$ (their Fig.4), in
agreement with the value Donato & al. find. This was interpreted by Milgrom &
Sanders (2005) as indicating a maximum halo acceleration as suggested by Brada
& Milgrom (1999), because the sample used was devoid of truly low-surface
brightness galaxies, for which “halo” accelerations are supposedly lower.
Here I will show, as a new result, that MOND does indeed predict a quasi-
universal value for the CHSD of the imaginary, or phantom, dark matter (DM),
but only for baryonic systems that are, by and large, in the Newtonian regime,
having mean internal accelerations of order $a_{\scriptscriptstyle 0}$ or
larger. In contradistinction, MOND predicts that, in principle, we can have
galaxies with arbitrarily small values of $\Sigma_{c}$, if the baryonic
surface density is low enough. However, the predicted CHSD scales as the
square root of the baryonic surface density, and so will have a rather
contracted span in a given sample.
Of course, each of the objects in the sample studied can, and should, be used
to subject MOND to a detailed, individual test. Inasmuch as MOND passes theses
tests, as it seems to do quite well, we can deduce that there is an acceptable
halo model whose analog of $\Sigma_{c}$ agrees with the MOND prediction. If
other halo models do not agree with the MOND prediction, it only shows that
there is a range of acceptable halo parameters, within the uncertainties in
the model parameters or assumptions (assumed density law for the halo, stellar
$M/L$ values, etc.).
Individual tests are, collectively, more decisive than tests of general rules,
which they subsume. Nevertheless, deducing and testing such general rules,
such as the mass-rotational-speed relation (aka the baryonic Tully-Fisher
relation), or the MOND prediction underlying the Faber-Jackson relation, have
obvious merits of their own, as they focus attention on certain unifying
principles. In this light it is important to consider the prediction of a
quasi-universal CHSD in itself.
In section 2, I explain how the quasi-universal CHSD arises in MOND, for high-
acceleration systems. In section 3, I treat systems with low surface density;
in particular, I show from results in the literature that disk galaxies with
the lowest surface densities analyzed to date, do have $\Sigma_{c}$ values
that fall systematically below the quasi-universal value. The discussion
section 4 deals with the special significance of the prediction at hand, in
comparison with other MOND predictions.
## 2 The emergence of a quasi-universal “halo” central surface density in
high acceleration systems
I shall be using the formulation of MOND as modified gravity put forth by
Bekenstein & Milgrom (1984). In this theory the MOND gravitational potential,
$\phi$, is determined by a nonlinear generalization of the Poisson equation
$\vec{\nabla}\cdot[\mu(|\vec{\nabla}\phi|/a_{\scriptscriptstyle
0})\vec{\nabla}\phi]=4\pi G\rho,$ (1)
$\rho$ being the true (“baryonic”) matter density. Here $\mu(x)$ is the
interpolating function characterizing the theory, and $a_{\scriptscriptstyle
0}$ is the MOND acceleration constant, known from various analyses to be
$a_{\scriptscriptstyle 0}\approx 1.2\times 10^{-8}{\rm cm~{}s^{-2}}$ (see,
e.g., Stark, McGaugh, & Swaters 2009 who find that gas dominated galaxies
satisfy the mass-asymptotic-rotational-velocity relation predicted by MOND,
$M=a_{\scriptscriptstyle 0}^{-1}G^{-1}V^{4}_{\infty}$, with this value of
$a_{\scriptscriptstyle 0}$). Similar results will follow from the pristine,
algebraic formulation of MOND (Milgrom 1983). Also, if the halo properties are
derived from rotation-curve analysis, the same results will follow in modified
inertia theories, since these theories predict the algebraic relation between
the Newtonian and MOND accelerations for circular orbits. We do not know
exactly what these modified inertia theories say about gravitational lensing,
but we expect similar results from this as well. Regarding lensing, the
existing relativistic extension of the modified-Poisson theory, TeVeS (see
Bekenstein 2006 and Skordis 2009 for reviews), says that we can use the halo
as deduced from the modified Poisson theory to derive lensing in the standard
way, at least when we can assume approximate spherical symmetry. Weak-lensing
halo properties can thus be compared directly with the predictions of this
theory.
When interpreted by a Newtonist, the departure predicted by MOND, and
encapsulated in the difference between the MOND acceleration field
$\vec{\nabla}\phi$ and the Newtonian one, is explained by the presence of
“dark matter”, or “phantom matter” whose density is (Milgrom 1986)
$\rho_{p}={1\over 4\pi G}\Delta\phi-\rho.$ (2)
Using the field equation (1) we can write
$\rho_{p}=-{1\over 4\pi Ga_{\scriptscriptstyle
0}}(\mu^{\prime}/\mu)\vec{\nabla}|\vec{\nabla}\phi|\cdot\vec{\nabla}\phi+(\mu^{-1}-1)\rho,$
(3)
which can be cast in another form
$\rho_{p}=-{a_{\scriptscriptstyle 0}\over 4\pi G}{\bf
e}\cdot\vec{\nabla}\mathcal{U}(|\vec{\nabla}\phi|/a_{\scriptscriptstyle
0})+(\mu^{-1}-1)\rho,$ (4)
where $\mathcal{U}(x)=\int L(x)dx$, with $L=x\mu^{\prime}/\mu$ the logarithmic
derivative of $\mu$, and ${\bf e}$ is a unit vector in the direction of
$\vec{\nabla}\phi$. This form is particularly useful for calculating column
densities of $\rho_{p}$ along field lines, as we want to do here. This
relation is exact. Expression (4), with $\vec{\nabla}\phi$ replaced by $-{\bf
g}$, holds exactly in the more primitive, algebraic formulation, whereby the
MOND acceleration ${\bf g}$ is given by $\mu(|{\bf g}|/a_{\scriptscriptstyle
0}){\bf g}={\bf g}_{\scriptscriptstyle N}$; ${\bf g}_{\scriptscriptstyle N}$
being the Newtonian acceleration; ${\bf g}$ is not generally derivable from a
potential.
Consider first an arbitrary point mass, and integrate expression (4) along a
line through the point mass. This gives the central surface density of the
phantom matter halo surrounding the mass, $\Sigma(0)$. Inside the mass
$\mu\approx 1$ so the second term does not contribute. The integral is
performed in two segments: from $-\infty$ to the point mass (where ${\bf e}$
is opposite the direction of integration) and from the other side of the point
mass to $\infty$. The two combined give
$\Sigma(0)=\int_{-\infty}^{\infty}\rho_{p}dz=\Sigma_{\scriptscriptstyle
M}[\mathcal{U}(\infty)-\mathcal{U}(0)]=\Sigma_{\scriptscriptstyle
M}\int_{0}^{\infty}L(x)dx\equiv\lambda\Sigma_{\scriptscriptstyle M},$ (5)
where,
$\Sigma_{\scriptscriptstyle M}\equiv{a_{\scriptscriptstyle 0}\over 2\pi G}$
(6)
is the relevant surface density proxy for $a_{\scriptscriptstyle 0}$ in the
present context. In the deep MOND regime ($x\ll 1$) $L(x)\approx 1$ , and far
outside the MOND regime $L(x)\approx 0$; so $\lambda$ is of order 1, and
depends only on the interpolating function $\mu(x)$.
I am dealing all along with central column density
$\Sigma(0)=2\int_{0}^{\infty}\rho dr$ of the MOND phantom halo. For a Burkert
halo this column density is related to the quantity $\Sigma_{c}$, used by
Donato & al., by $\Sigma(0)=(\pi/2)\Sigma_{c}$. So, translating the column
density to the MOND analog of $\Sigma_{c}$, call it $\Sigma_{c}^{*}$,
$\Sigma_{c}^{*}=(2\lambda/\pi)\Sigma_{\scriptscriptstyle
M}\equiv\gamma\Sigma_{\scriptscriptstyle M}.$ (7)
We have
$\Sigma_{\scriptscriptstyle M}=138(a_{\scriptscriptstyle 0}/1.2\times
10^{-8}{\rm cm~{}s^{-2}})M_{\scriptscriptstyle\odot}{\rm pc}^{-2},$ (8)
or, for the nominal value of $a_{\scriptscriptstyle 0}$,
$\log(\Sigma_{\scriptscriptstyle M}/M_{\scriptscriptstyle\odot}{\rm
pc}^{-2})=2.14$, compared with the value
$\log(\Sigma_{c}/M_{\scriptscriptstyle\odot}{\rm pc}^{-2})=2.15\pm 0.2$ found
by Donato et al.111The predicted MOND “halo” of an isolated system is not well
described by a Burkert profile: The MOND “halo” density behaves asymptotically
as $r^{-2}$, not $r^{-3}$, and it is expected to have a depression around the
center not a decreasing density profile everywhere. Nevertheless, these
differences are expected to produce only differences by a factor of order 1 in
the resulting $\Sigma_{c}$. The very near equality of
$\Sigma_{\scriptscriptstyle M}$ and the central value found by Donato & al. is
thus somewhat fortuitous..
For the limiting form of $\mu(x)$–with $\mu(x)=x$, for $x\leq 1$, and
$\mu(x)=1$, for $x>1$–we have $\lambda=1$, and $\gamma=2/\pi$. For
$\mu(x)=x(1+x^{2})^{-1/2}$, we have $\lambda=\pi/2$, and $\gamma=1$. Values of
$\lambda$ for other forms of $\mu$ can be read off Fig. 3 of Milgrom & Sanders
(2008) (where they were deduced numerically, and appear for other purposes).
One sees that $1\lesssim\lambda\lesssim 3$, and so $0.7\lesssim\gamma\lesssim
2$ for the range of $\mu$ forms studied there222The coefficient $\lambda$
diverges if $1-\mu(x)$ behaves at large $x$ as $x^{-1}$ or slower. The
divergence does not occur in the MOND regime, but comes from the Newtonian
regime very near the point mass. Such a behavior of $\mu$ is, however excluded
strongly from solar system constraints, and I preclude it..
Equations (5)-(7) are our basic result, around which all else in the paper
revolves. They tell us that for the simple case of a point mass a universal
value of $\Sigma_{c}$ is indeed predicted by MOND; its value is
$\approx\Sigma_{\scriptscriptstyle M}$, which agrees very well with the value
found by Donato & al..
Consider now an extended mass, $M$. If the mass is well contained within its
MOND transition radius, $R_{\scriptscriptstyle M}=(MG/a_{\scriptscriptstyle
0})^{1/2}$, namely if the Newtonian accelerations, and hence the MOND
accelerations, are high everywhere within the mass, then the procedure we
followed for a point mass applied approximately, and we get again
$\Sigma_{c}^{*}\approx\gamma\Sigma_{\scriptscriptstyle M}$.
Here I have to pause, and comment on a subtlety in the use of eq.(4), and in
interpreting the results thereof. This I demonstrate with two examples. First
consider a mass of finite extent whose density does not increase towards its
center as $r^{-1}$ or faster. In this case, the Newtonian acceleration, and so
also the MOND acceleration, goes to zero at the center, even if these
accelerations are much higher than $a_{\scriptscriptstyle 0}$ in most of the
bulk. In other words, there are two MOND regimes: one within some small sphere
of radius $r_{1}$ around the center, and another beyond the MOND transition
radius, $R_{\scriptscriptstyle M}=(MG/a_{\scriptscriptstyle 0})^{1/2}$. The
small $r$ region contributes to $\Sigma(0)$ through the first term in eq.(4),
an amount $-\Sigma_{\scriptscriptstyle M}\int_{0}^{X_{0}}L(x)dx$, where
$X_{0}$ is the maximum (MOND) acceleration in units of $a_{\scriptscriptstyle
0}$. This contribution is $\approx-\lambda\Sigma_{\scriptscriptstyle M}$ for
$X_{0}\gg 1$. The outer region contributes a positive quantity of the same
magnitude. In addition, the inner region contributes through the second term
in eq.(4), and its total contribution is positive (the phantom density is
always positive in the spherical case). The inner region of phantom mass, even
if it contributes to $\Sigma(0)$, has only little mass, is dynamically
unimportant, at large, and should not be included when comparing with results
for global halo parameters. I shall thus ignore it, and take
$\lambda\approx\int_{0}^{X_{0}}L(x)dx$. When the baryonic surface density is
low, the central, low-acceleration region is expanded and encompasses the
whole mass. The contribution of the first term in eq.(4) then can, indeed, be
taken to vanish, and the contribution to $\Sigma(0)$ comes from the second
term.
In another example, consider two arbitrary point masses along the line of
sight. Integrating the phantom density in eq.(4) along the line of sight now
gives $\Sigma(0)=2\lambda\Sigma_{\scriptscriptstyle M}$. (We now have to
integrate over four segments over which ${\bf e}$ changes sign: from $-\infty$
to the first mass, from there to the zero-field point somewhere between the
masses, from there to the second mass, and from there to infinity). This value
is exact and independent of the distance between the masses. How is this
consistent then with our deduction that
$\Sigma(0)\approx\lambda\Sigma_{\scriptscriptstyle M}$ for all systems well
within their transition radius? When the two masses are well separated, by
more then their joint transition radius, there is an extended halo surrounding
each of the masses, each halo with its own
$\Sigma(0)\approx\lambda\Sigma_{\scriptscriptstyle M}$, and the two column
densities add up. When the two masses are near each other, well within their
joint $R_{\scriptscriptstyle M}$, there will be a common halo of phantom
matter residing roughly beyond $R_{\scriptscriptstyle M}$, and this indeed has
$\Sigma(0)\approx\lambda\Sigma_{\scriptscriptstyle M}$ [arising from
integrating eq(4) in the outer two segments]. In addition, there is a small
region around the point of zero field between the two masses, which
contributes the same amount to the central column density, but which contains
little mass, is dynamically unimportant in the present context, and should be
eliminated from the result that is to be compared with the observations.
Keeping these caveats in mind, the reasoning leading to eq.(5) can be applied
not only to spherical systems. For example, for a disk galaxy with a high
central surface (baryonic) density,
$\Sigma_{b}(0)\gg\Sigma_{\scriptscriptstyle M}$, we can use this equation to
calculate the column density either along the symmetry axis, or along a
diameter in the plane of the disc (in both cases the field is always parallel
or antiparallel to the line of integration). If we ignore the small region of
phantom matter near the very center (or if we add a small matter cusp that
prevents the acceleration from vanishing at the center) we again get
$\Sigma(0)=\lambda\Sigma_{\scriptscriptstyle M}$.
Take now, more generally the extent of our mass to be $R$, and its mean
density $\rho$, and define $\Sigma_{b}=\rho R$, the baryonic equivalent of
$\Sigma_{c}$. The second term in eq.(4) can be estimated to contribute to
$\Sigma(0)$
$\approx 2\rho R[\mu^{-1}(g/a_{\scriptscriptstyle 0})-1],$ (9)
where $g$ is the MOND mean acceleration inside the mass, and is given by
$(g/a_{\scriptscriptstyle 0})\mu(g/a_{\scriptscriptstyle 0})\approx{4\pi\over
3}{\rho RG\over a_{\scriptscriptstyle 0}}={2\over
3}{\Sigma_{b}\over\Sigma_{\scriptscriptstyle M}}.$ (10)
The first term in eq.(4) is taken to contribute
$\approx\Sigma_{\scriptscriptstyle M}\int_{0}^{X_{0}}L(x)dx$, where
$X_{0}=g/a_{\scriptscriptstyle 0}$. Thus, we can write
$\Sigma_{c}^{*}=(2/\pi)\Sigma(0)\approx\Sigma_{\scriptscriptstyle
M}\\{(6/\pi)X_{0}[1-\mu(X_{0})]+\int_{0}^{X_{0}}L(x)dx\\}.$ (11)
For $X_{0}\gg 1$ this gives
$\Sigma_{c}^{*}\approx\gamma\Sigma_{\scriptscriptstyle M}$, again.
## 3 Low surface density systems
MOND does permit arbitrarily low values of $\Sigma_{c}$ for phantom halos in
low acceleration systems. When $X_{0}\ll 1$, namely, when the maximum (MOND)
acceleration in the system is much smaller than $a_{\scriptscriptstyle 0}$, we
get from eq.(11), to lowest order in $X_{0}$,
$\Sigma_{c}^{*}\approx(6/\pi+1)\Sigma_{\scriptscriptstyle M}X_{0}\approx
2.4\left({\Sigma_{b}\over\Sigma_{\scriptscriptstyle
M}}\right)^{1/2}\Sigma_{\scriptscriptstyle M}.$ (12)
Such low acceleration systems are characterized by low baryonic surface
densities $\Sigma_{b}/\Sigma_{\scriptscriptstyle M}\ll 1$. Note, however, that
the departure from the universal $\Sigma_{c}^{*}$ sets in at rather low
baryonic surface densities, since $\Sigma_{c}^{*}/\Sigma_{\scriptscriptstyle
M}$ scales as the square root of $\Sigma_{b}/\Sigma_{\scriptscriptstyle M}$.
The lowest acceleration disc galaxies studied to date have $X_{0}$ values only
down to 0.1-0.2; and we see from eq.(12) that even for values of $X_{0}$ as
low as 1/5 we get $\Sigma_{c}^{*}\approx 0.6\Sigma_{\scriptscriptstyle M}$.
Clearly, however, MOND does predicts that, for extremely low baryonic surface
density galaxies, the CHSD falls increasingly below the quasi-universal value.
To superficially check this expectation, I looked (rather randomly) in the
literature for derived halo parameters for the lowest acceleration disk
galaxies with rotation-curve analysis. Three such galaxies were analyzed in
light of MOND by Milgrom & Sanders (2007), showing rather satisfactory
agreement. These were also analyzed earlier in terms of cored isothermal
halos: For KK98 250 and KK98 251, I find in Begum & Chengalur (2005) best-fit
parameters that give $\Sigma_{c}=56$, and $66M_{\scriptscriptstyle\odot}/{\rm
pc}^{2}$, respectively. For NGC 3741, Begum & al. (2005) find parameters that
yield $\Sigma_{c}=56M_{\scriptscriptstyle\odot}/{\rm pc}^{2}$. All three
values fall substantially below the nominal quasi-universal value of
$\Sigma_{c}=140M_{\scriptscriptstyle\odot}/{\rm pc}^{2}$, and are consistent,
within the uncertainties, with our rough estimate (12), having $X_{0}$ values
of between 0.1 and 0.3. The first two galaxies were not included in the Donato
& al. analysis; but NGC 3741 was included, based on the analysis of Gentile &
al. (2007) (assuming a Burkert, not a cored isothermal halo), whose results
give $\Sigma_{c}=74$. This value is higher than the result of Begum & al.
(2005) (though consistent within the uncertainties), but still only about half
the quasi-universal value.
Another low acceleration galaxy that is worth analyzing in detail (and is not
included in the Donato & al. sample) is the dwarf Andromeda IV. Its rotation
curve is given in Chengalur & al. (2007). To my knowledge, its photometry and
HI distribution are not yet available publicly for rotation curve analysis.
However, according to Chengalur et &. (2007) it is heavily dominated by gas
with $M_{\scriptscriptstyle gas}/L\approx 18$, and it shows a very strong mass
discrepancy with $M_{\scriptscriptstyle dyn}/M_{\scriptscriptstyle gas}\approx
14$ at the last measured point. In deriving a cored isothermal halo parameters
we can thus approximately ignore the baryons and fit the rotation curve with
the halo alone. Doing this, I find, tentatively, $\Sigma_{c}\sim
45M_{\scriptscriptstyle\odot}/{\rm pc}^{2}$, about three times lower than
$\Sigma_{\scriptscriptstyle M}$. Since in this case $X_{0}\sim 0.1-0.15$, this
is also in agreement with the estimate of eq.(12).
Why then do Donato & al. suggest that the quasi-universal value of
$\Sigma_{c}$ applies to all galaxies, including the very low-acceleration
ones? This is based mostly on the analysis of dwarf spheroidal satellites of
the Galaxy. Their analysis includes only one well studied low-acceleration
disk, the above mentioned, NGC 3741–for which, as we saw, the actual
$\Sigma_{c}$ could be lower–and one somewhat higher acceleration galaxy, DDO
47. As regards the dwarf spheroidal Milky-Way satellites, MOND would indeed
predict lower values of $\Sigma_{c}$ than adopted by Donato & al.. However, as
Donato & al. emphasize themselves, the analysis of these systems is beset by
uncertainties in the model assumptions (e.g., assumptions on orbital
anisotropies), leading to non-unique results. Angus (2008) has analyzed these
dwarf spheroidals in MOND, and found that, with two exceptions perhaps, they
can be well explained by MOND, assuming appropriate orbit anisotropy
distributions. This would mean, as I stressed above, that there are acceptable
“halo” models that are consistent with the predictions of MOND. The disparate
values adopted by Donato & al. only demonstrate the non-uniqueness of the
halo-parameter determination.
## 4 Discussion
I have shown that the acceleration constant of MOND $a_{\scriptscriptstyle 0}$
defines a special surface density parameter $\Sigma_{\scriptscriptstyle
M}=a_{\scriptscriptstyle 0}/2\pi G$. This serves as a quasi-universal central
surface density of phantom halos around objects of all masses and structures,
provided they are themselves in the Newtonian regime (i.e., with bulk
accelerations of order $a_{\scriptscriptstyle 0}$ or higher).
This is a particularly interesting prediction of MOND, because most of the
other salient MOND predictions relate properties of the true matter (baryons)
to those of the putative dark matter halo. This is the case for the mass-
velocity (baryonic Tully-Fisher) relation, the Faber-Jackson relation, the
transition from baryon dominance to DM dominance at a fixed acceleration, the
full prediction of rotation curves, the necessity of a disk component of DM,
in disk galaxies, in addition to a spheroidal halo, etc. (see Milgrom 2008 for
a more detailed list, and explanations). Here, however, we have a prediction
that speaks of a property of halos themselves, without regard to the true mass
that engenders them, apart from the requirement that the baryons be well
concentrated. $\Sigma_{\scriptscriptstyle M}$ may also be viewed as an upper
limit, and accumulation value, for “halo” central surface densities,
irrespective of baryonic properties. It is clear then, that the
$a_{\scriptscriptstyle 0}$ that appears in this prediction need have nothing
to do with the $a_{\scriptscriptstyle 0}$ that appears in other relations, in
the framework of the DM doctrine. We could have a sample of halos all
satisfying the present prediction, and add to them baryons arbitrarily, so as
not to satisfy, e.g., the baryonic-mass-velocity relation
$MGa_{\scriptscriptstyle 0}=V^{4}$, which also revolves around some
acceleration constant. The fact that the $a_{\scriptscriptstyle 0}$ emerging
from the phenomenology here is the same as that appearing in the other
phenomenological relation should be viewed as another triumph of MOND.
There are two other MOND predictions that speak of properties of the halo
alone. The first is that the density profile of the “halo” of any isolated
object behaves asymptotically as $r^{-2}$ (asymptotic flatness of rotation
curves). The other such prediction is the maximally allowed acceleration (of
order $a_{\scriptscriptstyle 0}$) that a halo can produce (Brada & Milgrom
1999). This is simply a reflection of the MOND tenet that the phantom mass
cannot be present where accelerations are higher than roughly
$a_{\scriptscriptstyle 0}$. The prediction I discuss here can be understood,
qualitatively, as a result of the above two: On the asymptotic, $r^{-2}$, tail
of the “halo”, the acceleration it produces is $g_{h}\approx 4\pi G\rho r$.
Going inward, the maximum-acceleration prediction tells us that this behavior
can continue only down to a radius where $g_{h}\sim a_{\scriptscriptstyle 0}$.
Below this radius the halo density profile must become shallow and produce a
core. This gives $\rho_{0}r_{0}\sim a_{\scriptscriptstyle 0}/4\pi G$. It is
this that underlies our more quantitative result here.
However, there is nothing in MOND to forbid the halo density profile from
becoming shallow within a radius much larger then that where $g_{h}\sim
a_{\scriptscriptstyle 0}$. This can happen at arbitrarily large radii,
producing arbitrarily small values of $\Sigma_{c}^{*}$, as indeed our detailed
analysis shows.
## Acknowledgements
This research was supported by a center of excellence grant from the Israel
Science Foundation.
## References
* Angus (2008) Angus, G.W. 2008, MNRAS, 387, 1481
* (2) Begum, A. & Chengalur, J.N. 2005, AA, 424, 509
* Begum & al. (2005) Begum, A., Chengalur, J., & Karachentsev, I.D. 2005, AA 433, L1
* Bekenstein (2006) Bekenstein, J. 2006, Contemp. Phys., 47, 387
* Bekenstein & Milgrom (1984) Bekenstein, J. & Milgrom, M. 1984, ApJ, 286, 7
* Brada & Milgrom (1999) Brada, R. & Milgrom, M. 199, ApJL 512, L17
* Chengalur (2007) Chengalur, J.N., Begum, A., Karachentsev, I.D., Sharina, M., & Kaisin, S.S. 2007, Proceedings of ”Galaxies in the Local Volume”, ed. B. Koribalski, H. Jerjen, arXiv:0711.1807
* Donato & al. (2009) Donato, F., Gentile, G., Salucci, P., Frigerio Martins, C., Wilkinson, M.I., Gilmore, G., Grebel, E.K., Koch, A., & Wyse, R. 2009, MNRAS, 397, 1169
* Gentile & al. (2007) Gentile, G., Salucci, P., Klein, U., & Granato, G.L. 2007, MNRAS, 375, 199
* Milgrom (1983) Milgrom, M. 1983, ApJ, 270, 365
* Milgrom (1986) Milgrom, M. 1986, ApJ, 306, 9
* Milgrom (2008) Milgrom, M., 2008, In Proceedings XIX Rencontres de Blois; arXiv:0801.3133
* Milgrom and Sanders (2005) Milgrom, M., Sanders, R.H. 2005, MNRAS, 357, 45
* Milgrom and Sanders (2007) Milgrom, M., Sanders, R.H. 2007, ApJ, 658L, 17
* Milgrom and Sanders (2008) Milgrom, M., Sanders, R.H. 2008, ApJ., 678, 131
* Skordis (2009) Skordis, C. 2009, Class. Quant. Grav. 26 (14), 143001
* Stark & al. (2009) Stark, D.V., McGaugh, S.S., & Swaters, R.A. 2009, AJ, 138, 392
|
arxiv-papers
| 2009-09-28T20:00:04 |
2024-09-04T02:49:05.520431
|
{
"license": "Public Domain",
"authors": "Mordehai Milgrom (Weizmann Institute)",
"submitter": "Mordehai Milgrom",
"url": "https://arxiv.org/abs/0909.5184"
}
|
0909.5258
|
# OBSERVATIONAL CONSTRAINTS ON THE GENERALIZED CHAPLYGIN GAS
SERGIO DEL CAMPO Instituto de Física, Pontificia Universidad de Católica de
Valparaíso, Casilla 4950
Valparaíso , Chile
sdelcamp@ucv.cl J.R.VILLANUEVA Instituto de Física, Pontificia Universidad
de Católica de Valparaíso, Casilla 4950
Valparaíso , Chile
jose.villanueva.l@mail.ucv.cl
###### Abstract
In this paper we study a quintessence cosmological model in which the dark
energy component is considered to be the Generalized Chaplygin Gas and the
curvature of the three-geometry is taken into account. Two parameters
characterize this sort of fluid, the $\nu$ and the $\alpha$ parameters. We use
different astronomical data for restricting these parameters. It is shown that
the constraint $\nu\lesssim\alpha$ agrees enough well with the astronomical
observations.
###### keywords:
Dark Energy; exotic fluid.
## 1 Introduction
Current measurements of redshift and luminosity-distance relations of Type Ia
Supernovae (SNe) indicate that the expansion of the Universe presents an
accelerated phase [1, 2]. In fact, the astronomical measurements showed that
Type Ia SNe at a redshift of $z\sim 0.5$ were systematically fainted which
could be attributed to an acceleration of the universe caused by a non-zero
vacuum energy density. This gives as a result that the pressure and the energy
density of the universe should violate the strong energy condition,
$\rho_{X}+3\,p_{X}\,>\,0$, where $\rho_{X}$ and $p_{X}$ are energy density and
pressure of some matter denominated dark energy, respectively. A direct
consequence of this, it is that the pressure must be negative. However,
although fundamental for our understanding of the evolution of the universe,
its nature remains a completely open question nowadays.
Various models of dark energy have been proposed so far. Perhaps, the most
traditional candidate to be considered is a non-vanishing cosmological
constant [3, 4]. Other possibilities are quintessence [5, 6], k-essence [7, 8,
9], phantom field [10, 11, 12], holographic dark energy [13, 14], etc. (see
ref. D09 for model-independent description of the properties of the dark
energy and ref. S09 for possible alternatives).
One of the possible candidate for dark energy that would like to consider here
is the so-called Chaplygin gas (CG) [17]. This is a fluid described by a quite
unusual equation of state, whose characteristic is that it behaves as a
pressureless fluid at the early stages of the evolution of the universe and as
a cosmological constant at late times. Actually, in ref. K01, it was
recognized its relevance to the detected cosmic acceleration. They found that
the CG model exhibits excellent agreement with observations. From this time,
the cosmological implications of the CG model have been intensively
investigated in the literature [19, 20, 21, 22]. Subsequently, it was notice
that this model can be generalized, which now it is called the generalize
Chaplygin gas (GCG). This GCG model was introduced in ref. K01 and elaborated
in ref. B02. After these works, the cosmological implications of the GCG model
have been intensively investigated in the literature [24, 25, 26, 27, 28, 29,
30, 31, 32]. There are claims that it does not pass the test connected with
structure formation because of predicted but not observed strong oscillations
of the matter power spectrum [33]. It should be mentioned, however, the
oscillations in the Chaplygin gas component do not necessarily imply
corresponding oscillations in the observed baryonic power spectrum [34]. This
is a topic that requires much more studies. It is was realized that these kind
of models have a clearly stated connection with high-dimension theories [35].
Here, the GCG appears as an effective fluid associated with d-branes. Also, at
the fundamental level, it could be derived from the Born-Infeld action [36].
On the other hand, today we do not know precisely the geometry of the
universe, since we do not know the exact amount of matter present in the
Universe. Various tests of cosmological models, including space time geometry,
galaxy peculiar velocities, structure formation and very early universe
descriptions (related to the Guth s inflationary universe model [37]) support
a flat universe scenario. Specifically, by using the five-year Wilkinson
Microwave Anisotropy Probe (WMAP) data combined with measurements of Type Ia
supernovae (SN) and Baryon Acoustic Oscillations (BAO) in the galaxy
distribution, was reported the following value for the total matter density
parameter, $\Omega_{T}$, at the 68% CL uncertainties, $\Omega_{T}=1.02\pm
0.02$ [38].
In this respects we wish to study universe models that have curvature and are
composed by two matter components. One of these components is the usual
nonrelativistic dark matter (dust); the other component corresponds to dark
energy which is supposed to be a sort of quintessence-type matter, described
by a Chaplygin gas-type, or more specifically the GCG.
We should mention that in what concern with the Bayesian analysis the
cosmological constant is favored over GCG [39, 40, 41]. However, in ref. [42],
it was shown that the GCG models, proposed as candidates of the unified dark
matter-dark energy (UDME), are tested with the look-back time (LT) redshift
data. They found that the LT data only give a very weak constraint on the
parameters. But, when they combine the LT redshift data with the baryonic
acoustic oscillation peak the GCG appears as a viable candidate for dark
energy. On the other hand, the GCG model has been constrained with the
integrated Sach-Wolf effect. Recently, a gauge-invariant analysis of the
baryonic matter power spectrum for GCG cosmologies was shown to be compatible
with the data [43, 32, 44]. This result seems to strengthen the role of
Chaplygin gas type models as competitive candidates for the dark sector.
Our paper is organized as follow: In section II we present the main
characteristic properties and we introduce some definition related to the GCG.
In section III we study the kinematics of our model. Here, we take quantities
such that the modulus distance, luminosity distance, angular size, among
others. In section IV we proceed to describe the so-called shift parameter
which is related to the position of the first acoustic peak in the power
spectrum of the temperature anisotropies of the cosmic microwave background
(CMB) anisotropies. We give our conclusions in Section V.
## 2 The Generalized Chaplygin Gas (GCG)
Let us star by considering the equation of state (EOS) corresponding to the
GCG
$p_{gcg}=-\nu\frac{\Xi}{\rho_{gcg}^{\alpha}}\,.$ (1)
Here, $p_{gcg}$ and $\rho_{gcg}$ are the pressure and the energy density
related to the GCG, respectively. $\nu$ is the square of the actual speed of
sound in the GCG and $\alpha$ is the GCG index. $\Xi$ is a function of
$\alpha$ and $\rho_{gcg}^{(0)}$ (the present value of the energy density of
the GCG), and it is given by
$\Xi\equiv\Xi(\rho_{gcg}^{(0)}\mid\alpha)=\frac{1}{\alpha}\left(\rho_{gcg}^{(0)}\right)^{1+\alpha}.$
(2)
The dimensionless energy density related to the GCG
$f_{gcg}(z;\nu,\alpha)\equiv\rho_{gcg}(z;\nu,\alpha)/\rho_{gcg}^{(0)}$ becomes
given as a function of the red shift, $z$, and the parameters $\alpha$ and
$\nu$ as follows
$f_{gcg}(z;\nu,\alpha)=\left[\frac{\nu}{\alpha}+\left(1-\frac{\nu}{\alpha}\right)(1+z)^{3(1+\alpha)}\right]^{\frac{1}{1+\alpha}}.$
(3)
Here, we have considering a Friedmann-Robertson-Walker (FRW) metric, and we
have used the energy conservation equation:
${\frac{d\rho_{gcg}}{dt}+3H\left(\rho_{gcg}+p_{gcg}\right)=0}$, where $H$
represents the Hubble factor. Note that if $\alpha=\nu$, we get that
$f_{gcg}(z,\alpha)=1$ (the same happen for $z=0$), which means that the energy
density related to the GCG corresponds to a cosmological constant.
In Fig.1 we plot $f_{gcg}(z;\nu,\alpha)$ as a function of the red shift, $z$.
Note that this function is highly sensitive to the difference between the
values of $\alpha$ and $\nu$.
file=Fig01.eps,width=10cm
Figure 1: Plot of the function $f_{gcg}(z;\nu,\alpha)$ as a function of the
red shift, $z$ for the cases $\alpha>\nu$ ( $\alpha=0.9$ ; $\nu=0.1$, blue
line) and $\alpha\approx\nu$ ($\alpha=0.9$; $\nu=0.88$, red line). These two
cases are compared with that corresponding to the cosmological constant,
$\Lambda$, case (dashed line).
The derivative of the function $f_{gcg}(z;\nu,\alpha)$ with respect to the
redshift, $z$, becomes given by
$\frac{df_{gcg}(z;\nu,\alpha)}{dz}\equiv
f^{\prime}_{gcg}(z;\nu,\alpha)=3\left(1-\frac{\nu}{\alpha}\right)\frac{(1+z)^{3\alpha+2}}{f^{\alpha}(z;\nu,\alpha)}\,.$
Note that the sign of this function depends on the values that the constants
$\nu$ and $\alpha$ could take . For $\nu\gtrless\alpha$ we have that
$f^{\prime}_{gcg}(z;\nu,\alpha)\lessgtr 0$. We need to have that the function
$f_{gcg}(z;\nu,\alpha)$ to be greater that zero, since it corresponds to an
energy density in an expanding universe. Thus, we expect that the case
$\nu<\alpha$ be relevant for our study.
We can write the EOS related to the GCG in the barotropic form as follows
$p_{gcg}=\omega_{gcg}\rho_{gcg},$ (4)
where the equation of state parameter, $\omega_{gcg}(z)$, becomes given by
$\omega_{gcg}(z)=-\frac{\nu/\alpha}{\frac{\nu}{\alpha}+\left(1-\frac{\nu}{\alpha}\right)(1+z)^{3(1+\alpha)}}.$
(5)
Of course, for $\nu=\alpha=0$ we get $\omega_{gcg}=-1$, corresponding to the
cosmological constant case. In Fig.2 we have plotted the EOS parameter,
$\omega(z;\nu,\alpha)$, as a function of the red shift, $z$. Note that for
$z_{c}=\left(\frac{1}{1-\alpha/\nu}\right)^{\frac{1}{3(1+\alpha)}}-1$, with
$\alpha<\nu$, the EOS parameter, $\omega(z_{c};\nu,\alpha)$, goes to minus
(plus) infinity, i.e $\omega(z_{c};\nu,\alpha)\longrightarrow\mp\infty$. The
minus (plus) sign corresponds to the $z<z_{c}$ ($z>z_{c}$) branch. These
situations are represented in Fig.2 by the blue lines. For an accelerating
phase of the universe we need to take into account the $z<z_{c}$ branch only,
since it gives the right negative sign for the EOS parameter. For
$\alpha>\nu$, the EOS parameter always is negative, i.e.
$-\frac{\nu}{\alpha}\leq\omega_{gcg}<0$. Summarizing, we can see from the
latter equation that for $\nu>\alpha$ we have
$-1<-\nu/\alpha\leq\omega_{gcg}<0$ and for $\nu<\alpha$ we find that
$-1>-\nu/\alpha\geq\omega_{gcg}>-\infty$.
A Taylor expansion of the EOS parameter, $\omega_{gcg}(z)$, around $z=0$
becomes
$\omega_{gcg}(z)=-\beta+3\beta(1-\beta)(1+\alpha)z-3\beta(1-\beta)(1+\alpha)\left[3(1-2\beta)(1+\alpha)+1\right]z^{2}+O(z^{3}),$
(6)
where $\beta=\frac{\nu}{\alpha}$.
In a spatially flat universe, the combination of WMAP and the Supernova Legacy
Survey (SNLS) data leads to a significant constraint on the equation of state
parameter for the dark energy $w(0)=-0.967^{+0.073}_{-0.072}$ [45]. This
constraint restricts the value of the ratio $\frac{\nu}{\alpha}$. The value of
this ratio used above, (see Fig. 1), lies inside the observational
astronomical range of the parameter $\omega(0)$. The case in which the EOS
parameter is a linear function of the redshift was studied in [46, 47]. This,
it is a good parametrization at a low redshift.
Phenomenological models of a specific time dependent parametrization of the
EOS, together with a constant speed of sound have being described in the
literature. A simple example is the parametrization expressed by the EOS [48,
49] $\omega(z)=\omega(0)+\frac{d\omega(z)}{dz}\biggr{|}_{0}\frac{z}{(1+z)}$
corresponding to non-interacting dark energy. By matching this parametrization
with our expression at low redshift we find that the parameter
$\frac{d\omega(z)}{dz}\biggr{|}_{0}$ and $3\beta(1-\beta)(1+\alpha)$
coincides. The determination of the dynamical character of the EOS parameter,
$\omega(z)$, becomes important in future experiments. This relevance has been
notice by the The Dark Energy Task Force (DETF) [50]. The coming decade will
be an exciting period for dark energy research.
file=Fig02.eps,width=10cm
Figure 2: Plot of the EOS parameter, $\omega(z;\nu,\alpha)$, as a function of
the red shift, $z$. This function for $\nu<\beta$ lies in the range between
$-\nu/\beta$ (for $z=0$) and $0$ (for $z\longrightarrow\infty$). For
$\nu=\beta$ this parameter gets the value $-1$, and for $\nu>\beta$ this
parameter present two branches (one positive and the other negative). It
becomes $\omega_{gcg}\longrightarrow\mp\infty$ at some specific value of the
red shift, $z=z_{c}$.
## 3 KINEMATICS OF THE MODEL
In order to describe some important distances we introduce the dimensionless
Hubble function, $E(z)=\frac{H(z)}{H_{0}}$, reads as
$E^{2}(z;\nu,\alpha)=\Omega_{cdm}^{(0)}(1+z)^{3}+\Omega_{k}^{(0)}(1+z)^{2}+\Omega_{gcg}^{(0)}f_{gcg}(z;\nu,\alpha),$
(7)
where $\Omega_{k}^{(0)}=-k/H_{0}^{2}$, and $\Omega_{cdm}^{(0)}$ and
$\Omega_{k}^{(0)}$ represent the present cold dark matter and curvature
density parameters, respectively. Here. the parameter $k$ takes the values
$-1$, $0$ or $+1$, for open, flat or closed geometries, respectively.
$H_{0}\equiv H(0)=100h\,km\,s^{-1}Mpc^{-1}$ is the current value of the Hubble
parameter. The $E(z;\nu,\alpha)$ quantity depends on the values of the
parameters $\alpha$ and $\nu$, apart of the actual values of the density
parameters, $\Omega_{k}^{(0)}$, $\Omega_{cdm}^{(0)}$ and $\Omega_{gcg}^{(0)}$.
Note that these latter parameters satisfy the constraint
$\Omega_{k}^{(0)}+\Omega_{cdm}^{(0)}+\Omega_{gcg}^{(0)}=1$. On the other hand,
astronomical measurements will constraint the $\alpha$ and $\nu$ parameters,
as we will see.
In FIG.3 we have taken $\Omega_{gcg}^{(0)}=0.725$ and
$\Omega_{cdm}^{(0)}=0.275$ with $\Omega_{k}^{(0)}=0$ for the theoretical
curves, and we have introduced the observational values for the Hubble
parameter from Ref. S06. The curves were plotted for both regimes,
$\alpha\gtrsim\nu$ ( $\alpha=0.9$ and $\nu=0.88$) and $\alpha\gg\nu$
($\alpha=0.9$ and $\nu=0.1$). In order to compare these curves with the
standard model we have included the $\Lambda$CDM model, also. Note that the
curve with $\alpha\gtrsim\nu$ is closed to the observational data than that
curve corresponding to $\alpha\gg\nu$. Thus, when curvature is present into
the cosmological model, the curve with $\alpha\gtrsim\nu$ competes with the
standard cosmology (the $\Lambda$CDM model), in this respect. Note also that
no difference between the $\Lambda$CDM model and that model were a GCG is
included together with the curvature is found for low redshift.
file=Fig03.eps,width=10cm
Figure 3: Plot of the Hubble parameter, $H(z;\nu,\alpha)$, as a function of
the redshift, $z$. Here, we have introduced the observational values for the
Hubble’s parameter (see ref. S09) . The analytical curves were determined by
using $H_{0}=73[\textrm{Mpc}^{-1}\textrm{Km}/\textrm{s}]$ for the present
value of the Hubble’s parameter and we have taken $\Omega_{gcg}^{(0)}=0.725$
and $\Omega_{cdm}^{(0)}=0.275$ for a flat geometry. The two GCG curves (small
and large dashing) were plotted by taking $\alpha=0.9\gg\nu=0.1$ and
$\alpha=0.9\gtrsim\nu=0.88$. The solid line represents the $\Lambda$CDM model.
### 3.1 Luminosity distance - redshift
One of the more important observable magnitudes that we will consider here
will be luminous distance, $d_{L}$. This is defined as the ratio of the
emitted energy per unit time, $\mathcal{L}$, and the energy received per unit
time $\mathcal{F}$ [52]
$d_{L}=\frac{\mathcal{L}}{4\pi\mathcal{F}}.$ (8)
In this way, the luminosity distance can be written as
$d_{L}(z;\nu,\alpha)=H_{0}^{-1}(1+z)y(z;\nu,\alpha),$ (9)
where the function $y(z;\nu,\alpha)$ becomes given by
$y(z;\nu,\alpha)=\frac{1}{\sqrt{\left|\Omega_{k}^{(0)}\right|}}\;S_{k}\left\\{\sqrt{\left|\Omega_{k}^{(0)}\right|}\int_{0}^{z}\frac{dz^{\prime}}{E(z^{\prime};\nu,\alpha)}\right\\},$
(10)
and $S_{k}(x)$ takes the following expression for the different values of the
parameter $k$,
$S_{k}(x)\left\\{\begin{array}[]{ll}\sin(x),&k=+1;\\\ x,&k=0;\\\
\sinh(x),&k=-1.\\\ \end{array}\right.$ (11)
file=y.eps,width=10cm
Figure 4: Plots of the theoretical curves for $y(z;\nu,\alpha)$ as a function
of the redshift, $z$ for two different regimes: $\nu=0.1\ll\alpha=0.9$ and
$\nu=0.88\lesssim\alpha=0.9$. These curves are compared with astronomical data
extracted from Daly et al 2007; Left top: 192 Supernovas (Sn); Right top: 30
Radio Galaxies (RG); Left down: 38 Galaxy Clusters (CL); Right down: 192 Sn +
30 RG + 38 CL. Here, we have taken the values $\Omega_{k}^{(0)}=0.0045$,
$\Omega_{gcg}^{(0)}=0.7165$ and $\Omega_{cdm}^{(0)}=0.2790$.
By using the samples of $192$ supernova standard candles, $30$ radio galaxy
and $38$ cluster standard rulers, presented in ref. D07, we check our model
described by Eq. (10). This check is done under the assumption that the
curvature density parameter, $\Omega_{k}^{(0)}$ takes the value
$\Omega_{k}^{(0)}=0.0045$ and the other parameters are
$\Omega_{gcg}^{(0)}=0.7165$ and $\Omega_{cdm}^{(0)}=0.2790$. Fig.4 shows some
curves related to our model. It is clear that the range of parameters for the
GCG, as before, it is near to the limit $\alpha\gtrsim\nu$, better that the
limit $\alpha\gg\nu$. Nevertheless, we cannot discriminate with facility when
we compare our curves (for the $\alpha\gtrsim\nu$ case) with that
corresponding to the $\Lambda$CDM model. However, this comparison becomes
indistinguishable for small redshift, i.e. $z\lesssim 0.7$.
One interesting quantity related to the luminosity distance, $d_{L}$, is the
distance modulus, $\mu$, which is defined as [54]
$\mu={5}\;\log_{10}[d_{L}/(1\hbox{Mpc})]+25.$ (12)
In Fig.5 we have plotted $\mu$ as a function of the redshifts, $z$. The values
for the different GCG parameters are the two set: $\alpha=0.9$ and $\nu=0.88$,
and $\alpha=0.9$ and $\nu=0.01$. In each case we have considered that
$\Omega_{cdm}^{(0)}=0.279$, $\Omega_{gcg}^{(0)}=0.7255$ and
$\Omega_{k}^{(0)}=-0.0045$. Also, we have included in this plot the
$\Lambda$CDM model, with $\Omega_{\Lambda}^{(0)}=\Omega_{gcg}^{(0)}=0.7255$.
The data included in this graph were taken from ref. R04. Note that the case
for $\nu\lesssim\alpha$ becomes practically indistinguishable from that
corresponding to the $\Lambda$CDM model.
file=mu.eps,width=10cm
Figure 5: Graphic representing the magnitude $\mu(z;\nu,\alpha)$ as a function
of the redshifts, $z$. Here we have plotted two curves, one for
$\nu\lesssim\alpha$ ($\alpha=0.9$ and $\nu=0.88$) and the other one for
$\nu\ll\alpha$ ($\nu=0.01$ and $\alpha=0.9$). Here, we have taken the values
$\Omega_{gcg}^{(0)}=0.7255$ and $\Omega_{cdm}^{(0)}=0.279$. Also, we have
included in this plot the $\Lambda$CDM model, with
$\Omega_{\Lambda}^{(0)}=\Omega_{gcg}^{(0)}=0.7255$. The data were taken from
Riess et al 2004.
### 3.2 Angular size - redshift
The angular size, $\Theta$, is defined as the ratio of an object s physical
transverse size, $l$, to the angular diameter distance ,$d_{A}$. This latter
distance is related to the luminosity distance, $d_{L}$ by mean of the
relation $d_{A}=d_{L}/(1+z)^{2}$. Therefore, we have
$\Theta(z;\nu,\alpha)\equiv\frac{l}{d_{A}(z;\nu,\alpha)}=\kappa\frac{1+z}{y(z;\nu,\alpha)},$
(13)
Here, $l=l_{0}h^{-1}$, with $l_{0}$ the linear size scaling factor and
$\kappa=lH_{0}/c=0.432l_{0}[mas/pc]$.
Following our treatment of the comparison of the chaplygin gas with the
available data, we use the ref. G99 compilation into 12 bins with 12-13
sources which satisfies the conditions in which the spectral index lies in the
range $-0.38\leq\eta\leq 0.18$ and a total radio luminosity, $L$, which
satisfies the constraint, $Lh^{2}\geq 10^{26}[W/Hz]$.
This points are showed in FIG.6 together with the curves determined by taking
the values $l_{0}=4.86[pc]$ $\Omega_{cdm}^{(0)}=0.2790$,
$\Omega_{gcg}^{(0)}=0.7255$ $\Omega_{k}^{(0)}=-0.0045$. Note once again that
the case for which $\alpha\gtrsim\nu$ becomes favored than that the case
corresponding to $\alpha\gg\nu$. Here, as before we have include the case
corresponding to the $\Lambda$CDM model specified by a continuous line.
file=Fig05.eps,width=10cm
Figure 6: The angular size, $\Theta$, as a function of the redshift, $z$. The
curves were determined by using the value $l_{0}=4.86[pc]$ and
$\Omega_{cdm}^{(0)}=0.2790$, $\Omega_{gcg}^{(0)}=0.7255$
$\Omega_{k}^{(0)}=-0.0045$. The data correspond to 145 sources compiled by
Gurvits et al 1999.
### 3.3 Deceleration, jerk, and snap parameters - redshift
The luminosity distance, $d_{L}$, could be expanded in such a way that the
first Taylor coefficients of this expansion are related to the parameters
denominated deceleration ($q$), jerk ($j$), and snap ($s$) parameters
evaluated at present time. These three parameters are defined in term of the
second, third, and fourth derivatives of the scale factor with respect to
time, respectively. The expansion of $d_{L}$ in term of the redshift, $z$,
reads[54]
$\displaystyle\displaystyle d_{L}(z)=$
$\displaystyle\frac{cz}{H_{0}}\left\\{1+\frac{1}{2}\left[1-q_{0}\right]z-\frac{1}{6}\left[1-q_{0}-3q_{0}^{2}+j_{0}\right.\right.$
(14)
$\displaystyle\left.+\frac{kc^{2}}{H_{0}^{2}a_{0}^{2}}\right]z^{2}+\frac{1}{24}\left[2-2q_{0}-15q_{0}^{2}-15\,q_{0}^{3}+5j_{0}+10\,q_{0}j_{0}\right.$
$\displaystyle\left.\left.+s_{0}+\frac{kc^{2}(1+3q_{0})}{H_{0}^{2}a_{0}^{2}}\right]z^{3}+O(z^{4})\right\\}.$
For our model the deceleration parameter, $q(z;\nu,\alpha)$ becomes given by
$\displaystyle q(z;\nu,\alpha)$ $\displaystyle=$
$\displaystyle-1+\frac{(1+z)E^{\prime}(z;\nu,\alpha)}{E(z;\nu,\alpha)}$ (15)
$\displaystyle=\frac{1}{2}\left[1-\frac{3\frac{\nu}{\alpha}\Omega_{gcg}^{(0)}f^{-\alpha}(z;\nu,\alpha)+\Omega_{k}^{(0)}(1+z)^{2}}{E^{2}(z;\nu,\alpha)}\right].$
The present value of this parameter becomes
$q(0;\nu,\alpha)\equiv
q_{0}(\nu,\alpha)=\frac{1}{2}\left[\Omega_{cdm}^{(0)}-\left(3\frac{\nu}{\alpha}-1\right)\Omega_{gcg}^{(0)}\right].$
(16)
In order to describe an accelerating universe, we need to satisfy the
constraint
$\frac{\nu}{\alpha}>\frac{1}{3}\left(1+\frac{\Omega_{cdm}^{(0)}}{\Omega_{gcg}^{(0)}}\right).$
Taking the ratio
$\frac{\Omega_{cdm}^{(0)}}{\Omega_{gcg}^{(0)}}\approx\frac{3}{7}$ we get that
the $\nu$ and $\alpha$ parameters must satisfy the bound
$\frac{\nu}{\alpha}>\frac{10}{21}$. Note that the values of this ratio that
better agree with the astronomical data described previously satisfy this
restriction, since in most of them we have taken $\nu=0.88\lesssim\alpha=0.9$.
With respect to the jerk, $j$, parameter we have that this becomes given by
$j(z;\nu,\alpha)=3q^{2}(z;\nu,\alpha)+\frac{(1+z)^{2}E^{\prime\prime}(z;\nu,\alpha)}{E(z;\nu,\alpha)},$
(17)
which, at present time, i.e. $z=0$, it becomes
$j(0;\nu,\alpha)=1-\Omega_{k}^{(0)}+\frac{9\nu}{2}\left(1-\frac{\nu}{\alpha}\right)\Omega_{gcg}^{(0)}.$
(18)
In getting this latter expression we have made use of the constraint
$\Omega_{gcg}^{(0)}+\Omega_{cdm}^{(0)}+\Omega_{k}^{(0)}=1$.
This parameter contains information regarding the sound speed of the dark
matter component [57]. Also, the use of the jerk formalism infuses the
kinematical analysis with a feature in that all $\Lambda$CDM models are
represented by a single value of the jerk parameter $j=1$. Therefore, the jerk
formalism enables us to constrain and facilitates simple tests for departures
from the $\Lambda$CDM model in the kinematical manner [58]. In this reference
( R07 and references therein) it is reported the following values for the jerk
parameter: from the type Ia supernovae (SNIa) data of the Supernova Legacy
Survey project gives $j=1.32^{+1.37}_{-1.21}$, the X-ray galaxy cluster
distance measurements gives $j=0.51^{+2.55}_{-2.00}$, the gold SNIa sample
data yields a larger value $j=2.75^{+1.22}_{-1.10}$, and the combination of
all these three data set gives $j=2.16^{+0.81}_{-0.75}$.
file=grafj.eps,width=10cm
Figure 7: This plot presents the jerk, $j$, parameter as a function of the
redshifts, $z$. Here, we have taken the following set of
parameters:($\Omega_{gcg}^{(0)}=0.7255$ and $\Omega_{k}^{(0)}=-0.0045$),
($\Omega_{gcg}^{(0)}=0.721$ and $\Omega_{k}^{(0)}=0$) and
($\Omega_{gcg}^{(0)}=0.7165$ and $\Omega_{k}^{(0)}=0.0045$). These three set
of values have being plotted for the two cases $\nu=0.1\ll\alpha=0.9$ and
$\nu=0.88\lesssim\alpha=0.9$.
In Fig. 7 we have plotted the jerk, $j$, parameter as a function of the
redshifts, $z$, for the set of parameters ($\Omega_{gcg}^{(0)}=0.7255$ and
$\Omega_{k}^{(0)}=-0.0045$), ($\Omega_{gcg}^{(0)}=0.721$ and
$\Omega_{k}^{(0)}=0$) and ($\Omega_{gcg}^{(0)}=0.7165$ and
$\Omega_{k}^{(0)}=0.0045$). These three set of parameters have being plotted
for the two cases $\nu=0.1\ll\alpha=0.9.$ and $\nu=0.88\lesssim\alpha=0.9$.
Note that for the latter case the jerk function present a maximum which is not
present in the other case, when $\nu\ll\alpha$. Note also that for
$z\longrightarrow\infty$ the jerk parameter goes to the value corresponding to
the $\Lambda$CDM case. Also, we do not observe much differences for the
different type of geometries, since the curves are very similar.
With respect to the snap parameter $s$ we have that this parameter becomes
given by
$\displaystyle s(z;\nu,\alpha)$ $\displaystyle=$ $\displaystyle
15q^{3}(z;\nu,\alpha)+9q^{2}(z;\nu,\alpha)$ (19)
$\displaystyle-10q(z;\nu,\alpha)j(z;\nu,\alpha)-3j(z;\nu,\alpha)$
$\displaystyle\hskip
56.9055pt-\frac{(1+z)^{3}E^{\prime\prime\prime}(z;\nu,\alpha)}{E(z;\nu,\alpha)}.$
For a $\Lambda$CDM-universe the present expression for the snap parameter
becomes
$s_{0}=1-\frac{9}{2}\Omega_{cdm},$
and in our case it becomes at present, i.e. $z=0$,
$\displaystyle s(0;\nu,\alpha)$ $\displaystyle=$
$\displaystyle\frac{9\nu\Omega_{gcg}^{(0)}}{4\alpha^{2}}\left[6\alpha^{3}+\alpha^{2}(1-18\nu)+3\nu^{2}(2-\Omega_{gcg}^{(0)})\right.$
$\displaystyle+$
$\displaystyle\left.\alpha(2+\nu(3\Omega_{gcg}^{(0)}-5+12\nu))\right]-\frac{7}{2}$
$\displaystyle+$
$\displaystyle\frac{\Omega_{k}^{(0)}}{4}\left[16+9\nu\Omega_{gcg}^{(0)}-2\Omega_{k}^{(0)}-3\frac{\nu}{\alpha}(2+3\nu)\Omega_{gcg}^{(0)}\right].$
Here, we have used the constraint
$\Omega_{gcg}^{(0)}+\Omega_{cdm}^{(0)}+\Omega_{k}^{(0)}=1$ also.
In Ref. C08 was reported that the actual value of the snap parameter, $s_{0}$,
gets the value $s_{0}=3.39\pm 17.13$ for the fit by using the LZ relation [60]
and the value $s_{0}=8.32\pm 12.16$ for the fit by taking the GGL one [61].
file=grafs.eps,width=10cm
Figure 8: This plot presents the snap, $s$, parameter as a function of the
redshifts, $z$. Here, as before, we have taken the following set of
parameters:($\Omega_{gcg}^{(0)}=0.7255$ and $\Omega_{k}^{(0)}=-0.0045$),
($\Omega_{gcg}^{(0)}=0.721$ and $\Omega_{k}^{(0)}=0$) and
($\Omega_{gcg}^{(0)}=0.7165$ and $\Omega_{k}^{(0)}=0.0045$). These three set
of values have being plotted for the two cases $\nu=0.1\ll\alpha=0.9$ and
$\nu=0.88\lesssim\alpha=0.9$.
## 4 The first Doppler peak of the CMB spectrum and the shift parameter $R$
In this section, we are going to describe the position of the first Doppler
peak ($l_{LS}^{gcg}$) for the model studied in the previous section. The
scales that are important in determining the shape of the CMB anisotropy
spectrum are the sound horizon $d_{s}$ at the time of recombination, and the
previously introduced angular diameter distance $d_{A}^{LS}$ to the last
scattering surface. The former defines the physical scales for the Doppler
peak structure that depends on the physical matter density
($\Omega_{cdm}^{(0)}$), but not on the value of the GCG matter density
($\Omega_{gcg}^{(0)}$ ) or spatial curvature ($\Omega_{k}^{(0)}$), since these
are dynamically negligible at the time of recombination [62]. The latter
depends practically on all of the parameters and is given by
$d_{A}^{LS}=\frac{1}{H_{0}(1+z_{LS})}y(z_{LS};\nu,\alpha)$ (21)
where $y(z_{LS};\nu,\alpha)$ becomes given by (see Eq. 10)
$y(z_{LS};\nu,\alpha)=\frac{1}{\sqrt{\left|\Omega_{k}^{(0)}\right|}}\,S_{k}\left\\{\sqrt{\left|\Omega_{k}^{(0)}\right|}\int_{0}^{z_{LS}}\frac{dz^{\prime}}{E(z^{\prime};\nu,\alpha)}\right\\}.$
(22)
We may write for the localization of the first Doppler peak
$l_{LS}\propto\frac{d_{A}^{LS}}{d_{s}}$ (23)
where the constant of proportionality depends on both the shape of the
primordial power spectrum and the Doppler peak number [63]. Since we are going
to keep the $\Omega_{cdm}^{(0)}$ parameter fixed, we shall take $l_{LS}\approx
d_{A}^{LS}$, up to a factor that depends on $\Omega_{cdm}^{(0)}$ and $z_{LS}$
only
By using that $\Omega_{k}^{(0)}=1-\Omega_{cdm}^{(0)}-\Omega_{gcg}^{(0)}$ and
following ref. W00 and ref. d03 we can write for the position of the first
Doppler peak ($l_{LS}^{gcg}$)
$l_{LS}^{gcg}\sim\Omega_{T}^{-\eta},$ (24)
where $\Omega_{T}=\Omega_{k}^{(0)}+\Omega_{cdm}^{(0)}+\Omega_{gcg}^{(0)}$ and
$\eta=\frac{1}{6}I_{1}^{2}-\frac{1}{2}\frac{I_{2}}{I_{1}},$ (25)
with
$I_{1}=\int_{0}^{1}\frac{dx}{\sqrt{(1-\Omega^{(0)}_{cdm})x^{4}f_{gcg}(1/x-1;\nu,\alpha)+\Omega^{(0)}_{cdm}x}},$
(26)
and
$I_{2}=\int_{0}^{1}\frac{x^{4}f_{gcg}(1/x-1;\nu,\alpha)dx}{\sqrt{\left[(1-\Omega^{(0)}_{cdm})x^{4}f_{gcg}(1/x-1;\nu,\alpha)+\Omega^{(0)}_{cdm}x\right]^{3}}}.$
(27)
where $x=1/(1+z)$.
Note that the model $\Lambda CDM$ it is obtained when $f_{gcg}=1$, which
corresponds to take the values $\alpha=\nu=0$ [65].
In FIG.9 we show the parameter $\eta$ as a function of the
$\Omega_{cdm}^{(0)}$ parameter. Here, we have taken two different set of
values for the gcg parameters, $\alpha=0.9;\nu=0.88$ and
$\alpha=0.9;\nu=0.01$. In order to make a comparison we have included in this
plot the $\Lambda$CDM model.
file=eta.eps,width=10cm
Figure 9: This graph shows the parameter $\eta$ as a function of the
$\Omega_{cdm}^{(0)}$ parameter. We have considered two different set of values
for the gcg parameters: the set ($\alpha=0.9;\nu=0.88$) and the set
($\alpha=0.9;\nu=0.01$). Here, we have included the $\Lambda$CDM case.
One important parameter that describes the dependence of the first Doppler
peak position on the different parameters that characterize any model is the
shift parameter $R$. More specific, it gives the position of the first Doppler
peak with respect to its location in a flat reference model with
$\Omega_{cdm}^{(0)}=1$ [66, 67]. This becomes
$R(\Omega_{cdm}^{(0)},\Omega_{gcg}^{(0)};\nu,\alpha)=\sqrt{\frac{\Omega_{cdm}^{(0)}}{|\Omega_{k}^{(0)}|}}S_{k}\left[\sqrt{|\Omega_{k}^{(0)}|}\int^{1}_{0}\frac{dx}{x^{2}E(x;\nu,\alpha)}\right],$
where $\Omega_{k}^{(0)}=1-(\Omega_{cdm}^{(0)}+\Omega_{gcg}^{(0)})$. Note that
the initial point is common for the same value of the parameter with different
curvature, and the final point is common for the same curvature with different
value of the parameters. Note also that if we choose $\Omega_{k}^{(0)}=0$ and
$\alpha=\nu=0$ ($f_{gcg}(z;0,0)\rightarrow 1$) the $\Lambda$CDM case is
recuperated.
file=R03F.eps,width=10cm
Figure 10: Contour Plot in the $\Omega_{gcg}^{(0)}-\Omega_{cdm}^{(0)}$ plane
with $R=0.3$ for two set of values for the parameters $\nu$ and $\alpha$, i.e.
$\nu=0.1$ and $\nu=0.88$ for $\alpha=0.9$. Here, we have considered positive
and negative curvature.
## 5 Conclusions
In this paper we have described and study a cosmological model in which, apart
from the usual cold dark matter component, we have included a GCG associated
to the dark energy component. In this kind of model we have described the
properties of the GCG. The characterization of the GCG comes from the
determination of the GCG parameters, $\nu$ and $\alpha$ related to the
velocity of sound of the fluid and the power appearing in the EOS of the GCG,
respectively. By taking into account some observational astronomical data,
such that the Hubble parameter, the $y$-parameter, the angular size and the
luminosity distance we were able to restrict these parameters. All of them
agree with the condition $\nu\lesssim\alpha$. We have also described the
deceleration, the jerk and the snap parameters for our model. We expect that
with an appropriate data of these parameters will be possible to restrict the
parameters of the GCG fluid.
As an applicability of the GCG model described above, we have determined the
position of the first Doppler peak together with the shift parameter R. These
cases were compared with that corresponding the $\Lambda$CDM model.
We may conclude that, as far as we are concerned with the observed
acceleration detected in the universe and the location of the first Doppler
peak, we will be able to utilize a GCG model to describe the Universe we live
in.
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|
arxiv-papers
| 2009-09-29T03:48:48 |
2024-09-04T02:49:05.527427
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Sergio del Campo and Jos\\'e Villanueva",
"submitter": "Jose Villanueva mr.",
"url": "https://arxiv.org/abs/0909.5258"
}
|
0909.5313
|
201059-70Nancy, France 59
V. Arvind Srikanth Srinivasan
# The Remote Point Problem, Small Bias Spaces, and Expanding Generator Sets
V. Arvind and S. Srinivasan The Institute of Mathematical Sciences,
C.I.T. Campus, Chennai 600 113, India. arvind@imsc.res.in
srikanth@imsc.res.in
###### Abstract.
Using $\varepsilon$-bias spaces over $\mathbb{F}_{2}$, we show that the Remote
Point Problem (RPP), introduced by Alon et al [APY09], has an $\mbox{\rm
NC}^{2}$ algorithm (achieving the same parameters as [APY09]). We study a
generalization of the Remote Point Problem to groups: we replace
$\mathbb{F}_{2}^{n}$ by $\mathcal{G}^{n}$ for an arbitrary fixed group
$\mathcal{G}$. When $\mathcal{G}$ is Abelian we give an $\mbox{\rm NC}^{2}$
algorithm for RPP, again using $\varepsilon$-bias spaces. For nonabelian
$\mathcal{G}$, we give a deterministic polynomial-time algorithm for RPP. We
also show the connection to construction of expanding generator sets for the
group $\mathcal{G}^{n}$. All our algorithms for the RPP achieve essentially
the same parameters as [APY09].
###### Key words and phrases:
Small Bias Spaces, Expander Graphs, Cayley Graphs, Remote Point Problem.
###### 1991 Mathematics Subject Classification:
Algorithms and Complexity Theory
## 1\. Introduction
Valiant, in his celebrated work [V77] on circuit lower bounds for computing
linear transformations $A:\mathbb{F}^{n}\longrightarrow\mathbb{F}^{m}$ for a
field $\mathbb{F}$, initiated the study of rigid matrices. If explicit rigid
matrices of certain parameters can be constructed it would result in
superlinear lower bounds for logarithmic depth linear circuits over
$\mathbb{F}$. This problem and the construction of such rigid matrices has
remained elusive for over three decades.
Alon, Panigrahy and Yekhanin [APY09] recently proposed a problem that appears
to be of intermediate difficulty. Given a subspace $L$ of $\mathbb{F}_{2}^{n}$
by its basis and a number $r\in[n]$ as input, the problem is to compute in
deterministic polynomial time a point $v\in\mathbb{F}_{2}^{n}$ such that
$\Delta(u,v)\geq r$ for all $u\in L$, where $\Delta(u,v)$ is the Hamming
distance. They call this the _Remote Point Problem_. The point $v$ is said to
be $r$-far from the subspace $L$.
Alon et al [APY09] give a nice polynomial time-bounded (in $n$) algorithm for
computing a $v\in\mathbb{F}_{2}^{n}$ that is $c\log n$-far from a given
subspace $L$ of dimension $n/2$ and $c$ is a fixed constant. For $L$ such that
$\dim(L)=k<n/2$ they give a polynomial-time algorithm for computing a point
$v\in\mathbb{F}_{2}^{n}$ that is $\frac{cn\log k}{k}$-far from $L$.
#### Results of this paper
In [AS09a] we recently investigated the problem of proving circuit lower
bounds in the presence of help functions. Specifically, one of the problems we
consider is proving lower bounds for constant-depth Boolean circuits which can
take a given set of (arbitrary) help functions
$\\{h_{1},h_{2},\cdots,h_{m}\\}$ at the input level, where
$h_{i}:\\{0,1\\}^{n}\longrightarrow\\{0,1\\}$ for each $i$. Proving explicit
lower bounds for this model would allow us to separate EXP from the
polynomial-time many-one closure of nonuniform $\mbox{\rm AC}^{0}$. We show
that it suffices to find a polynomial-time solution to the Remote Point
Problem for parameters $k=2^{(\log\log n)^{c}}$ and $r=\frac{n}{2^{(\log\log
n)^{d}}}$ for all constants $c$ and $d$. Unfortunately, the parameters of the
Alon et al algorithm are inadequate for our application.
However, motivated by this connection, in the present paper we carry out a
more detailed study of the Remote Point Problem as an algorithmic question. We
briefly summarize our results.
1. The first question we address is whether we can give a deterministic parallel (i.e. NC) algorithm for the problem — Alon et al’s algorithm is inherently sequential as it is based on the method of conditional probabilities and pessimistic estimators.
It turns out an element of an $\varepsilon$-bias space for suitably chosen
$\varepsilon$ is a solution to the Remote Point Problem which gives us an NC
algorithm quite easily.
2. Since the RPP for $\mathbb{F}_{2}^{n}$ can be solved using small bias spaces, it naturally leads us to address the problem in a more general group-theoretic setting.
In the generalization we study we will replace $\mathbb{F}_{2}$ with an
arbitrary fixed finite group $\mathcal{G}$ such that $|\mathcal{G}|\geq 2$.
Hence we will have the $n$-fold product group $\mathcal{G}^{n}$ instead of the
vector space $\mathbb{F}_{2}^{n}$.
Given elements $x=(x_{1},x_{2},\ldots,x_{n}),y=(y_{1},y_{2},\ldots,y_{n})$ of
$\mathcal{G}^{n}$, let $\Delta(x,y)=|\\{i\mid x_{i}\neq y_{i}\\}|$. I.e.
$\Delta(x,y)$ is the _Hamming distance_ between $x$ and $y$. Furthermore, for
$S\subseteq\mathcal{G}^{n}$, let $\Delta(x,S)$ denote $\min_{y\in
S}\Delta(x,y)$.
We now define the _Remote Point Problem (RPP) over a finite group
$\mathcal{G}$_. The input is a subgroup $\mathcal{H}$ of $\mathcal{G}^{n}$,
where $\mathcal{H}$ is given by a generating set, and a number $r\in[n]$. The
problem is to compute in deterministic polynomial (in $n$) time an element
$x\in\mathcal{G}^{n}$ such that $\Delta(x,H)>r$. The results we show in this
general setting are the following.
* (a)
The Remote Point Problem over any _Abelian group_ $\mathcal{G}$ has an
$\mbox{\rm NC}^{2}$ algorithm for $r=O(\frac{n\log k}{k})$ and $k\leq n/2$,
where $k=\log_{|\mathcal{G}|}|\mathcal{H}|$.
* (b)
Over an arbitrary group $\mathcal{G}$ the Remote point problem has a
polynomial-time algorithm for $r=O(\frac{n\log k}{k})$ and $k\leq n/2$, where
$k=\log_{|\mathcal{G}|}|\mathcal{H}|$.
The parallel algorithm stated in part(a) above is based on $\varepsilon$-bias
space constructions for finite Abelian groups described in Azar et al [AMN98].
The sequential algorithm stated in part(b) above is a group-theoretic
generalization of the Alon et al algorithm for $\mathbb{F}_{2}^{n}$ [APY09].
Due to lack of space, some proofs have been omitted. They may be found in the
full version which has been published as an ECCC report [AS09b].
## 2\. Preliminaries
Fix a finite group $\mathcal{G}$ such that $|\mathcal{G}|\geq 2$. Given any
$x\in\mathcal{G}^{n}$, let $wt(x)$ denote the number of coordinates $i$ such
that $x_{i}\neq 1$, where $1$ is the identity of the group $\mathcal{G}$. By
$B(r)$, we will refer to the set of $x\in\mathcal{G}^{n}$ such that $wt(x)\leq
r$. Given a subset $S$ of $\mathcal{G}^{n}$, $B(S,r)$ will denote the set
$S\cdot B(r)=\left\\{sx\>\middle|\>s\in S,x\in B(r)\right\\}$. Clearly, for
any $S\subseteq\mathcal{G}^{n}$ and any $x\in\mathcal{G}^{n}$, $x\in B(S,r)$
if and only if $\Delta(x,S)\leq r$. We say that $x$ is _$r$ -close_ to $S$ if
$x\in B(S,r)$ and _$r$ -far_ from $S$ if $x\notin B(S,r)$.
The _Remote Point Problem (RPP) over $\mathcal{G}$_ is defined to be the
following algorithmic problem:
* INPUT: A subgroup $\mathcal{H}$ of $\mathcal{G}^{n}$ (given by its generators) and an $r\in\mathbb{N}$.
* OUTPUT: An $x\in\mathcal{G}^{n}$ such that $x\notin B(\mathcal{H},r)$.
Clearly, there are inputs to the above problem where no solution can be found.
But the input instances of the kind that we will study will clearly have a
solution (in fact, a random point of $\mathcal{G}^{n}$ will be a solution with
high probability).
Given a subgroup $\mathcal{H}$ of $\mathcal{G}^{n}$, denote by
$\delta(\mathcal{H})$ the quantity $\log_{|\mathcal{G}|}|\mathcal{H}|$. We
will call $\delta(\mathcal{H})$ the _dimension of $\mathcal{H}$ in
$\mathcal{G}^{n}$_.
We say that the RPP over $\mathcal{G}$ has a $(k(n),r(n))$-algorithm if there
is an efficient algorithm that solves the Remote Point Problem when given as
input a subgroup $\mathcal{H}$ of $\mathcal{G}^{n}$ of dimension at most
$k(n)$ and an $r$ that is bounded by $r(n)$. (Here, ‘efficient’ can correspond
to polynomial time or some smaller complexity class.)
A simple counting argument shows that there is a valid solution to the RPP
over $\mathcal{G}$ on inputs $(\mathcal{H},r)$ where
$\delta(\mathcal{H})+r\leq n(1-\frac{H(r/n)}{\log|G|}-\varepsilon)$, for any
fixed $\varepsilon>0$ (where $H(\cdot)$ denotes the binary entropy function).
However, the best known deterministic solution to the RPP – from [APY09] – is
a polynomial time $(k,\frac{cn\log k}{k})$-algorithm which works over
$\mathbb{F}_{2}^{n}$ (i.e, the group $\mathcal{G}$ involved is the additive
group of the field $\mathbb{F}_{2}$).
### 2.1. Some Group-Theoretic Algorithms
We introduce basic definitions and review some group-theoretic algorithms. Let
$\mathrm{Sym}(\Omega)$ denote the group of all permutations on a finite set
$\Omega$ of size $m$. In this section we use $G,H$ etc. to denote _permutation
groups on $\Omega$_, which are simply subgroups of $\mathrm{Sym}(\Omega)$.
Let $G$ be a subgroup of $\mathrm{Sym}(\Omega)$. For a subset
$\Delta\subseteq\Omega$ denote by $G_{\\{\Delta\\}}$ the _point-wise
stabilizer_ of $\Delta$. I.e $G_{\\{\Delta\\}}$ is the subgroup consisting of
exactly those elements of $G$ that fix each element of $\Delta$.
###### Theorem 2.1 (Schreier-Sims).
[Lu93]
1. (1)
If a subgroup $G$ of $\mathrm{Sym}(\Omega)$ is given by a generating set as
input along with the subset $\Delta$ there is a polynomial-time (sequential)
algorithm for computing a generator set for $G_{\\{\Delta\\}}$.
2. (2)
If a subgroup $G$ of $\mathrm{Sym}(\Omega)$ is given by a generating set as
input, then there is a polynomial time algorithm for computing $|G|$.
3. (3)
Given as input a permutation $\sigma\in\mathrm{Sym}(\Omega)$ and a generator
set for a subgroup $G$ of $\mathrm{Sym}(\Omega)$, we can test in deterministic
polynomial time if $\sigma$ is an element of $G$.
We are also interested in a special case of this problem which we now define.
A subset $\Gamma\subseteq\Omega$ is an _orbit_ of $G$ if
$\Gamma=\left\\{\sigma(i)\>\middle|\>\sigma\in G\right\\}$ for some
$i\in\Omega$. Any subgroup $G$ of $\mathrm{Sym}(\Omega)$ partitions $\Omega$
into orbits (called $G$-orbits).
For a constant $b>0$, a subgroup $G$ of $\mathrm{Sym}(\Omega)$ is defined to
be a _$b$ -bounded permutation group_ if every $G$-orbit is of size at most
$b$.
In [MC87], McKenzie and Cook studied the parallel complexity of _Abelian_
permutation group problems. Specifically, they gave an $\mbox{\rm NC}^{3}$
algorithm for testing membership in an Abelian permutation group given by a
generator set and for computing the order of an Abelian permutation group.
When restricted to $b$-bounded Abelian permutation groups, the algorithms of
[MC87] for these problems are actually $\mbox{\rm NC}^{2}$ algorithms. We
formally state their result and derive a consequence.
###### Theorem 2.2 ([MC87]).
There is an $\mbox{\rm NC}^{2}$ algorithm for membership testing in a
$b$-bounded Abelian permutation group $G$ given by a generator set.
We now consider problems over $\mathcal{G}^{n}$, for a fixed finite group
$\mathcal{G}$. We know from basic group theory that every group $\mathcal{G}$
is a permutation group acting on itself. I.e. every $\mathcal{G}$ can be seen
as a subgroup of $\mathrm{Sym}(\mathcal{G})$, where $\mathcal{G}$ acts on
itself by left (or right) multiplication. Therefore, $\mathcal{G}^{n}$ can be
easily seen as a permutation group on the set $\Omega=\mathcal{G}\times[n]$
and hence, $\mathcal{G}^{n}$ can be considered a subgroup of
$\mathrm{Sym}(\Omega)$. Furthermore, notice that each subset
$\mathcal{G}\times\\{i\\}$ is an orbit of this group $\mathcal{G}^{n}$. Hence,
$\mathcal{G}^{n}$ is a $b$-bounded permutation group contained in
$\mathrm{Sym}(\Omega)$, where $b=|\mathcal{G}|$. Finally, if $\mathcal{G}$ is
an Abelian group, then so is this subgroup of $\mathrm{Sym}(\Omega)$. We have
the following lemma as an easy consequence of Theorem 2.2.
###### Lemma 2.3.
Let $\mathcal{G}$ be Abelian. There is an $\mbox{\rm NC}^{2}$ algorithm that
takes as input a generator set for some subgroup $\mathcal{H}$ of
$\mathcal{G}^{n}$ and an $x\in\mathcal{G}^{n}$, and accepts iff
$x\in\mathcal{H}$.
Given any $y=(y_{1},y_{2},\ldots,y_{i})\in\mathcal{G}^{i}$ with $1\leq i\leq
n$ and any $S\subseteq\mathcal{G}^{n}$, let $S_{y}$ denote the set
$\left\\{x\in S\>\middle|\>x_{j}=y_{j}\text{ for }1\leq j\leq i\right\\}$.
###### Lemma 2.4.
Let $\mathcal{G}$ be any fixed finite group. There is a polynomial time
algorithm that takes as input a subgroup $\mathcal{H}$ of $\mathcal{G}^{n}$,
where $\mathcal{H}$ is given by generators, and a $y\in\mathcal{G}^{i}$ with
$1\leq i\leq n$, and computes $|\mathcal{H}_{y}|$.
###### Proof 2.5.
Let $\mathcal{K}=\\{(x_{1},x_{2},\ldots,x_{n})\in\mathcal{H}\mid
x_{1}=x_{2}=\cdots=x_{i}=1\\}$, where $1$ denotes the identity element of
$\mathcal{G}$. Clearly, $\mathcal{K}$ is a subgroup of $\mathcal{H}$. The set
$\mathcal{H}_{y}$, if nonempty, is simply a coset of $\mathcal{K}$ and thus,
we have $|\mathcal{H}_{y}|=|\mathcal{K}|$. To check if $\mathcal{H}_{y}$ is
nonempty, we consider the map
$\pi_{i}:\mathcal{G}^{n}\rightarrow\mathcal{G}^{i}$ that projects its input
onto its first $i$ coordinates; note that $\mathcal{H}_{y}$ is nonempty iff
the subgroup $\pi_{i}(\mathcal{H})$ contains $y$, which can be checked in
polynomial time by point ($3$) of Theorem 2.1 (here, we are identifying
$\mathcal{G}^{n}$ with a subgroup of $\mathrm{Sym}(\mathcal{G}\times[n])$ as
above). If $y\notin\pi_{i}(\mathcal{H})$, the algorithm outputs $0$.
Otherwise, we have $|\mathcal{H}_{y}|=|\mathcal{K}|$ and it suffices to
compute $|\mathcal{K}|$. But $\mathcal{K}$ is simply the point-wise stabilizer
of the set $\mathcal{G}\times[i]$ in $\mathcal{H}$, and hence $|\mathcal{K}|$
can be computed in polynomial time by points ($1$) and ($2$) of Theorem 2.1.
## 3\. Expanding Cayley Graphs and the Remote Point Problem
Fix a group $\mathcal{G}$ such that $|\mathcal{G}|\geq 2$, and consider an
instance of the RPP over $\mathcal{G}$. The main idea that we develop in this
section is that if we have a (symmetric) expanding generator set $S$ for the
group $\mathcal{G}^{n}$ with appropriate expansion parameters then for a
subgroup $\mathcal{H}$ of $\mathcal{G}^{n}$ such that $\delta(\mathcal{H})\leq
k$ some element of $S$ will be $r$-far from $H$, for suitable $k$ and $r$.
We review some definitions related to expander graphs (e.g. see the survey of
Hoory, Linial, and Wigderson [HLW06]). An undirected multigraph $G=(V,E)$ is
an $(n,d,\alpha)$-graph for $n,d\in\mathbb{N}$ and $\alpha>0$ if $|V|=n$, the
degree of each vertex is $d$, and the second largest value $\lambda(G)$ from
among the absolute values of eigenvalues of $A(G)$ – the adjacency matrix of
the graph $G$ – is bounded by $\alpha d$.
A _random walk_ of length $t\in\mathbb{N}$ on an $(n,d,\alpha)$-graph
$G=(V,E)$ is the output of the following random process: a vertex $v_{0}\in V$
of picked uniformly at random, and for $0\leq i<t$, if $v_{i}$ has been
picked, then $v_{i+1}$ is obtained by selecting a neighbour $v_{i+1}$
uniformly at random (i.e a random edge out of $v_{i}$ is picked, and $v_{i+1}$
is chosen to be the other endpoint of the edge); the output of the process is
$(v_{0},v_{1},\ldots,v_{t})$. We now state an important result regarding
random walks on expanders (see [HLW06, Theorem 3.6] for details).
###### Lemma 3.1.
Let $G=(V,E)$ be an $(n,d,\alpha)$-graph and $B\subseteq V$ with $|B|\leq\beta
n$. Then, the probability that a random walk $(v_{0},v_{1},\ldots,v_{t})$ is
entirely contained inside $B$ (i.e, $v_{i}\in B$ for each $i$) is bounded by
$(\beta+\alpha)^{t}$.
Let $\mathcal{H}$ be a group and $S$ a _symmetric_ multiset of elements from
$\mathcal{H}$. I.e. there is a bijection of multisets $\varphi:S\rightarrow S$
such that $\varphi(s)=s^{-1}$ for each $s\in S$. We define the Cayley graph
$C(\mathcal{H},S)$ to be the (multi)graph $G$ with vertex set $\mathcal{H}$
and edges of the form $(x,xs)$ for each $x\in\mathcal{H}$ and each $s\in S$;
since $S$ is symmetric, we consider $C(\mathcal{H},S)$ to be an undirected
graph by identifying the edges $(x,xs)$ and $(xs,(xs)\varphi(s))$, for each
$x$ and $s$.
We now show a lemma that will help relate generators of expanding Cayley
graphs on $\mathcal{G}^{n}$ and the RPP over $\mathcal{G}$. In what follows,
let $S$ be a symmetric multiset of elements from $\mathcal{G}^{n}$; let $G$
denote the Cayley graph $C(\mathcal{G}^{n},S)$; and let $N,D$ denote
$|\mathcal{G}|^{n}$ and $|S|$ (counted with repetitions) respectively.
###### Lemma 3.2.
Assume $S$ as above is such that $G$ is an $(N,D,\alpha)$-graph, where
$\alpha\leq\frac{1}{n^{d}}$, for some fixed $d>0$. Then, given any subgroup
$\mathcal{H}$ of $\mathcal{G}^{n}$ such that $\delta(\mathcal{H})\leq 2n/3$,
we have $\frac{|S\cap\mathcal{H}|}{|S|}\leq\frac{1}{n^{d/2}}$ for large enough
$n$ (where the elements of $S\cap\mathcal{H}$ are counted with repetitions).
###### Proof 3.3.
Let $S^{\prime}=S\cap\mathcal{H}$ and let $\eta=|S^{\prime}|/|S|$. We want an
upper bound on $\eta$. Consider a random walk $(x_{0},x_{1},\ldots,x_{t})$ of
length $t$ on the graph $G$ (the exact value of $t$ will be fixed later). Let
$\mathcal{B}$ denote the following event: there is a $y\in\mathcal{G}^{n}$
such that all the vertices $x_{0},x_{1},\ldots,x_{t}$ are all contained in the
coset $y\mathcal{H}$ of $\mathcal{H}$. Let $p$ denote the probability that
$\mathcal{B}$ occurs.
We will first lower bound $p$. At each step of the random walk, a random
$s_{i}\in S$ is chosen and $x_{i+1}$ is set to $x_{i}s_{i}$. If these $s_{i}$
all happen to belong to $S^{\prime}$, then the cosets $x_{i}\mathcal{H}$ and
$x_{i+1}\mathcal{H}$ are the same for all $i$ and hence, the event
$\mathcal{B}$ does occur. Hence, $p\geq\eta^{t}$.
We now upper bound $p$. Fix any coset $y\mathcal{H}$ of the subgroup
$\mathcal{H}$. Since the dimension of $\mathcal{H}$ in $\mathcal{G}^{n}$ is
bounded by $2n/3$, we have
$|y\mathcal{H}|=|\mathcal{H}|\leq|\mathcal{G}|^{2n/3}\leq
2^{-n/3}|\mathcal{G}^{n}|$. That is, the coset $y\mathcal{H}$ is a very small
subset of $\mathcal{G}^{n}$. Applying Lemma 3.1, we see that the probability
that the random walk $(x_{0},x_{1},\ldots,x_{t})$ is completely contained
inside this coset is bounded by
$(2^{-n/3}+n^{-d})^{t}\leq\frac{2^{t}}{n^{dt}}$, for large enough $n$. As the
total number of cosets of $\mathcal{H}$ is bounded by $|\mathcal{G}|^{n}$, an
application of the union bound tells us that $p$ is upper bounded by
$|\mathcal{G}|^{n}\frac{2^{t}}{n^{dt}}\leq\frac{|\mathcal{G}|^{n+t}}{n^{dt}}$.
Setting $t=\frac{2n}{d\log_{|G|}n-2}$ we see that $p$ is at most
$\frac{1}{n^{dt/2}}$.
Putting the upper and lower bounds together, we see that
$\eta^{t}\leq\frac{1}{n^{dt/2}}$ and hence, $\eta\leq\frac{1}{n^{d/2}}$. This
completes the proof.
We follow the structure of the algorithm for the RPP over $\mathbb{F}_{2}$ in
[APY09]. We first describe their $(n/2,c\log n)$-algorithm for the RPP,
followed by our own algorithm. We then describe how they extend this algorithm
to a $(k,\frac{cn\log k}{k})$-algorithm for any $k\leq n/2$; the same
procedure works for our algorithm also.
The $(n/2,c\log n)$-algorithm proceeds as follows. On an input instance
consisting of a subgroup $V$ (which is a subspace of $\mathbb{F}_{2}^{n}$) of
dimension at most $n/2$ and an $r\leq c\log n$,
1. (1)
The algorithm first computes a collection of $m=n^{O(c)}$ subspaces
$V_{1},V_{2},\ldots,V_{m}$, each of dimension at most $2n/3$ such that
$B(V,c\log n)\subseteq\bigcup_{i=1}^{m}V_{i}$.
2. (2)
The algorithm then finds an $x\in\mathbb{F}_{2}^{n}$ such that
$x\notin\bigcup_{i}V_{i}$. (This is done using a method similar to the method
of pessimistic estimators introduced by Raghavan [Rag88].)
Our algorithm will proceed exactly as the above algorithm in the first step.
The second step of our algorithm will be different (assuming that the group
$\mathcal{G}$ is Abelian). We first state Step 1 of the algorithm of [APY09]
in greater generality:
###### Lemma 3.4.
Let $\mathcal{G}$ be any fixed finite group with $|\mathcal{G}|\geq 2$. For
any constant $c>0$ and large enough $n$, the following holds. Given any
subgroup $\mathcal{H}$ of $\mathcal{G}^{n}$ such that
$\delta(\mathcal{H})\leq\frac{n}{2}$, there is a collection of $m\leq n^{10c}$
subgroups $\mathcal{H}_{1},\mathcal{H}_{2},\ldots,\mathcal{H}_{m}$ such that
$B(\mathcal{H},c\log n)\subseteq\bigcup_{i=1}^{m}\mathcal{H}_{i}$, and
$\delta(\mathcal{H}_{i})\leq 2n/3$ for each $i$. Moreover, there is a logspace
algorithm that, when given as input $\mathcal{H}$ as a set of generators,
produces generators for the subgroups
$\mathcal{H}_{1},\mathcal{H}_{2},\ldots,\mathcal{H}_{m}$.
###### Proof 3.5.
The proof follows exactly as in [APY09]. We reproduce it here for completeness
and to analyze the complexity of the procedure.
Let $1$ denote the identity element of $\mathcal{G}$. For each
$S\subseteq[n]$, let $\mathcal{G}^{n}(S)$ denote the subgroup of
$\mathcal{G}^{n}$ consisting of those $x$ such that $x_{i}=1$ for each
$i\notin S$. Note that $\delta(\mathcal{G}^{n}(S))=|S|$. Also note that for
each $S\subseteq[n]$, the group $\mathcal{G}^{n}(S)$ is a normal subgroup; in
particular, this implies that the set $\mathcal{K}\cdot\mathcal{G}^{n}(S)$ is
a subgroup of $\mathcal{G}^{n}$ whenever $\mathcal{K}$ is a subgroup of
$\mathcal{G}^{n}$.
Partition the set $[n]$ into $\ell\leq 10c\log n$ sets of size at most
$\lceil\frac{n}{10c\log n}\rceil$ each – we will call these sets
$S_{1},S_{2},\ldots,S_{\ell}$. For each $A\subseteq[\ell]$ of size $\lceil
c\log n\rceil$, let $\mathcal{K}_{A}$ denote the subgroup
$\mathcal{G}^{n}(\bigcup_{i\in A}S_{i})$. Note that the number of such
subgroups is at most $2^{\ell}\leq n^{10c}$. Also, for each $A$ as above,
$\delta(\mathcal{K}_{A})=|\bigcup_{i\in A}S_{i}|\leq\left(\frac{n}{10c\log
n}+1\right)(c\log n+1)<\frac{n}{9}$, for large enough $n$.
Consider any $x\in B(c\log n)$ (i.e, an element $x$ of $\mathcal{G}^{n}$ s.t
$wt(x)\leq c\log n$). We know that $x\in\mathcal{G}^{n}(S)$ for some $S$ of
size at most $c\log n$. Hence, it can be seen that
$x\in\mathcal{G}^{n}(\bigcup_{i\in A}S_{i})$ for some $A$ of size $\lceil
c\log n\rceil$; this shows that $B(c\log
n)\subseteq\bigcup_{A}\mathcal{K}_{A}$. Therefore, we see that
$B(\mathcal{H},c\log n)=\mathcal{H}B(c\log
n)\subseteq\bigcup_{A}\mathcal{H}\mathcal{K}_{A}$.
For each $A\subseteq[\ell]$ of size $\lceil c\log n\rceil$, let
$\mathcal{H}_{A}$ denote the subgroup $\mathcal{H}\mathcal{K}_{A}$ (note that
this is indeed a subgroup, since $\mathcal{K}_{A}$ is a normal subgroup).
Moreover, the cardinality of this subgroup is bounded by
$|\mathcal{H}|\cdot|\mathcal{K}_{A}|\leq|\mathcal{G}|^{n/2}|\mathcal{G}|^{n/9}<|\mathcal{G}|^{2n/3}$;
hence, $\delta(\mathcal{H}_{A})\leq 2n/3$. Thus, the collection of subgroups
$\\{\mathcal{H}_{A}\\}_{A}$ satisfies all the properties mentioned in the
statement of the lemma. That a set of generators for this subgroup can be
computed in deterministic logspace – for some suitable choice of
$S_{1},S_{2},\ldots,S_{\ell}$ – is a routine check from the definition of the
subgroups $\\{\mathcal{K}_{A}\\}_{A}$. This completes the proof of the lemma.
Using Lemma 3.4, we are able to efficiently “cover” $B(\mathcal{H},c\log n)$
for any small subgroup $\mathcal{H}$ of $\mathcal{G}^{n}$ by a union of small
subgroups. Therefore, to find a point that is $c\log n$-far from
$\mathcal{H}$, it suffices to find a point $x\in\mathcal{G}^{n}$ not contained
in any of the covering subgroups. To do this, we note that if $S$ is a
multiset containing elements from $\mathcal{G}^{n}$ such that
$C(\mathcal{G}^{n},S)$ is a Cayley graph with good expansion, then $S$ must
contain such an element. This is formally stated below.
###### Lemma 3.6.
For any constant $c>0$ and large enough $n\in\mathbb{N}$, the following holds.
Let $S$ be any multiset of elements of $\mathcal{G}^{n}$ such that
$\lambda(C(\mathcal{G}^{n},S))<\frac{1}{n^{20c}}$. Then, for $m\leq n^{10c}$
and any collection $\mathcal{H}_{1},\mathcal{H}_{2},\ldots,\mathcal{H}_{m}$ of
subgroups such that $\delta(\mathcal{H}_{i})\leq 2n/3$ for each $i$, there is
some $s\in S$ such that $s\notin\bigcup_{i}\mathcal{H}_{i}$.
###### Proof 3.7.
The proof follows easily from Lemma 3.2. Given any $i\in[m]$, we know, from
Lemma 3.2, that $|S\cap\mathcal{H}_{i}|<\frac{|S|}{n^{10c}}$ (where the
elements of the multisets are counted with repetitions). Hence,
$|S\cap\bigcup_{i}\mathcal{H}_{i}|\leq\sum_{i}|S\cap\mathcal{H}_{i}|<\frac{m|S|}{n^{10c}}\leq|S|$.
Therefore, there must be some $s\in S$ such that
$s\notin\bigcup_{i}\mathcal{H}_{i}$.
Therefore, to find a point $x$ that is $c\log n$-far from the subspace
$\mathcal{H}$, it suffices to construct an $S$ such that
$C(\mathcal{G}^{n},S)$ is a sufficiently good expander, find the covering
subgroups $\mathcal{H}_{i}$ ($i\in[m[$), and then to find an $s\in S$ that
does not lie in any of the $\mathcal{H}_{i}$. We follow the above approach to
give an efficient parallel algorithm for the RPP in the case that
$\mathcal{G}$ is an Abelian group. For arbitrary groups, we show that the
method of [APY09] yields a polynomial time algorithm.
## 4\. Remote Point Problem for Abelian Groups
Fix an Abelian group $\mathcal{G}$. Recall that a _character_ $\chi$ of
$\mathcal{G}^{n}$ is a homomorphism from $\mathcal{G}^{n}$ to
$\mathbb{C}^{*}_{1}$, the multiplicative subgroup of the complex numbers of
absolute value $1$. For $\varepsilon>0$, a distribution $\mu$ over
$\mathcal{G}^{n}$ is said to be $\varepsilon$-biased if, given any non-trivial
character $\chi$ of $\mathcal{G}^{n}$,
$\left|\mathop{\textbf{E}}_{x\sim\mu}[\chi(x)]\right|\leq\varepsilon$.
A multiset $S$ consisting of elements from $\mathcal{G}^{n}$ is said to be an
_$\varepsilon$ -biased space in $\mathcal{G}^{n}$_ if the uniform distribution
over $S$ is an $\varepsilon$-biased distribution.
It can be checked that a multiset consisting of
$(\frac{n}{\varepsilon})^{O(1)}$ independent, uniformly random elements from
$\mathcal{G}^{n}$ form an $\varepsilon$-biased space with high probability.
Explicit $\varepsilon$-biased spaces were constructed for the group
$\mathbb{F}_{2}^{n}$ by Naor and Naor in [NN93]; further constructions were
given by Alon et al. in [AGHP92]. Explicit constructions of
$\varepsilon$-biased spaces in $\mathbb{Z}_{d}^{n}$ were given by Azar et al.
in [AMN98]. We observe that this last construction yields a construction for
all Abelian groups $\mathcal{G}^{n}$, when $\mathcal{G}$ is of constant size.
We first state the result of [AMN98] in a form that we will find suitable.
###### Theorem 4.1.
For any fixed $d$, there is an $\mbox{\rm NC}^{2}$ algorithm that does the
following. On input $n$ and $\varepsilon>0$ (both in unary), the algorithm
produces a symmetric multiset $S\subseteq\mathbb{Z}_{d}^{n}$ of size
$O((\frac{n}{\varepsilon})^{2})$ such that $S$ is an $\varepsilon$-biased
space in $\mathbb{Z}_{d}^{n}$.
###### Proof 4.2.
It is easy to see that the $\varepsilon$-biased space construction in [AMN98]
can be implemented in deterministic logspace (and hence in $\mbox{\rm
NC}^{2}$). If the space $S$ obtained is not symmetric, we can consider the
multiset that is the disjoint union of $S$ and $S^{-1}$, which is also easily
seen to be $\varepsilon$-biased.
###### Remark 4.3.
We note that the definition of small bias spaces in [AMN98] differs somewhat
from our own definition above. But it is easy to see that an
$\varepsilon$-bias space in $\mathbb{Z}_{d}^{n}$ in the sense of [AMN98] is a
$(d\varepsilon)$-bias space according to our definition above.
###### Remark 4.4.
In a recent paper, Meka and Zuckerman [MZ09] observe, as we do below, that the
construction of [AMN98] gives small bias spaces for any arbitrary Abelian
group $\mathcal{G}$. Nevertheless, we present our own proof of this fact,
since the small bias spaces that follow from our proof are of _smaller_ size.
Specifically, our proof shows how to explicitly construct sample spaces of
size $O\left(\frac{n^{2}}{\varepsilon^{2}}\right)$, whereas the relevant
result in [MZ09] only produces small bias spaces of size
$O\left((\frac{n}{\varepsilon})^{b}\right)$, where $b$ is some constant that
depends on $\mathcal{G}$ (and can be as large as $\Omega(\log|\mathcal{G}|)$).
###### Lemma 4.5.
For any fixed group $\mathcal{G}$, there is an $\mbox{\rm NC}^{2}$ algorithm
which, on input $n$ and $\varepsilon>0$ in unary, produces a symmetric
multiset $S\subseteq\mathcal{G}^{n}$ of size $O((\frac{n}{\varepsilon})^{2})$
such that $S$ is an $\varepsilon$-biased space in $\mathcal{G}^{n}$.
###### Proof 4.6.
By the Fundamental Theorem of finite Abelian groups,
$\mathcal{G}\cong\mathbb{Z}_{d_{1}}\oplus\mathbb{Z}_{d_{2}}\oplus\cdots\oplus\mathbb{Z}_{d_{k}}$,
for positive integers $d_{1},d_{2},\ldots,d_{k}$ such that $d_{1}\mid
d_{2}\mid\cdots\mid d_{k}$. Let $\mathcal{G}_{0}$ denote
$\mathbb{Z}_{d_{k}}^{k}$. Note that for any $s,t\in\mathbb{N}$,
$\mathbb{Z}_{s}\cong\mathbb{Z}_{st}/\mathbb{Z}_{t}$. Hence, we see that that
$\mathcal{G}\cong\mathcal{G}_{0}/\mathcal{H}$, where $\mathcal{H}$ is the
subgroup
$\mathbb{Z}_{e_{1}}\oplus\mathbb{Z}_{e_{2}}\oplus\cdots\oplus\mathbb{Z}_{e_{k}}$,
and $e_{i}=d_{k}/d_{i}$ for each $i\in[k]$. Therefore,
$\mathcal{G}^{n}\cong\mathcal{G}_{0}^{n}/\mathcal{H}^{n}$. Let
$\pi:\mathcal{G}_{0}^{n}\rightarrow\mathcal{G}^{n}$ be the natural onto
homomorphism with kernel $\mathcal{H}^{n}$. Note that $\pi$ is just the
projection map and can easily be computed in $\mbox{\rm NC}^{2}$.
Since $\mathcal{G}_{0}^{n}\cong\mathbb{Z}_{d_{k}}^{nk}$, by Theorem 4.1, there
is an $\mbox{\rm NC}^{2}$ algorithm that constructs a symmetric multiset
$S_{0}\subseteq\mathcal{G}_{0}^{n}$ of size
$O(\left(\frac{kn}{\varepsilon}\right)^{2})$ such that $S_{0}$ is an
$\varepsilon$-biased space in $\mathcal{G}_{0}^{n}$. We claim that the
multiset $S=\pi(S_{0})$ is a symmetric $\varepsilon$-biased space in
$\mathcal{G}^{n}$. To see this, consider any non-trivial character $\chi$ of
$\mathcal{G}^{n}$; note that $\chi_{0}=\chi\circ\pi$ is a non-trivial
character of $\mathcal{G}_{0}^{n}$. We have
$\left|\mathop{\textbf{E}}_{x\sim
S}[\chi(x)]\right|=\left|\mathop{\textbf{E}}_{x_{0}\sim
S_{0}}[\chi(\pi(x_{0}))]\right|=\left|\mathop{\textbf{E}}_{x_{0}\sim
S_{0}}[\chi_{0}(x)]\right|\leq\varepsilon$
where the first equality follows from the definition of $S$, and the last
inequality follows from the fact that $S_{0}$ is an $\varepsilon$-biased space
in $\mathcal{G}_{0}^{n}$. Since $\chi$ was an arbitrary non-trivial character
of $\mathcal{G}^{n}$, we have proved that $S$ is indeed an
$\varepsilon$-biased space in $\mathcal{G}^{n}$. It is easy to see that $S$ is
symmetric. Finally, note that $S$ can be computed in $\mbox{\rm NC}^{2}$. This
completes the proof.
Finally, we mention a well-known connection between small bias spaces in
$\mathcal{G}^{n}$ and Cayley graphs over $\mathcal{G}^{n}$ (e.g. see Alon and
Roichman [AR94]).
###### Lemma 4.7.
Given any symmetric multiset $S\subseteq\mathcal{G}^{n}$, the Cayley graph
$C(\mathcal{G}^{n},S)$ is an $(|\mathcal{G}|^{n},|S|,\alpha)$-graph iff $S$ is
an $\alpha$-biased space.
Lemmas 4.7 and 4.5 have the following easy consequence:
###### Lemma 4.8.
For any Abelian group $\mathcal{G}$, there is an $\mbox{\rm NC}^{2}$ algorithm
which, on unary inputs $n$ and $\alpha>0$, produces a symmetric multiset
$S\subseteq\mathcal{G}^{n}$ of size $O((\frac{n}{\alpha})^{2})$ such that
$C(\mathcal{G}^{n},S)$ is a $(|\mathcal{G}|^{n},|S|,\alpha)$-graph.
Putting the above statement together with the results of Section 3, we have
the following.
###### Theorem 4.9.
For any constant $c>0$, the RPP over $\mathcal{G}$ has an $\mbox{\rm NC}^{2}$
$(n/2,c\log n)$-algorithm.
###### Proof 4.10.
Let $\mathcal{H}$ denote the input subgroup. By Lemma 3.4, there is a logspace
(and hence $\mbox{\rm NC}^{2}$) algorithm that computes a collection of
$m=n^{O(c)}$ many subgroups
$\mathcal{H}_{1},\mathcal{H}_{2},\ldots,\mathcal{H}_{m}$ such that
$B(\mathcal{H},c\log n)\subseteq\bigcup_{i=1}^{m}\mathcal{H}_{i}$ and
$\delta(\mathcal{H}_{i})\leq 2n/3$ for each $i\in[m]$. Now, fix any multiset
$S\subseteq\mathcal{G}^{n}$ such that the Cayley graph $C(\mathcal{G}^{n},S)$
is a $(|\mathcal{G}|^{n},|S|,\alpha)$-graph, where
$\alpha=\frac{1}{2n^{20c}}$; by Lemma 4.8, such an $S$ can be constructed in
$\mbox{\rm NC}^{2}$. It follows from Lemma 3.6 that there is some $s\in S$
such that $s\notin\bigcup_{i=1}^{m}\mathcal{H}_{i}$. Finally, by Lemma 2.3,
there is an $\mbox{\rm NC}^{2}$ algorithm to test if each $s\in S$ belongs to
$\mathcal{H}_{i}$, for any $i\in[m]$. Hence, we can find out (in parallel)
exactly which $s\in S$ do not belong to any of the $\mathcal{H}_{i}$ and
output one of them. The output element $s$ is surely $c\log n$-far from
$\mathcal{H}$.
Let $\mathcal{G}$ be Abelian. We observe that a method of [APY09], coupled
with Theorem 4.9, yields an efficient $(k,\frac{cn\log k}{k})$-algorithm for
any constant $c>0$, and $k\leq n/2$.
###### Theorem 4.11.
Let $c>0$ be any constant. If $\mathcal{G}$ is an Abelian group, then the RPP
over $\mathcal{G}$ has an $\mbox{\rm NC}^{2}$ $(k,\frac{cn\log
k}{k})$-algorithm for any $k\leq n/2$.
###### Proof 4.12.
Given as input a subgroup $\mathcal{H}$ such that $\delta(\mathcal{H})=k\leq
n/2$, the algorithm partitions $[n]$ as $[n]=\bigcup_{i=1}^{m}T_{i}$, where
$2k\leq|T_{i}|<4k$ for each $i$; note that $m\geq n/4k$. Let $\mathcal{H}_{i}$
denote the subgroup obtained when $\mathcal{H}$ is projected onto the
coordinates in $T_{i}$. Since $\delta(\mathcal{H}_{i})\leq k\leq|T_{i}|/2$, we
can, by Theorem 4.9, efficiently find a point $x_{i}\in\mathcal{G}^{|T_{i}|}$
that is at least $4c\log k$-far from $\mathcal{H}_{i}$. Putting these $x_{i}$
together in the natural way, we obtain an $x\in\mathcal{G}^{n}$ that is
$\frac{cn\log k}{k}$-far from the subgroup $\mathcal{H}$.
Since $\mathcal{G}$ is Abelian, using the algorithm of Theorem 4.9, the
$x_{i}$ can all be computed in parallel in $\mbox{\rm NC}^{2}$. Hence, the
entire procedure can be performed in $\mbox{\rm NC}^{2}$.
## 5\. RPP over General Groups
Let $\mathcal{G}$ denote some fixed finite group. We can generalize the
polynomial-time algorithm of [APY09], described for $\mathbb{F}_{2}$, to
compute a point $x\in\mathcal{G}^{n}$ that is $c\log n$-far from a given input
subgroup $\mathcal{H}$ such that $\delta(\mathcal{H})\leq n/2$. We only state
this result below and refer the interested reader to the full version [AS09b]
for details.
###### Theorem 5.1.
For any constant $c>0$, the RPP over $\mathcal{G}$ has a polynomial time
$(n/2,c\log n)$-algorithm.
Analogous to Theorem 4.11, we have the following solution to RPP for general
groups.
###### Theorem 5.2.
Let $c>0$ be any constant. For any $\mathcal{G}$, the RPP over $\mathcal{G}$
has a polynomial time $(k,\frac{cn\log k}{k})$-algorithm for any $k\leq n/2$.
###### Proof 5.3.
The construction is exactly the same as in the proof of Theorem 4.11. The only
difference is that we will apply the algorithm of Theorem 5.1. In this case,
the $x_{i}$ can all be found in deterministic polynomial time. Hence, the
entire procedure gives us a polynomial-time algorithm.
## 6\. Limitations of expanding sets
In the previous sections, we have shown how generators for expanding Cayley
graphs on $\mathcal{G}^{n}$, where $\mathcal{G}$ is a fixed finite group, can
help solve the RPP over $\mathcal{G}$. In particular, we have the following
easy consequence of Lemmas 3.4 and 3.6.
###### Corollary 6.1.
For any constant $c>0$, large enough $n$, and any symmetric multiset
$S\subseteq\mathcal{G}^{n}$ such that
$\lambda(C(\mathcal{G}^{n},S))<\frac{1}{n^{20c}}$, the following holds. If
$\mathcal{H}$ is any subgroup of $\mathcal{G}^{n}$ such that
$\delta(\mathcal{H})\leq n/2$, there is some $s\in S$ such that $s\notin
B(\mathcal{H},c\log n)$.
It makes sense to ask if the parameters in Corollary 6.1 are far from optimal.
Is it true that any polynomial-sized symmetric multiset
$S\subseteq\mathcal{G}^{n}$ with good enough expansion properties is
$\omega(\log n)$-far from every subgroup of dimension at most $n/2$? We can
show that this is not true. Formally, we can prove:
###### Theorem 6.2.
For any constant $c>0$ and large enough $n$, there is a symmetric multiset
$S\subseteq\mathbb{F}_{2}^{n}$ such that
$\lambda(C(\mathbb{F}_{2}^{n},S))\leq\frac{1}{n^{c}}$ but there is a subspace
$L$ of dimension $n/2$ such that $S\subseteq B(L,20c\log n)$.
It is well known that for any family of $d$-regular multigraphs $G$
$\lambda(G)=\Omega(1/\sqrt{d})$ (see e.g. [HLW06, Theorem 5.3]). As a
consequence of this lower bound it follows for any fixed group $\mathcal{G}$
and any multiset $S\subseteq\mathcal{G}^{n}$ that
$\lambda(C(\mathcal{G},S))=\Omega(1/\sqrt{|S|})$. Hence, the above theorem
tells us that just the expansion properties of $C(\mathbb{F}_{2}^{n},S)$ for
any $\mathop{\mathrm{poly}}(n)$-sized $S$ are not sufficient to guarantee
$\omega(\log n)$-distance from every subspace of dimension $n/2$. The proof of
the above statement can be found in the full version [AS09b].
## 7\. Discussion
For the remote point problem over an Abelian group $\mathcal{G}$, we have
shown how expanding generating sets for Cayley graphs of $\mathcal{G}^{n}$ can
be used to obtain deterministic $\mbox{\rm NC}^{2}$ algorithms. A natural
question is whether we can obtain a similar algorithm for non-Abelian
$\mathcal{G}$. Note that Lemma 3.6 holds in the non-Abelian setting too.
Hence, in order to obtain an $\mbox{\rm NC}^{2}$-algorithm for the RPP over
arbitrary non-Abelian $\mathcal{G}$ along the lines of our algorithm for
Abelian groups, we need to be able to check (in $\mbox{\rm NC}^{2}$) for
membership in $\mathcal{G}^{n}$, and we need to be able to construct small
multisets $S$ of $\mathcal{G}^{n}$ such that $C(\mathcal{G}^{n},S)$ has
sufficiently good expansion properties. Luks’ work [Lu86] yields an $\mbox{\rm
NC}^{4}$ test for membership in $\mathcal{G}^{n}$ for arbitrary $\mathcal{G}$.
Building on that, there is also an $\mbox{\rm NC}^{2}$ membership test for
$\mathcal{G}^{n}$ [AKV05]. However, we are unable to compute a (good enough)
expanding generator set for the group $\mathcal{G}^{n}$ in deterministic NC or
even in deterministic polynomial time.
## Acknowledgements
We are grateful to Noga Alon and Sergey Yekhanin for interesting comments. In
particular, Alon pointed out to us that Lemma 3.2 has an alternative proof
using the expander mixing lemma. We thank the anonymous referees for their
comments and suggestions.
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arxiv-papers
| 2009-09-29T11:21:20 |
2024-09-04T02:49:05.534263
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Vikraman Arvind and Srikanth Srinivasan",
"submitter": "Srikanth Srinivasan",
"url": "https://arxiv.org/abs/0909.5313"
}
|
0909.5318
|
2009 Vol. 9 No. XX, 000–000
11institutetext: Department of Astronomy, Nanjing University, Nanjing 210093,
China; hyf@nju.edu.cn
22institutetext: Purple Mountain Observatory, Chinese Academy of Sciences,
Nanjing 210008, China
Received [year] [month] [day]; accepted [year] [month] [day]
# Long-term Continuous Energy Injection in the Afterglow of GRB 060729
M. Xu 11 Y.-F. Huang 11 T. Lu 22
###### Abstract
A long plateau phase and an amazing brightness have been observed in the X-ray
afterglow of GRB 060729. This peculiar light curve is likely due to long-term
energy injection in external shock. Here we present a detailed numerical study
on the energy injection process of magnetic dipole radiation from a strongly
magnetized millisecond pulsar and model the multi-band afterglow observations.
It is found that this model can successfully explain the long plateaus in the
observed X-ray and optical afterglow light curves. The sharp break following
the plateaus should be due to the rapid decline of the emission power of the
central pulsar. At an even late time ($\sim 5\times 10^{6}s$), an obvious jet
break appears, which implies a relatively large half opening angle of
$\theta\sim 0.3$ for the GRB ejecta. Due to the energy injection, the Lorentz
factor of the outflow is still larger than two $10^{7}$s post the GRB trigger,
making the X-ray afterglow of this burst detectable by Chandra even 642 days
after the burst.
###### keywords:
gamma rays: bursts -ISM: jets and outflows
## 1 Introduction
GRB 970228 is the first gamma-ray burst (GRB) with an X-ray afterglow detected
(Costa et al. 1997). Optical (van Paradijs et al. 1997) and radio afterglow
(Frail et al. 1997) has also been unprecedently detected from this event. The
relativistic internal and external shock model is the most successful model to
explain these violent events ( Rees & Mészáros 1994; Piran 1999; Zhang 2007).
It is also widely believed that long GRBs should be due to the collapse of
massive stars (Woosley 1993; Paczyéski 1998; MacFadyen & Woosley 1999), and
short GRBs should be connected with the coalescence of two compact objects
(Eichler et al. 1989; Narayan et al. 1992; Gehrels et al. 2005; Nakar 2007).
The X-ray telescope (XRT) on board Swift reveals that the X-ray afterglows of
GRBs generally show a canonical behavior, with five components in the observed
X-ray afterglow light curves, i.e., steep decay phase, shallow decay phase,
normal decay phase, post jet break phase and X-ray flares (Zhang et al. 2006;
Nousek et al. 2006). The conventional models for shallow decay phase are
energy injection from strongly magnetized millisecond pulsar (Dai & Lu 1998;
Zhang & Mészáros 2001; Liang et al. 2007; Lyons et al. 2009) or from ejecta
with a highly dispersed Lorentz factor distribution (Rees & Mészáros 1998;
Sari & Mészáros 2000).
At $19:12:29$ UT of July 29, 2006, GRB 060729 triggered the $Swift$ Burst
Alert Telescope (BAT) and was quickly located (Grupe et al. 2006). This event
has a duration of $T_{90}=116\pm 10s$ (Parsons et al. 2006) and a redshift of
$z=0.54$ (Thoene et al. 2006). The isotropic energy release in the rest-frame
in $1keV-10MeV$ band was $E_{iso}=1.6\times 10^{52}ergs$ for a standard
cosmology model with $\Omega_{M}=0.27$, $\Omega_{\Lambda}=0.73$ and a Hubble
constant of $H_{0}=71km\cdot s^{-1}\cdot Mpc^{-1}$.
One of the distinguished properties of GRB 060729 is that it has a long flat
phase in the X-ray afterglow light curve (Grupe et al. 2007). Another
prominent character of GRB 060729 is its brightness. It can be observed by
$Chandra$ even $642$ days after the burst trigger (Grupe et al. 2009). Grupe
et al. (2009) compared the X-ray afterglow of GRB 060729 with other bright
X-ray afterglows and concluded that GRB 060729 was an exceptionally long-
lasting event. Actually, the brightness of the X-ray afterglow of GRB 060729
is not extraordinary at early time ($t<30000s$), but it becomes the brightest
one among all GRBs after $30000s$ since the trigger.
In view of the long plateau phase ($500s-30000s$) and the late time
($>30000s$) brightness of GRB 060729, a strong and long-term continuous energy
injection is implied (Liang et al. 2007; Grupe et al. 2007, 2009). Grupe et
al. (2007) presented an extensive study on this peculiar event and made a
detailed analysis on the pulsar-type energy injection for this plateau. But at
that time there was only 125 days of data and the jet break still did not
appear. In this paper, we use the energy injection model that involves the
dipole radiation from a strongly magnetized millisecond pulsar to explain the
special behavior of the multi-band afterglow of GRB 060729. The new data
observed by $Chandra$ (Grupe et al. 2009) will be incorporated. We detailedly
calculate the X-ray and optical (U-band, B-band and V-band) afterglow light
curves, and compare them with the observations. In Section 2, we briefly
describe the energy injection model. In Section 3 we present our detailed
numerical results. Finally, Section 4 is our conclusions and discussion.
## 2 Energy Injection from a Strongly Magnetized Millisecond Pulsar
Due to the strong magnetic field and rapid rotation, a new born millisecond
pulsar will radiate a huge amount of energy through magnetic dipole emission.
This energy can be comparable to or even larger than the initial energy of the
main GRB. Detailed discussions on this process have been given by Dai & Lu
(1998) and Zhang & Mészáros (2001).
Through magnetic dipole radiation, the new born pulsar in the center of the
GRB fireball will lose its rotational energy. The radiation power evolves with
time as
$L=L_{0}(1+\frac{t}{T})^{-2},$ (1)
where $L_{0}$ is the initial luminosity, i.e., the radiation power at the time
of $t=0$. $T$ is the characteristic spin-down timescale.
The initial luminosity depends on the parameters of the pulsar as
$L_{0}=4.0\times 10^{47}ergs\cdot
s^{-1}(B^{2}_{\bot,14}P^{-4}_{-3}R^{6}_{6}),$ (2)
where $B_{\bot,14}=B_{s}sin\vartheta/10^{14}G$, $B_{s}$ is the strength of the
dipole magnetic field at the surface of the pulsar, $\vartheta$ is the angle
between the rotation axis and the magnetic axis, $P_{-3}$ is the pulsar period
in units of $10^{-3}s$, and $R_{6}$ is the radius of the pulsar in units of
$10^{6}cm$.
The characteristic spin-down timescale of the pulsar can be calculated from
$T=5.0\times 10^{4}s(B^{-2}_{\bot,14}P^{2}_{-3}R^{-6}_{6}I_{45}),$ (3)
where $I_{45}$ is the moment of inertia of the pulsar in units of
$10^{45}g\cdot cm^{2}$.
The total energy of the magnetic dipole radiation can be derived by
integrating the emission power from $t=0$ to $t\rightarrow\infty$
$E_{total}=\int_{0}^{\infty}Ldt=\int_{0}^{\infty}[L_{0}(1+\frac{t}{T})^{-2}]dt=L_{0}T.$
(4)
## 3 Numerical Calculation and Results
A convenient method to describe the dynamics and radiation processes of GRB
afterglows has been proposed by Huang et al. (2000). It is appropriate for
both radiative and adiabatic blastwaves, and in both the ultra-relativistic
and the non-relativistic phases (Huang et al. 1999). Here we modify their
method accordingly so that it can be applicable to the energy injection
scenario.
### 3.1 Dynamics
The overall dynamical evolution of GRB afterglows has been described by Huang
et al. (1999, 2000). When the energy injection from a strongly magnetized
millisecond pulsar is included, the deceleration of the external shock is
mainly characterized by the following equation
$\frac{d\gamma}{dm}=\frac{-(\gamma^{2}-1)+d(Lt)/d(mc^{2})}{M_{\rm ej}+\epsilon
m+2(1-\epsilon)\gamma m},$ (5)
where $\gamma$ is the bulk Lorentz factor of the shocked medium, $m$ is the
swept-up mass, $M_{ej}$ is the initial ejecta mass, and $\epsilon$ is the
radiation efficiency.
For simplicity, here we only consider the synchrotron emission from shock-
accelerated electrons. To get the observed afterglow flux, we need to
integrate the emission power over the equal arrival time surface determined by
$\int\frac{1-\beta\cos\Theta}{\beta c}dR\equiv t,$ (6)
within the jet boundaries, where $\beta=\sqrt{\gamma^{2}-1}/\gamma$ and
$\Theta$ is the angle between the velocity of emitting material and the line
of sight.
### 3.2 Numerical Results
Inserting Eq. (1) into Eq. (5), we can conveniently calculate the evolution of
the external shock subject to the energy injection from a strongly magnetized
millisecond pulsar. In this section, we assume that the circum-burst medium is
homogeneous. We calculate the overall dynamical evolution of a uniform jet to
educe the X-ray and optical afterglow light curves, and try to give the best
fit to the observations of GRB 060729.
To get the best fit, we find that we need to set the parameters of the central
pulsar as follows. The radius is $R_{6}=1$. The rotation period is
$P_{-3}=1.49$. The magnetic field is $B_{\bot,14}=2.72$. The moment of inertia
is taken as $I_{45}\sim 2$, which is still typical for neutron stars (Datta
1988; Weber & Glendenning 1993). Then, according to equations $(2)$ and $(3)$,
the initial emission power and the spin-down timescale of the center pulsar
are $L_{0}=6.0\times 10^{47}ergs\cdot s^{-1}$, and $T=30000s$ respectively. So
the energy injection power is $L=6.0\times 10^{47}ergs\cdot
s^{-1}(1+\frac{t}{30000{\rm s}})^{-2}$.
In our calculations, we use the following parameters for the external shock of
GRB 060729: initial energy per solid angle $E_{0}=1.6\times 10^{52}/4\pi$
ergs, the initial Lorentz factor $\gamma_{0}=200$, the $ISM$ number density
$n=0.2cm^{-3}$, the power-law index of the energy distribution of electrons
$p=2.48$, the luminosity distance $D_{L}=3.12$ Gpc, the electron energy
fraction $\epsilon_{e}=0.15$, the magnetic energy fraction
$\epsilon_{B}=0.0002$, the half opening angle of the jet $\theta=0.3$, and the
observing angle $\theta_{obs}=0$. Here the observing angle is defined as the
angle between the line of sight and the jet axis.
Using the above parameter set, we can give a satisfactory fit to the multiband
afterglows of GRB 060729. In Fig. 1, we first show the evolution of the
Lorentz factor under the energy injection from a strongly magnetized
millisecond pulsar. We see that due to the continuous energy injection, the
Lorentz factor of the outflow is still larger than 2 after $10^{7}$s. It means
that the afterglow could be very bright even at very late stages.
Fig.2 illustrates the observed X-ray (0.3-10 keV) afterglow light curve of GRB
060729 and our best fit. We can see that the observed X-ray afterglow light
curve is fitted very well. Especially, the observed long plateau
($~{}500s-30000s$) is explained satisfactorily. This long flat phase is
resulted from the long-term continuous energy injection from the magnetic
dipole radiation of the strongly magnetized millisecond pulsar. After
$30000s$, the flat phase comes to the end and a break is seen in the light
curve. The reason is that the pulsar has consumed most of its rotation energy
on the spin-down timescale (T=30000s in our model), so that the power of
energy injection decreases sharply at that time. An obvious jet break is
presented at $t_{j}\sim 5\times 10^{6}s$. To produce such a late jet break, we
find that the half opening angle of the jet should be $\theta=0.3$, which is
relatively large among known GRBs.
Fig.3 illustrates our fit to the observed optical afterglows of GRB 060729 by
using the same parameters as in Figs. 1 and 2. All the data points are taken
from Grupe et al. (2007). We see that the observed optical afterglow can also
be satisfactorily explained.
## 4 Conclusions and Discussion
We have shown that the observed special behavior of the afterglow of GRB
060729 can be well explained by using the energy injection model. Our study
indicates that the central engine should be a strongly magnetized millisecond
pulsar, which continuously supplies energy to the GRB ejecta via magnetic
dipole radiation on a timescale of about 30000 s. The observed multi-band
afterglow light curves can be reproduced satisfactoriely by this model.
According to our calculations, the duration of the plateau phase in the
afterglow light curve should correspond to the spin-down timescale of the
pulsar ($T=30000s$). To further explain the observed jet break at $t_{j}\sim
5\times 10^{6}s$, we need a relatively large jet opening angle of
$\theta=0.3$.
From Equation (4), we can derive the total injected energy as
$E_{total}=L_{0}T=1.8\times 10^{52}ergs$. This energy is comparable to the
initial isotropic energy release in the main burst phase ($E_{iso}=1.6\times
10^{52}ergs$). The long-term continuous energy injection makes GRB 060729 the
brightest burst in X-ray band at late stages. In fact, the X-ray afterglow can
be observed even $642$ days after the trigger (Grupe et al. 2009).
In optical bands, the afterglow light curves show some similar properties as
in the X-ray band. For example, a flat stage is presented in the optical light
curves. Generally, our model can give a satisfactory explanation to the
optical afterglow. The time span of optical observations is very limited. We
do not have optical data for $t>10^{6}$ s, so that the jet break is still not
observed in optical band. However, note that the extensive analysises with
both the X-ray and optical data in recent years show that some of the jet-like
breaks in the afterglow light curves are chromatic (Panaitescu et al. 2006;
Liang et al. 2008). The nature of these breaks is then highly debatable. Thus
a long term monitoring of the multi-band afterglows is definitely necessary.
Also note that the early UV-optical afterglow light curves of GRB 060729 show
significant variations. This feature, however, is not seen in the X-ray light
curve. It indicates that other regions may also contribute to the optical
emission in this event.
In our current study, we have assumed that the energy injection is isotropic.
Magnetic dipole radiation actually should be anisotropic. However, this kind
of anisotropy is not significant and would not affect the final results
seriously.
According to our numerical results, the Lorentz factor of the jet is still
larger than 2 after $10^{7}$s. This is due to the continuous and long-term
energy injection. As a result, the time that the afterglow of GRB 060729
enters the Newtonian phase is significantly delayed.
The energy injection models were used to explain the afterglows of some GRBs,
such as GRB 010222 (Björnsson et al. 2002), GRB 021004 (Björnsson et al.
2004), GRB 030329 (Huang et al. 2006) and GRB 051221A (Fan & Xu 2006) etc. The
explanation of the afterglow from the short GRB 051221A also needs some kind
of energy injection from a magnetar (Fan & Xu 2006). However, we note that the
physical origin of the shallow decay segment is still highly debating (Zhang
2007). Generally speaking, while the achromatic breaks in both the X-ray and
the optical bands can be explained with conventional energy injection models,
the chromatic breaks of this segment observed in many events strongly
challenge these models (Liang et al. 2007). Alternative models that go beyond
the conventional ones were proposed (see Zhang 2007 for review). It is
interesting that a small fraction of XRT lightcurves show as a single power-
law without canonical feature (Liang et al. 2009). It was also argued that the
apparent difference of the canonical and single power-law XRT lightcurves may
be due to the improper zero time effect on the canonical XRT lightcurves
(Yamazaki 2009; Liang et al. 2009).
###### Acknowledgements.
We thank the anonymous referee for helpful suggestions. This work was
supported by the National Natural Science Foundation of China (Grant No.
10625313 & 10473023) and the National Basic Research Program of China (973
Program, grant 2009CB824800).
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Figure 1: Evolution of the bulk Lorentz factor of a jet with long-term energy
injection from a strongly magnetized millisecond pulsar. The parameters used
in this calculation have been given in Section 3.2 . Figure 2: Observed X-ray
afterglow light curve of GRB 060729 and our best fit by using the energy
injection model. The square points are observed data from _Swift_ and the
triangle points are observed data from _Chandra_ (Grupe et al. 2009). The tail
emission in the very early phase ($t<400$ s) is not considered in our fit.
Figure 3: Observed multi-band optical afterglow light curves of GRB 060729 and
our best fit by using the energy injection model. Observed data points are
taken from Grupe et al. (2007). The solid, dashed and dotted lines are our fit
to the observed light curves in the three bands, respectively. Note that the U
and V-band light curves have been shifted by 3 magnitudes for clarity.
|
arxiv-papers
| 2009-09-29T12:47:12 |
2024-09-04T02:49:05.541294
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ming Xu, Yong-Feng Huang, Tan Lu",
"submitter": "Ming Xu",
"url": "https://arxiv.org/abs/0909.5318"
}
|
0909.5508
|
# Lifetime difference in $D^{0}$-${\overline{D}}^{0}$ mixing within R-parity-
violating SUSY
Gagik K. Yeghiyan Department of Physics and Astronomy, Wayne State
University, Detroit, MI 48201, USA
###### Abstract
We re-examine constraints from the evidence for observation of the lifetime
difference in $D^{0}$-${\overline{D}}^{0}$ mixing on the parameters of
supersymmetric models with $R$-parity violation (RPV). We find that RPV SUSY
can give large negative contribution to the lifetime difference. We also
discuss the importance of the choice of weak or mass basis when placing the
constraints on RPV-violating couplings from flavor mixing experiments.
## I Introduction
Meson-antimeson mixing is an important vehicle for indirect search of New
Physics (NP) 48 . $D^{0}-\overline{D}{}^{0}$ mixing 36 is the only available
meson-antimeson mixing in the up-quark sector. The fact that the search is
indirect and complimentary to existing constraints from the bottom-quark
sector actually provides parameter space constraints for a large variety of NP
models 23 ; 6 .
One can write the normalized lifetime difference in $D^{0}-\overline{D}{}^{0}$
mixing, $y_{\rm D}\equiv\Delta\Gamma_{\rm D}/(2\Gamma_{\rm D})$, as an
absorptive part of the $D^{0}-\overline{D}{}^{0}$ mixing matrix Petrov:2003un
,
$y_{D}=\frac{1}{\Gamma_{\rm D}}\sum_{n}\rho_{n}\langle\overline{D}^{0}|{\cal
H}_{w}^{\Delta C=1}|n\rangle\langle n|{\cal H}_{w}^{\Delta C=1}|D^{0}\rangle,$
(1.1)
where $\rho_{n}$ is a phase space function that corresponds to a charmless
intermediate state $n$. This relation shows that $\Delta\Gamma_{\rm D}$ is
driven by transitions $D^{0},{\overline{D}}^{0}\to n$, i.e. physics of the
$\Delta C=1$ sector.
It was recently shown 6 that $D^{0}-\overline{D}{}^{0}$ mixing is a rather
unique system, where the lifetime difference can be used to constrain the
models of New Physics111A similar effect is possible in the bottom-quark
sector Badin:2007bv .. This stems from the fact that there is a well-defined
theoretical limit (the flavor $SU(3)$-limit) where the SM contribution
vanishes and the lifetime difference is dominated by the NP $\Delta C=1$
contributions. In real world, flavor $SU(3)$ is, of course, broken, so the SM
contribution is proportional to a (second) power of $m_{s}/\Lambda$, which is
a rather small number. If the NP contribution to $y_{\rm D}$ is non-zero in
the flavor $SU(3)$-limit, it can provide a large contribution to the mixing
amplitude.
To see this, consider a $D^{0}$ decay amplitude which includes a small NP
contribution, $A[D^{0}\to n]=A_{n}^{\rm(SM)}+A_{n}^{\rm(NP)}$. Experimental
data for D-meson decays are known to be in a decent agreement with the SM
estimates 47 ; 28 . Thus, $A_{n}^{\rm(NP)}$ should be smaller than (in sum)
the current theoretical and experimental uncertainties in predictions for
these decays.
One may rewrite equation (1.1) in the form (neglecting the effects of CP-
violation)
$\displaystyle y_{D}=\sum_{n}\frac{\rho_{n}}{\Gamma_{\rm
D}}A_{n}^{\rm(SM)}\bar{A}_{n}^{\rm(SM)}+2\sum_{n}\frac{\rho_{n}}{\Gamma_{\rm
D}}A_{n}^{\rm(NP)}\bar{A}_{n}^{\rm(SM)}+$
$\displaystyle+\sum_{n}\frac{\rho_{n}}{\Gamma_{\rm
D}}A_{n}^{\rm(NP)}\bar{A}_{n}^{\rm(NP)}\ \ .$ (1.2)
The first term in this equation corresponds to the SM contribution, which
vanishes in the $SU(3)$ limit. In ref. 6 , as well as in the superseding
papers Chen ; 37 , the last term in (1.2) has been neglected, thus the NP
contribution to $y_{\rm D}$ comes there solely from the second term, due to
interference of $A_{n}^{\rm(SM)}$ and $A_{n}^{\rm(NP)}$. While this
contribution is in general non-zero in the flavor $SU(3)$ limit, in a large
class of (popular) models it actually is 6 ; 37 . Then, in this limit, $y_{\rm
D}$ is completely dominated by pure $A_{n}^{\rm(NP)}$ contribution given by
the last term in eq. (1.2)! It is clear that the last term in equation (1.2)
needs more detailed and careful studies, at least within some of the NP
models.
Indeed, in reality, flavor $SU(3)$ symmetry is broken, so the first term in
Eq. (1.2) is not zero. It has been argued 29 that in fact the SM
$SU(3)$-violating contributions could be at a percent level, dominating the
experimental result, $y_{D}^{exp}=(0.73\pm 0.18)\%$ HFAG . The SM predictions
of $y_{D}$, stemming from evaluations of long-distance hadronic contributions,
are rather uncertain. While this precludes us from placing explicit
constraints on parameters of NP models, it has been argued that, even in this
situation, an upper bound on the NP contributions can be placed 23 by
displaying the NP contribution only, i.e. as if there were no SM contribution
at all. This procedure is similar to what was traditionally done in the
studies of NP contributions to $K^{0}-\overline{K}^{0}$ mixing, so we shall
employ it here too.
In order to evaluate importance of the NP contribution, as the flavor $SU(3$)
is broken, counting of suppression powers of $m_{s}/m_{c}$ for the SM
contribution versus those of $M_{W}^{2}/M_{NP}^{2}$ of the NP contribution
must be performed. For the last term in eq. (1.2) to be essential, the
following approximate rule applies:
$M_{W}^{4}/M_{NP}^{4}>m_{s}^{2}/m_{c}^{2}$. This term is of the primary
importance here: the second term in (1.2) is proven to be $\lesssim 10^{-4}$
in the most popular SM extensions 6 ; 37 ; 17 and, hence, negligible in
general.
The talk is based on the results presented in basic . We revisit the problem
of the NP contribution to $y_{\rm D}$ and provide constraints on R-parity-
violating supersymmetric (SUSY) models as a primary example. It has been
recently argued in 17 that within /R- SUSY models, new physics contribution
to $y_{\rm D}$ is rather small, mainly because of stringent constraints on the
relevant pair products of RPV coupling constants. However, this result has
been derived neglecting the transformation of these couplings from the weak
isospin basis to the quark mass basis. This approach seems to be quite
reasonable for the scenarios with the baryonic number violation. However, in
the scenarios with the leptonic number violation, transformation of the RPV
couplings from the weak eigenbasis to the quark mass eigenbasis turns to be
crucial, when applying the existing phenomenological constraints on these
couplings.
We show in that within R-parity-breaking supersymmetric models with the
leptonic number violation, new physics contribution to the lifetime difference
in $D^{0}-\overline{D}{}^{0}$ mixing may be large, due to the last term in eq.
(1.2). When being large, it is negative (if neglecting CP-violation), i.e.
opposite in sign to what is implied by the recent experimental evidence for
$D^{0}-\overline{D}{}^{0}$ mixing.
## II R-Parity Breaking Interactions: Weak vs Mass Eigenbases
We consider a general low-energy supersymmetric scenario with no assumptions
made on a SUSY breaking mechanism at the unification scales
$(\sim~{}(10^{16}~{}-~{}10^{18})GeV)$. The most general Yukawa superpotential
for an explicitly broken R-parity supersymmetric theory is given by
$\displaystyle
W_{/{R}}=\sum_{i,j,k}\Biggl{[}\frac{1}{2}\lambda_{ijk}L_{i}L_{j}E^{c}_{k}+\lambda^{\prime}_{ijk}L_{i}Q_{j}D^{c}_{k}+$
$\displaystyle+\frac{1}{2}\lambda^{\prime\prime}_{ijk}U^{c}_{i}D^{c}_{j}D^{c}_{k}\Biggr{]}$
(2.1)
where $L_{i}$, $Q_{j}$ are $SU(2)_{L}$ weak isodoublet lepton and quark
superfields, respectively; $E_{i}^{c}$, $U_{i}^{c}$, $D_{i}^{c}$ are $SU(2$)
singlet charged lepton, up- and down-quark superfields, respectively;
$\lambda_{ijk}$ and $\lambda^{\prime}_{ijk}$ are lepton number violating
Yukawa couplings, and $\lambda^{\prime\prime}_{ijk}$ is a baryon number
violating Yukawa coupling. To avoid rapid proton decay, we assume that
$\lambda^{\prime\prime}_{ijk}=0$ and work with a lepton number violating /R-
SUSY model.
For meson-to-antimeson oscillation processes, to the lowest order in the
perturbation theory, only the second term of (2.1) is of the importance. After
transforming quark fields from the weak isospin basis (used in eq. (2.1) to
the quark mass eigenbasis, the relevant R-parity breaking part of the
Lagrangian may be presented in a following form:
$\displaystyle{\cal
L}_{/{}R}=-\sum_{i,j,k}\widetilde{\lambda}^{\prime}_{ijk}\Big{[}\widetilde{e}_{i_{L}}\bar{d}_{k_{R}}u_{j_{L}}+\widetilde{u}_{j_{L}}\bar{d}_{k_{R}}e_{i_{L}}+$
$\displaystyle+\widetilde{d}^{*}_{k_{R}}\bar{e}_{i_{R}}^{c}u_{j_{L}}\Big{]}+\sum_{i,j,k}\lambda^{\prime}_{ijk}\Big{[}\widetilde{\nu}_{i_{L}}\bar{d}_{k_{R}}d_{j_{L}}+$
$\displaystyle\widetilde{d}_{j_{L}}\bar{d}_{k_{R}}\nu_{i_{L}}+\widetilde{d}^{*}_{k_{R}}\bar{\nu}_{i_{R}}^{c}d_{j_{L}}\Big{]}+h.c.$
(2.2)
where
$\widetilde{\lambda}^{\prime}_{ijk}\ =\ V^{*}_{jn}\ \lambda^{\prime}_{ink}$
(2.3)
with $V$ being the CKM matrix.
Very often in the literature (see e.g. 6 , 17 , 2 -5 ) one neglects the
difference between $\lambda^{\prime}$ and $\widetilde{\lambda}^{\prime}$,
based on the fact that diagonal elements of the CKM matrix dominate over non-
diagonal ones, i.e.
$V_{jn}=\delta_{jn}+O(\lambda)\qquad\mbox{so}\qquad\widetilde{\lambda}_{ijk}\approx\lambda^{\prime}_{ijk}+O(\lambda)$
(2.4)
where $\lambda=\sin\theta_{c}\sim 0.2$, with $\theta_{c}$ being the Cabibbo
angle.
Notice that relation (2.4) is valid if only there is no hierarchy in couplings
$\lambda^{\prime}$. On the other hand, the existing strong bounds on pair
products $\lambda^{\prime}\times\lambda^{\prime}$ (or
$\widetilde{\lambda}^{\prime}\times\widetilde{\lambda^{\prime}}$) 1 ; 2 ; 3
and relatively loose bounds on individual couplings $\lambda^{\prime}$ 1
suggest that such a hierarchy may exist. We have shown in the original work
basic that pair products
$\widetilde{\lambda}^{\prime}\times\widetilde{\lambda^{\prime}}$ may be orders
of magnitude greater than corresponding products
$\lambda^{\prime}\times\lambda^{\prime}$. This fact plays a crucial role in
our analysis.
In what follows, neglecting the transformation of RPV couplings from the weak
eigenbasis to the quark mass eigenbasis would lead to overestimate of existing
phenomenological bounds on these couplings. As a result, one would get that
within R-parity violating supersymmetric models, NP contribution to the
lifetime difference in $D^{0}-\overline{D}{}^{0}$ mixing is rather negligible
17 . Yet, this result is true if no hierarchy in the values of the relevant
RPV couplings exist. More generally, in presence of such hierarchy, due to
rather loose constraints on the relevant
$\widetilde{\lambda}^{\prime}\times\widetilde{\lambda^{\prime}}$ products, RPV
SUSY contribution to $y_{\rm D}$ may be of the same order or even exceed the
experimental value.
## III Dominant Contribution to $y_{\rm D}$
Figure 1: Diagrams giving the dominant contribution to $y_{\rm D}$ a) within
the full electroweak theory; b) within the low-energy effective theory.
Within R-parity violating SUSY models, the dominant contribution to the
lifetime difference in $D^{0}-\overline{D}{}^{0}$ mixing comes from the part
of $D^{0}-\overline{D}{}^{0}$ transition amplitude that occurs when both of
$\Delta C=1$ transitions are generated by NP interactions, due to exchange of
a charged slepton (see Fig. 1). This contribution to $y_{\rm D}$, denoted here
by $y_{\tilde{\ell}\tilde{\ell}}$, is given by the following formula:
$y_{\tilde{\ell}\tilde{\ell}}\approx\frac{-m_{c}^{2}f_{D}^{2}B_{D}m_{D}}{288\pi\Gamma_{D}m_{\tilde{\ell}}^{4}}\
\Biggl{[}\frac{1}{2}+\frac{5}{8}\frac{\bar{B}_{D}^{S}}{B_{D}}\Biggr{]}\left[\
\lambda_{ss}^{2}+\lambda_{dd}^{2}\right]$ (3.1)
where $f_{D}$ is D-meson decay constant, $B_{D}$ and $\bar{B}_{D}^{S}$ are
vacuum saturation factors 23 and
$\lambda_{ss}\equiv\sum_{i}\ \widetilde{\lambda}^{\prime*}_{i12}\
\widetilde{\lambda}^{\prime}_{i22},\ \ \lambda_{dd}\equiv\sum_{i}\
\widetilde{\lambda}^{\prime*}_{i11}\ \widetilde{\lambda}^{\prime}_{i21}$ (3.2)
To simplify the calculations, we assumed that all the sleptons are nearly
degenerate, i.e. $m_{\tilde{\ell}_{i}}=m_{\tilde{\ell}}$.
Note that $y_{\tilde{\ell}\tilde{\ell}}$ is non-vanishing in the exact flavor
$SU(3)$ limit. Also, present experimental data still allow for the slepton
masses to be $\sim 100~{}GeV$ 15 . Finally, present phenomenological
constraints on the coupling pair products $\lambda_{ss}$ and $\lambda_{dd}$
are rather loose, when taking into account the transformation of RPV couplings
from the weak eigenbasis to the quark mass eigenbasis (see basic for more
details). One has $|\lambda_{ss}|<0.29$, $|\lambda_{dd}|<0.29$ or
$\lambda_{ss}^{2}<0.0841$, $\lambda_{dd}^{2}<0.0841$. Thus, as it follows from
our discussion above, $y_{\tilde{\ell}\tilde{\ell}}$ may be quite large.
Indeed, the numerical analysis yields
$-0.12\left(\frac{100GeV}{m_{\tilde{\ell}}}\right)^{4}\leq
y_{\tilde{\ell}\tilde{\ell}}<0$ (3.3)
In other words, $|y_{\tilde{\ell}\tilde{\ell}}|$ may be $\sim 10^{-1}$, if
$m_{\tilde{\ell}}=100$ GeV.
Thus, within R-parity breaking supersymmetric models with the lepton number
violation, new physics contribution to $D^{0}-\bar{D}^{0}$ lifetime difference
is predominantly negative and may exceed in absolute value the experimentally
allowed interval. In order to avoid a contradiction with the experiment
($y_{D}^{exp}=(0.73\pm 0.18)\%$ HFAG ), one must either have a large positive
contribution from the Standard Model, or place severe restrictions on the
values of RPV couplings. As it follows from 29 , $y_{SM}$ may be as large as
$\sim 1\%$. In what follows, $|y_{new}|$ must be $\sim 1\%$ or smaller as
well. If $|y_{new}|\sim 1\%$, then, imposing condition
$-0.01\leq y_{new}\approx y_{\tilde{\ell}\tilde{\ell}}$ (3.4)
one obtains that either $m_{\tilde{\ell}}>185$GeV, or if $m_{\tilde{\ell}}\leq
185$GeV, condition (3.4) implies new bounds on $\lambda_{ss}$ and
$\lambda_{dd}$:
$\displaystyle|\lambda_{ss}|\leq
0.082\left(\frac{m_{\tilde{\ell}}}{100GeV}\right)^{2}$ (3.5)
$\displaystyle|\lambda_{dd}|\leq
0.082\left(\frac{m_{\tilde{\ell}}}{100GeV}\right)^{2}$ (3.6)
It is interesting to compare the restrictions on $\lambda_{ss}$ and
$\lambda_{dd}$, given by (3.5), (3.6), with those derived in 23 from study of
$D^{0}-\bar{D}^{0}$ mass difference. Bounds of 23 on $\lambda_{ss}$ and
$\lambda_{dd}$ turn to be about 20 times stronger than our ones. On the other
hand, constraints of ref. 23 on the RPV coupling products are derived in the
limit when the pure MSSM contribution to $\Delta m_{D}$ is negligible.
Generally speaking, the MSSM contribution to $D^{0}-\bar{D}^{0}$ mass
difference is significant even for the squark masses being about 2TeV. In what
follows, the destructive interference of the pure MSSM and R-parity violating
sector contributions may distort bounds of ref. 23 , making them inessential
as compared to (3.5), (3.6).
Contrary to this, pure MSSM contributes to $\Delta\Gamma_{D}$ only in the
next-to-leading order via two-loop dipenguin diagrams. Naturally, this
contribution is expected to be small. In what follows, unlike those of ref. 23
, our constraints on the RPV coupling products $\lambda_{ss}$ and
$\lambda_{dd}$, given by (3.5), (3.6), seem to be insensitive or weakly
sensitive to assumptions on the pure MSSM sector of the theory.
Thus, our main result is that within R-parity breaking supersymmetric theories
with the leptonic number violation, new physics contribution to
$\Delta\Gamma_{D}$ may be quite large and is predominantly negative.
## IV Conclusion
We computed a possible contribution from R-parity-violating SUSY models to the
lifetime difference in $D^{0}-\overline{D}{}^{0}$ mixing. The contribution
from RPV SUSY models with the leptonic number violation is found to be
negative, i.e. opposite in sign to what is implied by recent experimental
evidence, and possibly quite large, which implies stronger constraints on the
size of relevant RPV couplings.
We discussed currently available constraints on those couplings (especially on
the products of them), available from kaon mixing and rare kaon decays. We
emphasize that the use of these data in charm mixing has to be done carefully
separating the constraints on RPV couplings taken in the mass and weak
eigenbases, given the gauge and CKM structure of $D^{0}-\overline{D}{}^{0}$
mixing amplitudes.
###### Acknowledgements.
Author is grateful to S. Pakvasa and X. Tata for valuable discussions. This
work has been supported by the grants NSF PHY-0547794 and DOE DE-
FGO2-96ER41005.
## References
* (1) See e.g. L. B. Okun’ ”Leptony i Kvarki” (Leptons and Quarks), Moscow: Nauka, (1981) [Traslated into English, Amsterdam: North-Holland, (1984)].
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* (5) A. A. Petrov, In the Proceedings of Flavor Physics and CP Violation (FPCP 2003), Paris, France, 3-6 Jun 2003, pp MEC05 [arXiv:hep-ph/0311371].
* (6) A. Badin, F. Gabbiani and A. A. Petrov, Phys. Lett. B 653, 230 (2007).
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* (8) G. Burdman et al., Phys. Rev. D 66, 014009 (2002).
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* (11) A. F. Falk, Y. Grossman, Z. Ligeti, Y. Nir and A. A. Petrov, Phys.Rev. D 69, 114021 (2004);
A. F. Falk, Y. Grossman, Z. Ligeti and A. A. Petrov, Phys. Rev. D 65, 054034
(2002).
* (12) E. Barberio et al. [Heavy Flavor Averaging Group], arXiv:0808.1297 [hep-ex].
* (13) S. L. Chen, X. G. He, A. Hovhannisyan and H. C. Tsai, JHEP 09, 044 (2007) [arXiv:hep-ph/0706.1100].
* (14) A. A. Petrov, G. K. Yeghiyan, Phys. Rev. D 77, 034018 (2008).
* (15) A. Kundu, J. P. Saha, Phys. Rev, D 70, 096002 (2004).
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|
arxiv-papers
| 2009-09-30T04:45:42 |
2024-09-04T02:49:05.548690
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Gagik K. Yeghiyan",
"submitter": "Gagik Yeghiyan",
"url": "https://arxiv.org/abs/0909.5508"
}
|
0910.0034
|
# A simple electrostatic model applicable to biomolecular recognition
T. P. Doerr doerr@ncbi.nlm.nih.gov Yi-Kuo Yu yyu@ncbi.nlm.nih.gov National
Center for Biotechnology Information, National Library of Medicine, National
Institutes of Health, 8600 Rockville Pike MSC 6075, Bethesda, MD 20894-6075
###### Abstract
An exact, analytic solution for a simple electrostatic model applicable to
biomolecular recognition is presented. In the model, a layer of high
dielectric constant material (representative of the solvent, water) whose
thickness may vary separates two regions of low dielectric constant material
(representative of proteins, DNA, RNA, or similar materials), in each of which
is embedded a point charge. For identical charges, the presence of the
screening layer always lowers the energy compared to the case of point charges
in an infinite medium of low dielectric constant. Somewhat surprisingly, the
presence of a sufficiently thick screening layer also lowers the energy
compared to the case of point charges in an infinite medium of high dielectric
constant. For charges of opposite sign, the screening layer always lowers the
energy compared to the case of point charges in an infinite medium of either
high or low dielectric constant. The behavior of the energy leads to a
substantially increased repulsive force between charges of the same sign. The
repulsive force between charges of opposite signs is weaker than in an
infinite medium of low dielectric constant material but stronger than in an
infinite medium of high dielectric constant material. The presence of this
behavior, which we name asymmetric screening, in the simple system presented
here confirms the generality of the behavior that was established in a more
complicated system of an arbitrary number of charged dielectric spheres in an
infinite solvent.
###### pacs:
41.20.Cv,87.10.Ca
## I Introduction
The proper functioning of biomolecular systems depends upon the aggregation of
multiple molecules embedded in a high dielectric constant solvent (water).
From the medical point of view, there are both normal complexes (such as
ribosomes) and abnormal complexes (such as amyloid formations). Understanding
the microscopic mechanisms involved in the aggregation process would
illuminate both normal and abnormal states, and could aid the modification of
existing complexes or the design of new ones. This work examines the
electrostatic interaction, among the most important interactions in
biomolecular systems. Kauzmann -Chaplin2006
In previous research that developed a scheme for computing to known precision
the energy and forces in a system of an arbitrary number of charged dielectric
spheres embedded in an infinite solvent tpd06 , an effect that was called
asymmetric screening was observed. Namely, the magnitude of attractive
electrostatic interactions was decreased (relative to point charges in an
infinite solvent) while the magnitude of repulsive electrostatic interactions
was increased (again, relative to point charges in an infinite solvent). It
was speculated that this effect might aid biomolecules such as proteins in the
adoption of correct conformations and in intermolecular recognition.
This paper presents further studies of this effect in a simplified system that
is amenable to complete and thorough analytic examination. The simplicity of
the model is an advantage in this case because one wishes to examine in more
detail an effect that is already known to occur in the more general and less
symmetric system of spheres mentioned above. The system studied here can be
considered a simplified model of two molecular surfaces during the process of
binding or aggregation. Instead of spheres, consider two half-spaces, each
with a single point charge embedded, separated by an infinite slab of high
dielectric constant material (water, for example). If the dielectric constants
are swapped, then one would have a model of, for example, a membrane in water.
Separation of variables is used to obatin the potential, and from that the
energy and the force between the two half-spaces. It is more convenient to use
the surface charge method tpd06 -tpd04 to obtain the density of surface
charge induced on the two surfaces.
## II The General Situation
Consider a slab of material of thickness $2d$, infinite in the other
directions, with dielectric constant $\varepsilon_{0}$ sandwiched between two
half-spaces filled with materials of dielectric constant $\varepsilon_{1}$ and
$\varepsilon_{2}$ respectively. A charge $q_{1}$ lies within the external
material with dielctric constant $\varepsilon_{1}$ a distance $s_{1}$ from the
internal material (dielectric constant $\varepsilon_{0}$); a charge $q_{2}$
lies within the other external material (with dielctric constant
$\varepsilon_{2}$) a distance $s_{2}$ from the internal material and a
distance $s_{1}+s_{2}+2d$ from the charge $q_{1}$. Place the origin of
coordinates half way between the two charges. Place the $z$ axis through the
line joining the two charges, perpendicular to the surfaces of the internal
slab of material, and with the positive $z$ axis passing through the charge
$q_{1}$, as in Fig. 1. Because of the symmetry of the system, cylindrical
coordinates ($\rho$, $\phi$, and $z$) will be used.
Figure 1: The most general situation under consideration. The shaded region
is infinite in the $x$ and $y$ directions, has thickness $2d$ in the $z$
direction, and is filled with a material with dielectric constant
$\varepsilon_{0}$. The origin is chosen so that the distance from the origin
to each surface of the shaded region is $d$. The unshaded region entirely in
the $z>0$ half-space is filled with a material with dielectric constant
$\varepsilon_{1}$ and contains a charge $q_{1}$ on the positive $z$ axis a
distance $d+s_{1}$ from the origin and a fixed distance $s_{1}$ from the
surface of the shaded region. The unshaded region entirely in the $z<0$ half-
space is filled with a material with dielectric constant $\varepsilon_{2}$ and
contains a charge $q_{2}$ on the negative $z$ axis a distance $d+s_{2}$ from
the origin and a fixed distance $s_{2}$ from the surface of the shaded region.
We wish to find the electric potential ($\Phi$), the electrostatic energy
($U$), and the force ($\vec{F}$) required to pull the external materials
apart. We begin by determining the potential in the general case. Azimuthal
symmetry implies that the potential $\Phi$ is independent of $\phi$. The
symbols $\Phi_{0}$, $\Phi_{1}$, and $\Phi_{2}$ will be used to indicate the
potential in the interior material, in the material entirely in the positive
$z$ region, and in the material entirely in the negative $z$ region
respectively. The boundary conditions are
1. 1.
$\Phi\rightarrow 0$ as $z\rightarrow\pm\infty$
2. 2.
$\Phi_{0}(z=d)=\Phi_{1}(z=d)$
3. 3.
$\Phi_{2}(z=-d)=\Phi_{0}(z=-d)$
4. 4.
$\varepsilon_{0}\frac{\partial\Phi_{0}}{\partial
z}\left.\right|_{z=d}=\varepsilon_{1}\frac{\partial\Phi_{1}}{\partial
z}\left.\right|_{z=d}$
5. 5.
$\varepsilon_{2}\frac{\partial\Phi_{2}}{\partial
z}\left.\right|_{z=-d}=\varepsilon_{0}\frac{\partial\Phi_{0}}{\partial
z}\left.\right|_{z=-d}$
.
The appropriate general solution of Laplace’s equation is
$\Phi=\sum_{m=0}^{\infty}\int_{0}^{\infty}J_{m}(k\rho)(ae^{kz}+be^{-kz})(c\sin
m\phi+d\cos
m\phi)\,\mathrm{d}k\rightarrow\int_{0}^{\infty}J_{0}(k\rho)(ae^{kz}+be^{-kz})\,\mathrm{d}k,$
because of the azimuthal symmetry. The appropriate form of the potential of a
point charge at $\rho=0$ and $z=z^{\prime}$ is jdj
$\frac{1}{\sqrt{\rho^{2}+(z-z^{\prime})^{2}}}=\int_{0}^{\infty}e^{-k|z-z^{\prime}|}J_{0}(k\rho)\,\mathrm{d}k.$
The potential in the positive $z$ region of exterior material is a solution of
Laplace’s equation plus the potential of the screened point source:
$\Phi_{1}=\int_{0}^{\infty}B_{1}(k)e^{-kz}J_{0}(k\rho)\,\mathrm{d}k+\frac{q_{1}}{\varepsilon_{1}}\int_{0}^{\infty}e^{-k|z-d-
s_{1}|}J_{0}(k\rho)\,\mathrm{d}k,$ (1)
where boundary condition 1 has deleted one of the exponentials in the solution
of Laplace’s equation. Similarly, the potential in the negative $z$ region of
exterior material is
$\Phi_{2}=\int_{0}^{\infty}A_{2}(k)e^{kz}J_{0}(k\rho)\,\mathrm{d}k+\frac{q_{2}}{\varepsilon_{2}}\int_{0}^{\infty}e^{-k|z+d+s_{2}|}J_{0}(k\rho)\,\mathrm{d}k.$
(2)
The potential in the interior material is
$\Phi_{0}=\int_{0}^{\infty}(A_{0}(k)e^{kz}+B_{0}(k)e^{-kz})J_{0}(k\rho)\,\mathrm{d}k.$
(3)
Boundary conditions 2-5 determine the coefficients:
$\displaystyle B_{1}(k)$ $\displaystyle=$ $\displaystyle
e^{k(d-s_{1}-s_{2})}\frac{e^{ks_{2}}(\varepsilon_{0}+\varepsilon_{1})(\varepsilon_{0}-\varepsilon_{2})q_{1}-e^{k(4d+s_{2})}(\varepsilon_{0}-\varepsilon_{1})(\varepsilon_{0}+\varepsilon_{2})q_{1}+4e^{k(2d+s_{1})}\varepsilon_{0}\varepsilon_{1}q_{2}}{-(\varepsilon_{0}-\varepsilon_{1})\varepsilon_{1}(\varepsilon_{0}-\varepsilon_{2})+e^{4kd}\varepsilon_{1}(\varepsilon_{0}+\varepsilon_{1})(\varepsilon_{0}+\varepsilon_{2})}$
(4a) $\displaystyle A_{0}(k)$ $\displaystyle=$ $\displaystyle
2e^{k(d-s_{1}-s_{2})}\frac{e^{k(2d+s_{2})}(\varepsilon_{0}+\varepsilon_{2})q_{1}+e^{ks_{1}}(\varepsilon_{0}-\varepsilon_{1})q_{2}}{-(\varepsilon_{0}-\varepsilon_{1})(\varepsilon_{0}-\varepsilon_{2})+e^{4kd}(\varepsilon_{0}+\varepsilon_{1})(\varepsilon_{0}+\varepsilon_{2})}$
(4b) $\displaystyle B_{0}(k)$ $\displaystyle=$ $\displaystyle
2e^{k(d-s_{1}-s_{2})}\frac{e^{ks_{2}}(\varepsilon_{0}-\varepsilon_{2})q_{1}+e^{k(2d+s_{1})}(\varepsilon_{0}+\varepsilon_{1})q_{2}}{-(\varepsilon_{0}-\varepsilon_{1})(\varepsilon_{0}-\varepsilon_{2})+e^{4kd}(\varepsilon_{0}+\varepsilon_{1})(\varepsilon_{0}+\varepsilon_{2})}$
(4c) $\displaystyle A_{2}(k)$ $\displaystyle=$ $\displaystyle
e^{k(d-s_{1}-s_{2})}\frac{4e^{k(2d+s_{2})}\varepsilon_{0}\varepsilon_{2}q_{1}-e^{k(4d+s_{1})}(\varepsilon_{0}+\varepsilon_{1})(\varepsilon_{0}-\varepsilon_{2})q_{2}+e^{ks_{1}}(\varepsilon_{0}-\varepsilon_{1})(\varepsilon_{0}+\varepsilon_{2})q_{2}}{-(\varepsilon_{0}-\varepsilon_{1})\varepsilon_{2}(\varepsilon_{0}-\varepsilon_{2})+e^{4kd}\varepsilon_{2}(\varepsilon_{0}+\varepsilon_{1})(\varepsilon_{0}+\varepsilon_{2})}.$
(4d)
Not surprisingly, interchanging the indices 1 and 2 in the expression for
$B_{1}$ turns it into $A_{2}$.
The distribution of free charge (the two point charges) and the potential
determine the energy:
$U=\frac{1}{2}\int\rho_{f}\Phi=\frac{q_{1}}{2}\Phi^{\prime}_{1}(\rho=0,z=d+s_{1})+\frac{q_{2}}{2}\Phi^{\prime}_{2}(\rho=0,z=-d-s_{2}),$
(5)
where the primes on the potentials indicate that the potential of the point
charge in the corresponding region has been subtracted out in order to avoid
infinite self-energies. Substitution of Eq. (1), Eq. (2), and Eq. (4) into Eq.
(5) yields
$\displaystyle U$ $\displaystyle=$
$\displaystyle\frac{4q_{1}q_{2}\varepsilon_{0}}{(\varepsilon_{0}+\varepsilon_{1})(\varepsilon_{0}+\varepsilon_{2})}\int_{0}^{\infty}\frac{e^{-k(2d+s_{1}+s_{2})}}{1-\alpha_{1}\alpha_{2}e^{-4kd}}\,\mathrm{d}k+\frac{q_{1}^{2}}{2\varepsilon_{1}}\int_{0}^{\infty}\frac{e^{-2ks_{1}}(e^{-4kd}\alpha_{2}-\alpha_{1})}{1-\alpha_{1}\alpha_{2}e^{-4kd}}\,\mathrm{d}k$
(6)
$\displaystyle+\frac{q_{2}^{2}}{2\varepsilon_{2}}\int_{0}^{\infty}\frac{e^{-2ks_{2}}(e^{-4kd}\alpha_{1}-\alpha_{2})}{1-\alpha_{1}\alpha_{2}e^{-4kd}}\,\mathrm{d}k,$
where
$\alpha_{1}\equiv(\varepsilon_{0}-\varepsilon_{1})/(\varepsilon_{0}+\varepsilon_{1})$
and
$\alpha_{2}\equiv(\varepsilon_{0}-\varepsilon_{2})/(\varepsilon_{0}+\varepsilon_{2})$.
Because we imagine this situation to be a simplified model of two molecular
surfaces separated by a layer of water, the force should be obtained by
imagining that the charges are fixed with respect to the materials in which
they are embedded, but the thickness of the interior slab is allowed to vary.
In other words, the force we are considering is the negative of the derivative
of the energy with respect to $2d$:
$\vec{F}=-\frac{\partial U}{\partial(2d)}\hat{z},$
or in scalar form for the magnitude
$F=-\frac{1}{2}\frac{\partial U}{\partial d}.$
Clearly, this simple model neglects any internal rearrangement of the
molecules during the process of interaction, an effect that is believed to be
important in many cases. However, while a model designed to capture the
behavior of specific molecules would need to include such an effect, our
purpose is only to investigate one particular interaction, the very important
electrostatic interaction, and so this point is not a concern here. The force
is
$\displaystyle F$ $\displaystyle=$
$\displaystyle\frac{4q_{1}q_{2}\varepsilon_{0}}{(\varepsilon_{0}+\varepsilon_{1})(\varepsilon_{0}+\varepsilon_{2})}\int_{0}^{\infty}e^{-k(2d+s_{1}+s_{2})}k\frac{1+\alpha_{1}\alpha_{2}e^{-4kd}}{(1-\alpha_{1}\alpha_{2}e^{-4kd})^{2}}\,\mathrm{d}k$
(7)
$\displaystyle+\frac{q_{1}^{2}}{\varepsilon_{1}}\alpha_{2}(1-\alpha_{1}^{2})\int_{0}^{\infty}\frac{e^{-k(2s_{1}+4d)}k}{(1-\alpha_{1}\alpha_{2}e^{-4kd})^{2}}\,\mathrm{d}k$
$\displaystyle+\frac{q_{2}^{2}}{\varepsilon_{2}}\alpha_{1}(1-\alpha_{2}^{2})\int_{0}^{\infty}\frac{e^{-k(2s_{2}+4d)}k}{(1-\alpha_{1}\alpha_{2}e^{-4kd})^{2}}\,\mathrm{d}k.$
We now examine two particular cases.
## III Two Identical Charges in Identical Media
Let $q_{1}=q_{2}\equiv q$,
$\varepsilon_{1}=\varepsilon_{2}\equiv\varepsilon_{\mathrm{e}}$,
$\varepsilon_{0}\equiv\varepsilon_{\mathrm{i}}$, and $s_{1}=s_{2}\equiv s$. We
are now considering a slab of material (thickness $2d$ and infinite in the
other directions) with dielectric constant $\varepsilon_{\mathrm{i}}$
sandwiched between two half-spaces filled with a material of dielctric
constant $\varepsilon_{\mathrm{e}}$. (Internal material is indicated by the
subscript ‘i’, and external material is indicated by subscript ‘e’.) A charge
$q$ lies in the external material a distance $s$ from the internal material.
An identical charge $q$ lies in the other semi-infinite external material a
distance $s$ from the internal material and a distance $2s+2d$ from the other
charge. See Fig. 2, with the positive charge chosen.
Figure 2: A simplified situation considered in detail. The charges are now of
equal magnitute and are constrained to be the same distance from the origin.
The cases of identical charges and of opposite charges are both considered.
Both unshaded regions have the same dielectric constant, referred to as
$\varepsilon_{\mathrm{e}}$. The dielectric constant of the shaded slab is now
referred to as $\varepsilon_{\mathrm{i}}$.
The potential, the energy, and the force follow upon making the appropriate
substitutions in Eqs. (1-3), Eq. (6), and Eq. (7) respectively.
(Alternatively, it is a simple matter to set up and solve the boundary value
problem for this particular situation.)
Making the appropriate substitutions in Eq. (6), letting
$\alpha=(\varepsilon_{\mathrm{i}}-\varepsilon_{\mathrm{e}})/(\varepsilon_{\mathrm{i}}+\varepsilon_{\mathrm{e}})$,
and using the identity
$(4\varepsilon_{\mathrm{i}}\varepsilon_{\mathrm{e}}/(\varepsilon_{\mathrm{i}}+\varepsilon_{\mathrm{e}})^{2})=1-\alpha^{2}$,
one finds the energy:
$\displaystyle U$ $\displaystyle=$
$\displaystyle\frac{q^{2}}{\varepsilon_{\mathrm{e}}}\int_{0}^{\infty}e^{-2ks}\frac{e^{-2kd}(4\varepsilon_{\mathrm{i}}\varepsilon_{\mathrm{e}}/(\varepsilon_{\mathrm{i}}-\varepsilon_{\mathrm{e}})^{2})+\alpha(e^{-4kd}-1)}{1-\alpha^{2}e^{-4kd}}\,\mathrm{d}k$
(8) $\displaystyle=$
$\displaystyle\frac{q^{2}}{\varepsilon_{\mathrm{e}}}\int_{0}^{\infty}e^{-2k(s+d)}\frac{1-\alpha
e^{2kd}}{1-\alpha e^{-2kd}}\,\mathrm{d}k.$
One may evaluate the integral by expanding the denominator in a series:
$\displaystyle U$ $\displaystyle=$
$\displaystyle\frac{q^{2}}{\varepsilon_{\mathrm{e}}}\int_{0}^{\infty}\sum_{n=0}^{\infty}\left(\alpha^{n}e^{-2k(s+(n+1)d)}-\alpha^{n+1}e^{-2k(s+nd)}\right)\,\mathrm{d}k$
(9) $\displaystyle=$
$\displaystyle\frac{q^{2}}{2\varepsilon_{\mathrm{e}}}\sum_{n=0}^{\infty}\left(\frac{\alpha^{n}}{s+(n+1)d}-\frac{\alpha^{n+1}}{s+nd}\right)$
$\displaystyle=$
$\displaystyle\frac{q^{2}(1-\alpha^{2})}{2\varepsilon_{\mathrm{e}}\alpha}\sum_{n=0}^{\infty}\frac{\alpha^{n}}{s+nd}-\frac{q^{2}}{2\varepsilon_{\mathrm{e}}\alpha
s}$ $\displaystyle=$
$\displaystyle\frac{q^{2}(1-\alpha^{2})}{2\varepsilon_{\mathrm{e}}\alpha
s}{}_{2}F_{1}\left(\frac{s}{d},1;\frac{s}{d}+1;\alpha\right)-\frac{q^{2}}{2\varepsilon_{\mathrm{e}}\alpha
s}.$
where ${}_{2}F_{1}$ is a Gauss hypergeometric function.
Even though the series in Eq. (9) was obtained by separation of variables, it
can be interpreted as the effect of an infinite sequence of image charges. The
charges have separations $2s+2nd$ for $n=0,1,2,\ldots$. The magnitude of the
image charges can be read off from the coefficients of
$q/(\varepsilon_{\mathrm{e}}(2s+2nd))$ with appropriate care taken to separate
out the direct interaction of the free charges. This interpretation brings to
mind recent work that used an approximate series of image charges to study a
pair of membranes in a solvent of water and ions.Pincus2008
Because the dielectric constant of water ($\approx 80$crc ) is much larger
than the dielectric constant of protein ($\approx 4$honig86 ), we are most
interested in screening situation: $0\leq\alpha\leq 1$. In the limit
$\alpha\rightarrow 1$, the interior slab becomes metallic. In this case we
find that $U=-q^{2}/(\varepsilon_{\mathrm{e}}2s)$, which is just the
interaction energy of each free charge with its image charge due to the metal;
the two free charges do not ‘feel’ each other. If the media all have the same
dielectric constant, then $\alpha=0$ and
$U=q^{2}/(\varepsilon_{\mathrm{e}}2(s+d))$, which is simply the energy of two
charges in an infinite dielectric medium. Similarly, if $d=0$ we find the
obvious result $U=q^{2}/(\varepsilon_{\mathrm{e}}2s)$. Finally, in the limit
that $d\rightarrow\infty$,
$U\rightarrow-(q^{2}\alpha)/(\varepsilon_{\mathrm{e}}2s)<0$. In this case, the
two fixed charges do not see each other, but each point charge can still
induce a charge density on the nearby surface, and this process will always
reduce the energy. Therefore $U$ is negative in this limit. The behavior just
summarized can be seen in Fig. 3 and Fig. 4.
Figure 3: Graphs of the energy as a function of separation, both for
identical charges and for opposite charges. For comparison, the energy of
point charges, both identical and opposite, in an infinite uniform medium
(both $\varepsilon_{\mathrm{e}}$ and $\varepsilon_{\mathrm{i}}$) is shown. The
calculations are for $\varepsilon_{\mathrm{e}}=1$,
$\varepsilon_{\mathrm{i}}=80$, $s=1$, and $q=1$. For opposite charges
separated by a high dielectric layer, the energy varies little. For like
charges separated by a high dielectric layer, the energy at small separations
changes rapidly. Figure 4: Graphs of the energy as a function of
$\varepsilon_{\mathrm{i}}$, both for identical charges and for opposite
charges. For comparison, the energy of point charges, both identical and
opposite, in an infinite uniform medium (both $\varepsilon_{\mathrm{e}}$ and
$\varepsilon_{\mathrm{i}}$) is shown. The calculations are for $2s+2d=5$,
$\varepsilon_{\mathrm{e}}=1$, $s=1$, and $q=1$.
Making the appropriate substitutions in Eq. (7) and again using the identity
$(4\varepsilon_{\mathrm{i}}\varepsilon_{\mathrm{e}}/(\varepsilon_{\mathrm{i}}+\varepsilon_{\mathrm{e}})^{2})=1-\alpha^{2}$,
one finds the force:
$F=\frac{q^{2}(1-\alpha^{2})}{\varepsilon_{\mathrm{e}}}\int_{0}^{\infty}\frac{ke^{-2k(d+s)}}{(1-\alpha
e^{-2kd})^{2}}\,\mathrm{d}k.$ (10)
Rather than performing a similar procedure with series to evaluate the
integral, one may simply differentiate the series for $U$:
$\displaystyle F$ $\displaystyle=$
$\displaystyle-\frac{q^{2}\alpha(1-\alpha^{-2})}{\varepsilon_{\mathrm{e}}}\sum_{n=0}^{\infty}\frac{n\alpha^{n}}{(2s+n2d)^{2}}$
(11) $\displaystyle=$
$\displaystyle\frac{q^{2}(1-\alpha^{2})}{4\varepsilon_{\mathrm{e}}\alpha}\sum_{n=0}^{\infty}\frac{n\alpha^{n}}{(s+nd)^{2}}.$
As noted above, for the case of complete screening (i.e., $\alpha=1$) the free
charges do not ‘feel’ each other. As expected, the force vanishes in this
case. If the media all have the same dielectric constant, then $\alpha=0$ and
$F=q^{2}/(\varepsilon_{\mathrm{e}}(2s+2d)^{2})$, the force between two
identical charges in an infinite dielectric medium. On the other hand, if
$d=0$ we find the curious result
$F=(q^{2}\varepsilon_{\mathrm{i}})/(\varepsilon_{\mathrm{e}}^{2}4s^{2})$. When
$d=0$ one might expect $F$ not to depend on $\varepsilon_{\mathrm{i}}$.
However, $F(d)$ samples $U(d)$ in the vicinity of $d$, and even when $d=0$ a
dependence is generated on $\varepsilon_{\mathrm{i}}$, which characterizes the
material that would fill the gap if one were to draw the two outer regions
apart. Indeed, for $d=0$ and $\varepsilon_{\mathrm{i}}\rightarrow 1$, the
force becomes infinite, i.e., the energy changes discontinuously at $d=0$ if
$\alpha=1$. The behavior just summarized can be seen in Fig. 5 and Fig. 6.
Figure 5: Graphs of the force as a function of separation, both for identical
charges and for opposite charges. For comparison, the force between point
charges, both identical and opposite, in an infinite uniform medium (both
$\varepsilon_{\mathrm{e}}$ and $\varepsilon_{\mathrm{i}}$) is shown. The
calculations are for $\varepsilon_{\mathrm{e}}=1$,
$\varepsilon_{\mathrm{i}}=80$, $s=1$, and $q=1$. The inset is a close-up of
the three curves near the $x$ axis for small separations. Figure 6: Graphs of
the force as a function of $\varepsilon_{\mathrm{i}}$, both for identical
charges and for opposite charges. For comparison, the force between point
charges, both identical and opposite, in an infinite uniform medium (both
$\varepsilon_{\mathrm{e}}$ and $\varepsilon_{\mathrm{i}}$) is shown. The
calculations are for $2s+2d=5$ $\varepsilon_{\mathrm{e}}=1$, $s=1$, and $q=1$.
The difference between $U$ and the energy of two point charges in an infinite
medium of dielectric constant $\varepsilon_{\mathrm{e}}$ is defined to be
$\Delta U$. (This could not be calculated in the general case because in that
case there is no single exterior material.) One finds
$\Delta
U=\frac{q^{2}}{\varepsilon_{\mathrm{e}}}\int_{0}^{\infty}e^{-2k(s+d)}\left(\frac{1-\alpha
e^{2kd}}{1-\alpha
e^{-2kd}}-1\right)\,\mathrm{d}k=-\frac{q^{2}\alpha}{\varepsilon_{\mathrm{e}}}\int_{0}^{\infty}e^{-2ks}\frac{1-e^{-4kd}}{1-\alpha
e^{-2kd}}\,\mathrm{d}k.$ (12)
Notice that $\Delta U\leq 0$ in the case of screening ($\alpha>0$), which
makes sense because the energy should be lowered by replacing a portion of the
low dielectric constant material with higher dielectric constant material. If
$\alpha=0$, the energy $U$ is the same as the term we have just subtracted
off, so $\Delta U=0$. Similarly, if $d=0$, then $\Delta U=0$.
The force difference $\Delta F$ corresponding to $\Delta U$ can be obtained
either from the expression for $\Delta U$ or the expression for $F$:
$\Delta
F=\frac{q^{2}}{\varepsilon_{\mathrm{e}}}\int_{0}^{\infty}ke^{-2k(d+s)}\left(\frac{(1-\alpha^{2})}{(1-\alpha
e^{-2kd})^{2}}-1\right)\,\mathrm{d}k.$ (13)
In the case of $d=0$ we find that $\Delta
F=\frac{q^{2}\alpha}{2\varepsilon_{\mathrm{e}}s^{2}(1-\alpha)}$. If
$\alpha=1$, then $\Delta F=-q^{2}/(\varepsilon_{\mathrm{e}}(2s+2d)^{2})<0$
which, as expected, is just the term we subtracted off to form $\Delta F$.
Clearly $\Delta F=0$ if $\alpha=0$. The behavior of $\Delta F$ for small but
non-zero $\alpha$ may be deduced from the series expression for $F$:
$\displaystyle\Delta F$ $\displaystyle=$
$\displaystyle\frac{q^{2}}{4\varepsilon_{\mathrm{e}}}\sum_{n=1}^{\infty}\frac{n(1-\alpha^{2})\alpha^{n}}{\alpha(s+nd)^{2}}-\frac{q^{2}}{4\varepsilon_{\mathrm{e}}(s+d)^{2}}$
$\displaystyle>$
$\displaystyle\frac{q^{2}}{4\varepsilon_{\mathrm{e}}}\sum_{n=1}^{\infty}\frac{n(1-\alpha^{2})\alpha^{n}}{\alpha(ns+nd)^{2}}-\frac{q^{2}}{4\varepsilon_{\mathrm{e}}(s+d)^{2}}$
$\displaystyle=$
$\displaystyle\frac{q^{2}}{4\varepsilon_{\mathrm{e}}(s+d)^{2}}\frac{(1-\alpha^{2})}{\alpha}\sum_{n=1}^{\infty}\frac{\alpha^{n}}{n}-\frac{q^{2}}{4\varepsilon_{\mathrm{e}}(s+d)^{2}}$
$\displaystyle=$
$\displaystyle\frac{q^{2}}{4\varepsilon_{\mathrm{e}}(s+d)^{2}}\frac{\alpha}{2}+{\cal
O}(\alpha^{2}).$
When $\Delta F>0$, the repulsion between identical charges is stronger than
the case when both identical charges are in one uniform medium with dielectric
constant $\varepsilon_{\mathrm{e}}$. Upon letting
$\varepsilon_{\mathrm{e}}\rightarrow 1$ (see Fig. 5 and Fig. 6), we see that
one can have a repulsion larger than in vacuum, a counter-intuitive
conclusion. The origin of this behavior can be deduced by returning to Eq.
(6), the energy for the more general situation first described. Setting
$q_{1}=0$, $q_{2}=q$, and $s_{2}=s$ but retaining distinct dielectric
constants in each region, we find
$\displaystyle U$ $\displaystyle=$
$\displaystyle\frac{q^{2}}{2\varepsilon_{2}}\int_{0}^{\infty}\frac{e^{-2ks}(e^{-4kd}\alpha_{1}-\alpha_{2})}{1-\alpha_{1}\alpha_{2}e^{-4kd}}\,\mathrm{d}k$
$\displaystyle=$
$\displaystyle\frac{q^{2}}{2\varepsilon_{2}}\sum_{n=0}^{\infty}\alpha_{1}^{n}\alpha_{2}^{n}\left[\frac{\alpha_{1}}{2s+4(n+1)d}-\frac{\alpha_{2}}{2s+4nd}\right],$
and
$F=\frac{q^{2}}{\varepsilon_{2}}\sum_{n=0}^{\infty}\alpha_{1}^{n}\alpha_{2}^{n}\left[\frac{(n+1)\alpha_{1}}{(2s+4(n+1)d)^{2}}-\frac{n\alpha_{2}}{(2s+4nd)^{2}}\right].$
Each factor of $\alpha_{1}$ ($\alpha_{2}$) indicates an image reflection
across the surface of the material with dielectric constant $\varepsilon_{1}$
($\varepsilon_{2}$). Notice that the induced charge of the leading term
(proportional to $\alpha_{1}$) is the same sign as the free charge because the
image charge is located on the low dielectric side of the interface. If the
image charge were located on the high dielectric side of the interface
($\varepsilon_{0}<\varepsilon_{1}$ and $\varepsilon_{0}<\varepsilon_{2}$) then
the induced charge would have the opposite sign leading to an attractive force
similar to the more familiar case of a charge near a conductor.
Now consider the energy and force differences ($\widetilde{\Delta U}$ and
$\widetilde{\Delta F}$) when the comparison is made to the interaction with
the $\varepsilon_{\mathrm{i}}$ material everywhere. The energy difference in
this case, $\widetilde{\Delta U}$, is
$\displaystyle\widetilde{\Delta U}$ $\displaystyle=$
$\displaystyle\frac{q^{2}}{\varepsilon_{\mathrm{e}}}\int_{0}^{\infty}e^{-2k(s+d)}\left(\frac{1-\alpha
e^{2kd}}{1-\alpha
e^{-2kd}}-\frac{\varepsilon_{\mathrm{e}}}{\varepsilon_{\mathrm{i}}}\right)\,\mathrm{d}k$
(14) $\displaystyle=$
$\displaystyle\frac{q^{2}}{\varepsilon_{\mathrm{i}}\varepsilon_{\mathrm{e}}}\int_{0}^{\infty}e^{-2k(s+d)}\frac{(\varepsilon_{\mathrm{i}}-\varepsilon_{\mathrm{e}})+\alpha(\varepsilon_{\mathrm{e}}e^{-2kd}-\varepsilon_{\mathrm{i}}e^{2kd})}{1-\alpha
e^{-2kd}}\,\mathrm{d}k.$
In order to understand the behavior of $\widetilde{\Delta U}$, we observe that
$\widetilde{\Delta
U}(d=0)=(q^{2}(\varepsilon_{\mathrm{i}}-\varepsilon_{\mathrm{e}}))/(\varepsilon_{\mathrm{i}}\varepsilon_{\mathrm{e}}2s)\geq
0$ with equality when $\varepsilon_{\mathrm{i}}=\varepsilon_{\mathrm{e}}$
(i.e., $\alpha=0$). However, as $d\rightarrow\infty$, $\widetilde{\Delta
U}\rightarrow-(q^{2}\alpha)/(2\varepsilon_{\mathrm{e}}s)\leq 0$. Evidently,
for any positive $\alpha$, $\widetilde{\Delta U}$ is positive for small $d$
and becomes negative for sufficiently large $d$. This behavior can be inferred
from Fig. 3. Given that
$(\varepsilon_{\mathrm{i}}/\varepsilon_{\mathrm{e}})(1-\alpha)=(1+\alpha$),
$\widetilde{\Delta F}$ is
$\displaystyle\widetilde{\Delta F}$ $\displaystyle=$
$\displaystyle\frac{q^{2}}{\varepsilon_{\mathrm{i}}}\int_{0}^{\infty}ke^{-2k(s+d)}\left[\frac{(\varepsilon_{\mathrm{i}}/\varepsilon_{\mathrm{e}})(1-\alpha^{2})}{(1-\alpha
e^{-2kd})^{2}}-1\right]\,\mathrm{d}k$ $\displaystyle=$
$\displaystyle\frac{q^{2}}{\varepsilon_{\mathrm{i}}}\int_{0}^{\infty}ke^{-2k(s+d)}\left[\frac{(1+\alpha)^{2}}{(1-\alpha
e^{-2kd})^{2}}-1\right]\,\mathrm{d}k$ $\displaystyle\geq$
$\displaystyle\frac{q^{2}}{\varepsilon_{\mathrm{i}}}\int_{0}^{\infty}ke^{-2k(s+d)}[(1+\alpha)^{2}-1]\,\mathrm{d}k\geq
0,$
which guarantees that $\widetilde{\Delta F}\geq 0$, as would be expected based
upon Figs. 5 and 6.
## IV Two Opposite Charges in Identical Media
Consider the same situation as in the previous section except that the two
charges are of opposite sign. Namely, let $q_{1}\equiv q$, $q_{2}\equiv-q$,
$\varepsilon_{1}=\varepsilon_{2}\equiv\varepsilon_{\mathrm{e}}$,
$\varepsilon_{0}\equiv\varepsilon_{\mathrm{i}}$, and $s_{1}=s_{2}\equiv s$.
See Fig. 2, with the negative charge chosen. The potential, the energy, and
the force follow upon making the appropriate substitutions in Eqs. (1-3), Eq.
(6), and Eq. (7) respectively. (Alternatively, it is a simply matter to set up
and solve the boundary value problem for this particular situation.)
Making the appropriate substitutions in Eq. (6), letting
$\alpha=(\varepsilon_{\mathrm{i}}-\varepsilon_{\mathrm{e}})/(\varepsilon_{\mathrm{i}}+\varepsilon_{\mathrm{e}})$,
and using the identity
$(4\varepsilon_{\mathrm{i}}\varepsilon_{\mathrm{e}}/(\varepsilon_{\mathrm{i}}-\varepsilon_{\mathrm{e}})^{2})=1-\alpha^{2}$,
one finds the energy:
$\displaystyle U$ $\displaystyle=$
$\displaystyle-\frac{q^{2}}{\varepsilon_{\mathrm{e}}}\int_{0}^{\infty}e^{-2ks}\frac{e^{-2kd}(1-\alpha^{2})-\alpha(e^{-4kd}-1)}{1-\alpha^{2}e^{-4kd}}\,\mathrm{d}k$
(16) $\displaystyle=$
$\displaystyle-\frac{q^{2}}{\varepsilon_{\mathrm{e}}}\int_{0}^{\infty}e^{-2k(s+d)}\frac{1+\alpha
e^{2kd}}{1+\alpha e^{-2kd}}\,\mathrm{d}k.$
Again, we are most interested in screening situation: $0\leq\alpha\leq 1$.
When $\alpha=1$ (perfect screening), we find that
$U=-q^{2}/(\varepsilon_{\mathrm{e}}2s)$, which is just the interaction energy
of each free charge with its image charge due to the metal; the two free
charges do not ‘feel’ each other. If the media all have the same dielectric
constant, then $\alpha=0$ and $U=-q^{2}/(\varepsilon_{\mathrm{e}}2(s+d))$,
which is simply the energy of two charges in an infinite dielectric medium.
Similarly, if $d=0$ we find the obvious result
$U=-q^{2}/(\varepsilon_{\mathrm{e}}2s)$. Finally, in the limit that
$d\rightarrow\infty$,
$U\rightarrow-(q^{2}\alpha)/(\varepsilon_{\mathrm{e}}2s)<0$. In this case, the
two fixed charges do not see each other, but each point charge can still
induce a charge density on the nearby surface, and this process will always
reduce the energy. Note that if $\alpha$ is close to unity (e.g., a water
solvent), $U$ varies little as $d$ goes from 0 to $\infty$. The behavior just
summarized can be seen in Fig. 3 and Fig. 4. Comparing Eq. (16) with Eq. (8),
one sees that the series for $U$ is the series for identical charges with an
overall minus sign and the substitution $\alpha\rightarrow-\alpha$.
Making the appropriate substitutions in Eq. (7) and again using the identity
$(4\varepsilon_{\mathrm{i}}\varepsilon_{\mathrm{e}}/(\varepsilon_{\mathrm{i}}-\varepsilon_{\mathrm{e}})^{2})=1-\alpha^{2}$,
one finds the force:
$F=-\frac{q^{2}(1-\alpha^{2})}{\varepsilon_{\mathrm{e}}}\int_{0}^{\infty}\frac{ke^{-2k(d+s)}}{(1+\alpha
e^{-2kd})^{2}}\,\mathrm{d}k.$ (17)
As noted above, for the case of complete screening (i.e., $\alpha=1$) the free
charges do not ‘feel’ each other. As expected, the force vanishes in this
case. If the media all have the same dielectric constant, then $\alpha=0$ and
$F=-q^{2}/(\varepsilon_{\mathrm{e}}(2s+2d)^{2})$, the force between two
opposite charges in an infinite dielectric medium. If $d=0$ we find the
somewhat non-obvious result $F=-q^{2}/(\varepsilon_{\mathrm{i}}4s^{2})$, the
explanation for which is the same as in the case of identical charges. The
behavior of the force in the case of opposite charges is more consistent with
naive intuition: the force with a high dielectric layer is somewhere in
between the force with low dielectric everywhere and the force with high
dielectric everywhere. The behavior just summarized can be seen in Fig. 5 and
Fig. 6. Comparing Eq. (17) with Eq. (10), one sees that the series for $F$ is
the series for identical charges with an overall minus sign and the
substitution $\alpha\rightarrow-\alpha$.
The energy difference $\Delta U$ is now calculated along the lines used in the
case of identical charges:
$\displaystyle\Delta U$ $\displaystyle=$
$\displaystyle\frac{q^{2}}{\varepsilon_{\mathrm{e}}}\int_{0}^{\infty}e^{-2k(s+d)}\left[1-\frac{1+\alpha
e^{2kd}}{1+\alpha e^{-2kd}}\right]\,\mathrm{d}k$ $\displaystyle=$
$\displaystyle-\frac{2q^{2}\alpha}{\varepsilon_{\mathrm{e}}}\int_{0}^{\infty}e^{-2k(s+d)}\frac{\sinh
2kd}{1+\alpha e^{-2kd}}\,\mathrm{d}k\leq 0.$
Since $U(d=0)=-q^{2}/(\varepsilon_{\mathrm{e}}(2s+2d))$ and
$U\rightarrow-q^{2}\alpha/(2\varepsilon_{\mathrm{e}}s)$ as
$d\rightarrow\infty$, it is clear that $\Delta U$ should be negative (see Fig.
3). As expected, the energy difference $\Delta U$ vanishes both for $d=0$ and
for $\alpha=0$. For $\alpha=1$, each charge interacts with its image charge,
and therefore $\Delta U=-(q^{2}d)/(2\varepsilon_{\mathrm{e}}s(s+d))$.
Now consider $\Delta F$ for opposite charges:
$\Delta
F=\frac{q^{2}}{\varepsilon_{\mathrm{e}}}\int_{0}^{\infty}ke^{-2k(s+d)}\left[\frac{\alpha^{2}-1}{(1+\alpha
e^{-2kd})^{2}}+1\right]\,\mathrm{d}k\geq 0.$ (19)
The magnitude of the attractive force between opposite charges with a
screening layer is always less than when both charges are in one uniform
dielectric medium with dielectric constant $\varepsilon_{\mathrm{e}}$. This
agrees with intuition upon letting $\varepsilon_{\mathrm{e}}\rightarrow 1$. As
expected, $\Delta F$ vanishes if $\alpha=0$. Also, $\Delta
F=q^{2}/(\varepsilon_{\mathrm{e}}(2s+2d)^{2})$ if $\alpha=1$, which confirms
that there is no force between charges that have a metal between them. For
$d=0$, the force difference $\Delta
F=(q^{2}\alpha)/(2\varepsilon_{\mathrm{e}}s^{2}(1+\alpha))$ depends on
$\alpha$ for the reason noted in the case of identical charges.
The energy difference when the comparison is made to the interaction with the
$\varepsilon_{\mathrm{i}}$ material everywhere is $\widetilde{\Delta U}$:
$\displaystyle\widetilde{\Delta U}$ $\displaystyle=$
$\displaystyle\frac{q^{2}}{\varepsilon_{\mathrm{e}}}\int_{0}^{\infty}e^{-2k(s+d)}\left[\frac{\varepsilon_{\mathrm{e}}}{\varepsilon_{\mathrm{i}}}-\frac{1+\alpha
e^{2kd}}{1+\alpha e^{-2kd}}\right]\,\mathrm{d}k$ (20) $\displaystyle=$
$\displaystyle\frac{q^{2}}{\varepsilon_{\mathrm{e}}\varepsilon_{\mathrm{i}}}\int_{0}^{\infty}e^{-2k(s+d)}\left[\frac{(\varepsilon_{\mathrm{e}}-\varepsilon_{\mathrm{i}})+\alpha(\varepsilon_{\mathrm{e}}e^{-2kd}-\varepsilon_{\mathrm{i}}e^{2kd})}{1+\alpha
e^{-2kd}}\right]\,\mathrm{d}k.$
As expected on the basis of Fig. 3), $\widetilde{\Delta U}$ is less than or
equal to 0 since both terms within the square brackets are less than or equal
to 0 in the case of screening ($0\leq\alpha\leq 1$). For $\alpha=0$, the
energy difference $\widetilde{\Delta U}$ vanishes, while for $\alpha=1$ and
$d\rightarrow\infty$, $\widetilde{\Delta
U}=-q^{2}/(\varepsilon_{\mathrm{e}}2s)$, the energy of interaction due to the
presence of image charges. For $d=0$, $\widetilde{\Delta
U}=q^{2}(\varepsilon_{\mathrm{e}}-\varepsilon_{\mathrm{i}})/2\varepsilon_{\mathrm{e}}\varepsilon_{\mathrm{i}}s$.
Now consider $\widetilde{\Delta F}$:
$\displaystyle\widetilde{\Delta F}$ $\displaystyle=$
$\displaystyle-\frac{q^{2}}{\varepsilon_{\mathrm{i}}}\int_{0}^{\infty}ke^{-2k(s+d)}\left[\frac{\varepsilon_{\mathrm{i}}(1-\alpha)(1+\alpha)}{\varepsilon_{\mathrm{e}}(1+\alpha
e^{-2kd})^{2}}-1\right]\,\mathrm{d}k$ (21) $\displaystyle=$
$\displaystyle-\frac{q^{2}}{\varepsilon_{\mathrm{i}}}\int_{0}^{\infty}ke^{-2k(s+d)}\left[\frac{(1+\alpha)^{2}}{(1+\alpha
e^{-2kd})^{2}}-1\right]\,\mathrm{d}k\leq 0.$
The attraction between unlike charges in our setting is always stronger than
when the charges are in a uniform dielectric medium of dielectric constant
$\varepsilon_{\mathrm{i}}$. Clearly, $\widetilde{\Delta F}$ vanishes when
$\alpha=0$ and when $d=0$.
## V Comments
The energy and force for the case of two point charges in a dielectric medium
with a layer of differing dielectric between them has been compared with two
baselines: point charges in a uniform medium having the dielectric constant of
the separating layer and point charges in a uniform medium having the
dielectric constant of the exterior medium. In the latter case, we find that
for opposite charges, $\Delta F>0$ always, implying a weakened attraction when
compared to the baseline. For identical charges, however, there are cases for
which the repulsion is actually enhanced compared to this baseline. Since it
is possible to let $\varepsilon_{\mathrm{e}}\rightarrow 1$, this situation
corresponds to an effective repulsion that is stronger than the vacuum case, a
counter-intuitive result. We refer to this behavior as ‘asymmetric screening’.
When both repulsion and attraction are weakened compared to the
$\varepsilon_{\mathrm{e}}$ baseline, which one is reduced more? This question
is easily answered by considering
$\delta F\equiv\Delta F_{\mathrm{att}}-(-\Delta F_{\mathrm{rep}})=\Delta
F_{\mathrm{att}}+\Delta F_{\mathrm{rep}}.$
When $\delta F>0$, there is a larger reduction of the attraction than of the
repulsion, and vice versa. Using Eq. (13) for $\Delta F_{\mathrm{rep}}$ and
Eq. (19) for $\Delta F_{\mathrm{att}}$, we find
$\delta F=\Delta F_{\mathrm{att}}+\Delta
F_{\mathrm{rep}}=\frac{q^{2}}{\varepsilon_{\mathrm{e}}}\int_{0}^{\infty}ke^{-2k(s+d)}\left[\frac{1-\alpha^{2}}{(1-\alpha
e^{-2kd})^{2}}-\frac{1-\alpha^{2}}{(1+\alpha
e^{-2kd})^{2}}\right]\,\mathrm{d}k\geq 0.$
For the case of the $\varepsilon_{\mathrm{i}}$ baseline, we see that
$\widetilde{\Delta F}$ is always negative for opposite charges. This indicates
an enhanced attraction compared to the baseline (when both charges are in a
uniform medium of dielectric constant $\varepsilon_{\mathrm{i}}$). For
identical charges we have $\widetilde{\Delta F}>0$, implying that the
repulsion is always enhanced when compared to this baseline. One can consider
$\widetilde{\delta F}\equiv\widetilde{\Delta
F}_{\mathrm{att}}-(-\widetilde{\Delta F}_{\mathrm{rep}})=\widetilde{\Delta
F}_{\mathrm{att}}+\widetilde{\Delta F}_{\mathrm{rep}}.$
When $\widetilde{\delta F}>0$, the repulsion of identical charges is enhanced
more then the attraction of opposite charges is. Using Eq. (III) for
$\widetilde{\Delta F}_{\mathrm{rep}}$ and Eq. (21) for $\widetilde{\Delta
F}_{\mathrm{att}}$, we find
$\widetilde{\delta F}=\widetilde{\Delta F}_{\mathrm{att}}+\widetilde{\Delta
F}_{\mathrm{rep}}=\frac{q^{2}}{\varepsilon_{\mathrm{i}}}\int_{0}^{\infty}ke^{-2k(s+d)}\left[\frac{(1+\alpha)^{2}}{(1-\alpha
e^{-2kd})^{2}}-\frac{(1+\alpha)^{2}}{(1+\alpha
e^{-2kd})^{2}}\right]\,\mathrm{d}k\geq 0.$
According to Fig. 5, asymmetric screening is quite pronounced at short ranges,
and we expect the phenomenon to play an important role in biomolecular
recognition and in the adoption of the native conformation of proteins.
Particularly pronounced is the enhanced repulsion between charges of the same
sign. This behavior should exert a rather strong veto on poor matching of
charges as one part of a molecule interacts with another part or as two
molecules interact with each other. Therefore, accurate calculation of
electrostatic interaction is essential when considering biomolecular systems.
## Acknowledgements
This research was supported by the Intramural Research Program of the NIH,
National Library of Medicine.
*
## Appendix A Surface Charge Method
The surface charge methodtpd06 -tpd04 provides a relatively easy path to the
induced surface charge. In the case of two identical charges, symmetry implies
that the induced surface charge densities on the two surfaces are identical
functions in the plane. Therefore we may write
$\Phi=\frac{q}{\varepsilon_{\mathrm{e}}|\vec{r}-(d+s)\hat{z}|}+\frac{q}{\varepsilon_{\mathrm{e}}|\vec{r}+(d+s)\hat{z}|}+\int_{z^{\prime}=+d}\frac{\sigma(\rho^{\prime})}{|\vec{r}-\vec{r}^{\prime}|}\,\mathrm{d}S^{\prime}+\int_{z^{\prime}=-d}\frac{\sigma(\rho^{\prime})}{|\vec{r}-\vec{r}^{\prime}|}\,\mathrm{d}S^{\prime}.$
(22)
The induced surface charge density $\sigma(\rho)$ is unknown, but can be
expanded in a complete set of functions. Because of the cylindrical symmetry,
Bessel functions are the obvious choice in this case. Any reasonably well-
behaved function $f(\rho)$ gives rise to the pair of transformsarfken1
$\displaystyle f(\rho)$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}a(\beta)J_{\nu}(\beta\rho)\,\mathrm{d}\beta$
$\displaystyle a(\beta)$ $\displaystyle=$
$\displaystyle\beta\int_{0}^{\infty}f(\rho)J_{\nu}(\beta\rho)\rho\,\mathrm{d}\rho,$
allowing us to write the surface charge as
$\sigma(\rho)=\int_{0}^{\infty}S(\beta)J_{\nu}(\beta\rho)\,\mathrm{d}\beta.$
Furthermore, the denominator of the integrals in Eq. (22) can also be expanded
in Bessel functions jdj :
$\frac{1}{|\vec{r}-\vec{r}^{\prime}|}=\sum_{m=-\infty}^{\infty}\int_{0}^{\infty}\mathrm{d}k\,e^{im(\phi-\phi^{\prime})}J_{m}(k\rho)J_{m}(k\rho^{\prime})e^{-k(z_{>}-z_{<})},$
where $z_{>}=\max\\{z,z^{\prime}\\}$ and $z_{>}=\min\\{z,z^{\prime}\\}$.
In the vicinity of the surfaces, the potentials of the point charges are
$\frac{q}{\varepsilon_{\mathrm{e}}|\vec{r}-(d+s)\hat{z}|}=\frac{q}{\varepsilon_{\mathrm{e}}}\int_{0}^{\infty}\mathrm{d}k\,J_{0}(k\rho)e^{-k(d+s-z)}$
and
$\frac{q}{\varepsilon_{\mathrm{e}}|\vec{r}+(d+s)\hat{z}|}=\frac{q}{\varepsilon_{\mathrm{e}}}\int_{0}^{\infty}\mathrm{d}k\,J_{0}(k\rho)e^{-k(z+d+s)}.$
The potential near the boundary at $z=d$ due to the induced surface charge at
$z=d$ is
$\displaystyle\int_{z^{\prime}=+d}\frac{\sigma(\rho^{\prime})}{|\vec{r}-\vec{r}^{\prime}|}\,\mathrm{d}S^{\prime}$
$\displaystyle=$
$\displaystyle\int(\rho^{\prime}\mathrm{d}\phi^{\prime}\mathrm{d}\rho^{\prime})\left[\int_{0}^{\infty}{\cal
S}(\beta)J_{\nu}(\beta\rho^{\prime})\,\mathrm{d}\beta\right]$
$\displaystyle\times\left[\sum_{m=-\infty}^{\infty}\int_{0}^{\infty}\mathrm{d}k\,e^{im(\phi-\phi^{\prime})}J_{m}(k\rho)J_{m}(k\rho^{\prime})e^{-k(z_{>}-z_{<})}\right]$
$\displaystyle=$ $\displaystyle\int_{0}^{\infty}\mathrm{d}\beta\,{\cal
S}(\beta)\int_{0}^{\infty}\mathrm{d}k\,e^{-k(z_{>}-z_{<})}\sum_{m=-\infty}^{\infty}J_{m}(k\rho)\left[\int\mathrm{d}\phi^{\prime}e^{im(\phi-\phi^{\prime})}\right]$
$\displaystyle\times\left[\int\mathrm{d}\rho^{\prime}\rho^{\prime}J_{\nu}(\beta\rho^{\prime})J_{m}(k\rho^{\prime})\right]$
$\displaystyle=$ $\displaystyle 2\pi\int_{0}^{\infty}\mathrm{d}\beta\,{\cal
S}(\beta)\int_{0}^{\infty}\mathrm{d}k\,e^{-k(z_{>}-z_{<})}J_{0}(k\rho)\left[\int\mathrm{d}\rho^{\prime}\rho^{\prime}J_{\nu}(\beta\rho^{\prime})J_{0}(k\rho^{\prime})\right].$
Letting $\nu=0$ turns the $\rho^{\prime}$ integral into a standard one,
arfken2
$\int_{0}^{\infty}J_{\nu}(\beta\rho)J_{\nu}(\beta^{\prime}\rho)\rho\,\mathrm{d}\rho=\frac{\delta(\beta-\beta^{\prime})}{\beta}\qquad(v>-1/2),$
and therefore
$\int_{z^{\prime}=+d}\frac{\sigma(\rho^{\prime})}{|\vec{r}-\vec{r}^{\prime}|}\,\mathrm{d}S^{\prime}=2\pi\int_{0}^{\infty}\mathrm{d}k\,e^{-k(z_{>}-z_{<})}J_{0}(k\rho){\cal
S}(k)/k.$
So for $z^{\prime}=+d$ and $z>d$ (just above the top interface)
$2\pi\int_{0}^{\infty}\mathrm{d}k\,e^{-k(z-d)}J_{0}(k\rho){\cal S}(k)/k.$
For $z^{\prime}=+d$ and $z<d$ (just below the top interface)
$2\pi\int_{0}^{\infty}\mathrm{d}k\,e^{-k(d-z)}J_{0}(k\rho){\cal S}(k)/k.$
For $z^{\prime}=-d$ and $z$ near $d$ one finds a similar formula that is valid
either above or below interface:
$2\pi\int_{0}^{\infty}\mathrm{d}k\,e^{-k(z+d)}J_{0}(k\rho){\cal S}(k)/k.$
The boundary condition at $z=d$ is
$\varepsilon_{\mathrm{i}}\left.\frac{\partial\Phi_{z\leq d}}{\partial
z}\right|_{z=d}=\varepsilon_{\mathrm{e}}\left.\frac{\partial\Phi_{z\geq
d}}{\partial z}\right|_{z=d}$
for every value of $\rho$, which leads to an equation easily solved for ${\cal
S}(k)$:
${\cal S}(k)=\frac{qk\alpha
e^{-ks}(e^{-2kd}-1)}{2\pi\varepsilon_{\mathrm{e}}(1-\alpha e^{-2kd})}.$
Therefore
$\displaystyle\sigma(\rho)$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}J_{0}(k\rho){\cal S}(k)\,\mathrm{d}k$
$\displaystyle=$
$\displaystyle\frac{q\alpha}{2\pi\varepsilon_{\mathrm{e}}}\int_{0}^{\infty}J_{0}(k\rho)\frac{ke^{-ks}(e^{-2kd}-1)}{(1-\alpha
e^{-2kd})}\,\mathrm{d}k$ $\displaystyle=$
$\displaystyle-\frac{q\alpha}{2\pi\varepsilon_{\mathrm{e}}}\int_{0}^{\infty}J_{0}(k\rho)ke^{-ks}\left(1+\frac{(\alpha-1)e^{-2kd}}{(1-\alpha
e^{-2kd})}\right)\,\mathrm{d}k$ $\displaystyle=$
$\displaystyle-\frac{q\alpha}{2\pi\varepsilon_{\mathrm{e}}}\left[\int_{0}^{\infty}J_{0}(k\rho)ke^{-ks}\,\mathrm{d}k+\int_{0}^{\infty}J_{0}(k\rho)ke^{-k(s+2d)}(\alpha-1)\sum_{n=0}^{\infty}\alpha^{n}e^{-2knd}\,\mathrm{d}k\right]$
$\displaystyle=$
$\displaystyle-\frac{q\alpha}{2\pi\varepsilon_{\mathrm{e}}}\left[\int_{0}^{\infty}J_{0}(k\rho)ke^{-ks}\,\mathrm{d}k+\sum_{n=0}^{\infty}\alpha^{n}(\alpha-1)\int_{0}^{\infty}J_{0}(k\rho)ke^{-k(s+2(n+1)d)}\,\mathrm{d}k\right].$
Since all variables are real, we make use of the following integralGR1
$\int_{0}^{\infty}e^{-\alpha x}J_{\nu}(\beta
x)x^{\nu+1}\,\mathrm{d}x=\frac{(2\alpha)(2\beta)^{\nu}\Gamma(\nu+(3/2))}{\sqrt{\pi}(\alpha^{2}+\beta^{2})^{\nu+(3/2)}}$
for $\nu>-1$ and $\alpha>0$. Recall that
$\Gamma(n+(1/2))=\sqrt{\pi}(2n-1)!!2^{-n}$, so that
$\Gamma(3/2)=\sqrt{\pi}/2$. Therefore, the surface charge density is
$\sigma(\rho)=-\frac{q\alpha}{2\pi\varepsilon_{\mathrm{e}}}\left[\frac{s}{(s^{2}+\rho^{2})^{3/2}}+\sum_{n=0}^{\infty}\alpha^{n}(\alpha-1)\frac{s+2(n+1)d}{\left[(s+2(n+1)d)^{2}+\rho^{2}\right]^{3/2}}\right],$
from which it is easy to verify that
$\int\sigma(\rho)2\pi\rho\mathrm{d}\rho=0$.
This charge density can be used to recover same energy and force as before. To
compute the energy (and then the force), $\Phi(\rho=0,z=s+d)$ must be computed
from $\sigma(\rho)$. For $z^{\prime}=d$, $\rho=0$, and $z=s+d$,
$|\vec{r}-\vec{r}^{\prime}|^{2}=s^{2}+\rho^{\prime 2}$. Therefore
$\displaystyle\int\frac{\sigma(\rho^{\prime})}{|\vec{r}-\vec{r}^{\prime}|}\,\mathrm{d}S^{\prime}$
$\displaystyle=$ $\displaystyle
2\pi\int\frac{\rho^{\prime}\sigma(\rho^{\prime})}{(s^{2}+\rho^{\prime
2})^{1/2}}\,\mathrm{d}\rho^{\prime}$ $\displaystyle=$ $\displaystyle
2\pi\int\left({\cal
S}(k)\int\frac{\rho^{\prime}J_{0}(k\rho^{\prime})}{(s^{2}+\rho^{\prime
2})^{1/2}}\,\mathrm{d}\rho^{\prime}\right)\,\mathrm{d}k.$
The $\rho^{\prime}$ integral is found in tablesGR2 to be
$\int_{0}^{\infty}\frac{xJ_{0}(xy)}{(a^{2}+x^{2})^{1/2}}\,\mathrm{d}x=\frac{e^{-ay}}{y},$
and so
$\int\frac{\sigma(\rho^{\prime})}{|\vec{r}-\vec{r}^{\prime}|}\,\mathrm{d}S^{\prime}=\frac{q\alpha}{\varepsilon_{\mathrm{e}}}\int_{0}^{\infty}e^{-2ks}\frac{e^{-2kd}-1}{1-\alpha
e^{-2kd}}\,\mathrm{d}k.$
For $z^{\prime}=-d$, $\rho=0$, and $z=s+d$,
$|\vec{r}-\vec{r}^{\prime}|^{2}=(s+2d)^{2}+\rho^{\prime 2}$. The contribution
to the potential from the induced surface charge at $z^{\prime}=-d$ is
$\int\frac{\sigma(\rho^{\prime})}{|\vec{r}-\vec{r}^{\prime}|}\,\mathrm{d}S^{\prime}=\frac{q\alpha}{\varepsilon_{\mathrm{e}}}\int_{0}^{\infty}e^{-2k(s+d)}\frac{e^{-2kd}-1}{1-\alpha
e^{-2kd}}\,\mathrm{d}k.$
The potential at $\rho=0$ and $z=s+d$ is
$\displaystyle\Phi(\rho=0,z=s+d)$ $\displaystyle=$
$\displaystyle\frac{q}{\varepsilon_{\mathrm{e}}}\int_{0}^{\infty}\\!\left(e^{-k(2s+2d)}+\frac{\alpha
e^{-2ks}(e^{-2kd}-1)}{1-\alpha e^{-2kd}}+\frac{\alpha
e^{-k(2s+2d)}(e^{-2kd}-1)}{1-\alpha e^{-2kd}}\right)\mathrm{d}k$
$\displaystyle=$
$\displaystyle\frac{q}{\varepsilon_{\mathrm{e}}}\int_{0}^{\infty}e^{-2k(s+d)}\frac{1-\alpha
e^{2kd}}{1-\alpha e^{-2kd}}\,\mathrm{d}k,$
and therefore
$U=\frac{q^{2}}{\varepsilon_{\mathrm{e}}}\int_{0}^{\infty}e^{-2k(s+d)}\frac{1-\alpha
e^{2kd}}{1-\alpha e^{-2kd}}\,\mathrm{d}k,$
in agreement with Section III. Because $U$ agrees, everything that follows
from $U$ must also agree.
For opposite charges
$\Phi=\frac{q}{\varepsilon_{\mathrm{e}}|\vec{r}-(d+s)\hat{z}|}-\frac{q}{\varepsilon_{\mathrm{e}}|\vec{r}+(d+s)\hat{z}|}+\int_{z^{\prime}=+d}\frac{\sigma_{+}(\rho^{\prime})}{|\vec{r}-\vec{r}^{\prime}|}\,\mathrm{d}S^{\prime}+\int_{z^{\prime}=-d}\frac{\sigma_{-}(\rho^{\prime})}{|\vec{r}-\vec{r}^{\prime}|}\,\mathrm{d}S^{\prime}$
However, by symmetry $\sigma_{+}=-\sigma_{-}\equiv\sigma$. The boundary
condition yields
${\cal S}(k)=\frac{qk\alpha
e^{-ks}(e^{-2kd}+1)}{2\pi\varepsilon_{\mathrm{e}}(1+\alpha e^{-2kd})}$
The surface charge density becomes
$\sigma(\rho)=\frac{q\alpha}{2\pi\varepsilon_{\mathrm{e}}}\left[\frac{s}{(s^{2}+\rho^{2})^{3/2}}+\sum_{n=0}^{\infty}(-\alpha)^{n}(1-\alpha)\frac{s+2(n+1)d}{\left[(s+2(n+1)d)^{2}+\rho^{2}\right]^{3/2}}\right]$
Again, it is easy to verify that $\int\sigma(\rho)2\pi\rho\mathrm{d}\rho=0$
and that the energy $U$ reproduces the result in Section IV.
## References
* (1) W. Kauzmann, Adv. Protein Chem. 14, 1 (1959).
* (2) A. Parsegian, Nature 221, 844 (1969).
* (3) A. Ben-Naim, Hydrophobic Interactions (Plenum Press, New York, 1980).
* (4) B. Honig and A Nicholls, Science 268, 1144 (1995).
* (5) D. Chandler, Nature 437, 640 (2005).
* (6) M. Chaplin, Nature Reviews: Molecular and Cell Biology 7, 861 (2006).
* (7) T. P. Doerr and Y.-K. Yu, Phys. Rev. E 73, 061902 (2006).
* (8) Y.-K. Yu, Physica A 326, 522 (2003).
* (9) T. P. Doerr and Y.-K. Yu, Am. J. Phys. 72, 190 (2004).
* (10) J. D. Jackson, Classical Electrodynamics, Second Ed., (John Wiley & Sons, New York, 1975) p. 131.
* (11) Y. S. Jho, M. W. Kim, P. A. Pincus, and F. L. H. Brown, J. Chem. Phys. 129, 134511 (2008).
* (12) D. R. Lide, ed., CRC Handbook of Chemistry and Physics (CRC Press, 2003).
* (13) B. H. Honig, W. L. Hubbell, and R. F. Flewelling, Ann. Rev. Biophys. Biophys. Chem. 15, 163 (1986).
* (14) G. B. Arfken and H. J. Weber Mathematical Methods for Physicists, Fourth Ed., (Academic Press, San Diego, 1995) p. 650.
* (15) G. B. Arfken and H. J. Weber Mathematical Methods for Physicists, Fourth Ed., (Academic Press, San Diego, 1995) p. 648.
* (16) I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, (Academic Press, San Diego, 1980) 6.623 #2.
* (17) I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, (Academic Press, San Diego, 1980) 6.554 #1.
|
arxiv-papers
| 2009-09-30T21:29:30 |
2024-09-04T02:49:05.563391
|
{
"license": "Public Domain",
"authors": "T. P. Doerr and Yi-Kuo Yu",
"submitter": "Timothy Doerr",
"url": "https://arxiv.org/abs/0910.0034"
}
|
0910.0097
|
# Scalable Database Access Technologies
for ATLAS Distributed Computing
A. Vaniachine, for the ATLAS Collaboration ANL, Argonne, IL 60439, USA
###### Abstract
ATLAS event data processing requires access to non-event data (detector
conditions, calibrations, etc.) stored in relational databases. The database-
resident data are crucial for the event data reconstruction processing steps
and often required for user analysis. A main focus of ATLAS database
operations is on the worldwide distribution of the Conditions DB data, which
are necessary for every ATLAS data processing job. Since Conditions DB access
is critical for operations with real data, we have developed the system where
a different technology can be used as a redundant backup. Redundant database
operations infrastructure fully satisfies the requirements of ATLAS
reprocessing, which has been proven on a scale of one billion database queries
during two reprocessing campaigns of 0.5 PB of single-beam and cosmics data on
the Grid. To collect experience and provide input for a best choice of
technologies, several promising options for efficient database access in user
analysis were evaluated successfully. We present ATLAS experience with
scalable database access technologies and describe our approach for prevention
of database access bottlenecks in a Grid computing environment.
## I Introduction
A starting point for any ATLAS physics analysis is data reconstruction. ATLAS
event data reconstruction requires access to non-event data (detector
conditions, calibrations, etc.) stored in relational databases. These
database-resident data are crucial for the event data reconstruction steps and
often required for user analysis. Because Conditions DB access is critical for
operations with real data, we have developed the system where a different
technology can be used as a redundant backup.
A main focus of ATLAS database operations is on the worldwide distribution of
the Conditions DB data, which are necessary for every ATLAS data
reconstruction job. To support bulk data reconstruction operations of
petabytes of ATLAS raw events, the technologies selected for database access
in data reconstruction must be scalable. Since our Conditions DB mirrors the
complexity of the ATLAS detector 1 , the deployment of a redundant
infrastructure for Conditions DB access is a non-trivial task.
## II Managing Complexity
Driven by the complexity of the ATLAS detector, the Conditions DB organization
and access is complex (Figure 1). To manage this complexity, ATLAS adopted a
Conditions DB technology called COOL 2 . COOL was designed as a common
technology for experiments at the Large Hadron Collider (LHC). The LHC
Computing Grid (LCG) project developed COOL—Conditions Of Objects for LCG—as a
subproject of an LCG project on data persistency called POOL—Pool Of
persistent Objects for LHC POOL . The main technology for POOL data storage is
ROOT ROOT .
Figure 1: Software for transparent access to several Conditions DB
implementation technologies. Software for access to database-resident
information is called CORAL, software for access to ROOT files is called POOL.
In COOL the conditions are characterized by the interval-of-validity metadata
and an optional version tag. ATLAS Conditions DB contains both database-
resident information and external data in separate files that are referenced
by the database-resident data. These files are in a POOL/ROOT format. ATLAS
database-resident information exists in its entirety in Oracle but can be
distributed in smaller “slices” of data using SQLite—a file-based technology.
Figure 2: Subdetectors of the ATLAS detector.
The complexity of the Conditions DB organization is reflected in database
access statistics by data reconstruction jobs. These jobs access a slice of
Conditions DB data organized in sixteen database schemas: two global schemas
(online and offline) plus one or two schemas per each subdetector (Figure 2).
Jobs access 747 tables, which are grouped in 122 “folders” plus some system
tables. There are 35 distinct database-resident data types ranging from 32 bit
to 16 MB in size and referencing 64 external POOL files. To process a 2 GB
file with 1000 raw events a typical reconstruction job makes $\sim$2000
queries reading $\sim$40 MB of database-resident data, with some jobs read
tens of MB extra. In addition, about the same volume of data is read from the
external POOL files.
## III Data Reconstruction
Data reconstruction is a starting point for any ATLAS data analysis. Figure 3
shows simplified flow of raw events and conditions data in reconstruction.
Figure 3: Simplified flow of data from the detector (Fig. 2) used in
reconstruction at CERN and Tier-1 sites.
### III.1 First-pass processing at CERN
Scalable access to Conditions DB is critical for data reconstruction at CERN
using alignment and calibration constants produced within 24 hours—the “first-
pass” processing. Two solutions assure scalability:
* •
replicated AFS volume for POOL files,
* •
throttling of job submission at Tier-0.
The physics discovery potential of the Tier-0 processing results is limited
because the reconstruction at CERN is conservative in scope and uses
calibration and alignment constants that will need to be modified as analysis
of the data proceeds. As our knowledge of the detector improves, it is
necessary to rerun the reconstruction—the “reprocessing.” The reprocessing
uses enhanced software and revised conditions for improved reconstruction
quality. Since the Tier-0 is generally fully occupied with first-pass
reconstruction, the reprocessing uses the shared computing resources, which
are distributed worldwide—the Grid.
### III.2 Reprocessing on the Grid
Figure 4: Database Release build is on a critical path in ATLAS reprocessing
workflow.
ATLAS uses three Grids (each with a different interface) split in ten
“clouds”. Each cloud consists of a large computing center with tape data
storage (Tier-1 site) and associated 5–6 smaller computing centers (Tier-2
sites). There are also Tier-3 sites—these are physicist’s own computing
facilities at the university or the department.
Reprocessing improves the particle identification and measurements over the
first-pass processing at CERN, since the reprocessing uses enhanced software
and revised conditions. Figure 4 shows reprocessing workflow that includes
build of software and database releases. To make sure that the results are of
the highest quality obtainable, the full reprocessing campaigns on large
fractions of the total data sample require months of preparation—these are the
data that will be used in conferences and publications. As a result, most of
the time in full reprocessing campaigns is occupied with validation of
software and database releases, not actual running.
To give faster feedback to subdetector groups we are doing reprocessing of
smaller amounts of data, much quicker, to allow small modifications in
software and conditions to be applied to previously processed data or as a
contingency in case the Tier-0 ends up with a backlog of work. This is called
“fast” reprocessing. It is also possible to do reprocessing not of the raw
data but of the reconstructed data made during the last reprocessing campaign.
This is called ESD reprocessing. The fast and ESD reprocessing are also
performed on the Grid, in exactly the same way as “full” reprocessing.
## IV Database Access on the Grid
### IV.1 Database Release
None of Tier-0 solutions for scalable database access is available on the
Grid. To overcome scalability limitations of distributed database access 4 ,
we use the Database Release technology for deployment of the Conditions DB
data on the Grid. Similarly to ATLAS software release packaging for
distribution on the Grid, the Database Release integrates all necessary data
in a single tar file:
* •
the Geometry DB snapshot as an SQLite file,
* •
selected Conditions DB data as an SQLite file,
* •
corresponding Conditions DB POOL files and their POOL File Catalogue (Figure
5).
Years of experience resulted in continuous improvements in the Database
Release technology, which is used for ATLAS Monte Carlo simulations on the
Grid. In 2007 the Database Release technology was proposed as a backup for
database access in reprocessing at Tier-1 sites.
Figure 5: Database Release technology hides the complexity of Conditions DB
access (Fig. 1).
### IV.2 Challenges in Conditions DB Access
In addition to Database Releases, Conditions DB data are delivered to all ten
Tier-1 sites via continuous updates using Oracle Streams technology 5 . To
assure scalable database access during reprocessing we stress-tested Oracle
servers at the Tier-1 sites. As a result of stress-tests, we realized that the
original model, where reprocessing jobs would run only at Tier-1 sites and
access directly their Oracle servers, causes unnecessary restrictions to the
reprocessing throughput and most likely overload all Oracle servers when many
jobs start at once.
In the first reprocessing campaign, the main problem with Oracle overload was
exacerbated by additional scalability challenges. Frirst, the reprocessing
jobs for the cosmics data are five time faser than the baseline jobs
reconstructing the LHC collision data, resulting in a fivefold increase in the
Oracle load. Second, having data on Tier-1s disks increases Oracle load
sixfold (in contrast with the original model of reprocessing data from tapes).
Combined with other limitations, these factors required increase in
scalability by orders of magnitude. To overcome the Conditions DB scalability
challenges in reprocessing on the Grid, the Database Release technology,
originally developed as a backup, was selected as a baseline.
### IV.3 Conditions DB Release
To overcome scalability limitations in Oracle access on the Grid, the
following strategic decisions were made:
* •
read most of database-resident data from SQLite,
* •
optimize SQLite access and reduce volume of SQLite replicas,
* •
maintain access to Oracle (to assure a working backup technology, when
required).
As a result of these decisions, the Conditions DB Release technology fully
satisfies reprocessing requirements, which has been proven on a scale of one
billion database queries during two reprocessing campaigns of 0.5 PB of
single-beam and cosmics data on the Grid 3 . By enabling reprocessing at the
Tier-2 sites, the Conditions DB Release technology effectively doubled CPU
capacities at the BNL Tier-1 site during the first ATLAS reprocessing
campaign.
Conditions DB Release optimization for the second reprocessing campaign
eliminated bottlenecks experienced earlier at few Tier-1 sites with limited
local network capabilities. This Conditions DB Release was also used in user
analysis of the reprocessed data on the Grid and during a successful world-
wide LCG exercise called STEP’09. In a recent fast reprocessing campaign, the
Conditions DB Release integrated in a 1 GB dataset a slice of the Conditions
DB data from two-weeks of data taking during this summer. The dataset was
“frozen” to guarantee reproducibility of the reprocessing results. During the
latest ESD reprocessing campaign, further optimizations fit in a 1.4 GB volume
a slice of Conditions DB for the data taking period of $0.23\cdot 10^{7}$ s,
which is about one quarter of the nominal LHC year.
To automate Conditions DB Release build sequence, we are developing the db-on-
demand services (Figure 6). Recently these services were extended to support
new requirements of the fast and ESD reprocessing that included check for
missing interval-of-validity metadata.
Figure 6: Architecture of db-on-demand components automating Conditions DB
Release build.
### IV.4 Direct Oracle Access
For years ATLAS Monte Carlo simulations jobs used SQLite replicas for access
to simulated Conditions DB data. Recently Monte Carlo simulations are becoming
more realistic by using access to real Conditions DB data. This new type of
simulation jobs requires access to Oracle servers. More realistic simulations
provided an important new use case that validates our software for database
access in a production environment. First realistic simulations used the
software that has not yet been fully optimized for direct Oracle access. Thus
the experience collected during summer was mixed: finished jobs peaked above
5000 per day; however, during remote database access some jobs used 1 min of
CPU per hour, and others had transient segmentation faults and required
several attempts to finish. There is a room for significant performance
improvements with the software optimized for direct Oracle access 2 .
To prevent bottlenecks in direct Oracle access in a Grid computing
environment, we are developing a Pilot Query system for throttling job
submission on the Grid. Figure 7 shows the proof-of-principle demonstration of
the Pilot Query approach at the Tier-1 site in Lyon. Development of the next
generation Pilot Query system is now complete and ready for testing.
Figure 7: Throttling Oracle server load on the Grid: (a) first batch of 300
jobs submitted; (b) monitoring shows Oracle load is limited by the Pilot Query
technology as we set ATLAS application-specific Oracle load limit at 4 (c).
## V Database Access Strategy
Because Conditions DB access is crucial for operations with LHC data, we are
developing the system where a different technology can be used as a redundant
backup, in case of problems with a baseline technology. While direct access to
Oracle databases gives in theory the most flexible system, it is better to use
the technology that is best suited to each use case 6 :
* •
Monte Carlo simulations: continue using the DB Release;
* •
first-pass processing: continue using direct Oracle access at CERN;
* •
reprocessing: continue using the Conditions DB Release;
* •
user analysis:
* –
Grid jobs with large conditions data need: use the Frontier/Squid servers;
* –
local jobs with stable conditions data: use the Conditions DB Release.
Status of late-coming components for database access in user analysis is
described below.
### V.1 db-on-demand
In user analysis, automated db-on-demand services eliminate the need for a
central bookkeeping of database releases, since these will be created “on-
demand” (Figure 6). In order to have a user-friendly system, we will develop a
web interface with user authentication based on secure technology for database
access 7 , where each user would submit the request for a Conditions DB
Release including all data needed to analyse a given set of events.
### V.2 DoubleCheck
Frontier is a system for access to database-resident data via http protocol
used by the CDF and CMS experiments 8 . To achieve scalability, the system
deploys multiple layers of hardware and software between a database server and
a client: the Frontier Java servlet running within a Tomcat servlet container
and the Squid—a single-threaded http proxy/caching server. In 2006 ATLAS tests
done in collaboration with LCG found that Frontier does not maintain Squid
cache consistency, which does not guarantee that ATLAS jobs obtain
reproducible results in case of continuous updates to Conditions DB. In 2008
ATLAS resumed Frontier development and testing following recent breakthrough
in addressing the Frontier cache consistency problem 9 .
In CMS case the cache consistency solution works for queries to a single table
at a time. This does not work for ATLAS, as most our queries are for two
tables. Hence the name DoubleCheck is chosen for a solution to the cache
consistency problem developed for ATLAS. A major milestone in DoubleCheck
development was achieved in July—the proof-of-principle test demonstrated that
the LCG cache consistency solution developed for CMS can be extended to work
for ATLAS. Further tests validated DoubleCheck for our major use case—updates
of Conditions DB tables with the interval-of-validity metadata. DoubleCheck
guarantees Frontier cache consistency within 15 minutes, which is close to
delays observed in data propagation via Oracle Streams.
With no showstoppers in sight, ATLAS is now developing a plan and schedule for
deployment, validation, and stress-testing of Frontier/Squid for database
access in user analysis on the Grid.
## VI Conclusions
ATLAS has a well-defined strategy for redundant deployment of critical
database-resident data. For each use case the most suited technology is chosen
as a baseline:
* •
Oracle for the first-pass processing at Tier-0;
* •
Database Release for simulations and reprocessing on the Grid;
* •
Frontier for user analysis on the Grid.
The redundancy assures that an alternative technology can be used when
necessary.
ATLAS experience demonstrated that this strategy worked well as new
unanticipated requirements emerged. For example, the Conditions DB Release
technology, originally developed as a backup, was choosen as a baseline to
assure scalability of database access on the Grid. The baseline thechnology
fully satisfies the requirements of several reprocessing procedures developed
by the ATLAS collaboration. Steps are being taken to assure that Oracle can be
used as a backup in case of unexpected problems with the baseline thechnology.
Each major ATLAS use case is functionally covered by more than one of the
available technologies, so that we can achieve a redundant and robust data
access system, ready for the challenge of the first impact with LHC collision
data.
###### Acknowledgements.
I wish to thank all my collaborators who contributed to ATLAS database
operations activities. This work is supported in part by the U.S. Department
of Energy, Division of High Energy Physics, under Contract DE-AC02-06CH11357.
## References
* (1) G. Aad et al. [ATLAS Collaboration], “The ATLAS Experiment at the CERN Large Hadron Collider,” JINST 3, S08003 (2008).
* (2) A. Valassi et al. “COOL Performance Optimization and Scalability Tests,” CHEP09, to be published in J. Phys. Conf. Ser.
* (3) R. Chytracek et al. “POOL development status and production experience,” IEEE Trans. Nucl. Sci. 52, 2827 (2005).
* (4) R. Brun and F. Rademakers, “ROOT: An object oriented data analysis framework,” Nucl. Instrum. Meth. A 389, 81 (1997).
* (5) R. Basset et al. “Advanced Technologies for Scalable ATLAS Conditions Database Access on the Grid,” CHEP09, to be published in J. Phys. Conf. Ser.
* (6) A. Vaniachine, D. Malon and M. Vranicar, “Advanced technologies for distributed database services hyperinfrastructure,” Int. J. Mod. Phys. A 20, 3877 (2005).
* (7) A. V. Vaniachine and J. G. von der Schmitt, “Development, deployment and operations of ATLAS databases,” J. Phys. Conf. Ser. 119, 072031 (2008).
* (8) D. Barberis et al. “Strategy for Remote Access to ATLAS Database Resident Information,” August 2009, 9pp.
* (9) A. Vaniachine, S. Eckmann and W. Deng “Securing distributed data management with SHIELDS,” in the Proceedings of XXI International Symposium on Nuclear Electronics and Computing, NEC’2007, Varna, 10-17 Sep 2007, pp 446-449.
* (10) B. J. Blumenfeld et al. “CMS conditions data access using FroNTier,” J. Phys. Conf. Ser. 119, 072007 (2008).
* (11) D. Dykstra and L. Lueking, “Greatly improved cache update times for conditions data with Frontier/Squid,” CHEP09, preprint FERMILAB-CONF-09-231-CD-CMS, May 2009, 6pp.
|
arxiv-papers
| 2009-10-01T07:20:54 |
2024-09-04T02:49:05.571617
|
{
"license": "Public Domain",
"authors": "A. Vaniachine (for the ATLAS Collaboration)",
"submitter": "Alexandre Vaniachine",
"url": "https://arxiv.org/abs/0910.0097"
}
|
0910.0213
|
# Real-Time Scanning Charged-Particle Microscope Image Composition with
Correction of Drift
Petr Cizmar, András E. Vladár, and Michael T. Postek National Institute of
Standards and Technology111 Contribution of the National Institute of
Standards and Technology; not subject to copyright. Certain commercial
equipment is identified in this report to adequately describe the experimental
procedure. Such identification does not imply recommendation or endorsement by
the National Institute of Standards and Technology, nor does it imply that the
equipment identified is necessarily the best available for the purpose., 100
Bureau Drive, Gaithersburg, MD 20899
###### Abstract
In this article, a new scanning electron microscopy (SEM) image composition
technique is described, which can significantly reduce drift related image
corruptions. Drift-distortion commonly causes blur and distortions in the SEM
images. Such corruption ordinarily appears when conventional image-acquisition
methods, i.e. “slow scan” and “fast scan”, are applied. The damage is often
very significant; it may render images unusable for metrology applications,
especially, where sub-nanometer accuracy is required. The described correction
technique works with a large number of quickly taken frames, which are
properly aligned and then composed into a single image. Such image contains
much less noise than the individual frames, whilst the blur and deformation is
minimized. This technique also provides useful information about changes of
the sample position in time, which may be applied to investigate the drift
properties of the instrument without a need of additional equipment.
(a) (b)
Figure 1: Illustration of drift-distortion-related image corruption on
simulated cizmar-simim-scanning “slow scan” SEM images of a gold-on-carbon
sample. (a) Ideal, undistorted image. (b) Typical corrupted image.
## Introduction
Advances in fundamental nano-science, development of nano-materials, and
eventually manufacturing of nanometer-scale products all depend to some extent
on the capability to accurately and reproducibly measure dimensions,
properties, and performance characteristics at the nano-scale. Scanning
electron microscopes (SEMs) have been used in this application for many years
postek-advanced , postek-photomask . Since progress in nano-science and nano-
technology has been rather rapid recently, the dimensions of nano-structures
and nano-objects have shrunk significantly. Consequently, accurate SEM imaging
has been emphasized. The dimensions or distances have been measured from SEM
images or line-scans. Current imaging methods in SEM are often incapable of
achieving the desired accuracy, because the SEM images, at such high
magnifications, often suffer from drift-related distortion. In many cases, the
drift is significant and the SEM images exhibit deformations or blur. The same
problem is also experienced in other fields, e.g. scanning probe microscopies.
(a) (b)
Figure 2: Illustration of the composition (averaging) of displaced image
frames. (a) Composition of a few image frames, (b) composition of a large
number of frames, the image exhibits excessive blur.
(a) (b)
(c) (d)
Figure 3: Illustration of the cross-correlation displacement detection with
noise filtering. (a) Original frame, (b) noise-reduced frame, (c) unprocessed
cross-correlation function, and (d) correlation function of two noise-reduced
frames.
Several correction methods are being developed that compensate for these
effects. Some work on correcting the time-dependent drift distortions has been
performed in fields similar to scanning electron microscopy kawasaki-drift ,
mantooth , chang-wang , xu-li . A research in drift-distortion evaluation and
correction in SEMs has been published in sutton1 , sutton2 , sutton3 .
Technique described in these papers covers correction in images with slow
drift and low magnification. The overall imaging times are high, reaching tens
of minutes. The magnification does not exceed 10000. Technique for very fast
SEM or Scanning helium-ion beam microscopy, where signal-to-noise ratio (SNR)
may drop below $5\times 10^{-1}$, is still needed. This manuscript describes a
possible correction method based on composition of drift-distortion corrected
SEM images. The technique uses cross-correlation for displacement detection.
It not only provides more accurate images, but also sample position
information, which can be successfully employed in diagnostic applications.
The method is implemented as a software program in the C programming language.
With this approach, the solution is fast, multi-platform, multi-processor
capable, and moreover can be easily integrated into the majority of the SEM
software.
## Drift Effect on Images
In the SEM, the image is formed by scanning over the sample and acquisition of
an intensity value at each location on the sample corresponding to a pixel in
the image. The intensity value $\xi(\vec{r})$ depends on the landing position
of the electron beam (on the sample) $\vec{r}$. Most SEMs use the raster
pattern for scanning over the sample. Let the raster pattern be defined by the
time-dependent vector function:
$\displaystyle\vec{r}_{r}(t)$ $\displaystyle=$ $\displaystyle
M\left(x(t)\vec{e}_{x}+y(t)\vec{e}_{y}\right),$ (1) $\displaystyle t_{p}$
$\displaystyle=$ $\displaystyle t_{D}+t_{d},$ $\displaystyle y(t)$
$\displaystyle=$
$\displaystyle\left\lfloor\frac{t}{Xt_{p}+t_{j}}\right\rfloor,$ (2)
$\displaystyle x(t)$ $\displaystyle=$
$\displaystyle\left\lfloor\frac{t}{t_{p}}\right\rfloor-Xy(t),$ (3)
$\displaystyle 0\leq$ $\displaystyle t$ $\displaystyle\leq Y(Xt_{p}+t_{j}),$
where $t$ is time, $M$ is a constant of the length on sample corresponding to
a single-pixel step. $x$ and $y$ are column and row indexes in the SEM image.
$\vec{e_{x}}$ and $\vec{e_{y}}$ are the unit vectors in x- and y-direction,
$t_{D}$ is the dwell time of one pixel, $t_{d}$ is the dead time between two
pixels, $t_{j}$ is the time needed to move the beam to the beginning of the
new line. $\lfloor q\rfloor$ is a symbol for the ${\rm floor}(q)$ function as
used in programming languages. $X$ and $Y$ are the pixel-width and pixel-
height of the SEM image. These equation are in agreement with those published
in sutton1 .
(a) (b)
(c) (d)
(e) (f)
Figure 4: Demonstration of the method on real SEM images of the gold-on-carbon
resolution sample. Horizontal field-of-view is 441 nm for all images, (a)
single acquired image with the pixel dwell time 50 ns, (b) composition of 10
images, (c) composition of 20 images, (d) composition of 40 images, (e)
corrected composition of 120 images, and (f) Plain average of the same 120
images.
Imaging in the SEM may be defined as a relation between the intensity map of
the sample $\xi(\vec{r})$ and the SEM image $I(x,y)$:
$I(x(t),y(t))=K\xi(\vec{r}(t)).$ (4)
The relation between $I$ and $\xi$ may in practice be very general. For
simplicity, let $K$ be a constant in this paper, since this does not affect
generality of the described technique. In the ideal case
$\vec{r}(t)=\vec{r}_{r}(t)$; however, drift and space distortions are always
present in scanning microscopes and they often significantly affect the
position $\vec{r}$:
$\vec{r}(t)=\vec{r}_{r}(t)+\vec{D}_{d}(t)+\vec{D}_{s}(\vec{r}_{r}).$ (5)
The space distortion $\vec{D}_{s}$ is constant in time and may be simply
compensated for, when its function is known. This kind of distortion is caused
by non-linearities in deflection amplifiers and appears mostly at low
magnifications. On the other hand, the drift distortion $\vec{D}_{d}$ is
changing in time, its function is usually unknown, and it most significantly
affects the high-magnification images. The drift distortion may arise from
several sources; e.g. translational motion of the sample, tilt or deformation
of the electron-optical column, outer forces and vibrations, or temperature
expansion. High-magnification images are very sensitive to drift distortion,
since microscopic displacements, tilts, or temperature changes can easily
cause nanometer distortions and displacements, which can significantly impair
the SEM image and its usability for nanometer-scale measurements.
The drift-distortion function is generally unknown, however, since it
characterizes motion of physical bodies, it must be continuous and thus
square-integrable. Therefore, drift-distortion function may be expanded to
Fourier series:
$\displaystyle D_{cd}(t)$ $\displaystyle=$
$\displaystyle\sum\limits_{n=-\infty}^{\infty}c_{n}{\rm e}^{-{\rm i}nt},$ (6)
$\displaystyle\vec{D}_{d}$ $\displaystyle=$
$\displaystyle\Re(D_{cd})\vec{e}_{x}+\Im(D_{cd})\vec{e}_{y},$ (7)
$\displaystyle U$ $\displaystyle\propto$
$\displaystyle\sum_{n=-\infty}^{\infty}c_{n}^{2}n^{2},$ (8)
where $c_{n}$ are the (complex) Fourier coefficients, $U$ is the overall
energy of the drifting system. Since $U$ is limited, for high $n$ the
coefficients $c_{n}$ must be nearing zero. In practice, $c_{n}$ for
frequencies higher than 200 Hz correspond to noise only and are negligible.
This approximate number is based on experimental values. Therefore, the
$D_{cd}(t)$ can be written:
$D_{cd}(t)\approx\sum\limits_{n=-N}^{N}c_{n}{\rm e}^{-{\rm i}nt},\\\ $ (9)
where $N$ represents the highest significant angular frequency.
Figure 5: Illustration of the image composition procedure. Boxes denote
entities like images or numbers, arrows indicate processes. The letters
(a)—(m) represent individual steps in the procedure.
## SEM Imaging Methods
In the SEM, the acquired intensity signal always contains noise. The intensity
function is thus a superposition of real signal and noise:
$\xi(\vec{r},t)=\xi_{s}(\vec{r})+\xi_{n}(t),$ (10)
where $\xi_{s}$ is the position-dependent real signal and $\xi_{n}$ is the
time-dependent noise. This noise is a superposition of Poisson noise,
originating from the electron source and the secondary emission, noise
originating from the amplifier and electronics, quantization-error noise, etc.
Due to the central limit theorem, it is legitimate to suppose that the mean
value of this noise is zero:
$<\xi_{n}(t)>=0.$ (11)
In order to obtain a SEM image with a desired level of noise, the overall
pixel dwell-time $t_{D}$ must be sufficiently high. Unfortunately, the
electron yield is usually low and the $t_{D}$ must often be set to times
ranging from tens to several hundreds of $\mu$s.
There are two common techniques to achieve this in the SEM, i.e “slow-scan”
and “fast scan”. With the “slow-scan” method, the image is acquired within a
single scan. The acquired value is in this case:
$\displaystyle I(x(t_{0}),y(t_{0}))$ $\displaystyle=$
$\displaystyle\frac{K}{t_{D}}\int\limits_{t_{0}}^{t_{0}+t_{D}}\left(\xi_{s}(\vec{r}(t))+\xi_{n}(t)\right){\rm
d}\kern-0.56905ptt,$ (13)
$\displaystyle\int\limits_{t_{0}}^{t_{0}+t_{D}}\xi_{n}(t){\rm
d}\kern-0.56905ptt\approx 0,$ $\displaystyle\vec{r}(t)$ $\displaystyle=$
$\displaystyle\vec{r}_{r}(t_{0})+\vec{D}_{s}(\vec{r}_{r}(t_{0}))+\vec{D}_{d}(t)=$
(14) $\displaystyle=$ $\displaystyle{\rm const}+\vec{D}_{d}(t).$
The noise is reduced by long integration as shown in Eq. (13). Required level
of noise determines the dwell-time $t_{D}$. Since the desired beam position
does not change during the acquisition of a single pixel, the only changing
component of the position (Eq. (5)) is the $D_{d}$ as stated in Eq. (14). In
practice, the drift-distortion-related displacements are not significant
between two pixels, because the time $t_{p}$ is still not long enough. However
the line-acquisition time $Xt_{p}+t_{j}$ may already be much larger than the
period of the highest frequencies. Therefore, if the “slow-scan” technique is
employed, the line-scans and thus also the images may be significantly
distorted (See Fig. 1). The distortion is time-dependent, and thus different
for each line and image and cannot be corrected, unless additional information
about the drift-distortion function $\vec{D}_{d}(t)$ is provided.
The other common imaging method in SEM is the “fast-scan”. The image is
composed of multiple ($N_{i}$) frames, for which averaging is the mostly
applied technique. The frames are acquired with the lowest possible pixel-
dwell time $t_{D}$. Since, in practice, the $t_{D}$ can be set to as low as 25
ns, the change in the drift-distortion function during this time is negligible
and the integral (13) can be approximated as constant. The image pixel value
is then an average of corresponding frame-pixel values:
$\displaystyle I_{k}(x(t_{0}),y(t_{0}))$ $\displaystyle=$ $\displaystyle
K\xi_{s}(\vec{r}(t_{0}+kt_{f}))+$ (15) $\displaystyle+$ $\displaystyle
K\xi_{n}(t_{0}+kt_{f}),$ $\displaystyle I(x,y)$ $\displaystyle=$
$\displaystyle\frac{1}{N_{i}}\sum_{k=0}^{N_{i}}I_{k}(x,y).$ (16)
$\displaystyle t_{f}$ $\displaystyle=$ $\displaystyle Y(Xt_{p}+t_{j})+t_{jj},$
(17)
$t_{f}$ is a time period between beginnings of acquisition of two following
frames, $t_{jj}$ is the dead time between the end of acquisition of one frame
and beginning of the next one. Considering Eq. (11), the higher $N_{i}$, the
lower noise level is present in the composed image. The required noise-level
thus determines the number of composed frames $N_{i}$. For high $N_{i}$:
$\sum_{k=0}^{N_{i}-1}\xi_{n}(t_{0}+kt_{f})\approx 0.\\\ $ (18)
Because the scanning raster pattern is constant for all frames,
$\vec{r}_{r}(t_{0}+kt_{f})=\vec{r}(t_{0}).\\\ $ (19)
Eq. (15) may be expanded:
$\displaystyle I(x(t_{0}),y(t_{0}))$ $\displaystyle=$
$\displaystyle\frac{K}{N_{i}}\sum_{k=0}^{N_{i}-1}\xi_{s}[\vec{r}_{r}(t_{0})+$
(20) $\displaystyle+$
$\displaystyle\vec{D}_{s}(\vec{r}_{r}(t_{0}))+\vec{D}_{d}(t_{0}+kt_{f})].$
With current SEMs, the frame-acquisition time $t_{f}$ can be much lower than
the period of even the highest drift-distortion frequencies. The drift-
distortion within the single-frame acquisition time is then minimal. However,
it becomes significant during acquisition of the whole image, especially, when
the dead times $t_{jj}$ are high, which is the case even with many current
instruments. Considering drift effects negligible within a single frame, the
drift affects all image pixels equally. Point-spread function (PSF) may be
constructed from the function $\vec{D}_{d}(t)$. Such a PSF consists of
multiple separate points, which produces blurry images similar to Fig. 2. The
PSF can not be used for deconvolution-based drift-distortion correction, since
it is unknown like the $\vec{D}_{d}$ itself.
## Inter-Frame Drift-Distortion Correction
The “fast-scan” method may be significantly improved using drift-distortion
correction. This is possible, when the frames are taken during short enough
times and $Y(Xt_{p}+t_{j})\ll 2\pi/N$. Since the space-distortion
$\vec{D}_{s}$ is much less pronounced and much smaller that the drift-
distortion $\vec{D}_{d}$ at very high magnifications, it will be neglected
from now on. The Eq. (20) then becomes:
$\displaystyle I(x,y)$ $\displaystyle=$
$\displaystyle\frac{K}{N_{i}}\sum_{k=0}^{N_{i}-1}\xi_{s}[\vec{r}_{r}+\vec{D}_{dk}],$
(21) $\displaystyle\vec{D}_{dk}$ $\displaystyle=$
$\displaystyle\vec{D}_{d}(t_{0}+kt_{f}).$ (22)
The image is in this case the mean value of $N_{i}$ displaced images.
Fortunately, under certain conditions, it is possible to find the displacement
vectors of the images, which are equal to the drift-distortion values
$\vec{D}_{dk}$. These vectors then can be compensated for and thus the drift-
distortion can be corrected. One possible approach is a cross-correlation-
based displacement detection, which is used in this work. However, choice of
the method determines the requirements, which may include low-enough noise,
well-pronounced image features, etc. The complete set of requirements will be
addressed in future publications.
## Cross-Correlation with Noise Reduction
If two image frames $f$ and $g$ contain similar features at different
positions, the cross-correlation integral has a large value at the vector
corresponding to the displacement of the features. The SEM digital image
frames are in this application represented by discrete two-dimensional real
functions. Therefore, the two-dimensional discrete cross-correlation is
applied.
If the image frames are noisy, the peak in the cross-correlation function
becomes overridden by numerous other peaks, corresponding to random
correlation of noise (Fig. 3c) This often makes finding the displacement
vector impossible. This could be tackled by low-pass frequency filtering. This
can be performed in the frequency domain. The cut-off frequency is determined
by the filter-radius $R$. Although, this filter significantly wipes out all
high-frequency features from the image, and therefore it is inapplicable for
general reduction of noise, it still works very well for the total-maximum
search of a two-dimensional function. The maximum of the cross-correlation
function becomes higher above the background and it is easy to find it. (See
Fig. 3d).
According to the cross-correlation theorem papoulis-cc , the cross-correlation
can be calculated using the Fourier transform. The widely used FFTW3 frigo-
fftw3 algorithm is applied for Fourier transform calculations. In order to
speed up the calculation, the cross-correlation is combined with the frequency
filtering. (See Fig. 5e.) The conjugation and multiplication is done only in
the central circle (the lowest frequencies) of the Fourier image, while the
rest is zeroed:
$\displaystyle J$ $\displaystyle=$ $\displaystyle\left\\{\begin{array}[]{l
l}[{F}(f(\vec{r}))]^{*}\cdot{F}(g(\vec{r}))&\quad\mbox{if $|\vec{r}|\leq
R$,}\\\ 0&\quad\mbox{otherwise.}\end{array}\right.$ (25) $\displaystyle I$
$\displaystyle=$ $\displaystyle F^{-1}(J)$ (26)
Then, the inverse Fourier transform is applied and the noise-reduced cross-
correlation image is obtained. For every pair of frames, only two forward and
one inverse Fourier transforms are needed, while one of them is also part of
the next pair.
(a) (b)
(c)
Figure 6: The results of the corrected composition compared with the
uncorrected “slow scan” demonstrated on a real SEM image of a gold-on-carbon
resolution sample. Horizontal field of view is 422 nm. (a) Slow-scan image
with the pixel dwell time 300 $\mu\rm{s}$. (b) The result with the new
technique. (c) Difference between the two images.
## Sub-Pixel-Accuracy Displacement Detection
In order to find the displacement vectors, the maximum of the cross-
correlation is searched for. Since this method is sample-dependent, there may
appear problems finding the displacement vector corresponding to the
transition. For example, if the image contains periodic features, there are
several maxima in the cross-correlation image. In case of large blur or noise,
the peak may be very wide and the uncertainty in the position of the maximum
may be very high.
Plain search for maximum provides just single-pixel accuracy of the
displacement-vector. The peak may be interpolated by a suitable function,
which enables for calculating the displacement vectors with sub-pixel
accuracy. The third-order two-dimensional polynomial function with
coefficients $k_{0}$—$k_{9}$ was chosen as a suitable function for the
interpolation. The coefficients may be found by polynomial fitting using the
common least squares method, which is widely used in similar applications. The
Cholesky factorization gentle-cholesky is applied to speed the calculation.
The maximum is then numerically found; its position is the searched
displacement-vector with sub-pixel accuracy. However, the exact accuracy-value
calculation is not covered in this article.
Since the displacement-vectors are calculated with sub-pixel accuracy, it is
reasonable to shift the images with sub-pixel accuracy as well. Let the
displacement-vector be $\vec{s}=s_{x}\vec{e}_{x}+s_{y}\vec{e}_{y}$. The shift
can be performed with sub-pixel accuracy using the Fourier Transform, because
$F_{s}(\omega,\phi)=F(\omega,\phi)\exp(-i\omega x_{s}-i\phi y_{s}).$ (27)
The obtained displacement-vectors with the sub-pixel accuracy obtained in the
previous step are used for shifting, which compensates for the inter-frame
drift-distortion. The corrected images are then pixel-by-pixel averaged
together, however, other composition methods, e.g. median filtering, can be
also used.
(a) (b)
Figure 7: Example of a drift tracking. Subsequent displacement vectors are
displayed in the graph. (a) Drift-distortion of a sample placed on a fixed
stage. Frames were taken every 60 s. (b) Drift distortion of the sample used
in Fig. 4. Frames were taken every 1 s.
## Composition Procedure
The procedure of the method is illustrated in Fig. 5. The process starts with
acquisition of two frames; A and B (Figs. 5a and b). In order to minimize the
influence drift-distortion on the frames, these must be taken with minimum
pixel-dwell time possible, which is usually limited by the instrument. Both
frames are then converted into the frequency domain (Figs. 5c and d). These
frequency-domain images are then conjugated and multiplied, which is combined
with frequency filtering (Fig. 5e). This results in a cross-correlation image
in the frequency domain. Then, the cross-correlation image in space domain is
obtained (Fig. 5f). The cross-correlation image is interpolated by the third-
order two-dimensional polynomial function. This enables finding the
displacement vectors with sub-pixel accuracy (Fig. 5g). The coordinates of the
maximum denote the found displacement vector (Fig. 5h). This displacement
vector is used to shift the image B in its frequency-domain representation,
which enables the sub-pixel accuracy alignment (Fig. 5i). Shifted image B is
converted into the space domain (Fig. 5j) and averaged with the image A (Fig.
5k). Image A has (except in the first iteration) higher information weight, as
it already represents a sum of multiple image frames. If the SNR is not
sufficient (Fig. 5l), the composed image is copied into the frame A (Fig. 5m)
and a new frame B is acquired. The process then repeats until the SNR is
sufficient, or the software runs out of frames.
## Results
The discussed image-composition method was tested on gold-on-carbon resolution
images (Fig. 4) and on artificial images. A Mac Pro computer with two dual-
core Intel Xeon Central Processor Units (CPUs) and 4 GB of Random Access
Memory (RAM) was employed for the calculations. The 64-bit edition of Gentoo
Linux Operating System (OS) was installed on the computer.
For the real images, the pixel dwell time was set to the lowest instrument
setting (50 ns). The frame-rate was 1 frame per second, which was also the
fastest setting. Single acquired frame (Fig. 4a) was very noisy; only the most
prominent features (about 200 nm in diameter) were clearly visible. A
composition of 10 frames (Fig. 4b) already contained visible features in the
background (about 20 nm in diameter); some inner structure of the grains
(sized about 5-10 nm) became observable. Compositions of 20 (Fig. 4c) and 40
(Fig. 4d) frames embodied some more detail. The inner structure of the grains,
as well as all the background features are clearly visible. The composition of
120 frames from the new composition method (Fig. 4e) and the existing
averaging method (Fig. 4f) are included for a comparison of the new and the
traditional averaging methods. The traditionally averaged image was
significantly more blurred than the image composed using the described method.
Both images have similar SNR. The final image (Fig. 4e) exhibits low noise and
high detail whilst preserving the shapes and dimensions.
The slow-scan image is displayed in Fig. 6a. The pixel dwell time was 300
$\mu$s, which is a common choice for such imaging. The image looked clean and
noise free on inspection; however, the difference between the slow-scan image
and the image obtained with the new composition method, as shown in Fig. 6c
exhibited a significant difference, which is believed to be associated with
the distortions in the slow-scan image. The image was acquired for 6000 times
longer time than the image in Fig. 4a and the drift-distortion affected the
shapes significantly.
The obtained sequences of displacement-vectors were also used to track the
sample position with respect to the beam (Fig. 7). This information was very
usable for drift investigation. In the case of the sample of the fixed stage,
displayed in Fig. 7a, there was a roughly 70-nm-long straight start-up drift
followed by a periodical circular drift, which was caused by periodical
temperature changes inside the electron-optical column.
On the other hand, the Fig. 7b shows the displacement-vector sequence
associated with a typical drift in the SEM. Assuming the obtained curve to be
associated with a relative trajectory of physical bodies with a position noise
superimposed to it, it is possible to estimate the accuracy of the
displacement-vector searching to be approximately 0.5 nm, which corresponds to
0.5 pixels.
Speed of calculation is another important aspect of this method and its
implementation. It was not possible to try the technique in a real-time
imaging application, since this would require integration of the technique
into the SEM software, which was not possible. However, the calculation times
were measured and on 512$\times$512-pixel-large images, a single search for a
displacement-vector took in average 0.08 s, while the times of individual
frame compositions are very consistent.
## Conclusion
The technique is implemented as a computer program written in C language,
which is the advantage due to its optimization possibilities and ease of
possible incorporation into SEM software. On reasonably fast computers, this
program is capable of real-time processing. The algorithm is well
distributable, thus, it is suitable for running on computer clusters or multi-
core or multi-processor environments, including graphics processing units
(GPUs). The method has been verified on real and artificial SEM images
demonstrating its usability for true-shape imaging and for drift investigation
applications. It was also tested for the calculation speed, which is high
enough for real-time processing, when integrated into the SEM software.
Since the power of this method strongly depends on many factors, e.g. sample-
feature shapes, noise, image size, etc., its limits should be throughly
examined. Calculation of accuracy and confidence intervals, influence of
sample charging and contamination are still under investigation. These issues
will be addressed in future works on this project.
## References
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* [5] T. Kawasaki, H. Utsuro, Y. Takai, and R. Shimizu. Evaluation of image drift correction by three-dimensional Fourier analysis. Journal of Electron Microscopy, 48(1):35–37, 1999.
* [6] B. A. Mantooth, Z. J. Donhauser, K. F. Kelly, and P. S. Weiss. Cross-correlation image tracking for drift correction and adsorbate analysis. Review of Scientific Instruments, 73(2, Part 1):313–317, FEB 2002.
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* [9] M. T. Postek, A. E. Vladar, M. H. Bennett, T. Rice, and R. Knowles. Photomask dimensional metrology in the scanning electron microscope, part II: High-pressure/environmental scanning electron microscope. Journal of Microlithography Microfabrication and Microsystems, 3(2):224–231, APR 2004.
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|
arxiv-papers
| 2009-10-01T16:33:44 |
2024-09-04T02:49:05.579922
|
{
"license": "Public Domain",
"authors": "Petr Cizmar, Andras E. Vladar, and Michael T. Postek",
"submitter": "Petr Cizmar",
"url": "https://arxiv.org/abs/0910.0213"
}
|
0910.0281
|
# Hypergraphic LP Relaxations for Steiner Trees
Deeparnab Chakrabarty Jochen Könemann David Pritchard
(University of Waterloo 111Supported by NSERC grant no. 288340 and by an Early
Research Award. Email: (deepc, jochen, dagpritc @uwaterloo.ca))
###### Abstract
We investigate hypergraphic LP relaxations for the Steiner tree problem,
primarily the partition LP relaxation introduced by Könemann et al. [Math.
Programming, 2009]. Specifically, we are interested in proving upper bounds on
the integrality gap of this LP, and studying its relation to other linear
relaxations. Our results are the following.
Structural results: We extend the technique of uncrossing, usually applied to
families of sets, to families of partitions. As a consequence we show that any
basic feasible solution to the partition LP formulation has sparse support.
Although the number of variables could be exponential, the number of positive
variables is at most the number of terminals.
Relations with other relaxations: We show the equivalence of the partition LP
relaxation with other known hypergraphic relaxations. We also show that these
hypergraphic relaxations are equivalent to the well studied bidirected cut
relaxation, if the instance is quasibipartite.
Integrality gap upper bounds: We show an upper bound of $\sqrt{3}\doteq 1.729$
on the integrality gap of these hypergraph relaxations in general graphs. In
the special case of uniformly quasibipartite instances, we show an improved
upper bound of $73/60\doteq 1.216$. By our equivalence theorem, the latter
result implies an improved upper bound for the bidirected cut relaxation as
well.
## 1 Introduction
In the Steiner tree problem, we are given an undirected graph $G=(V,E)$, non-
negative costs $c_{e}$ for all edges $e\in E$, and a set of terminal vertices
$R\subseteq V$. The goal is to find a minimum-cost tree $T$ spanning $R$, and
possibly some Steiner vertices from $V\setminus R$. We can assume that the
graph is complete and that the costs induce a metric. The problem takes a
central place in the theory of combinatorial optimization and has numerous
practical applications. Since the Steiner tree problem is
$\mathsf{NP}$-hard222Chlebík and Chlebíková show that no
$(96/95-\epsilon)$-approximation algorithm can exist for any positive
$\epsilon$ unless $\mathsf{P}$=$\mathsf{NP}$ [5]. we are interested in
approximation algorithms for it. The best published approximation algorithm
for the Steiner tree problem is due to Robins and Zelikovsky [29], which for
any fixed $\epsilon>0$, achieves a performance ratio of $1+\frac{\ln
3}{2}+\epsilon\doteq 1.55$ in polynomial time; an improvement is currently in
press [3], see also Remark 1.1.
In this paper, we study linear programming (LP) relaxations for the Steiner
tree problem, and their properties. Numerous such formulations are known
(e.g., see [1, 7, 8, 10, 11, 18, 24, 25, 35, 36]), and their study has led to
impressive running time improvements for integer programming based methods.
Despite the significant body of work in this area, none of the known
relaxations is known to exhibit an integrality gap provably
smaller333Achieving an integrality gap of $2$ is relatively easy for most
relaxations by showing that the minimum spanning tree restricted on the
terminals is within a factor $2$ of the LP. than $2$. The integrality gap of a
relaxation is the maximum ratio of the cost of integral and fractional optima,
over all instances. It is commonly regarded as a measure of strength of a
formulation. One of the contributions of this paper are improved bounds on the
integrality gap for a number of Steiner tree LP relaxations.
A Steiner tree relaxation of particular interest is the bidirected cut
relaxation [11, 36] (precise definitions will follow in Section 1.2). This
relaxation has a flow formulation using $O(|E||R|)$ variables and constraints,
which is much more compact than the other relaxations we study. Also, it is
also widely believed to have an integrality gap significantly smaller than $2$
(e.g., see [4, 28, 34]). The largest lower bound on the integrality gap known
is $8/7$ (by Martin Skutella, reported in [23]), and Chakrabarty et al. [4]
prove an upper bound of $4/3$ in so called quasi-bipartite instances (where
Steiner vertices form an independent set).
Another class of formulations are the so called hypergraphic LP relaxations
for the Steiner tree problem. These relaxations are inspired by the
observation that the minimum Steiner tree problem can be encoded as a minimum
cost hyper-spanning tree (see Section 1.2.2) of a certain hypergraph on the
terminals. They are known to be stronger than the bidirected cut relaxation
[26], and it is therefore natural to try to use them to get better
approximation algorithms, by drawing on the large corpus of known LP
techniques. In this paper, we focus on one hypergraphic LP in particular: the
partition LP of Könemann et al. [23].
### 1.1 Our Results and Techniques
There are three classes of results in this paper: structural results,
equivalence results, and integrality gap upper bounds.
Structural results, Section 2: We extend the powerful technique of uncrossing,
traditionally applied to families of sets, to families of partitions. Set
uncrossing has been very successful in obtaining exact and approximate
algorithms for a variety of problems (for instance, [13, 21, 31]). Using
partition uncrossing, we show that any basic feasible solution to the
partition LP has at most $(|R|-1)$ positive variables (even though it can have
an exponentially large number of variables and constraints).
Equivalence results, Section 3: In addition to the partition LP, two other
hypergraphic LPs have been studied before: one based on _subtour elimination_
due to Warme [35], and a _directed hypergraph relaxation_ of Polzin and
Vahdati Daneshmand [26]; these two are known to be equivalent [26]. We prove
that in fact _all three hypergraphic relaxations are equivalent_ (that is,
they have the same objective value for any Steiner tree instance). We give two
proofs (for completeness and to demonstrate our new techniques), one showing
the equivalence of the partition LP and the subtour LP via partition
uncrossing, and one showing the equivalence of the partition LP to the
directed LP via hypergraph orientation results of Frank et al. [14].
We also show that, on quasibipartite instances, the hypergraphic and the
bidirected cut LP relaxations are equivalent. We find this surprising for the
following reasons. Firstly, some instances are known where the hypergraph
relaxations is strictly stronger than the bidirected cut relaxation [26].
Secondly, the bidirected cut relaxations seems to resist uncrossing
techniques; e.g. even in quasi-bipartite graphs extreme points for bidirected
cut can have as many as $\Omega(|V|^{2})$ positive variables [27, Sec. 4.9].
Thirdly, the known approaches to exploiting the bidirected cut relaxation
(mostly primal-dual and local search algorithms [28, 4]) are very different
from the combinatorial hypergraphic algorithms for the Steiner tree problem
(almost all of them employ greedy strategies). In short, there is no
qualitative similarity to suggest why the two relaxations should be
equivalent! We believe a better understanding of the bidirected cut relaxation
is important because it is central in theory _and_ practical for
implementation.
Improved integrality gap upper bounds, Section 4: For _uniformly
quasibipartite instances_ (quasibipartite instances where for each Steiner
vertex, all incident edges have the same cost), we show that the integrality
gap of the hypergraphic LP relaxations is upper bounded by $73/60\doteq
1.216$. Our proof uses the approximation algorithm of Gröpl et al. [20] which
achieves the same ratio with respect to the (integral) optimum. We show, via a
simple dual fitting argument, that this ratio is also valid with respect to
the LP value. To the best of our knowledge this is the only nontrivial class
of instances where the best currently known approximation ratio and
integrality gap upper bound are the same.
For general graphs, we give simple upper bounds of $2\sqrt{2}-1\doteq 1.83$
and $\sqrt{3}\doteq 1.729$ on the integrality gap of the hypergraph
relaxation. Call a graph gainless if the minimum spanning tree of the
terminals is the optimal Steiner tree. To obtain these integrality gap upper
bounds, we use the following key property of the hypergraphic relaxation which
was implicit in [23]: on gainless instances (instances where the optimum
terminal spanning tree is the optimal Steiner tree), the LP value equals the
minimum spanning tree and the integrality gap is 1. Such a theorem was known
for quasibipartite instances and the bidirected cut relaxation (implicitly in
[28], explicitly in [4]); we extend techniques of [4] to obtain improved
integrality gaps on all instances.
###### Remark 1.1.
The recent independent work of Byrka et al. [3], which gives an improved
approximation for Steiner trees in general graphs, also shows an integrality
gap bound of $1.55$ on the hypergraphic directed cut LP. This is stronger than
our integrality gap bounds and was obtained prior to the completion of our
paper; yet we include our bounds because they are obtained using fairly
different methods which might be of independent interest in certain settings.
The proof in [3] can be easily modified to show an integrality gap upper bound
of $1.28$ in quasibipartite instances. Then using our equivalence result, we
get an integrality gap upper bound of $1.28$ for the bidirected cut relaxation
on quasibipartite instances, improving the previous best of $4/3$.
### 1.2 Bidirected Cut and Hypergraphic Relaxations
#### 1.2.1 The Bidirected Cut Relaxation
The first bidirected LP was given by Edmonds [11] as an exact formulation for
the spanning tree problem. Wong [36] later extended this to obtain the
bidirected cut relaxation for the Steiner tree problem, and gave a dual ascent
heuristic based on the relaxation. For this relaxation, introduce two arcs
$(u,v)$ and $(v,u)$ for each edge $uv\in E$, and let both of their costs be
$c_{uv}$. Fix an arbitrary terminal $r\in R$ as the root. Call a subset
$U\subseteq V$ valid if it contains a terminal but not the root, and let
$\mathrm{valid}(V)$ be the family of all valid sets. Clearly, the in-tree
rooted at $r$ (the directed tree with all vertices but the root having out-
degree exactly $1$) of a Steiner tree $T$ must have at least one arc with tail
in $U$ and head outside $U$, for all valid $U$. This leads to the bidirected
cut relaxation ($\mathcal{B}$) (shown in Figure 1 on page 1 with dual) which
has a variable for each arc $a\in A$, and a constraint for every valid set
$U$. Here and later, $\delta^{\mathrm{out}}(U)$ denotes the set of arcs in $A$
whose tail is in $U$ and whose head lies in $V\setminus U$. When there are no
Steiner vertices, Edmonds’ work [11] implies this relaxation is exact.
$\displaystyle\min\sum_{a\in A}c_{a}x_{a}:\quad$ $\displaystyle
x\in\mathbf{R}^{A}_{\geq 0}$ ($\mathcal{B}$)
$\displaystyle\sum_{a\in\delta^{\mathrm{out}}(U)}x_{a}\geq 1,\quad$
$\displaystyle\forall U\in{\mathrm{valid}(V)}$ (1)
$\displaystyle\max\sum_{U}z_{U}:\quad$ $\displaystyle
z\in\mathbf{R}^{\mathrm{valid}(V)}_{\geq 0}$ ($\mathcal{B}_{D}$)
$\displaystyle\sum_{U:a\in\delta^{\mathrm{out}}(U)}z_{U}\leq c_{a},\quad$
$\displaystyle\forall a\in A$ (2)
Figure 1: The bidirected cut relaxation ($\mathcal{B}$) and its dual
($\mathcal{B}_{D}$).
Goemans & Myung [18] made significant progress in understanding the LP, by
showing that the bidirected cut LP has the same value independent of which
terminal is chosen as the root, and by showing that a whole “catalogue” of
very different-looking LPs also has the same value; later Goemans [17] showed
that if the graph is series-parallel, the relaxation is exact. Rajagopalan and
Vazirani [28] were the first to show a non-trivial integrality gap upper bound
of $3/2$ on quasibipartite graphs; this was subsequently improved to $4/3$ by
Chakrabarty et al. [4], who gave another alternate formulation for
($\mathcal{B}$).
#### 1.2.2 Hypergraphic Relaxations
Given a Steiner tree $T$, a _full component_ of $T$ is a maximal subtree of
$T$ all of whose leaves are terminals and all of whose internal nodes are
Steiner nodes. The edge set of any Steiner tree can be partitioned in a unique
way into full components by splitting at internal terminals; see Figure 2 on
page 2 for an example.
Figure 2: Black nodes are terminals and white nodes are Steiner nodes. Left: a
Steiner tree for this instance. Middle: the Steiner tree’s edges are
partitioned into full components; there are four full components. Right: the
hyperedges corresponding to these full components.
Let $\mathcal{K}$ be the set of all nonempty subsets of terminals
(_hyperedges_). We associate with each $K\in\mathcal{K}$ a fixed full
component spanning the terminals in $K$, and let $C_{K}$ be its cost444We
choose the minimum cost full component if there are many. If there is no full
component spanning $K$, we let $C_{K}$ be infinity. Such a minimum cost
component can be found in polynomial time, if $|K|$ is a constant.. The
problem of finding a minimum-cost Steiner tree spanning $R$ now reduces to
that of finding a minimum-cost hyper-spanning tree in the hypergraph
$(R,\mathcal{K})$.
Spanning trees in (normal) graphs are well understood and there are many
different exact LP relaxations for this problem. These exact LP relaxations
for spanning trees in graphs inspire the hypergraphic relaxations for the
Steiner tree problem. Such relaxations have a variable $x_{K}$ for
every555Observe that there could be exponentially many hyperedges. This
computational issue is circumvented by considering hyperedges of size at most
$r$, for some constant $r$. By a result of Borchers and Du [2], this leads to
only a $(1+\Theta(1/\log r))$ factor increase in the optimal Steiner tree
cost. $K\in\mathcal{K}$, and the different relaxations are based on the
constraints used to capture a hyper-spanning tree, just as constraints on
edges are used to capture a spanning tree in a graph.
The oldest hypergraphic LP relaxation is the subtour LP introduced by Warme
[35] which is inspired by Edmonds’ subtour elimination LP relaxation [12] for
the spanning tree polytope. This LP relaxation uses the fact that there are no
hypercycles in a hyper-spanning tree, and that it is spanning. More formally,
let $\rho(X):=\max(0,|X|-1)$ be the rank of a set $X$ of vertices. Then a sub-
hypergraph $(R,\mathcal{K}^{\prime})$ is a hyper-spanning tree iff
$\sum_{K\in\mathcal{K}^{\prime}}\rho(K)=\rho(R)$ and
$\sum_{K\in\mathcal{K}^{\prime}}\rho(K\cap S)\leq\rho(S)$ for every subset $S$
of $R$. The corresponding LP relaxation, denoted below as ($\mathcal{S}$), is
called the subtour elimination LP relaxation.
$\displaystyle\min\Big{\\{}\sum_{K\in\mathcal{K}}C_{K}x_{K}:~{}$
$\displaystyle x\in\mathbf{R}^{\mathcal{K}}_{\geq
0},~{}\sum_{K\in\mathcal{K}}x_{K}\rho(K)=\rho(R),$ ($\mathcal{S}$)
$\displaystyle\sum_{K\in\mathcal{K}}x_{K}\rho(K\cap S)\leq\rho(S),~{}\forall
S\subset R\Big{\\}}$
Warme showed that if the maximum hyperedge size $r$ is bounded by a constant,
the LP can be solved in polynomial time.
The next hypergraphic LP introduced for Steiner tree was a directed hypergraph
formulation ($\mathcal{D}$), introduced by Polzin and Vahdati Daneshmand [26],
and inspired by the bidirected cut relaxation. Given a full component $K$ and
a terminal $i\in K$, let $K^{i}$ denote the arborescence obtained by directing
all the edges of $K$ towards $i$. Think of this as directing the hyperedge $K$
towards $i$ to get the directed hyperedge $K^{i}$. Vertex $i$ is called the
_head_ of $K^{i}$ while the terminals in $K\setminus i$ are the _tails_ of
$K$. The cost of each directed hyperedge $K^{i}$ is the cost of the
corresponding undirected hyperedge $K$. In the directed hypergraph
formulation, there is a variable $x_{K^{i}}$ for every directed hyperedge
$K^{i}$. As in the bidirected cut relaxation, there is a vertex $r\in R$ which
is a root, and as described above, a subset $U\subseteq R$ of terminals is
valid if it does not contain the root but contains at least one vertex in $R$.
We let $\Delta^{\mbox{\scriptsize{$\mathrm{out}$}}}(U)$ be the set of directed
full components coming out of $U$, that is all $K^{i}$ such that $U\cap
K\neq\varnothing$ but $i\notin U$. Let $\overrightarrow{\mathcal{K}}$ be the
set of all directed hyperedges. We show the directed hypergraph relaxation and
its dual in Figure 3.
$\displaystyle\min\Big{\\{}\sum_{K\in\mathcal{K},i\in K}C_{K}x_{K^{i}}:$
$\displaystyle\,\,x\in\mathbf{R}^{\overrightarrow{\mathcal{K}}}_{\geq 0}$
($\mathcal{D}$)
$\displaystyle\sum_{K^{i}\in\Delta^{\mbox{\scriptsize{$\mathrm{out}$}}}(U)}x_{K^{i}}\geq
1,\quad$ $\displaystyle\forall\mbox{ valid }~{}U\subseteq R\Big{\\}}$ (3)
$\displaystyle\max\Big{\\{}\sum_{U}z_{U}:~{}~{}~{}~{}\quad$ $\displaystyle
z\in\mathbf{R}^{\textrm{valid}(R)}_{\geq 0}\\!\\!\\!\\!$ ($\mathcal{D}_{D}$)
$\displaystyle\sum_{U:K\cap U\neq\varnothing,i\notin U}z_{U}\leq C_{K},\quad$
$\displaystyle\forall K\in\mathcal{K},\forall i\in K\Big{\\}}$ (4)
Figure 3: The directed hypergraph relaxation ($\mathcal{D}$) and its dual
($\mathcal{D}_{D}$).
Polzin & Vahdati Daneshmand [26] showed that
$\mathop{\mathrm{OPT}}\eqref{eq:LP-
PUDir}=\mathop{\mathrm{OPT}}\eqref{eq:LP-S}$. Moreover they observed that this
directed hypergraphic relaxation strengthens the bidirected cut relaxation.
###### Lemma 1.2 ([26]).
For any instance, $\mathop{\mathrm{OPT}}\eqref{eq:LP-
PUDir}\geq\mathop{\mathrm{OPT}}\eqref{eq:LP-B}$.
###### Proof sketch..
It suffices to show that any solution $x$ of ($\mathcal{D}$) can be converted
to a feasible solution $x^{\prime}$ of ($\mathcal{B}$) of the same cost. For
each arc $a$, let $x^{\prime}_{a}$ be the sum of $x_{K^{i}}$ over all directed
full components $K^{i}$ that (when viewed as an arborescence) contain $a$. Now
for any valid subset $U$ of $V$, it is not hard to see that every directed
full component leaving $R\cap U$ has at least one arc leaving $U$, hence
$\sum_{a\in\delta^{\mathrm{out}}(U)}{x^{\prime}}_{a}\geq\sum_{K^{i}\in\Delta^{\mbox{\scriptsize{$\mathrm{out}$}}}(R\cap
U)}x_{K^{i}}\geq 1$ and $x^{\prime}$ is feasible as needed. ∎
See [26] for an example where the strict inequality
$\mathop{\mathrm{OPT}}\eqref{eq:LP-
PUDir}>\mathop{\mathrm{OPT}}\eqref{eq:LP-B}$ holds.
Könemann et al. [23], inspired by the work of Chopra [6], described a
partition-based relaxation which captures that given any partition of the
terminals, any hyper-spanning tree must have sufficiently many “cross
hyperedges”. More formally, a partition, $\pi$, is a collection of pairwise
disjoint nonempty terminal sets $(\pi_{1},\ldots,\pi_{q})$ whose union equals
$R$. The number of parts $q$ of $\pi$ is referred to as the partition’s rank
and denoted as $r(\pi)$. Let $\Pi_{R}$ be the set of all partitions of $R$.
Given a partition $\pi=\\{\pi_{1},\ldots,\pi_{q}\\}$, define the rank
contribution $\mathtt{rc}_{K}^{\pi}$ of hyperedge $K\in\mathcal{K}$ for $\pi$
as the rank reduction of $\pi$ obtained by merging the parts of $\pi$ that are
touched by $K$; i.e.,
$\mathtt{rc}_{K}^{\pi}:=|\\{i\,:\,K\cap\pi_{i}\neq\varnothing\\}|-1.$ Then a
hyper-spanning tree $(R,\mathcal{K}^{\prime})$ must satisfy
$\sum_{K\in\mathcal{K}^{\prime}}\mathtt{rc}^{\pi}_{K}\geq r(\pi)-1$. The
partition based LP of [23] and its dual are given in Figure 4 on page 4.
$\displaystyle\min\Big{\\{}\sum_{K\in\mathcal{K}}C_{K}x_{K}:\quad$
$\displaystyle x\in\mathbf{R}^{\mathcal{K}}_{\geq 0}\\!\\!\\!$ ($\mathcal{P}$)
$\displaystyle\sum_{K\in\mathcal{K}}x_{K}\mathtt{rc}_{K}^{\pi}\geq
r(\pi)-1,\quad$ $\displaystyle\forall\pi\in\Pi_{R}\Big{\\}}$ (5)
$\displaystyle\max\Big{\\{}\sum_{\pi}(r(\pi)-1)\cdot y_{\pi}:\quad$
$\displaystyle y\in\mathbf{R}^{\Pi_{R}}_{\geq 0}$ ($\mathcal{P}_{D}$)
$\displaystyle\sum_{\pi\in\Pi_{R}}y_{\pi}\mathtt{rc}_{K}^{\pi}\leq
C_{K},\quad$ $\displaystyle\forall K\in\mathcal{K}\Big{\\}}$ (6)
Figure 4: The unbounded partition relaxation ($\mathcal{P}$) and its dual
($\mathcal{P}_{D}$).
The feasible region of ($\mathcal{P}$) is _unbounded_ , since if $x$ is a
feasible solution for ($\mathcal{P}$) then so is any $x^{\prime}\geq x$. We
obtain a _bounded_ partition LP relaxation, denoted by
($\mathcal{P}^{\prime}$) and shown below, by adding a valid equality
constraint to the LP.
$\displaystyle\min\Big{\\{}\sum_{K\in\mathcal{K}}C_{K}x_{K}:x\in\eqref{eq:LP-
PU},\sum_{K\in\mathcal{K}}x_{K}(|K|-1)=|R|-1\Big{\\}}$
($\mathcal{P}^{\prime}$)
#### 1.2.3 Discussion of Computational Issues
The bidirected cut relaxation is very attractive from a perspective of
computational implementation. Although the formulation given in Section 1.2.1
has an exponential number of constraints, an equivalent compact flow
formulation with $O(|E||R|)$ variables and constraints is well-known.
What is known regarding solving the hypergraphic LPs? They are good enough to
get theoretical results but less attractive in practice, as we now explain.
Using a separation oracle, Warme showed [35] that for any chosen family
$\mathcal{K}$ of full components, the subtour LP can be optimized in time
$\textrm{poly}(|V|,|\mathcal{K}|)$. For the common _$r$ -restricted setting_
of $\mathcal{K}$ to be all possible full components of size at most $r$ for
constant $r$, we have $\mathcal{K}\leq\tbinom{|R|}{r}$. This is polynomial for
any fixed $r$, and the relative error caused by this choice of $r$ is at most
the _$r$ -Steiner ratio_ $\rho_{r}=1+\Theta(1/\log r)$ [2]. But this is not so
practical: to get relative error $1+\epsilon$, we apply the ellipsoid
algorithm to an LP with $|R|^{\exp(\Theta(1/\epsilon))}$ variables!
In the _unrestricted setting_ where $\mathcal{K}$ contains all possible full
components without regard to size, it is an open problem to optimize any of
the hypergraphic LPs exactly in polynomial time. We make some progress here:
in quasibipartite instances, the proof method of our hypergraphic-bidirected
equivalence theorem (Section 3.3) implies that one can exactly compute the LP
optimal value, and a dual optimal solution. Regarding this open problem, we
note that the $r$-restricted LP optimum is at most $\rho_{r}$ times the
unrestricted optimum, and wonder whether there might be some advantage gained
by using the fact that the hypergraphic LPs have sparse optima.
We reiterate our feeling that it is important to obtain practical algorithms
and understand the bidirected cut relaxation as well as possible, e.g. we know
now that it has an integrality gap of at most 1.28 on quasi-bipartite
instances, but obtaining such a bound directly could give new insights.
#### 1.2.4 Other Related Work
In the special case of $r$-restricted instances for $r=3$, the partition
hypergraphic LP is essentially a special case of an LP introduced by Vande
Vate [33] for matroid matching, which is totally dual half-integral [16].
Additional facts about the hypergraphic relaxations appear in the thesis of
the third author [27], e.g. a combinatorial “gainless tree formulation” for
the LPs similar in flavour to the “1-tree bound” for the Held-Karp TSP
relaxation.
## 2 Uncrossing Partitions
In this section we are interested in uncrossing a minimal set of tight
partitions that uniquely define a basic feasible solution to ($\mathcal{P}$).
We start with a few preliminaries necessary to state our result formally.
### 2.1 Preliminaries
We introduce some needed well-known properties of partitions that arise in
combinatorial lattice theory [32].
###### Definition 2.1.
We say that a partition $\pi^{\prime}$ _refines_ another partition $\pi$ if
each part of $\pi^{\prime}$ is contained in some part of $\pi$. We also say
$\pi$ coarsens $\pi^{\prime}$. Two partitions _cross_ if neither refines the
other. A family of partitions forms a _chain_ if no pair of them cross.
Equivalently, a chain is any family $\pi^{1},\pi^{2},\dotsc,\pi^{t}$ such that
$\pi^{i}$ refines $\pi^{i-1}$ for each $1<i\leq t$.
The family $\Pi_{R}$ of all partitions of $R$ forms a _lattice_ with a _meet
operator_ $\wedge:\Pi_{R}^{2}\to\Pi_{R}$ and a _join operator_
$\vee:\Pi_{R}^{2}\to\Pi_{R}$. The meet $\pi\wedge\pi^{\prime}$ is the coarsest
partition that refines both $\pi$ and $\pi^{\prime}$, and the join
$\pi\vee\pi^{\prime}$ is the most refined partition that coarsens both $\pi$
and $\pi^{\prime}$. See Figure 5 on page 5 for an illustration.
###### Definition 2.2 (Meet of partitions).
Let the parts of $\pi$ be $\pi_{1},\dotsc,\pi_{t}$ and let the parts of
$\pi^{\prime}$ be $\pi^{\prime}_{1},\dotsc,\pi^{\prime}_{u}$. Then the parts
of the meet $\pi\wedge\pi^{\prime}$ are the nonempty intersections of parts of
$\pi$ with parts of $\pi^{\prime}$,
$\pi\wedge\pi^{\prime}=\\{\pi_{i}\cap\pi^{\prime}_{j}\mid 1\leq i\leq t,1\leq
j\leq u\textrm{ and }\pi_{i}\cap\pi^{\prime}_{j}\neq\varnothing\\}.$
Given a graph $G$ and a partition $\pi$ of $V(G)$, we say that $G$ _induces_
$\pi$ if the parts of $\pi$ are the vertex sets of the connected components of
$G$.
###### Definition 2.3 (Join of partitions).
Let $(R,E)$ be a graph that induces $\pi$, and let $(R,E^{\prime})$ be a graph
that induces $\pi^{\prime}$. Then the graph $(R,E\cup E^{\prime})$ induces
$\pi\vee\pi^{\prime}$.
(a)
(b)
(c)
Figure 5: Illustrations of some partitions. The black dots are the terminal
set $R$. (a): two partitions; neither refines the other. (b): the meet of the
partitions from (a). (c): the join of the partitions from (a).
Given a feasible solution $x$ to ($\mathcal{P}$), a partition $\pi$ is _tight_
if $\sum_{K\in\mathcal{K}}x_{K}\mathtt{rc}^{\pi}_{K}=r(\pi)-1$. Let
$\mathop{{\tt tight}}(x)$ be the set of all tight partitions. We are
interested in uncrossing this set of partitions. More precisely, we wish to
find a cross-free set of partitions (chain) which uniquely defines $x$. One
way would be to prove the following.
###### Property 2.4.
If two crossing partitions $\pi$ and $\pi^{\prime}$ are in $\mathop{{\tt
tight}}(x)$, then so are $\pi\wedge\pi^{\prime}$ and $\pi\vee\pi^{\prime}$.
This type of property is already well-used [9, 13, 21, 31] for sets (with
meets and joins replaced by unions and intersections respectively), and the
standard approach is the following. The typical proof considers the
constraints in ($\mathcal{P}$) corresponding to $\pi$ and $\pi^{\prime}$ and
uses the “supermodularity” of the RHS and the “submodularity” of the
coefficients in the LHS. In particular, if the following is true,
$\displaystyle\forall\pi,\pi^{\prime}:~{}r(\pi\vee\pi^{\prime})+r(\pi\wedge\pi^{\prime})$
$\displaystyle~{}~{}~{}\geq~{}~{}~{}r(\pi)+r(\pi^{\prime})$ (7)
$\displaystyle\forall
K,\pi,\pi^{\prime}:~{}\mathtt{rc}_{K}^{\pi}+\mathtt{rc}_{K}^{\pi^{\prime}}$
$\displaystyle~{}~{}~{}\geq~{}~{}~{}\mathtt{rc}_{K}^{\pi\vee\pi^{\prime}}+\mathtt{rc}_{K}^{\pi\wedge\pi^{\prime}}$
(8)
then Property 2.4 can be proved easily by writing a string of
inequalities.666In this hypothetical scenario we get
$r(\pi)+r(\pi^{\prime})-2=\sum_{K}x_{K}(\mathtt{rc}_{K}^{\pi}+\mathtt{rc}_{K}^{\pi^{\prime}})\geq\sum_{K}x_{K}(\mathtt{rc}_{K}^{\pi\wedge\pi^{\prime}}+\mathtt{rc}_{K}^{\pi\vee\pi^{\prime}})\geq
r(\pi\wedge\pi^{\prime})+r(\pi\vee\pi^{\prime})-2\geq
r(\pi)+r(\pi^{\prime})-2$; thus the inequalities hold with equality, and the
middle one shows $\pi\wedge\pi^{\prime}$ and $\pi\vee\pi^{\prime}$ are tight.
Inequality (7) is indeed true (see, for example, [32]), but unfortunately
inequality (8) is not true in general, as the following example shows.
###### Example 2.5.
Let $R=\\{1,2,3,4\\}$, $\pi=\\{\\{1,2\\},\\{3,4\\}\\}$ and
$\pi^{\prime}=\\{\\{1,3\\},\\{2,4\\}\\}.$ Let $K$ denote the full component
$\\{1,2,3,4\\}$. Then
$\mathtt{rc}_{K}^{\pi}+\mathtt{rc}_{K}^{\pi^{\prime}}=1+1<0+3=\mathtt{rc}_{K}^{\pi\vee\pi^{\prime}}+\mathtt{rc}_{K}^{\pi\wedge\pi^{\prime}}.$
Nevertheless, Property 2.4 is true; its correct proof is given in Section 2.2
and depends on a simple though subtle extension of the usual approach. The
crux of the insight needed to fix the approach is not to consider _pairs_ of
constraints in ($\mathcal{P}$), but rather multi-sets which may contain more
than two inequalities. Using this uncrossing result, we can prove the
following theorem (details are given in Section 2.3). Here, we let
$\underline{\pi}$ denote $\\{R\\}$, the unique partition with (minimal) rank
1; later we use $\overline{\pi}$ to denote $\\{\\{r\\}\mid r\in R\\}$, the
unique partition with (maximal) rank $|R|$.
###### Theorem 1.
Let $x^{*}$ be a basic feasible solution of ($\mathcal{P}$), and let
$\mathcal{C}$ be an inclusion-wise maximal chain in $\mathop{{\tt
tight}}(x^{*})\backslash\underline{\pi}$. Then $x^{*}$ is uniquely defined by
$\sum_{K\in\mathcal{K}}\mathtt{rc}_{K}^{\pi}x^{*}_{K}=r(\pi)-1\quad\forall\pi\in\mathcal{C}.$
(9)
Any chain of distinct partitions of $R$ that does not contain
$\underline{\pi}$ has size at most $|R|-1$, and this is an upper bound on the
rank of the system in (9). Elementary linear programming theory immediately
yields the following corollary.
###### Corollary 2.6.
Any basic solution $x^{*}$ of ($\mathcal{P}$) has at most $|R|-1$ non-zero
coordinates.
### 2.2 Partition Uncrossing Inequalities
We start with the following definition.
###### Definition 2.7.
Let $\pi\in\Pi_{R}$ be a partition and let $S\subset R$. Define the _merged
partition_ $m(\pi,S)$ to be the most refined partition that coarsens $\pi$ and
contains all of $S$ in a single part. See Figure 6 on page 6 for an example.
Informally, $m(\pi,S)$ is obtained by merging all parts of $\pi$ which
intersect $S$. Formally, $m(\pi,S)$ equals the set of parts
$\\{\\{\pi_{j}\\}_{j:\pi_{j}\cap S=\varnothing},\bigcup_{j:\pi_{j}\cap
S\neq\varnothing}\pi_{j}\\}$.
Figure 6: Illustration of merging. The left figure shows a (solid) partition
$\pi$ along with a (dashed) set $S$. The right figure shows the merged
partition $m(\pi,S)$.
We will use the following straightforward fact later:
$\mathtt{rc}_{K}^{\pi}=r(\pi)-r(m(\pi,K)).$ (10)
We now state the (true) inequalities which replace the false inequality (8).
Later, we show how one uses these to obtain partition uncrossing, e.g. to
prove Property 2.4.
###### Lemma 2.8 (Partition Uncrossing Inequalities).
Let $\pi,\pi^{\prime}\in\Pi_{R}$ and let the parts of $\pi$ be
$\pi_{1},\pi_{2},\dotsc,\pi_{r(\pi)}$.
$\displaystyle r(\pi)\left[r(\pi^{\prime})-1\right]+\left[r(\pi)-1\right]$
$\displaystyle=$
$\displaystyle\left[r(\pi\wedge\pi^{\prime})-1\right]+\sum_{i=1}^{r(\pi)}\left[r(m(\pi^{\prime},\pi_{i}))-1\right]$
(11) $\displaystyle\forall K\in\mathcal{K}:\quad
r(\pi)\Bigl{[}\mathtt{rc}_{K}^{\pi^{\prime}}\Bigr{]}+\Bigl{[}\mathtt{rc}_{K}^{\pi}\Bigr{]}$
$\displaystyle\geq$
$\displaystyle\Bigl{[}\mathtt{rc}_{K}^{\pi\wedge\pi^{\prime}}\Bigr{]}+\sum_{i=1}^{r(\pi)}\Bigl{[}\mathtt{rc}_{K}^{m(\pi^{\prime},\pi_{i})}\Bigr{]}$
(12)
Before giving the proof of the above lemma, let us first show how it can be
used to prove the statement Property 2.4.
Proof of Property 2.4. Since $\pi$ and $\pi^{\prime}$ are tight,
$\displaystyle
r(\pi)[r(\pi^{\prime})-1]+[r(\pi)-1]=r(\pi)\Bigl{[}\sum_{K}x_{K}\mathtt{rc}_{K}^{\pi^{\prime}}\Bigr{]}+\Bigl{[}\sum_{K}x_{K}\mathtt{rc}_{K}^{\pi}\Bigr{]}=\sum_{K}x_{K}\biggl{(}r(\pi)\Bigl{[}\mathtt{rc}_{K}^{\pi^{\prime}}\Bigr{]}+\Bigl{[}\mathtt{rc}_{K}^{\pi}\Bigr{]}\biggr{)}$
$\displaystyle\geq\sum_{K}x_{K}\biggl{(}\Bigl{[}\mathtt{rc}_{K}^{\pi\wedge\pi^{\prime}}\Bigr{]}+\sum_{i=1}^{r(\pi)}\Bigl{[}\mathtt{rc}_{K}^{m(\pi^{\prime},\pi_{i})}\Bigr{]}\biggr{)}=\sum_{K}x_{K}\Bigl{[}\mathtt{rc}_{K}^{\pi\wedge\pi^{\prime}}\Bigr{]}+\sum_{i=1}^{r(\pi)}\sum_{K}x_{K}\Bigl{[}\mathtt{rc}_{K}^{m(\pi^{\prime},\pi_{i})}\Bigr{]}$
$\displaystyle\geq\left[r(\pi\wedge\pi^{\prime})-1\right]+\sum_{i=1}^{r(\pi)}\left[r(m(\pi^{\prime},\pi_{i}))-1\right]=r(\pi)\left[r(\pi^{\prime})-1\right]+\left[r(\pi)-1\right]$
where the first inequality follows from (12) and the second from (5) (as $x$
is feasible); the last equality is (11). Since the first and last terms are
equal, all the inequalities are equalities, in particular our application of
(5) shows that $\pi\wedge\pi^{\prime}$ and each $m(\pi^{\prime},\pi_{i})$ is
tight. Iterating the latter fact, we see that $m(\dotsb
m(m(\pi^{\prime},\pi_{1}),\pi_{2}),\dotsb)=\pi\vee\pi^{\prime}$ is also tight.
$\square$
To prove the inequalities in Lemma 2.8 we need the following lemma that
relates the rank of sets and the rank contribution of partitions. Recall
$\rho(X):=\max(0,|X|-1)$.
###### Lemma 2.9.
For a partition $\pi=\\{\pi_{1},\dotsc,\pi_{t}\\}$ of $R$, where $t=r(\pi)$,
and for any $K\subseteq R$, we have
$\rho(K)=\mathtt{rc}_{K}^{\pi}+\sum_{i=1}^{t}\rho(K\cap\pi_{i}).$
###### Proof.
By definition, $K\cap\pi_{i}\neq\varnothing$ for exactly
$1+\mathtt{rc}_{K}^{\pi}$ values of $i$. Also, $\rho(K\cap\pi_{i})=0$ for all
other $i$. Hence
$\sum_{i=1}^{t}\rho(K\cap\pi_{i})=\sum_{i:K\cap\pi_{i}\neq\varnothing}(|K\cap\pi_{i}|-1)=\left(\sum_{i:K\cap\pi_{i}\neq\varnothing}|K\cap\pi_{i}|\right)-(\mathtt{rc}_{K}^{\pi}+1).$
(13)
Observe that
$\sum_{i:K\cap\pi_{i}\neq\varnothing}|K\cap\pi_{i}|=|K|=\rho(K)+1$; using this
fact together with Equation (13) we obtain
$\sum_{i=1}^{t}\rho(K\cap\pi_{i})=\left(\sum_{i:K\cap\pi_{i}\neq\varnothing}|K\cap\pi_{i}|\right)-(\mathtt{rc}_{K}^{\pi}+1)=\rho(K)-1+(\mathtt{rc}_{K}^{\pi}+1).$
Rearranging, the proof of Lemma 2.9 is complete. ∎
Proof of Lemma 2.8. First, we argue that
$\pi\wedge\pi^{\prime}=\overline{\pi}$ holds without loss of generality. In
the general case, for each part $p$ of $\pi\wedge\pi^{\prime}$ with $|p|\geq
2$, contract $p$ into one pseudo-vertex and define the new $K$ to include the
pseudo-vertex corresponding to $p$ if and only if $K\cap p\neq\varnothing$.
This contraction does not affect the value of any of the terms in Equations
(12) and (11), so is without loss of generality. After contraction, for any
part $\pi_{i}$ of $\pi$ and part $\pi^{\prime}_{j}$ of $\pi^{\prime}$, we have
$|\pi_{i}\cap\pi^{\prime}_{j}|\leq 1$, so indeed
$\pi\wedge\pi^{\prime}=\overline{\pi}$.
###### Proof of Equation (11).
Fix $i$. Since $|\pi_{i}\cap\pi^{\prime}_{j}|\leq 1$ for all $j$, the rank
contribution $\mathtt{rc}_{\pi_{i}}^{\pi^{\prime}}$ is equal to $|\pi_{i}|-1.$
Then using Equation (10) we know that
$r(m(\pi^{\prime},\pi_{i}))=r(\pi^{\prime})-|\pi_{i}|+1$. Thus adding over all
$i$, the right-hand side of Equation (11) is equal to
$|R|-1+\sum_{i=1}^{r(\pi)}(r(\pi^{\prime})-|\pi_{i}|)=|R|-1+r(\pi)r(\pi^{\prime})-|R|$
and this is precisely the left-hand side of Equation (11). ∎
###### Proof of Equation (12).
Fix $i$. Since $|\pi_{i}\cap\pi^{\prime}_{j}|\leq 1$ for all $j$, we have
$\mathtt{rc}_{K}^{\pi^{\prime}}-\mathtt{rc}_{K}^{m(\pi^{\prime},\pi_{i})}\geq\rho(\pi_{i}\cap
K)$ (14)
because, when we merge the parts of $\pi^{\prime}$ intersecting $\pi_{i}$, we
make $K$ span at least $\rho(\pi_{i}\cap K)$ fewer parts. Note that the
inequality could be strict if both $\pi_{i}$ and $K$ intersect a part of
$\pi^{\prime}$ without having a common vertex in that part.
Adding the right-hand side of Equation (14) over all $i$ gives
$\sum_{i=1}^{r(\pi)}(\mathtt{rc}_{K}^{\pi^{\prime}}-\mathtt{rc}_{K}^{m(\pi^{\prime},\pi_{i})})\geq\sum_{i=1}^{r(\pi)}\rho(\pi_{i}\cap
K)=\rho(K)-\mathtt{rc}_{K}^{\pi}.$ (15)
where the last equality follows from Lemma 2.9. To finish the proof we observe
$\rho(K)=\mathtt{rc}_{K}^{\pi\wedge\pi^{\prime}}$, since
$\pi\wedge\pi^{\prime}=\overline{\pi}$. ∎
This completes the proof of Lemma 2.8. $\hfill\Box$
### 2.3 Sparsity of Basic Feasible Solutions: Proof of Theorem 1
###### Proof.
Let $\mathop{{\tt supp}}(x^{*})$ be the full components $K$ with
$x^{*}_{K}>0$. Consider the constraint submatrix with rows corresponding to
the tight partitions and columns corresponding to the full components in
$\mathop{{\tt supp}}(x^{*})$. Since $x^{*}$ is a basic feasible solution, any
full-rank subset of rows uniquely defines $x^{*}$. We now show that any
maximal chain $\mathcal{C}$ in $\mathop{{\tt tight}}(x^{*})$ corresponds to
such a subset.
Let ${\tt row}(\pi)\in\mathbf{R}^{\mathop{{\tt supp}}(x^{*})}$ denote the row
corresponding to partition $\pi$ of this matrix, i.e., ${\tt
row}(\pi)_{K}=\mathtt{rc}^{\pi}_{K}$, and given a collection $\mathcal{R}$ of
partitions (rows), let $\mathop{{\tt span}}(\mathcal{R})$ denote the linear
span of the rows in $\mathcal{R}$. We now prove that for any tight partition
$\pi\notin\mathcal{C}$, we have ${\tt row}(\pi)\in\mathop{{\tt
span}}(\mathcal{C})$; this will complete the proof of the theorem.
For sake of contradiction, suppose ${\tt row}(\pi)\not\in\mathop{{\tt
span}}(\mathcal{C})$. Choose $\pi$ to be the counterexample partition with
smallest rank $r(\pi)$. Firstly, since $\mathcal{C}$ is maximal, $\pi$ must
cross some partition $\sigma$ in $\mathcal{C}$. Choose $\sigma$ to be the most
refined partition in $\mathcal{C}$ which crosses $\pi$. Let the parts of
$\sigma$ be $(\sigma_{1},\ldots,\sigma_{t})$. The following claim uses the
partition uncrossing inequalities to derive a linear dependence between the
rows corresponding to $\sigma,\pi$ and the partitions formed by merging parts
of $\sigma$ with $\pi$.
###### Claim 2.10.
We have ${\tt row}(\sigma)+|r(\sigma)|\cdot{\tt row}(\pi)={\tt
row}(\pi\wedge\sigma)+\sum_{i=1}^{t}{\tt row}(m(\pi,\sigma_{i}))$.
###### Proof.
Since $\sigma$ and $\pi$ are both tight partitions, the proof of Property 2.4
shows that the partition inequality (12) holds with equality for all
$K\in\mathop{{\tt supp}}(x^{*})$, $\pi$ and $\sigma$, implying the claim. ∎
Let $\mathtt{cp}_{\pi}(\sigma)$ be the parts of $\sigma$ which intersect at
least two parts of $\pi$; i.e., merging the parts of $\pi$ that intersect
$\sigma_{i}$, for any $\sigma_{i}\in\mathtt{cp}_{\pi}(\sigma)$, decreases the
rank of $\pi$. Formally,
$\mathtt{cp}_{\pi}(\sigma):=\\{\sigma_{i}\in\sigma:~{}~{}m(\pi,\sigma_{i})\neq\pi\\}$
Note that one can modify Claim 2.10 by subtracting
$(r(\sigma)-|\mathtt{cp}_{\pi}(\sigma)|){\tt row}(\pi)$ from both sides to get
${\tt row}(\sigma)+|\mathtt{cp}_{\pi}(\sigma)|\cdot{\tt row}(\pi)={\tt
row}(\pi\wedge\sigma)+\sum_{\sigma_{i}\in\mathtt{cp}_{\pi}(\sigma)}{\tt
row}(m(\pi,\sigma_{i}))$ (16)
Now if ${\tt row}(\pi)\notin\mathop{{\tt span}}(\mathcal{C})$, we must have
either ${\tt row}(\pi\wedge\sigma)$ is not in $\mathop{{\tt
span}}(\mathcal{C})$ or ${\tt row}(m(\pi,\sigma_{i}))$ is not in $\mathop{{\tt
span}}(\mathcal{C})$ for some $i$. We show that either case leads to the
needed contradiction, which will prove the theorem.
Case 1:
${\tt row}(\pi\wedge\sigma)\notin\mathop{{\tt span}}(\mathcal{C})$. Note there
is $\sigma^{\prime}\in\mathcal{C}$ which crosses $\pi\wedge\sigma$, since
$\pi\wedge\sigma$ is not in the maximal chain $\mathcal{C}$. Since
$\sigma^{\prime},\sigma\in\mathcal{C}$ and by considering the refinement
order, it is easy to see that $\sigma^{\prime}$ (strictly) refines $\sigma$
and $\sigma^{\prime}$ crosses $\pi$. This contradicts our choice of $\sigma$
as the most refined partition in $\mathcal{C}$ crossing $\pi$, since
$\sigma^{\prime}$ was also a candidate.
Case 2:
${\tt row}(m(\pi,\sigma_{i}))\not\in\mathop{{\tt span}}(\mathcal{C})$. Note
$m(\pi,\sigma_{i})$ is also tight. Since
$\sigma_{i}\in\mathtt{cp}_{\pi}(\sigma)$, $m(\pi,\sigma_{i})$ has smaller rank
than $\pi$. This contradicts our choice of $\pi$.
This completes the proof of Theorem 1. ∎
## 3 Equivalence of Formulations
In this section we describe our equivalence results. A summary of the known
and new results is given in Figure 7 on page 7.
$\mathop{\mathrm{OPT}}\eqref{eq:LP-S}$$\mathop{\mathrm{OPT}}$($\mathcal{P}$)$\mathop{\mathrm{OPT}}$($\mathcal{P}^{\prime}$)$\mathop{\mathrm{OPT}}$($\mathcal{D}$)$\mathop{\mathrm{OPT}}$($\mathcal{B}$)$=$[Thm.
2]$=$[Thm. 4]$=$[26]$=$[Appendix A]$\geq$[Lemma 1.2],[26]$\leq$in quasi-
bipartite [Thm. 5] Figure 7: Summary of relations among various LP relaxations
As we mentioned in the introduction, we give a redundant set of proofs for
completeness and to demonstrate novel techniques. The proof that
($\mathcal{P}$) and ($\mathcal{D}$) have the same value, which appears in
Appendix A, is a consequence of hypergraph orientation results of Frank et al.
[14].
### 3.1 Bounded and Unbounded Partition Relaxations
###### Theorem 2.
The LPs ($\mathcal{P}^{\prime}$) and ($\mathcal{P}$) have the same optimal
value.
We actually prove a stronger statement.
###### Definition 3.1.
The collection $\mathcal{K}$ of hyperedges is _down-closed_ if whenever
$S\in\mathcal{K}$ and $\varnothing\neq T\subset S$, then $T\in\mathcal{K}.$
For down-closed $\mathcal{K}$, the cost function
$C:\mathcal{K}\to\mathbf{R}_{+}$ is _non-decreasing_ if $C_{S}\leq C_{T}$
whenever $S\subset T$.
###### Theorem 3.
If the set of hyperedges is down-closed and the cost function is non-
decreasing, then ($\mathcal{P}^{\prime}$) and ($\mathcal{P}$) have the same
optimal value.
Theorem 3 implies Theorem 2 since the hypergraph and cost function derived
from instances of the Steiner tree problem are down-closed and non-decreasing
(e.g. $C_{\\{k\\}}=0$ for every $k\in R$; we remark that the variables
$x_{\\{k\\}}$ act just as placeholders). Our proof of Theorem 2 relies on the
following operation which we call shrinking.
###### Definition 3.2.
Given an assignment $x:\mathcal{K}\to\mathbf{R}_{+}$ to the full components,
suppose $x_{K}>0$ for some $K$. The operation ${\tt
Shrink}(x,K,K^{\prime},\delta)$, where $K^{\prime}\subseteq K$,
$|K^{\prime}|=|K|-1$ and $0<\delta\leq x_{K}$, changes $x$ to $x^{\prime}$ by
decreasing $x^{\prime}_{K}:=x_{K}-\delta$ and increasing
$x^{\prime}_{K^{\prime}}:=x_{K^{\prime}}+\delta$.
Note that shrinking is defined only for down-closed hypergraphs. Also note
that on performing a shrinking operation, the cost of the solution cannot
increase, if the cost function is non-decreasing. The theorem is proved by
taking the optimum solution to ($\mathcal{P}$) which minimizes the sum
$\sum_{K\in\mathcal{K}}x_{K}|K|$, and then showing that this must satisfy the
equality in ($\mathcal{P}^{\prime}$), or a shrinking operation can be
performed. Now we give the details.
###### Proof of Theorem 3.
It suffices to exhibit an optimum solution of ($\mathcal{P}$) which satisfies
the equality in ($\mathcal{P}^{\prime}$). Let $x$ be an optimal solution to
($\mathcal{P}$) which minimizes the sum $\sum_{K\in\mathcal{K}}x_{K}|K|$.
###### Claim 3.3.
For every $K$ with $x_{K}>0$ and for every $r\in K$, there exists a tight
partition (w.r.t. $x$) $\pi$ such that the part of $\pi$ containing $r$
contains no other vertex of $K$.
###### Proof.
Let $K^{\prime}=K\setminus\\{r\\}$. If the above is not true, then this
implies that for every tight partition $\pi$, we have
$\mathtt{rc}_{K}^{\pi}=\mathtt{rc}_{K^{\prime}}^{\pi}$. We now claim that
there is a $\delta>0$ such that we can perform ${\tt
Shrink}(x,K,K^{\prime},\delta)$ while retaining feasibility in
($\mathcal{P}$). This is a contradiction since the shrink operation strictly
reduces $\sum_{K}|K|x_{K}$ and doesn’t increase cost. Specifically, take
$\delta:=\min\\{x_{K},\min_{\pi:\mathtt{rc}_{K^{\prime}}^{\pi}\neq\mathtt{rc}_{K}^{\pi}}\sum_{K}\mathtt{rc}_{K}^{\pi}x_{K}-r(\pi)+1\\}$
which is positive since for tight partitions we have
$\mathtt{rc}_{K}^{\pi}=\mathtt{rc}_{K^{\prime}}^{\pi}$. ∎
Let $\mathop{{\tt tight}}(x)$ be the set of tight partitions, and
$\pi^{*}:=\bigwedge\\{\pi\mid\pi\in\mathop{{\tt tight}}(x)\\}$ the meet of all
tight partitions. By Property 2.4, $\pi^{*}$ is tight. By Claim 3.3, for any
$K$ with $x_{K}>0$, we have $\mathtt{rc}_{K}^{\pi^{*}}=|K|-1$. Thus,
$r(\pi^{*})-1=\sum_{K\in\mathcal{K}}x_{K}\mathtt{rc}_{K}^{\pi^{*}}=\sum_{K\in\mathcal{K}}x_{K}(|K|-1)\geq
r(\overline{\pi})-1$. But since $\overline{\pi}$ is the unique maximal-rank
partition, this implies $\pi^{*}=\overline{\pi}$. Thus $\overline{\pi}$ is
tight. This implies $x\in\eqref{eq:LP-P2}$. ∎
### 3.2 Partition and Subtour Elimination Relaxations
###### Theorem 4.
The feasible regions of ($\mathcal{P}^{\prime}$) and ($\mathcal{S}$) are the
same.
###### Proof.
Let $x$ be any feasible solution to the LP ($\mathcal{S}$). Note that the
equality constraint of ($\mathcal{P}^{\prime}$) is the same as that of
($\mathcal{S}$). We now show that $x$ satisfies (5). Fix a partition
$\pi=\\{\pi_{1},\dotsc,\pi_{t}\\}$, so $t=r(\pi)$. For each $1\leq i\leq t$,
subtract the inequality constraint in ($\mathcal{S}$) with $S=\pi_{i}$, from
the equality constraint in ($\mathcal{S}$) to obtain
$\sum_{K\in\mathcal{K}}x_{K}\Bigl{(}\rho(K)-\sum_{i=1}^{t}\rho(K\cap\pi_{i})\Bigr{)}\geq\rho(R)-\sum_{i=1}^{t}\rho(\pi_{i}).$
(17)
From Lemma 2.9,
$\rho(K)-\sum_{i=1}^{t}\rho(K\cap\pi_{i})=\mathtt{rc}_{K}^{\pi}$. We also have
$\rho(R)-\sum_{i=1}^{t}\rho(\pi_{i})=|R|-1-(|R|-r(\pi))=r(\pi)-1$. Thus $x$ is
a feasible solution to the LP ($\mathcal{P}^{\prime}$).
Now, let $x$ be a feasible solution to ($\mathcal{P}^{\prime}$) and it
suffices to show that it satisfies the inequality constraints of
($\mathcal{S}$). Fix a set $S\subset R$. Note when $S=\varnothing$ that
inequality constraint is vacuously true so we may assume $S\neq\varnothing$.
Let $R\backslash S=\\{r_{1},\dotsc,r_{u}\\}$. Consider the partition
$\pi=\\{\\{r_{1}\\},\dotsc,\\{r_{u}\\},S\\}$. Subtract (5) for this $\pi$ from
the equality constraint in ($\mathcal{P}^{\prime}$), to obtain
$\sum_{K\in\mathcal{K}}x_{K}(\rho(K)-\mathtt{rc}_{K}^{\pi})\leq\rho(R)-r(\pi)+1.$
(18)
Using Lemma 2.9 and the fact that $\rho(K\cap\\{r_{j}\\})=0$ (the set is
either empty or a singleton), we get $\rho(K)-\mathtt{rc}_{K}^{\pi}=\rho(K\cap
S)$. Finally, as $\rho(R)-r(\pi)+1=|R|-1-(|R\backslash S|+1)+1=\rho(S),$ the
inequality (18) is the same as the constraint needed. Thus $x$ is a feasible
solution to ($\mathcal{S}$), proving the theorem. ∎
### 3.3 Partition and Bidirected Cut Relaxations in Quasibipartite Instances
###### Theorem 5.
On quasibipartite Steiner tree instances,
$\mathop{\mathrm{OPT}}\eqref{eq:LP-B}\geq\mathop{\mathrm{OPT}}\eqref{eq:LP-
PUDir}$.
To prove Theorem 5, we look at the duals of the two LPs and we show
$\mathop{\mathrm{OPT}}\eqref{eq:LP-
BD}\geq\mathop{\mathrm{OPT}}\eqref{eq:LP-A}$ in quasibipartite instances.
Recall that the support of a solution to ($\mathcal{D}_{D}$) is the family of
sets with positive $z_{U}$. A family of sets is called _laminar_ if for any
two of its sets $A,B$ we have $A\subseteq B,B\subseteq A$, or $A\cap
B=\varnothing$.
###### Lemma 3.4.
There exists an optimal solution to ($\mathcal{D}_{D}$) whose support is a
laminar family of sets.
###### Proof.
Choose an optimal solution $z$ to ($\mathcal{D}_{D}$) which maximizes
$\sum_{U}z_{U}|U|^{2}$ among all optimal solutions. We claim that the support
of this solution is laminar. Suppose not and there exists $U$ and $U^{\prime}$
with $U\cap U^{\prime}\neq\varnothing$ and $z_{U}>0$ and $z_{U^{\prime}}>0$.
Define $z^{\prime}$ to be the same as $z$ except
$z^{\prime}_{U}=z_{U}-\delta$,
$z^{\prime}_{U^{\prime}}=z_{U^{\prime}}-\delta$, $z^{\prime}_{U\cup
U^{\prime}}=z_{U\cup U^{\prime}}+\delta$ and $z^{\prime}_{U\cap
U^{\prime}}=z_{U\cap U^{\prime}}+\delta$; we will show for small $\delta>0$,
$z^{\prime}$ is feasible. Note that $U\cap U^{\prime}$ is not empty and $U\cup
U^{\prime}$ doesn’t contain $r$, and the objective value remains unchanged.
Also note that for any $K$ and $i\in K$, if $z_{U\cup U^{\prime}}$ or
$z_{U\cap U^{\prime}}$ appears in the summand of a constraint, then at least
one of $z_{U}$ or $z_{U^{\prime}}$ also appears. If both $z_{U\cup
U^{\prime}}$ and $z_{U\cap U^{\prime}}$ appears, then both $z_{U}$ and
$z_{U^{\prime}}$ appears. Thus $z^{\prime}$ is an optimal solution and
$\sum_{U}z^{\prime}_{U}|U|^{2}>\sum_{U}z_{U}|U|^{2}$, contradicting the choice
of $z$. ∎
###### Lemma 3.5.
For quasibipartite instances, given a solution of ($\mathcal{D}_{D}$) with
laminar support, we can get a feasible solution to ($\mathcal{B}_{D}$) of the
same value.
###### Proof.
This lemma is the heart of the theorem, and is a little technical to prove. We
first give a sketch of how we convert a feasible solution $z$ of
($\mathcal{D}_{D}$) into a feasible solution to ($\mathcal{B}_{D}$) of the
same value.
Comparing ($\mathcal{D}_{D}$) and ($\mathcal{B}_{D}$) one first notes that the
former has a variable for every valid subset of the terminals, while the
latter assigns values to all valid subsets of the entire vertex set. We say
that an edge $uv$ is _satisfied_ for a candidate solution $z$, if both a)
$\sum_{U:u\in U,v\notin U}z_{U}\leq c_{uv}$ and b) $\sum_{U:v\in U,u\notin
U}z_{U}\leq c_{uv}$ hold; $z$ is then feasible for ($\mathcal{B}_{D}$) if all
edges are satisfied.
Let $z$ be a feasible solution to ($\mathcal{D}_{D}$). One easily verifies
that all terminal-terminal edges are satisfied. On the other hand, terminal-
Steiner edges may initially not be satisfied. To see this consider the Steiner
vertex $v$ and its neighbours depicted in Figure 3.3 on page 3.3 below.
Initially, none of the sets in $z$’s support contains $v$, and the load on the
edges incident to $v$ is quite skewed: the left-hand side of condition a)
above may be large, while the left-hand side of condition b) is initially $0$.
To construct a valid solution for ($\mathcal{B}_{D}$), we therefore lift the
initial value $z_{S}$ of each terminal subset $S$ to supersets of $S$, by
adding Steiner vertices. The lifting procedure processes each Steiner vertex
$v$ one at a time; when processing $v$, we change $z$ by moving dual from some
sets $U$ to $U\cup\\{v\\}$. Such a dual transfer decreases the left-hand side
of condition a) for edge $uv$, and increases the (initially $0$) left-hand
sides of condition b) for edges connecting $v$ to neighbours other than $v$.
We will soon see that there is a way of carefully lifting duals around $v$
that ensures that all edges incident to $v$ become satisfied. The definition
of our procedure will ensure that these edges remain satisfied for the rest of
the lifting procedure. Since there are no Steiner-Steiner edges, all edges
will be satisfied once all Steiner vertices are processed.
Lifting variable $z_{U}$. (5.5cm,4.5cm)[fr]
Throughout the lifting procedure, we will maintain that $z$ remains unchanged,
when projected to the terminals. Formally, we maintain the following crucial
projection invariant:
The quantity $\sum_{U:S\subseteq U\subseteq S\cup(V\setminus R)}z_{U}$ remains
constant, for all terminal sets $S$. (PI)
This invariant leads to two observations: first, the constraint (4) is
satisfied by $z$ at all times, even when it is defined on subsets of all
vertices; second, $\sum_{U\subseteq V}z_{U}$ is constant throughout, and the
objective value of $z$ in ($\mathcal{B}_{D}$) is not affected by the lifting.
The existence of a lifting of duals around Steiner vertex $v$ such that (PI)
is maintained, and such that all edges incident to $v$ are satisfied can be
phrased as a feasibility problem for a linear system of inequalities. We will
use Farkas’ lemma and the feasibility of $z$ for (4) to complete the proof.
We now fill in the proof details. Let $\Gamma(v)$ denote the set of neighbours
of vertex $v$ in the given graph $G$. In each iteration, where we process
Steiner node $v$, let
$\mathcal{U}_{v}:=\\{U:z_{U}>0~{}~{}\textrm{and}~{}~{}U\cap\Gamma(v)\neq\varnothing\\}$
be the sets in $z$’s support that contain neighbours of $v$. Note that
$U\in\mathcal{U}_{v}$ could contain Steiner vertices on which the lifting
procedure has already taken place. However, by (PI) and by Lemma 3.4 the
multi-family $\\{U\cap R:U\in\mathcal{U}_{v}\\}$ is laminar. In the lifting
process, we will transfer $x_{U}$ units of the $z_{U}$ units of dual of each
set $U\in\mathcal{U}_{v}$ to the set $U^{\prime}=U\cup\\{v\\}$; this decreases
the dual load (LHS of (2)) on arcs from $U\cap\Gamma(v)$ to $v$ (e.g. $uv$ in
Figure 3.3 on page 3.3) and increases the dual load on arcs from $v$ to
$\Gamma(v)\backslash U$ (e.g. $vu^{\prime}$ in the figure). The following
system of inequalities describes the set of feasible liftings.
$\displaystyle\forall U\in\mathcal{U}_{v}:$ $\displaystyle\qquad x_{U}\leq
z_{U}$ (L1) $\displaystyle\forall u\in\Gamma(v):$
$\displaystyle\qquad\sum_{U:u\in U}(z_{U}-x_{U})\leq c_{uv}$ (L2)
$\displaystyle\forall u\in\Gamma(v):$ $\displaystyle\qquad\sum_{U:u\notin
U}x_{U}\leq c_{uv}$ (L3)
###### Claim 3.6.
If (L1), (L2), (L3) have a feasible solution $x\geq 0$, then the lifting
procedure can be performed at Steiner vertex $v$, while maintaining the
projection invariant property.
###### Proof.
Define the new solution to be $z_{U}:=z_{U}-x_{U}$, and, $z_{(U\cup
v)}:=x_{U}$, for all $U\in\mathcal{U}_{v}$, and $z_{U}$ remains unchanged for
all other $U$. It is easy to check that all edges which were satisfied remain
satisfied, and (L2) and (L3) imply that all edges incident to $v$ are
satisfied. Also note that the projection invariant property is maintained. ∎
By Farkas’ lemma, if (L1), (L2), (L3) do not have a feasible solution $x\geq
0$, then there exist non-negative multipliers — $\lambda_{U}$ for all
$U\in\mathcal{U}_{v}$, and $\alpha_{u},\beta_{u}$ for all $u\in\Gamma(v)$ —
satisfying the following dual set of linear inequalities:
$\displaystyle\sum_{U\in\mathcal{U}_{v}}\lambda_{U}z_{U}+\sum_{u\in\Gamma(v)}\alpha_{u}\bigl{(}c_{uv}-\sum_{U:u\in
U}z_{U}\bigr{)}+\sum_{u\in\Gamma(v)}\beta_{u}c_{uv}$ $\displaystyle\quad<\quad
0$ (D1) $\displaystyle\forall U\in\mathcal{U}_{v}:\lambda_{U}-\sum_{u\in
U}\alpha_{u}+\sum_{u\notin U}\beta_{u}$ $\displaystyle\quad\geq\quad 0$ (D2)
As a technicality, note that the sub-system
$\\{\eqref{eq:L1},\eqref{eq:L2},x\geq 0\\}$ is feasible — take $x=z$. Thus any
$\alpha,\beta,\lambda$ satisfying (D1) and (D2) has $\sum_{u}\beta_{u}>0$, so
by dividing all $\alpha,\beta,\lambda$ by $\sum_{i}\beta_{i}$, we may assume
without loss of generality that
$\displaystyle\sum_{u\in\Gamma(v)}\beta_{u}=1.$ (D3)
Subtracting (D3) from (D2) allows us to rewrite the latter set of constraints
conveniently as
$\displaystyle\forall U\in\mathcal{U}_{v}:$
$\displaystyle\qquad\lambda_{U}-\sum_{u\in U}(\alpha_{u}+\beta_{u})+1\geq 0.$
(D2’)
The following claim shows that (L1), (L2), (L3) does have a feasible solution,
and thus by Claim 3.6, lifting can be done, which completes the proof of Lemma
3.5.
###### Claim 3.7.
There exists no feasible solution to $\\{\alpha,\beta,\lambda\geq
0:\eqref{eq:D1},\eqref{eq:D2'},\textrm{and }\eqref{eq:D3}\\}$.
###### Proof.
Consider the linear program which minimizes the LHS of (D1) subject to the
constraints (D2’) and (D3). We show that the LP has value at least $0$, which
will complete the proof.
Let $(\lambda^{*},\alpha^{*},\beta^{*})$ be an optimal solution to the LP. In
Lemma 3.8 we will show that the constraint matrix of the LP is totally
unimodular; hence, since the right-hand side of the given system is integral,
we may assume that $\lambda^{*},\alpha^{*}$, and $\beta^{*}$ are non-negative
and integral. From (D3) we infer
There is a unique $\bar{u}\in\Gamma(v)$ for which $\beta^{*}_{\bar{u}}=1$; for
all $u\neq\bar{u}$, $\beta^{*}_{u}=0$. (19)
Moreover, since each $\lambda_{U}$ appears only in the two constraints (D2’)
and $\lambda_{U}\geq 0$, and since $\lambda_{U}$ has nonnegative coefficient
in the objective, we may assume
$\lambda^{*}_{U}=\lambda^{*}_{U}(\alpha^{*},\beta^{*}):=\max\\{\sum_{u\in
U}(\alpha^{*}_{u}+\beta^{*}_{u})-1,0\\}$ (20)
for all $U$.
Next, we establish the following:
$\alpha^{*}_{u}+\beta^{*}_{u}\in\\{0,1\\}$ for all $u\in\Gamma(v)$. (21)
Suppose for the sake of contradiction that property (21) does not hold for our
solution. Let $u$ be such that $\alpha^{*}_{u}+\beta^{*}_{u}\geq 2$. By (19),
$\alpha^{*}_{u}\geq 1$. We propose the following update to our solution:
decrease $\alpha^{*}_{u}$ by $1$ (which by (20) will decrease
$\lambda^{*}_{U}$ by $1$ for all $U\in\mathcal{U}_{v}$). This maintains the
feasibility of (D2’), and the objective value decreases by
$\sum_{U\in\mathcal{U}_{v}:u\in U}z_{U}+(c_{uv}-\sum_{u\in U}z_{U})$
which is non-negative as $c\geq 0$. By repeating this operation, we may
clearly ensure property (21).
Let $K\subseteq\Gamma(v)$ be the set
$\\{u\mid\alpha^{*}_{u}+\beta^{*}_{u}=1\\}$ and recall $\bar{u}$ is the unique
terminal with $\beta^{*}_{\bar{u}}=1$; $\bar{u}$ is clearly a member of $K$.
At $(\alpha^{*},\beta^{*},\lambda^{*})$, we evaluate the objective and collect
like terms to get value
$\displaystyle\sum_{U\in\mathcal{U}_{v}}z_{U}\rho(U\cap K)+\sum_{u\in
K\setminus\bar{u}}(c_{uv}-\sum_{U:u\in U}z_{U})+c_{\bar{u}v}$
$\displaystyle=\sum_{u\in K}c_{uv}+\sum_{U\in\mathcal{U}_{v}}z_{U}(\rho(U\cap
K)-|(K\backslash\bar{u})\cap U|)$ $\displaystyle=\sum_{u\in
K}c_{uv}-\sum_{U\in\mathcal{U}_{v}:U\cap K\neq\varnothing,\bar{u}\not\in
U}z_{U}$
where the last equality follows by considering cases. Finally, combining the
fact that $\sum_{u\in K}c_{uv}\geq C_{K}$ (since these edges form one possible
full component on terminal set $K$) together with (4) for the pair
$(K,\bar{u})$, it follows that the LP’s optimal value is non-negative as
needed. ∎
###### Lemma 3.8.
The incidence matrix defined by (D2’) and (D3) is totally unimodular.
###### Proof.
The incidence matrix has $|\mathcal{U}_{v}|+1$ rows ($|\mathcal{U}_{v}|$
corresponding to (D2’) and one last row corresponding to (D3)) and
$|\mathcal{U}_{v}|+2|\Gamma(v)|$ columns. Furthermore, the columns
corresponding to $\alpha_{u}$’s are same as those corresponding to
$\beta_{u}$’s, except for the last row, where there are $0$’s in the
$\alpha$-columns and $1$’s in the $\beta$-columns.
To show that this matrix is totally unimodular we use Ghouila-Houri’s
characterization of total unimodularity (e.g. see [30, Thm. 19.3]):
###### Theorem 6 (Ghouila-Houri 1962).
A matrix is totally unimodular iff the following holds for _every_ subset
$\mathcal{R}$ of rows: we can assign weights $w_{r}\in\\{-1,+1\\}$ to each row
$r\in\mathcal{R}$ such that $\sum_{r\in\mathcal{R}}w_{r}r$ is a $\\{0,\pm
1\\}$-vector.
Note that we can safely ignore the columns corresponding to variables
$\lambda_{U}$ for sets $U\in\mathcal{U}_{v}$, since each of them contains a
single $1$ occurring in constraint (D2’) for set $U$.
The row subset $\mathcal{R}$ corresponds to a subset of $\mathcal{U}_{v}$ —
which we will denote $\mathcal{R}\cap\mathcal{U}_{v}$ — plus possibly the
single row corresponding to (D3). Each row in $\mathcal{R}\cap\mathcal{U}_{v}$
has its values determined by the characteristic vector of $U\cap\Gamma(v)$. So
long as any set appears more than once in $\\{U\cap\Gamma(v)\mid
U\in\mathcal{R}\cap\mathcal{U}_{v}\\}$ we can assign one copy weight $+1$ and
the other copy weight $-1$; these rows cancel out. Thus, henceforth we assume
$\\{U\cap\Gamma(v)\mid U\in\mathcal{R}\cap\mathcal{U}_{v}\\}$ has no duplicate
sets.
There is a standard representation of a laminar family as a forest of rooted
trees, where there is a node corresponding to each set, with containment in
the family corresponding to ancestry in the forest. Given the forest for the
laminar family $\\{U\cap\Gamma(v)\mid U\in\mathcal{R}\cap\mathcal{U}_{v}\\}$,
the assignment of weights to the rows of the matrix is as follows. Let the
root nodes of all trees be at height $0$ with height increasing as one goes to
children nodes. Give weight $-1$ to rows corresponding to nodes at even
height, and weight $+1$ to rows corresponding to nodes at odd height. If
$\mathcal{R}$ contains the row corresponding to (D3), give it weight $+1$.
Finally, let us argue that these weights have the needed property. Consider
first a column corresponding to $\alpha_{u}$ for any $u$. The rows of
$\mathcal{R}$ with $1$ in this column form a path, from the largest set
containing $u$ (which is a root node) to the smallest set containing $u$. The
weighted sum in this column is an alternating sum $-1+1-1+1\dotsb$, which is
either $-1$ or $0$, which is in $\\{0,\pm 1\\}$ as needed. Second, in a column
for some $\beta_{u}$, if $\mathcal{R}$ doesn’t contain (resp. contains) the
row corresponding to (D3), the weighted sum is the same as for $\alpha_{u}$
(resp. plus 1); in either case its weighted sum is in $\\{0,\pm 1\\}$ as
needed. ∎
This finishes the proof of Lemma 3.5, and hence also that of Theorem 5. ∎
## 4 Improved Integrality Gap Upper Bounds
We first show the improved bound of $73/60$ for uniformly quasibipartite
graphs. We then show the $(2\sqrt{2}-1)\doteq 1.828$ upper bound on general
graphs, which contains the main ideas, and then end by giving a
$\sqrt{3}\doteq 1.729$ upper bound.
### 4.1 Uniformly Quasibipartite Instances
Uniformly quasibipartite instances of the Steiner tree problem are
quasibipartite graphs where the cost of edges incident on a Steiner vertex are
the same. They were first studied by Gröpl et al. [20], who gave a $73/60$
factor approximation algorithm. In the following, we show that the cost of the
returned tree is no more than than
$\frac{73}{60}\mathop{\mathrm{OPT}}\eqref{eq:LP-PU}$, which upper-bounds the
integrality gap by $\frac{73}{60}$.
We start by describing the algorithm of Gröpl et al. [20] in terms of full
components. A collection $\mathcal{K}^{\prime}$ of full components is acyclic
if there is no list of $t>1$ distinct terminals and hyperedges in
$\mathcal{K}^{\prime}$ of the form $r_{1}\in K_{1}\ni r_{2}\in K_{2}\dotsb\ni
r_{t}\in K_{t}\ni r_{1}$ — i.e. there are no _hypercycles_.
Procedure RatioGreedy 1: Initialize the set of acyclic components
$\mathcal{L}$ to $\varnothing$. 2: Let $L^{*}$ be a minimizer of
$\frac{C_{L}}{|L|-1}$ over all full components $L$ such that $|L|\geq 2$ and
$L\cup\mathcal{L}$ is acyclic. 3: Add $L^{*}$ to $\mathcal{L}$. 4: Continue
until $(R,\mathcal{L})$ is a hyper-spanning tree and return $\mathcal{L}$.
###### Theorem 7.
On a uniformly quasibipartite instance RatioGreedy returns a Steiner tree of
cost at most $\frac{73}{60}\mathop{\mathrm{OPT}}\eqref{eq:LP-PU}$.
###### Proof.
Let $t$ denote the number of iterations and
$\mathcal{L}:=\\{L_{1},\ldots,L_{t}\\}$ be the ordered sequence of full
components obtained. We now define a dual solution to ($\mathcal{P}_{D}$). Let
$\pi(i)$ denote the partition induced by the connected components of
$\\{L_{1},\dotsc,L_{i}\\}$. Let $\theta(i)$ denote $C_{L_{i}}/(|L_{i}|-1)$ and
note that $\theta$ is nondecreasing. Define $\theta(0)=0$ for convenience. We
define a dual solution $y$ with
$y_{\pi(i)}=\theta(i+1)-\theta(i)$
for $0\leq i<t$, and all other coordinates of $y$ set to zero; $y$ is not
generally feasible, but we will scale it down to make it so. By evaluating a
telescoping sum, it is not hard to find that
$\sum_{i}y_{\pi(i)}(r(\pi(i))-1)=C(\mathcal{L})$. In the rest of the proof we
will show for any $K\in\mathcal{K}$,
$\sum_{i}y_{\pi(i)}\mathtt{rc}^{\pi(i)}_{K}\leq 73/60\cdot C_{K}$ — by
scaling, this also proves that $\frac{60}{73}y$ is a feasible dual solution,
and hence completes the proof.
Fix any $K\in\mathcal{K}$ and let $|K|=k$. Since the instance in question is
uniformly quasi-bipartite, the full component $K$ is a star with a Steiner
centre and edges of a fixed cost $c$ to each terminal in $K$. For $1\leq i<k$,
let $\tau(i)$ denote the last iteration $j$ in which
$\mathtt{rc}_{K}^{\pi(j)}\geq k-i$. Let $K_{i}$ denote any subset of $K$ of
size $k-i+1$ such that $K_{i}$ contains at most one element from each part of
$\pi(\tau(i))$; i.e., $|K_{i}|=k-i+1$ and
$\mathtt{rc}_{K_{i}}^{\pi(\tau(i))}=k-i$.
Our analysis hinges on the fact that $K_{i}$ was a valid choice for
$L_{\tau(i)+1}$. More specifically, note that
$\\{L_{1},\dotsc,L_{\tau(i)},K_{i}\\}$ is acyclic, hence by the greedy nature
of the algorithm, for any $1\leq i<k,$
$\theta(\tau(i)+1)=C_{L_{\tau(i)+1}}/(|L_{\tau(i)+1}|-1)\leq
C_{K_{i}}/(|K_{i}|-1)\leq\frac{c\cdot(k-i+1)}{k-i}.$
Moreover, using the definition of $\tau$ and telescoping we compute
$\sum_{\pi}y_{\pi}\mathtt{rc}_{K}^{\pi}=\sum_{i=0}^{t-1}(\theta(i+1)-\theta(i))\mathtt{rc}_{K}^{\pi(i)}=\sum_{i=1}^{k-1}\theta(\tau(i)+1)\leq\sum_{i=1}^{k-1}\frac{c\cdot(k-i+1)}{k-i}=c\cdot(k-1+H(k-1)),$
where $H(\cdot)$ denotes the harmonic series. Finally, note that
$(k-1+H(k-1))\leq\frac{73}{60}k$ for all $k\geq 2$ (achieved at $k=5$).
Therefore, $\frac{60}{73}y$ is a valid solution to ($\mathcal{P}_{D}$). ∎
### 4.2 General graphs
We start with a few definitions and notations in order to prove the
$2\sqrt{2}-1$ and $\sqrt{3}$ integrality gap bounds on ($\mathcal{P}$). Both
results use similar algorithms, and the latter is a more complex version of
the former. For conciseness we let a “graph” be a triple $G=(V,E,R)$ where
$R\subset V$ are $G$’s terminals. In the following, we let
${\mathtt{mtst}}(G;c)$ denote the minimum _terminal spanning tree_ , i.e. the
minimum spanning tree of the terminal-induced subgraph $G[R]$ under edge-costs
$c:E\to\mathbf{R}$. We will abuse notation and let ${\mathtt{mtst}}(G;c)$ mean
both the tree and its cost under $c$.
When contracting an edge $uv$ in a graph, the new merged node resulting from
contraction is defined to be a terminal iff at least one of $u$ or $v$ was a
terminal; this is natural since a Steiner tree in the new graph is a minimal
set of edges which, together with $uv$, connects all terminals in the old
graph. Our algorithm performs contraction, which may introduce parallel edges,
but one may delete all but the cheapest edge from each parallel class without
affecting the analysis.
Our first algorithm proceeds in stages. In each stage we apply the operation
$G\mapsto G/K$ which denotes contracting all edges in some full component $K$.
To describe and analyze the algorithm we introduce some notation. For a
minimum terminal spanning tree $T={\mathtt{mtst}}(G;c)$ define ${\tt
drop}_{T}(K;c):=c(T)-{\mathtt{mtst}}(G/K;c)$. We also define ${\tt
gain}_{T}(K;c):={\tt drop}_{T}(K)-c(K)$, where $c(K)$ is the cost of full
component $K$. A tree $T$ is called _gainless_ if for every full component $K$
we have ${\tt gain}_{T}(K;c)\leq 0$. The following useful fact is implicit in
[23] (see also Appendix B).
###### Theorem 8 (Implicit in [23]).
If ${\mathtt{mtst}}(G;c)$ is gainless, then
$\mathop{\mathrm{OPT}}\eqref{eq:LP-PU}$ equals the cost of
${\mathtt{mtst}}(G;c)$.
We now give the first algorithm and its analysis, which uses a reduced cost
trick introduced by Chakrabarty et al.[4].
Procedure Reduced One-Pass Heuristic 1: Define costs $c^{\prime}_{e}$ by
$c^{\prime}_{e}:=c_{e}/\sqrt{2}$ for all terminal-terminal edges $e$, and
$c^{\prime}_{e}=c_{e}$ for all other edges. Let $G_{1}:=G,$
$T_{i}:={\mathtt{mtst}}(G_{i};c^{\prime})$, and $i:=1$. 2: The algorithm
considers the full components in any order. When we examine a full component
$K$, if ${\tt gain}_{T_{i}}(K;c^{\prime})>0$, let $K_{i}:=K$,
$G_{i+1}:=G_{i}/K_{i}$, $T_{i+1}:={\mathtt{mtst}}(G_{i+1};c^{\prime})$, and
$i:=i+1$. 3: Let $f$ be the final value of $i$. Return the tree
$T_{alg}:=T_{f}\cup\bigcup_{i=1}^{f-1}K_{i}$.
Note that the full components are scanned in any order and they are not
examined a priori. Hence the algorithm works just as well if the full
components arrive “online,” which might be useful for some applications.
###### Theorem 9.
$c(T_{alg})\leq(2\sqrt{2}-1)\mathop{\mathrm{OPT}}\eqref{eq:LP-PU}$.
###### Proof.
First we claim that ${\tt gain}_{T_{f}}(K;c^{\prime})\leq 0$ for all $K$. To
see this there are two cases. If $K=K_{i}$ for some $i$, then we immediately
see that ${\tt drop}_{T_{j}}(K)=0$ for all $j>i$ so ${\tt
gain}_{T_{f}}(K)=-c(K)\leq 0$. Otherwise (if for all $i,$ $K\neq K_{i}$) $K$
had nonpositive gain when examined by the algorithm; and the well-known
_contraction lemma_ (e.g., see [19, §1.5]) immediately implies that ${\tt
gain}_{T_{i}}(K)$ is nonincreasing in $i$, so ${\tt gain}_{T_{f}}(K)\leq 0$.
By Theorem 8, $c^{\prime}(T_{f})$ equals the value of ($\mathcal{P}$) on the
graph $G_{f}$ with costs $c^{\prime}$. Since $c^{\prime}\leq c$, and since at
each step we only contract terminals, the value of this optimum must be at
most $\mathop{\mathrm{OPT}}\eqref{eq:LP-PU}$. Using the fact that
$c(T_{f})=\sqrt{2}c^{\prime}(T_{f})$, we get
$\displaystyle
c(T_{f})=\sqrt{2}c^{\prime}(T_{f})\leq\sqrt{2}\mathop{\mathrm{OPT}}\eqref{eq:LP-
PU}$ (22)
Furthermore, for every $i$ we have ${\tt gain}_{T_{i}}(K_{i};c^{\prime})>0$,
that is, ${\tt drop}_{T_{i}}(K_{i};c^{\prime})>c^{\prime}(K)=c(K)$. The
equality follows since $K$ contains no terminal-terminal edges. However, ${\tt
drop}_{T_{i}}(K_{i};c^{\prime})=\frac{1}{\sqrt{2}}{\tt drop}_{T_{i}}(K_{i};c)$
because all edges of $T_{i}$ are terminal-terminal. Thus, we get for every
$i=1$ to $f$, ${\tt drop}_{T_{i}}(K_{i};c)>\sqrt{2}\cdot c(K_{i})$.
Since ${\tt
drop}_{T_{i}}(K_{i};c):={\mathtt{mtst}}(G_{i};c)-{\mathtt{mtst}}(G_{i+1};c)$,
we have
$\sum_{i=1}^{f-1}{\tt drop}_{T_{i}}(K_{i};c)={\mathtt{mtst}}(G;c)-c(T_{f}).$
Thus, we have
$\sum_{i=1}^{f-1}c(K_{i})\leq\frac{1}{\sqrt{2}}\sum_{i=1}^{f}{\tt
drop}_{T_{i}}(K_{i};c)=\frac{1}{\sqrt{2}}({\mathtt{mtst}}(G;c)-c(T_{f}))\leq\frac{1}{\sqrt{2}}(2\mathop{\mathrm{OPT}}\eqref{eq:LP-
PU}-c(T_{f}))$
where we use the fact that ${\mathtt{mtst}}(G,c)$ is at most twice
$\mathop{\mathrm{OPT}}\eqref{eq:LP-PU}$777This follows using standard
arguments, and can be seen, for instance, by applying Theorem 8 to the cost-
function with all terminal-terminal costs divided by 2, and using short-
cutting.. Therefore
$c(T_{alg})=c(T_{f})+\sum_{i=1}^{f-1}c(K_{i})\leq\Bigl{(}1-\frac{1}{\sqrt{2}}\Bigr{)}c(T_{f})+\sqrt{2}\mathop{\mathrm{OPT}}\eqref{eq:LP-
PU}.$
Finally, using $c(T_{f})\leq\sqrt{2}\mathop{\mathrm{OPT}}\eqref{eq:LP-PU}$
from (22), the proof of Theorem 9 is complete. ∎
#### 4.2.1 Improving to $\sqrt{3}$
To get the improved factor of $\sqrt{3}$, we use a more refined iterated
contraction approach. The crucial new concept is that of the loss of a full
component, introduced by Karpinski and Zelikovsky [22]. The intuition is as
follows. In each iteration, the $(2\sqrt{2}-1)$-factor algorithm contracts a
full component $K$, and thus commits to include $K$ in the final solution; the
new algorithm makes a smaller commitment, by contracting a _subset_ of $K$’s
edges, which allows for a possibility of better recovery later.
Given a full component $K$ (viewed as a tree with leaf set $K$ and internal
Steiner nodes), ${\tt loss}(K)$ is defined to be the minimum-cost subset of
$E(K)$ such that $(V(K),{\tt loss}(K))$ has at least one terminal per
connected component — i.e. the cheapest way in $K$ to connect each Steiner
node to the terminal set. We also use ${\tt loss}(K)$ to denote the total
_cost_ of these edges. Note that no two terminals are connected by ${\tt
loss}(K)$. A very useful theorem of Karpinski and Zelikovsky [22] is that for
any full component $K$, ${\tt loss}(K)\leq c(K)/2$.
Now we have the ingredients to give our new algorithm. In the description
below, $\alpha>1$ is a parameter (which will be set to $\sqrt{3}$). In each
iteration, the algorithm contracts the loss of a single full component $K$ (we
note it follows that the terminal set has constant size over all iterations).
Procedure Reduced One-Pass Loss-Contracting Heuristic 1: Initially $G_{1}:=G$,
$T_{1}:={\mathtt{mtst}}(G;c)$, and $i:=1$. 2: The algorithm considers the
full components in any order. When we examine a full component $K$, if ${\tt
gain}_{T_{i}}(K;c)>(\alpha-1){\tt loss}(K),$ let $K_{i}:=K$,
$G_{i+1}:=G_{i}/{\tt loss}(K_{i})$, $T_{i+1}:={\mathtt{mtst}}(G_{i+1};c)$, and
$i:=i+1$. 3: Let $f$ be the final value of $i$. Return the tree
$T_{alg}:=T_{f}\cup\bigcup_{i=1}^{f-1}{\tt loss}(K_{i}).$
We now analyze the algorithm.
###### Claim 4.1.
$c(T_{f})\leq(\frac{1+\alpha}{2})\mathop{\mathrm{OPT}}\eqref{eq:LP-PU}$.
###### Proof.
Using the contraction lemma again, ${\tt gain}_{T_{f}}(K;c)\leq(\alpha-1){\tt
loss}(K)$ for all $K$, so
$\displaystyle{\tt drop}_{T_{f}}(K;c)\leq c(K)+(\alpha-1){\tt
loss}(K)=c(K)+(\alpha-1){\tt loss}(K)\leq\Big{(}\frac{1+\alpha}{2}\Big{)}c(K)$
(23)
since ${\tt loss}(K)\leq c(K)/2$.
To finish the proof of Claim 4.1, we proceed as in the proof of Equation (22).
Define $c^{\prime}_{e}:=c_{e}/(\frac{1+\alpha}{2})$ for all edges $e$ which
join two vertices of the original terminal set $R$, and $c^{\prime}_{e}=c_{e}$
for all other edges. Note that (23) implies that $T_{f}$ is gainless with
respect to $c^{\prime}$. Thus, by Theorem 8, the value of LP ($\mathcal{P}$)
on $(G_{f},c^{\prime})$ equals $c^{\prime}(T_{f})$. Since we only reduce costs
(as $\alpha\geq 1$), this optimum is no more than the original
$\mathop{\mathrm{OPT}}\eqref{eq:LP-PU}$ giving us
$c^{\prime}(T_{f})\leq\mathop{\mathrm{OPT}}\eqref{eq:LP-PU}$. Now using the
definition of $c^{\prime}$, the proof of the claim is complete. ∎
###### Claim 4.2.
For any $i\geq 1$, we have $c(T_{i})-c(T_{i+1})\geq{\tt
gain}_{T_{i}}(K_{i};c)+{\tt loss}(K_{i})$.
###### Proof.
Recall that $T_{i+1}$ is a minimum terminal spanning tree of $G_{i+1}$ under
$c$. Consider the following other terminal spanning tree $T$ of $G_{i+1}$:
take $T$ to be the union of $K_{i}/{\tt loss}(K_{i})$ with
${\mathtt{mtst}}(G_{i}/K_{i};c)$. Hence $c(T_{i+1})\leq
c(T)={\mathtt{mtst}}(G_{i}/K_{i};c)+c(K_{i})-{\tt loss}(K_{i})$. Rearranging,
and using the definition of gain, we obtain:
$c(T_{i})-c(T_{i+1})\geq c(T_{i})-{\mathtt{mtst}}(G_{i}/K_{i};c)-c(K_{i})+{\tt
loss}(K_{i})={\tt gain}_{T_{i}}(K_{i};c)+{\tt loss}(K_{i}),$
and this completes the proof. ∎
Now we are ready to prove the integrality gap upper bound of $\sqrt{3}$.
###### Theorem 10.
$c(T_{alg})\leq\sqrt{3}\mathop{\mathrm{OPT}}\eqref{eq:LP-PU}$.
###### Proof.
By the algorithm, we have for all $i$ that ${\tt
gain}_{T_{i}}(K_{i})\geq(\alpha-1){\tt loss}(K_{i})$, and thus ${\tt
gain}_{T_{i}}(K_{i};c)+{\tt loss}(K_{i})\geq\alpha{\tt loss}(K_{i})$. Thus,
from Claim 4.2, we get
$\sum_{i=1}^{f-1}{\tt
loss}(K_{i})\leq\frac{1}{\alpha}\sum_{i=1}^{f-1}\Big{(}c(T_{i})-c(T_{i+1})\Big{)}$
The right-hand sum telescopes to give us
$c(T_{1})-c(T_{f})={\mathtt{mtst}}(G;c)-c(T_{f})$. Thus,
$\displaystyle c(T_{alg})$ $\displaystyle=c(T_{f})+\sum_{i=1}^{f-1}{\tt
loss}(K_{i})\leq
c(T_{f})+\frac{1}{\alpha}({\mathtt{mtst}}(G;c)-c(T_{f}))=\frac{1}{\alpha}{\mathtt{mtst}}(G;c)+\frac{\alpha-1}{\alpha}c(T_{f})$
$\displaystyle\leq\Big{(}\frac{2}{\alpha}+\frac{(\alpha-1)(1+\alpha)}{2\alpha}\Big{)}\mathop{\mathrm{OPT}}\eqref{eq:LP-
PU}=\Big{(}\frac{\alpha^{2}+3}{2\alpha}\Big{)}\mathop{\mathrm{OPT}}\eqref{eq:LP-
PU}$
which follows from ${\mathtt{mtst}}(G;c)\leq
2\mathop{\mathrm{OPT}}\eqref{eq:LP-PU}$ and Claim 4.1. Setting
$\alpha=\sqrt{3}$, the proof of the theorem is complete. ∎
## 5 Conclusion
In this paper we looked at several hypergraphic LP relaxations for the Steiner
tree problem, and showed they all have the same objective value. Furthermore,
we noted some connections to the bidirected cut relaxation for Steiner trees:
although hypergraphic relaxations are stronger than the bidirected cut
relaxation in general, in quasibipartite graphs all these relaxations are
equivalent. We obtained structural results about the hypergraphic relaxations
showing that basic feasible solutions have sparse support. We also showed
improved upper bounds on the integrality gaps on the hypergraphic relaxations
via simple algorithms.
Reiterating the comments in Section 1.2.3, the hypergraphic LPs are powerful
(e.g. as evidenced by Byrka et al. [3]) but may not be manageable for
computational implementation. Some interesting areas for future work include:
non-ellipsoid-based algorithms to solve the hypergraphic LPs in the
$r$-restricted setting; resolving the complexity of optimizing them in the
unrestricted setting; and directly using the bidirected cut relaxation to
achieve good results (e.g. in quasi-bipartite instances).
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## Appendix A Directed Hypergraph LP Relaxation
###### Theorem 11.
For any Steiner tree instance, $\mathop{\mathrm{OPT}}\eqref{eq:LP-
PU}=\mathop{\mathrm{OPT}}\eqref{eq:LP-PUDir}$.
###### Proof.
First, we show $\mathop{\mathrm{OPT}}\eqref{eq:LP-
PU}\leq\mathop{\mathrm{OPT}}\eqref{eq:LP-PUDir}$. Consider a feasible solution
$x$ to $\eqref{eq:LP-PUDir}$, and define a solution $x^{\prime}$ to
$\eqref{eq:LP-PU}$ by $x^{\prime}_{K}=\sum_{i\in K}x_{K^{i}}$; informally,
$x^{\prime}$ is obtained from $x$ by ignoring the orientation of the
hyperedges. Clearly $x^{\prime}$ and $x$ have the same objective value.
Further, $x^{\prime}$ is feasible for $\eqref{eq:LP-PU}$; to see this, for any
partition $\pi$, note that (5) is implied by the sum of constraints (3) over
$U$ set to those parts of $\pi$ not containing the root — any orientation of a
full component with rank contribution $t$ must leave at least $t$ parts.
To obtain the reverse direction $\mathop{\mathrm{OPT}}\eqref{eq:LP-
PUDir}\leq\mathop{\mathrm{OPT}}\eqref{eq:LP-PU}$, we use a similar strategy.
We require some notation and a hypergraph orientation theorem of Frank et al.
[14]. For any $U\subset R$ we say that a directed hyperedge _$K^{i}$ lies in
$\Delta^{\mbox{\scriptsize{$\mathrm{in}$}}}(U)$_ if $i\in U$ and $K\backslash
U\neq\varnothing$, i.e. if
$K^{i}\in\Delta^{\mbox{\scriptsize{$\mathrm{out}$}}}(R\backslash U)$. Two
subsets $U$ and $W$ of $R$ are called _crossing_ if all four sets $U\setminus
W$, $W\setminus U$, $U\cap W$, and $R\setminus(U\cup W)$ are non-empty. A set-
function $p:2^{R}\to{\mathbb{Z}}$ is a _crossing supermodular_ function if
$p(U)+p(W)\leq p(U\cap W)+p(U\cup W)$
for all crossing sets $U$ and $W$. A directed hypergraph is said to _cover_
$p$ if $|\Delta^{\mbox{\scriptsize{$\mathrm{in}$}}}(U)|\geq p(U)$ for all
$U\subset R$. Here is the needed result.
###### Theorem 12 (Frank, Király & Király [14]).
Given a hypergraph $H=(R,\mathcal{X})$, and a crossing supermodular function
$p$, the hypergraph has an orientation covering $p$ if and only if for every
partition $\pi$ of $R$,
_(a)_
$\sum_{K\in\mathcal{X}}\min\\{1,\mathtt{rc}^{\pi}_{K}\\}\geq\sum_{\pi_{i}\in\pi}p(\pi_{i})$,
and, _(b)_
$\sum_{K\in\mathcal{X}}\mathtt{rc}^{\pi}_{K}\geq\sum_{\pi_{i}\in\pi}p(R\setminus\pi_{i})$.
We will show every rational solution $x$ to $\eqref{eq:LP-PU}$ can be
fractionally oriented to get a feasible solution for $\eqref{eq:LP-PUDir}$,
which will complete the proof of Theorem 11. Let $M$ be the smallest integer
such that the vector $Mx$ is integral. Let $\mathcal{X}$ be a multi-set of
hyperedges which contains $Mx_{K}$ copies of each $K$. Define the function $p$
by $p(U)=M$ if $r\in U\neq R$, and $p(U)=0$ otherwise; i.e. $p(U)=M$ iff
$R\backslash U$ is valid.
###### Claim A.1.
$H=(R,\mathcal{X})$ satisfies conditions (a) and (b).
###### Proof.
Note $\sum_{\pi_{i}\in\pi}p(R\setminus\pi_{i})=M(r(\pi)-1)$ since all parts of
$\pi$ are valid except the part containing the root $r$. Thus condition (b),
upon scaling by $\frac{1}{M}$, is a restatement of constraint (5), which holds
since $x$ is feasible for ($\mathcal{P}$).
For this $p$, condition (a) follows from (b) in the following sense. Fix a
partition $\pi$, and let $\pi_{1}$ be the part of $\pi$ containing $r$. If
$\pi_{1}=R$ then (a) is vacuously true, so assume $\pi_{1}\neq R$. Let
$\sigma$ be the rank-2 partition $\\{\pi_{1},R\setminus\pi_{1}\\}$. Then it is
easy to check that
$\min\\{1,\mathtt{rc}^{\pi}_{K}\\}\geq\mathtt{rc}^{\sigma}_{K}$ for all $K$,
and consequently
$\sum_{K\in\mathcal{X}}\min\\{1,\mathtt{rc}^{\pi}_{K}\\}\geq\sum_{K\in\mathcal{X}}\mathtt{rc}^{\sigma}_{K}$
and
$\sum_{\pi_{i}\in\sigma}p(R\setminus\pi_{i})=M=\sum_{\pi_{i}\in\pi}p(\pi_{i})$.
Thus, (a) for $\pi$ follows from (b) for $\sigma$. ∎
It is not hard to check that $p$ is crossing supermodular. Now using Theorem
12, take an orientation of $\mathcal{X}$ that covers $p$.
For each $K\in\mathcal{K}$ and each $i\in K$, let $n_{K^{i}}$ denote the
number of the $Mx_{K}$ copies of $K$ that are oriented as $K^{i}$, i.e.
directed towards $i$. So, $\sum_{i\in K}n_{K^{i}}=Mx_{K}$. Let
$x^{\prime}_{K^{i}}:=\frac{n_{K^{i}}}{M}$ for all $K^{i}$. Hence
$\sum_{i}x^{\prime}_{K^{i}}=x_{K}$ and $x^{\prime}$ has the same objective
value as $x$.
To complete the proof, we show $x^{\prime}$ is feasible for ($\mathcal{D}$).
Fix a valid subset $U$ and consider condition (3) for a valid set $U$. Note
that $p(R\backslash U)=M$. Therefore, since the orientation covers $p$, we get
$\sum_{K^{i}\in\Delta^{\mbox{\scriptsize{$\mathrm{out}$}}}(U)}x^{\prime}_{K^{i}}=\frac{1}{M}\sum_{K^{i}\in\Delta^{\mbox{\scriptsize{$\mathrm{out}$}}}(U)}n_{K^{i}}=\frac{1}{M}\sum_{K^{i}\in\Delta^{\mbox{\scriptsize{$\mathrm{in}$}}}(R\backslash
U)}n_{K^{i}}\geq\frac{1}{M}p(R\backslash U)=\frac{1}{M}M=1$
as needed. ∎
## Appendix B Gainless MSTs and Hypergraphic Relaxations
###### Theorem 8 (Implicit in [23]).
If the MST induced by the terminals is gainless, then
$\mathop{\mathrm{OPT}}\eqref{eq:LP-PU}$ equals the cost of that MST.
###### Proof.
Let $\Pi$ be the set of all partitions of the terminal set. As before, we let
$r(\pi)$ be the rank of a partition $\pi\in\Pi$, and we use $E_{\pi}$ for the
set of edges in our graph that cross the partition; i.e., $E_{\pi}$ contains
all edges whose endpoints lie in different parts of $\pi$. Fulkerson’s [15]
formulation of the spanning tree polyhedron and its dual are as follows.
$\displaystyle\min\Big{\\{}\sum_{e\in E}c_{e}x_{e}:\quad$ $\displaystyle
x\in\mathbf{R}^{E}_{\geq 0}$ ($\mathcal{M}$) $\displaystyle\sum_{e\in
E_{\pi}}x_{e}\geq r(\pi)-1\quad$ $\displaystyle\forall\pi\in\Pi\Big{\\}}$ (24)
$\displaystyle\max\Big{\\{}\sum_{\pi}(r(\pi)-1)\cdot y_{\pi}:\quad$
$\displaystyle y\in\mathbf{R}^{\Pi}_{\geq 0}$ ($\mathcal{M}_{D}$)
$\displaystyle\sum_{\pi:e\in E_{\pi}}y_{\pi}\leq c_{e},\quad$
$\displaystyle\forall e\in E\Big{\\}}$ (25)
The high-level overview of the proof is as follows. We first give a brief
sketch of a folklore primal-dual interpretation of Kruskal’s minimum-spanning
tree algorithm with respect to Fulkerson’s LP (for more information see, e.g.,
[23]). Running Kruskal’s algorithm on the terminal set then returns a minimum
spanning tree $T$ and a feasible dual $y$ to Equation ($\mathcal{M}_{D}$) such
that
$c(T)=\sum_{\pi}(r(\pi)-1)y_{\pi}.$
The final step will be to show that, if the returned MST is gainless, then the
spanning tree dual $y$ is feasible for ($\mathcal{P}_{D}$), and its value is
$c(T)$ as well. Weak duality and the fact that the optimal value of
($\mathcal{P}$) is at most $c(T)$ imply the theorem.
Kruskal’s algorithm can be viewed as a process over time. For each time
$\tau\geq 0$, the algorithm keeps a forest $T^{\tau}$, and a feasible dual
solution $y^{\tau}$; initially $T^{0}=(V,\varnothing)$ and $y^{0}=0$. Let
$\pi^{\tau}$ be the partition induced by the connected components of
$T^{\tau}$. If $T^{\tau}$ is not a spanning tree, Kruskal’s algorithm grows
the dual variable $y_{\pi^{\tau}}$ corresponding to the current partition
until constraint Equation ($\mathcal{M}_{D}$)e: for some edge $e$ prevents any
further increase. The algorithm then adds $e$ to the partial tree and
continues. The algorithm stops at the first time $\tau^{*}$ where
$T^{\tau^{*}}$ is a spanning tree.
Let $T$ be the gainless spanning tree returned by Kruskal, and let $y$ be the
corresponding dual. We claim that $y$ is feasible for ($\mathcal{P}_{D}$). To
see this, consider a full component $K$. Clearly, the rank contribution
$\mathtt{rc}^{\pi^{0}}_{K}$ of $K$ to the initial partition $\pi^{0}$ is
$|K|-1$; similarly, the final rank contribution
$\mathtt{rc}^{\pi^{\tau^{*}}}_{K}$ is $0$. Every edge that is added during the
algorithm’s run either leaves the rank contribution of $K$ unchanged, or it
decreases it by $1$. Let $e_{1},\ldots,e_{|K|-1}$ be the edges of the final
tree $T$ whose addition to $T$ decreases $K$’s rank contribution. Also let
$0\leq\tau_{1}\leq\tau_{2}\leq\ldots\leq\tau_{|K|-1}\leq\tau^{*}$
be the times where these edges are added. Note that, by definition, we must
have $c_{e_{i}}=\tau_{i}$ for all $i$. We therefore have
$\sum_{i=1}^{|K|-1}c_{e_{i}}=\sum_{i=1}^{|K|-1}\tau_{i}.$ (26)
The right-hand side of this equality is easily checked to be equal to
$\int_{0}^{\tau^{*}}\mathtt{rc}^{\pi^{\tau}}_{K}d\tau,$
which in turn is equal to $\sum_{\pi}\mathtt{rc}^{\pi}_{K}y_{\pi}$, by the
definition of Kruskal’s algorithm. It is not hard to see that the left-hand
side of (26) is the drop ${\tt drop}_{T}(K)$ induced by $K$. Together with the
fact that $T$ is gainless, we obtain
$c_{K}\geq{\tt drop}_{T}(K)=\sum_{\pi}\mathtt{rc}^{\pi}_{K}y_{\pi}.$
Now observe that the right-hand side of this equation is the left-hand side of
(6). It follows that $y$ is feasible for ($\mathcal{P}_{D}$). ∎
|
arxiv-papers
| 2009-10-01T21:57:06 |
2024-09-04T02:49:05.588315
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Deeparnab Chakrabarty, Jochen Koenemann, David Pritchard",
"submitter": "David Pritchard",
"url": "https://arxiv.org/abs/0910.0281"
}
|
0910.0510
|
# Thermodynamics of interacting holographic dark energy
with apparent horizon as an IR cutoff
Ahmad Sheykhi 111 sheykhi@mail.uk.ac.ir Department of Physics, Shahid Bahonar
University, P.O. Box 76175, Kerman, Iran
Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), Maragha,
Iran
###### Abstract
As soon as an interaction between holographic dark energy and dark matter is
taken into account, the identification of IR cutoff with Hubble radius
$H^{-1}$, in flat universe, can simultaneously drive accelerated expansion and
solve the coincidence problem. Based on this, we demonstrate that in a non-
flat universe the natural choice for IR cutoff could be the apparent horizon
radius, $\tilde{r}_{A}={1}/{\sqrt{H^{2}+k/a^{2}}}$. We show that any
interaction of dark matter with holographic dark energy, whose infrared cutoff
is set by the apparent horizon radius, implies an accelerated expansion and a
constant ratio of the energy densities of both components thus solving the
coincidence problem. We also verify that for a universe filled with dark
energy and dark matter the Friedmann equation can be written in the form of
the modified first law of thermodynamics, $dE=T_{h}dS_{h}+WdV$, at apparent
horizon. In addition, the generalized second law of thermodynamics is
fulfilled in a region enclosed by the apparent horizon. These results hold
regardless of the specific form of dark energy and interaction term. Our study
might reveal that in an accelerating universe with spatial curvature, the
apparent horizon is a physical boundary from the thermodynamical point of
view.
## I Introduction
The combined analysis of cosmological observations reveal that nearly three
quarters of our universe consists of a mysterious energy component usually
dubbed “dark energy” which is responsible for the cosmic expansion, and the
remaining part consists of pressureless matter Rie . The nature of such
previously unforeseen energy still remains a complete mystery, except for the
fact that it has negative pressure. In this new conceptual set up, one of the
important questions concerns the thermodynamical behavior of the accelerated
expanding universe driven by dark energy. It is important to ask whether
thermodynamics in an accelerating universe can reveal some properties of dark
energy. The profound connection between thermodynamics and gravity has been
observed in the cosmological situations Cai2 ; Cai3 ; CaiKim ; Fro ; Wang ;
Cai4 ; Shey1 ; Shey2 ; Shey3 . This connection implies that the
thermodynamical properties can help understand the dark energy, which gives
strong motivation to study thermodynamics in the accelerating universe. It is
also of great interest to investigate the validity of the generalized second
law of thermodynamics in the accelerating universe driven by dark energy
wangb0 . The generalized second law of thermodynamics is an important
principle in governing the development of the nature.
An interesting attempt for probing the nature of dark energy within the
framework of quantum gravity, is the so-called “Holographic Dark Energy” (HDE)
proposal. This model which has arisen a lot of enthusiasm recently Coh ; Li ;
Huang ; Hsu ; HDE ; Setare1 ; wang0 ; wang1 , is motivated from the
holographic hypothesis Suss1 and has been tested and constrained by various
astronomical observations Xin . It is important to note that in the
literature, various scenarios of HDE have been studied via considering
different system’s IR cutoff. In the absence of interaction between dark
matter and dark energy in flat universe, Li Li discussed three choices for
the length scale $L$ which is supposed to provide an IR cutoff. The first
choice is the Hubble radius, $L=H^{-1}$ Hsu , which leads to a wrong equation
of state, namely that for dust. The second option is the particle horizon
radius. In this case it is impossible to obtain an accelerated expansion. Only
the third choice, the identification of $L$ with the radius of the future
event horizon gives the desired result, namely a sufficiently negative
equation of state to obtain an accelerated universe.
However, as soon as an interaction between dark energy and dark matter is
taken into account, the first choice, $L=H^{-1}$, in flat universe, can
simultaneously drive accelerated expansion and solve the coincidence problem
pav1 . Based on this, we demonstrate that in a non-flat universe the natural
choice for IR cutoff could be the apparent horizon radius. We show that any
interaction of pressureless dark matter with HDE, whose infrared cutoff is set
by the apparent horizon radius, implies a constant ratio of the energy
densities of both components thus solving the coincidence problem. Besides, it
was argued that for an accelerating universe inside the event horizon the
generalized second law does not satisfy, while the accelerating universe
enveloped by the Hubble horizon satisfies the generalized second law Jia .
This implies that the event horizon in an accelerating universe might not be a
physical boundary from the thermodynamical point of view. Thus, it looks that
we need to define a convenient horizon that satisfies all of our accepted
principles in a universe with any spacial curvature. In the next section, we
study the interacting HDE with apparent horizon as an IR cutoff. In section
III, we examine the first law of thermodynamics on the apparent horizon in an
accelerating universe with spacial curvature. In section IV, we investigate
the validity of the generalized second law of thermodynamics in a region
enclosed by the apparent horizon. The last section is devoted to conclusions.
## II Interacting HDE with apparent horizon as an IR cutoff
We consider a homogenous and isotropic Friedmann-Robertson-Walker (FRW)
universe which is described by the line element
$ds^{2}={h}_{\mu\nu}dx^{\mu}dx^{\nu}+\tilde{r}^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}),$ (1)
where $\tilde{r}=a(t)r$, $x^{0}=t,x^{1}=r$, the two dimensional metric
$h_{\mu\nu}$=diag $(-1,a^{2}/(1-kr^{2}))$. Here $k$ denotes the curvature of
space with $k=0,1,-1$ corresponding to open, flat, and closed universes,
respectively. A closed universe with a small positive curvature
($\Omega_{k}\simeq 0.01$) is compatible with observations spe . Then, the
dynamical apparent horizon, a marginally trapped surface with vanishing
expansion, is determined by the relation
$h^{\mu\nu}\partial_{\mu}\tilde{r}\partial_{\nu}\tilde{r}=0$, which implies
that the vector $\nabla\tilde{r}$ is null on the apparent horizon surface. The
apparent horizon was argued as a causal horizon for a dynamical spacetime and
is associated with gravitational entropy and surface gravity Hay2 ; Bak . A
simple calculation gives the apparent horizon radius for the FRW universe
$\tilde{r}_{A}=\frac{1}{\sqrt{H^{2}+k/a^{2}}}.$ (2)
The corresponding Friedmann equation takes the form
$\displaystyle H^{2}+\frac{k}{a^{2}}=\frac{8\pi
G}{3}\left(\rho_{m}+\rho_{D}\right),$ (3)
where $\rho_{m}$ and $\rho_{D}$ are the energy density of dark matter and dark
energy inside apparent horizon, respectively. Since we consider the
interaction between dark matter and dark energy, $\rho_{m}$ and $\rho_{D}$ do
not conserve separately; they must rather enter the energy balances
$\displaystyle\dot{\rho}_{m}+3H\rho_{m}=Q,$ (4)
$\displaystyle\dot{\rho}_{D}+3H\rho_{D}(1+w_{D})=-Q.$ (5)
where $w_{D}=p_{D}/\rho_{D}$ is the equation of state parameter of HDE, and
$Q$ stands for the interaction term. We also ignore the baryonic matter
($\Omega_{BM}\approx 0.04$) in comparison with dark matter and dark energy
($\Omega_{DM}+\Omega_{DE}\approx 0.96$). We shall assume the ansatz
$Q=\Gamma\rho_{D}$ with $\Gamma>0$ which means that there is an energy
transfer from the dark energy to dark matter. It is important to note that the
continuity equations imply that the interaction term should be a function of a
quantity with units of inverse of time (a first and natural choice can be the
Hubble factor $H$) multiplied with the energy density. Therefore, the
interaction term could be in any of the following forms: (i) $Q\propto
H\rho_{D}$, (ii) $Q\propto H\rho_{m}$, or (iii) $Q\propto
H(\rho_{m}+\rho_{D})$. However, we can present the above three forms in one
expression as $Q=\Gamma\rho_{D}$, where
$\displaystyle\begin{array}[]{ll}\Gamma=3b^{2}H\hskip 36.98866pt{\rm for}\ \
Q\propto H\rho_{D},&\\\ \Gamma=3b^{2}Hu\hskip 31.2982pt{\rm for}\ \ Q\propto
H\rho_{m},&\\\ \Gamma=3b^{2}H(1+u)\ \ {\rm for}\ \ Q\propto
H(\rho_{m}+\rho_{D}),&\end{array}$ (9)
with $b^{2}$ is a coupling constant and $u=\rho_{m}/\rho_{D}$ is the ratio of
energy densities. The freedom of choosing the specific form of the interaction
term $Q$ stems from our incognizance of the origin and nature of dark energy
as well as dark matter. Moreover, a microphysical model describing the
interaction between the dark components of the universe is not available
nowadays. If we introduce, as usual, the fractional energy densities such as
$\displaystyle\Omega_{m}=\frac{8\pi G\rho_{m}}{3H^{2}},\hskip
14.22636pt\Omega_{D}=\frac{8\pi G\rho_{D}}{3H^{2}},\hskip
14.22636pt\Omega_{k}=\frac{k}{H^{2}a^{2}},$ (10)
then, the Friedmann equation can be written as
$\Omega_{m}+\Omega_{D}=1+\Omega_{k}$. In terms of the apparent horizon radius,
we can rewrite the Friedmann equation as
$\frac{1}{\tilde{r}_{A}^{2}}=\frac{8\pi G}{3}\left(\rho_{m}+\rho_{D}\right).$
(11)
For completeness, we give the deceleration parameter
$q=-\frac{\ddot{a}}{aH^{2}}=-1-\frac{\dot{H}}{H^{2}},$ (12)
which combined with the Hubble parameter and the dimensionless density
parameters form a set of useful parameters for the description of the
astrophysical observations. It is a matter of calculation to show that
$q=-(1+\Omega_{k})+\frac{3}{2}\Omega_{D}(1+u+w_{D}).$ (13)
The evolution of $u$ is governed by
$\dot{u}=3Hu\left[w_{D}+\frac{1+u}{u}\frac{\Gamma}{3H}\right].$ (14)
We assume the HDE density has the form
$\rho_{D}=\frac{3c^{2}}{8\pi G\tilde{r}_{A}^{2}},$ (15)
where $c^{2}$ is a constant, the coefficient $3$ is for convenient, and we
have set the apparent horizon radius $L={\tilde{r}_{A}}$ as system’s IR cutoff
in holographic model of dark energy. Inserting Eq. (15) in Eq. (11)
immediately yields
$\rho_{m}=\frac{3(1-c^{2})}{8\pi G\tilde{r}_{A}^{2}}.$ (16)
Thus we reach
$u=\frac{\rho_{m}}{\rho_{D}}=\frac{1-c^{2}}{c^{2}}.$ (17)
This implies that the ratio of the energy densities is a constant; thus the
coincidence problem can be solved. Taking the derivative of Eq. (15) we get
$\dot{\rho}_{D}=-2\rho_{D}\frac{\dot{\tilde{r}_{A}}}{\tilde{r}_{A}}=-3c^{2}H\rho_{D}(1+u+w_{D}).$
(18)
where we have employed Eqs. (4), (5) and (11). Combining this equation with
(5) we obtain
$w_{D}=-\left(1+\frac{1}{u}\right)\frac{\Gamma}{3H}.$ (19)
Substituting $w_{D}$ into (13), we find
$q=-(1+\Omega_{k})-\frac{3}{2}\Omega_{D}(1+u)\left(\frac{\Gamma}{3Hu}-1\right).$
(20)
The interaction parameter $\frac{\Gamma}{3H}$ together with the energy density
ratio $u$ determine the equation of state parameter. In the absence of
interaction, we encounter dust with $w_{D}=0$. For the choice
$L=\tilde{r}_{A}$ an interaction is the only way to have an equation of state
different from that for dust. Any decay of the dark energy component into
pressureless matter is necessarily accompanied by an equation of state
$w_{D}<0$. The existence of an interaction has another interesting
consequence. Inserting expression $w_{D}$ into (14) leads to $\dot{u}=0$,
i.e., $u$ = const. Thus, any interaction of dark matter with HDE, whose
infrared cutoff is set by the apparent horizon radius, implies an accelerated
expansion and a constant ratio of the energy densities, irrespective of the
specific structure of the interaction. It is important to note that although
choosing $L=H^{-1}$, in a spatially flat universe, can drive accelerated
expansion and solve the coincidence problem pav1 , but taking into account the
spatial curvature term gives rise to an additional dynamics which implies a
small (compared with the Hubble rate) change of the energy density ratio; thus
the coincidence problem cannot be solved exactly (see pav2 for details). This
implies that in an accelerating universe with spacial curvature the Hubble
radius $H^{-1}$ is not a convenient choice.
In summary, in a universe with spacial curvature, the identification of IR
cutoff with apparent horizon radius $\tilde{r}_{A}$ is not only the most
obvious but also the simplest choice which can simultaneously drive
accelerated expansion and solve the coincidence problem. It is important to
note that the interaction is essential to simultaneously solve the coincidence
problem and have late acceleration. There is no non-interacting limit, since
in the absence of interaction, i.e., $\Gamma=0$, there is no acceleration.
## III First law of thermodynamics
In this section we are going to examine the first law of thermodynamics. In
particular, we show that for a closed universe filled with HDE and dark matter
the Friedmann equation can be written directly in the form of the modified
first law of thermodynamics at apparent horizon regardless of the specific
form of the dark energy. The associated temperature with the apparent horizon
can be defined as $T=\kappa/2\pi$, where $\kappa$ is the surface gravity
$\kappa=\frac{1}{\sqrt{-h}}\partial_{\mu}\left(\sqrt{-h}h^{\mu\nu}\partial_{\mu\nu}\tilde{r}\right).$
Then one can easily show that the surface gravity at the apparent horizon of
FRW universe can be written as
$\kappa=-\frac{1}{\tilde{r}_{A}}\left(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\right).$
(21)
When $\dot{\tilde{r}}_{A}\leq 2H\tilde{r}_{A}$, the surface gravity
$\kappa\leq 0$, which leads the temperature $T\leq 0$ if one defines the
temperature of the apparent horizon as $T=\kappa/2\pi$ . Physically it is not
easy to accept the negative temperature, the temperature on the apparent
horizon should be defined as $T=|\kappa|/2\pi$. Recently the connection
between temperature on the apparent horizon and the Hawking radiation has been
considered in cao , which gives more solid physical implication of the
temperature associated with the apparent horizon.
Taking differential form of equation (11) and using Eqs. (4) and (5), we can
get the differential form of the Friedmann equation
$\frac{1}{4\pi
G}\frac{d\tilde{r}_{A}}{\tilde{r}_{A}^{3}}=H\rho_{D}\left(1+u+w_{D}\right)dt.$
(22)
Multiplying both sides of the equation (22) by a factor
$4\pi\tilde{r}_{A}^{3}\left(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\right)$,
and using the expression (21) for the surface gravity, after some
simplification one can rewrite this equation in the form
$\displaystyle-\frac{\kappa}{2\pi}\frac{2\pi\tilde{r}_{A}d\tilde{r}_{A}}{G}$
$\displaystyle=$ $\displaystyle
4\pi\tilde{r}_{A}^{3}H\rho_{D}\left(1+u+w_{D}\right)dt-2\pi\tilde{r}_{A}^{2}\rho_{D}\left(1+u+w_{D}\right)d\tilde{r}_{A}.$
(23)
$E=(\rho_{m}+\rho_{D})V$ is the total energy content of the universe inside a
$3$-sphere of radius $\tilde{r}_{A}$, where
$V=\frac{4\pi}{3}\tilde{r}_{A}^{3}$ is the volume enveloped by 3-dimensional
sphere with the area of apparent horizon $A=4\pi\tilde{r}_{A}^{2}$. Taking
differential form of the relation
$E=(\rho_{m}+\rho_{D})\frac{4\pi}{3}\tilde{r}_{A}^{3}$ for the total matter
and energy inside the apparent horizon, we get
$dE=4\pi\tilde{r}_{A}^{2}(\rho_{m}+\rho_{D})d\tilde{r}_{A}+\frac{4\pi}{3}\tilde{r}_{A}^{3}(\dot{\rho}_{m}+\dot{\rho}_{D})dt.$
(24)
Using Eqs. (4) and (5), we obtain
$dE=4\pi\tilde{r}_{A}^{2}\rho_{D}(1+u)d\tilde{r}_{A}-4\pi\tilde{r}_{A}^{3}H\rho_{D}\left(1+u+w_{D}\right)dt.$
(25)
Substituting this relation into (23), and using the relation between
temperature and the surface gravity, we get the modified first law of
thermodynamics on the apparent horizon
$dE=T_{h}dS_{h}+WdV,$ (26)
where $S_{h}={A}/{4G}$ is the entropy associated to the apparent horizon, and
$W=\frac{1}{2}(\rho_{m}+\rho_{D}-p_{D})=\frac{1}{2}\rho_{D}\left(1+u-w_{D}\right)$
(27)
is the matter work density Hay2 . The work density term is regarded as the
work done by the change of the apparent horizon, which is used to replace the
negative pressure if compared with the standard first law of thermodynamics,
$dE=TdS-pdV$. For a pure de Sitter space, $\rho_{m}+\rho_{D}=-p_{D}$, then our
work term reduces to the standard $-p_{D}dV$ and we obtain exactly the first
law of thermodynamics.
## IV Generalized Second law of thermodynamics
In this section we turn to investigate the validity of the generalized second
law of thermodynamics in a region enclosed by the apparent horizon.
Differentiating Eq. (11) with respect to the cosmic time and using Eqs. (4)
and (5) we get
$\dot{\tilde{r}}_{A}=4\pi GH{\tilde{r}_{A}^{3}}\rho_{D}(1+u+w_{D}).$ (28)
One can see from the above equation that $\dot{\tilde{r}}_{A}>0$ provided
condition $w_{D}>-1-u$, holds. Let us now turn to find out $T_{h}\dot{S_{h}}$:
$T_{h}\dot{S_{h}}=\frac{1}{2\pi\tilde{r}_{A}}\left(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\right)\frac{d}{dt}\left(\frac{\pi\tilde{r}_{A}^{2}}{G}\right).$
(29)
After some simplification and using Eq. (28) we get
$T_{h}\dot{S_{h}}=4\pi
H{\tilde{r}_{A}^{3}}\rho_{D}(1+u+w_{D})\left(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\right).$
(30)
As we argued above the term
$\left(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\right)$ is positive to
ensure $T_{h}>0$, however, in an accelerating universe the equation of state
parameter of dark energy may cross the phantom divide, i.e., $w_{D}<-1-u$.
This indicates that the second law of thermodynamics, $\dot{S_{h}}\geq 0$,
does not hold on the apparent horizon. Then the question arises, “will the
generalized second law of thermodynamics,
$\dot{S_{h}}+\dot{S_{m}}+\dot{S_{D}}\geq 0$, can be satisfied in a region
enclosed by the apparent horizon?” The entropy of dark energy plus dark matter
inside the apparent horizon, $S=S_{m}+S_{D}$, can be related to the total
energy $E=(\rho_{m}+\rho_{D})V$ and pressure $p_{D}$ in the horizon by the
Gibbs equation Pavon2
$TdS=d[(\rho_{m}+\rho_{D})V]+p_{D}dV=V(d\rho_{m}+d\rho_{D})+\rho_{D}(1+u+w_{D})dV,$
(31)
where $T=T_{m}=T_{D}$ and $S=S_{m}+S_{D}$ are the temperature and the total
entropy of the energy and matter content inside the horizon, respectively.
Here we assumed that the temperature of both dark components are equal, due to
their mutual interaction. We also limit ourselves to the assumption that the
thermal system bounded by the apparent horizon remains in equilibrium so that
the temperature of the system must be uniform and the same as the temperature
of its boundary. This requires that the temperature $T$ of the energy content
inside the apparent horizon should be in equilibrium with the temperature
$T_{h}$ associated with the apparent horizon, so we have $T=T_{h}$Pavon2 .
This expression holds in the local equilibrium hypothesis. If the temperature
of the fluid differs much from that of the horizon, there will be spontaneous
heat flow between the horizon and the fluid and the local equilibrium
hypothesis will no longer hold. This is also at variance with the FRW
geometry. In general, when we consider the thermal equilibrium state of the
universe, the temperature of the universe is associated with the apparent
horizon. Therefore from the Gibbs equation (31) we can obtain
$T_{h}(\dot{S_{m}}+\dot{S_{D}})=4\pi{\tilde{r}_{A}^{2}}\rho_{D}(1+u+w_{D})\dot{\tilde{r}}_{A}-4\pi
H{\tilde{r}_{A}^{3}}\rho_{D}(1+u+w_{D}).$ (32)
To check the generalized second law of thermodynamics, we have to examine the
evolution of the total entropy $S_{h}+S_{m}+S_{D}$. Adding equations (30) and
(32), we get
$T_{h}(\dot{S}_{h}+\dot{S}_{m}+\dot{S}_{D})=2\pi{\tilde{r}_{A}^{2}}\rho_{D}(1+u+w_{D})\dot{\tilde{r}}_{A}=\frac{A}{2}\rho_{D}(1+u+w_{D})\dot{\tilde{r}}_{A}.$
(33)
where $A>0$ is the area of apparent horizon. Substituting
$\dot{\tilde{r}}_{A}$ from Eq. (28) into (33) we get
$T_{h}(\dot{S}_{h}+\dot{S}_{m}+\dot{S}_{D})=2\pi
GAH{\tilde{r}_{A}}^{3}\rho^{2}_{D}(1+u+w_{D})^{2}.$ (34)
The right hand side of the above equation cannot be negative throughout the
history of the universe, which means that
$\dot{S_{h}}+\dot{S_{m}}+\dot{S_{D}}\geq 0$ always holds. This indicates that
for a universe with spacial curvature filled with interacting dark components,
the generalized second law of thermodynamics is fulfilled in a region enclosed
by the apparent horizon.
## V conculusions
It is worthwhile to note that in the literature, various scenarios of HDE have
been studied via considering different system’s IR cutoff. In the absence of
interaction the convenient choice for the IR cutoff are the radial size of the
horizon $R_{h}$ and the radius of the event horizon measured on the sphere of
the horizon $L=ar(t)$ in spatially flat and curved universe, respectively.
Although, in these cases the HDE gives the observation value of dark energy in
the universe and can drive the universe to an accelerated expansion phase, but
an obvious drawback concerning causality appears. Event horizon is a global
concept of spacetime; existence of event horizon of the universe depends on
future evolution of the universe; and event horizon exists only for universe
with forever accelerated expansion. However, as soon as an interaction between
dark energy and dark matter is taken into account, the identification of $L$
with $H^{-1}$ in flat universe, can simultaneously drive accelerated expansion
and solve the coincidence problem pav1 . The Hubble radius is not only the
most obvious but also the simplest choice in flat universe.
In this paper, we demonstrated that in a universe with spacial curvature the
natural choice for IR cutoff could be the apparent horizon radius,
$\tilde{r}_{A}={1}/{\sqrt{H^{2}+k/a^{2}}}$. We showed that any interaction of
pressureless dark matter with HDE, whose infrared cutoff is set by the
apparent horizon radius, implies a constant ratio of the energy densities of
both dark components thus solving the coincidence problem. In addition, we
examined the validity of the first and the generalized second law of
thermodynamics for a universe filled with mutual interacting dark components
in a region enclosed by the apparent horizon. These results hold regardless of
the specific form of dark energy and interaction term $Q$. Our study further
supports that in a universe with spatial curvature, the apparent horizon is a
physical boundary from the thermodynamical point of view.
###### Acknowledgements.
I thank the anonymous referee for constructive comments. I am also grateful to
Prof. B. Wang for helpful discussions and reading the manuscript. This work
has been supported by Research Institute for Astronomy and Astrophysics of
Maragha.
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|
arxiv-papers
| 2009-10-03T03:45:09 |
2024-09-04T02:49:05.605894
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ahmad Sheykhi",
"submitter": "Ahmad Sheykhi",
"url": "https://arxiv.org/abs/0910.0510"
}
|
0910.0562
|
# Le problème de Yamabe avec singularités et la conjecture de Hebey–Vaugon
Farid Madani
Université Pierre et Marie Curie
École Doctorale de Sciences Mathématiques de Paris Centre
Thèse de doctorat
Discipline: Mathématiques
présentée par
Farid Madani
Le problème de Yamabe avec singularités et la conjecture de Hebey–Vaugon
dirigée par Thierry Aubin
Soutenue le 29 septembre 2009 devant le jury composé de :
M. Bernd Ammann | Universität Regensburg | Rapporteur
---|---|---
M. Emmanuel Hebey | Université de Cergy–Pontoise |
M. Frédéric Hélein | Université Paris-Diderot |
M. Emmanuel Humbert | Université Nancy I | Rapporteur
M. Michel Vaugon | Université Paris 6 | Directeur de thèse
Institut de Mathématiques de Jussieu | | École doctorale Paris centre Case 188
---|---|---
175, rue du chevaleret | | 4 place Jussieu
75013 Paris | | 75 252 Paris cedex 05
_À la mémoire de Thierry Aubin_.
### Remerciements
Je tiens tout d’abord à exprimer ma profonde gratitude et reconnaissance
envers mon directeur de thèse Thierry Aubin. J’ai eu la douleur de le perdre
au début de cette année. Il m’a introduit à la recherche mathématique, et j’ai
particulièrement apprécié son honnêteté mathématique et sa façon de raisonner.
J’aimerais aussi exprimer ma gratitude envers Michel Vaugon, qui a accepté de
reprendre la direction de ma thèse. En très peu de temps il a lu ma thèse et
fait beaucoup de précieux commentaires. Je le remercie pour sa disponibilité
et sa sympathie.
Bernd Ammann et Emmanuel Humbert ont accepté d’être rapporteurs de ma thèse et
de participer à mon jury. Je les remercie pour les remarques et suggestions
qu’ils ont faites sur mon travail.
Je remercie Emmanuel Hebey et Frédéric Hélein pour avoir accepté d’être
membres de mon jury. Un remerciement particulier pour Emmanuel Hebey pour ses
commentaires et suggestions pertinentes.
Je tiens à remercier Tien-Cuong Dinh et Elisha Falbel pour leur soutien et
leurs conseils au cours des ces trois années de thèse.
Durant ma thèse, j’ai partagé, avec les thésards du 7ème étage, pas mal de
déjeuners (presque tous les jours). Je les remercie pour les pauses de détente
que l’on a partagé à l’institut et même à l’extérieur. Je pense qu’ils se
reconnaissent sans les citer un par un. Je les remercie aussi pour ces
séminaires mathématiques, où l’on peut comprendre jusqu’à 100% du contenu. Un
remerciement spécial pour Johan, Julien, Nicolas et pour mon "frère d’armes"
Nabil.
Je salue chaleureusement tous mes amis qui sont toujours de mon coté. Enfin,
je remercie profondément tous les membres de ma famille pour leur soutien
constant durant toutes mes études. Ils occupent une place particulière au fond
de moi.
###### Contents
1. Notations
2. Introduction
3. Introduction (English version)
4. 1 Théorèmes de régularité et généralités
1. 1.1 Les courbures
2. 1.2 Le Laplacien
3. 1.3 Les espaces de Sobolev
1. 1.3.1 Théorèmes des espaces de Banach
4. 1.4 Inégalité de la meilleure constante
5. 1.5 L’inégalité de Hardy sur une variété compacte
6. 1.6 La régularité des solutions de l’équation de type Yamabe
5. 2 Étude d’équations de type Yamabe
1. 2.1 Existence de solutions sans présence de symétries
1. 2.1.1 Application
2. 2.2 Existence de solutions en présence de symétries
1. 2.2.1 Le groupe d’isométries et le groupe conforme
2. 2.2.2 Inégalité de la meilleure constante en présence de symétries
6. 3 Le problème de Yamabe avec singularités
1. 3.1 Le problème de Yamabe
2. 3.2 Choix de la métrique
3. 3.3 Le Laplacien conforme
1. 3.3.1 L’invariance conforme faible
4. 3.4 L’invariant conforme de Yamabe
5. 3.5 Fonction de Green
6. 3.6 La métrique de Cao–Günther
7. 3.7 Le théorème de la masse positive
8. 3.8 Théorème d’existence de solutions sans présence de symétries
9. 3.9 Unicité des solutions
10. 3.10 Application
11. 3.11 Le problème de Yamabe équivariant
1. 3.11.1 Le problème de Hebey–Vaugon
2. 3.11.2 L’invariant de Yamabe $\boldsymbol{G-}$conforme
12. 3.12 Théorème d’existence de solutions en présence de symétries
7. 4 Calculs techniques sur la courbure scalaire
1. 4.1 Calculs sur l’intégrale de la courbure scalaire
2. 4.2 Généralisation d’un théorème de T. Aubin
8. 5 Autour de la conjecture de Hebey–Vaugon
1. 5.1 La conjecture de Hebey–Vaugon
2. 5.2 Les travaux de Hebey–Vaugon
3. 5.3 Preuve du théorème principal
9. A Détails des calculs (Chapitre 4)
10. B Détails des calculs (Chapitre 5)
1. Le cas $\boldsymbol{8\leq\omega\leq 15}$
### Notations
$[q]$ la partie entière de $q$
---
$N=\frac{2n}{n-2}$
$[[1,n]]=\\{1,2,\cdots,n\\}$
$S_{n}$ la sphère unité de dimension $n$
$S_{n}(r)$ la sphère de rayon $r$
$g_{can}$ la métrique canonique sur $S_{n}$
$\mathcal{E}$ la métrique euclidienne
$\mathrm{d}\sigma$ l’élément de volume associé à $(S_{n-1},g_{can})$
$\mathrm{d}\sigma_{r}$ l’élément de volume de $S_{n-1}(r)$
$vol(M)$ volume de la variété $M$
$\omega_{n}$ volume de la sphère $S_{n}$
$\Delta_{g}$ le Laplacien de la métrique $g$
$\Delta_{\mathcal{E}}$ le Laplacien de la métrique euclidienne $\mathcal{E}$
$|\beta|=k$ si $\beta\in\mathbb{N}^{k}$
$K(n,2)^{-2}=\frac{1}{4}n(n-2)\omega_{n}^{2/n}$
$\nabla_{i}=\nabla_{\partial_{i}}$ la dérivée covariante
$\nabla_{\beta}=\nabla_{\beta_{1}}\cdots\nabla_{\beta_{k}}$
$R_{g}$ la courbure scalaire associée à $g$
$L_{g}=\Delta_{g}+\frac{n-2}{4(n-1)}R_{g}$ le Laplacien conforme
$G_{P}$ fonction de Green en $P$.
$T(M)$ l’espace tangent de $M$
$T^{*}M$ l’espace cotangent de $M$
$\Gamma(M)$ l’espace des champs de vecteurs $C^{\infty}$
$L^{p}(M)$ espace de Lebesgue sur $M$
$H^{p}_{q}(M)$ Espace de Sobolev
$H^{p}_{q,G}(M)$ Espace de Sobolev $G-$invariant
$H_{1}(M)=H^{2}_{1}(M)$, $H_{1,G}(M)=H^{2}_{1,G}(M)$
$\|\cdot\|_{p}$ norme sur $L^{p}$
$\|\cdot\|_{H_{1}}$ norme sur $H_{1}$
$(\cdot,\cdot)_{g,L^{2}}=(\cdot,\cdot)_{L^{2}}$ produit scalaire sur $L^{2}$
avec la métrique $g$
$(\cdot,\cdot)_{g,H_{1}}=(\cdot,\cdot)_{H_{1}}$ produit scalaire sur $H_{1}$
avec la métrique $g$
$\mu(g)=\mu_{N}(g)$ l’invariant conforme de Yamabe
$\mu_{G}(g)=\mu_{N,G}(g)$ l’invariant $G-$conforme de Yamabe
$E(\varphi)$ énergie de $\varphi$
$I_{g}$ La fonctionnelle de Yamabe
$I(M,g)$ le groupe d’isométries de $(M,g)$
$C(M,g)$ le groupe conforme de $(M,g)$
$G$ sous groupe de $I(M,g)$
## Introduction
Le travail présenté dans cette thèse est séparé en deux parties. La première
partie est consacrée à l’étude d’un certain type d’équations aux dérivées
partielles non linéaires sur une variété compacte. Ensuite, on donne une
signification géométrique de ces équations. La particularité ici est que l’un
des coefficients de ces équations n’a pas la régularité habituellement
supposée, ce qui permettra d’obtenir un "théorème de Yamabe" avec
singularités. La seconde partie est consacrée à l’étude d’une conjecture de
Hebey–Vaugon dans le cadre du problème de Yamabe équivariant.
#### Première partie
On considère une variété riemannienne $(M,g)$ compacte de dimension $n\geq 3$.
On note $R_{g}$ la courbure scalaire de $g$. Le problème de Yamabe est le
suivant:
###### Problème 0.1.
Existe-t-il une métrique conforme à $g$ de courbure scalaire constante?
On pose $\tilde{g}=\varphi^{\frac{4}{n-2}}g$, où $\varphi$ est une fonction
$C^{\infty}$ strictement positive. $\tilde{g}$ est une solution du problème de
Yamabe si et seulement si $\varphi$ est solution de l’équation suivante:
$\frac{4(n-1)}{n-2}\Delta_{g}\varphi+R_{g}\varphi=R_{\tilde{g}}\varphi^{\frac{n+2}{n-2}}$
(1)
où $\Delta_{g}=-\nabla^{i}\nabla_{i}$ est le Laplacien de $g$ et
$R_{\tilde{g}}$ est une constante qui joue le rôle de la courbure scalaire de
$\tilde{g}$. T. Aubin a ramené la résolution de ce problème à la résolution de
la conjecture suivante:
###### Conjecture 0.1 (T. Aubin [Aub]).
Si $(M,g)$ est une variété riemannienne compacte $C^{\infty}$ de dimension
$n\geq 3$ et non conformément difféomorphe à $(S_{n},g_{can})$ alors
$\mu(M,g)<\mu(S_{n},g_{can})$ (2)
où
$\mu(M,g)=\inf\biggl{\\{}\displaystyle\frac{\int_{M}|\nabla\psi|^{2}+\frac{n-2}{4(n-1)}R_{g}\psi^{2}\mathrm{d}v}{\|\psi\|^{2}_{\frac{2n}{n-2}}},\;\psi\in
H_{1}(M)-\\{0\\}\biggr{\\}}$.
Il est bien connu que $\mu(S_{n},g_{can})=\frac{1}{4}n(n-2)\omega_{n}^{2/n}$.
Les travaux de T. Aubin [Aub], R. Schoen [Schoen] et H. Yamabe [Yam] ont
montré que cette conjecture est toujours vraie, et le problème de Yamabe admet
toujours des solutions. En d’autres termes, dans chaque classe conforme $[g]$,
on peut toujours trouver une métrique à courbure scalaire constante.
On note par $I(M,g)$ et $C(M,g)$ le groupe d’isométries et le groupe conforme
de $(M,g)$ respectivement. Soit $G$ un sous groupe de $I(M,g)$. E. Hebey et M.
Vaugon [HV] ont étudié le problème de Yamabe équivariant, qui généralise le
problème de Yamabe, et que l’on peut exprimer de la manière suivante:
###### Problème 0.2.
Existe-t-il une métrique $g_{0}$, $G-$invariante qui minimise la fonctionnelle
$J(g^{\prime})=\frac{\int_{M}R_{g^{\prime}}\mathrm{d}v(g^{\prime})}{(\int_{M}\mathrm{d}v(g^{\prime}))^{\frac{n-2}{n}}}$
où $g^{\prime}$ appartient à la classe $G-$conforme de $g$:
$[g]^{G}:=\\{\tilde{g}=e^{f}g/f\in
C^{\infty}(M),\;\sigma^{*}\tilde{g}=\tilde{g}\quad\forall\sigma\in G\\}$
E. Hebey et M. Vaugon ont montré que ce problème à toujours des solutions, ce
qui a pour première conséquence l’existence d’une métrique $g_{0}$,
$G-$invariante et conforme à $g$, telle que la courbure scalaire de $g_{0}$
est constante. La deuxième conséquence est que la conjecture suivante est
démontrée.
###### Conjecture 0.2 (Lichnerowicz [Lic]).
Pour toute variété riemannienne $(M,g)$, compacte $C^{\infty}$, de dimension
$n$ et qui n’est pas conformément difféomorphe à $(S_{n},g_{can})$, il existe
une métrique $\tilde{g}$ conforme à $g$ de courbure scalaire $R_{\tilde{g}}$
constante et pour laquelle $I(M,\tilde{g})=C(M,g)$.
Le travail présenté dans la première partie de la thèse est l’étude du
problème de Yamabe 0.1 (sans et avec la présence de symétries), lorsque la
métrique $g$ n’est pas nécessairement $C^{\infty}$. On suppose que la métrique
$g$ est dans $H^{p}_{2}$, où $p>n$, l’espace de Sobolev des métriques dont on
donnera la définition plus loin. Grâce aux inclusions de Sobolev
$H^{p}_{2}\subset C^{1,\beta}$ (l’espace de Hölder d’exposant
$\beta\in]0,1[$), les métriques sont donc de classe $C^{1,\beta}$. Les
tenseurs de courbures de Riemann, de Ricci et la courbure scalaire sont dans
$L^{p}$. Plus précisément, si on suppose que $g$ satisfait l’hypothèse
suivante:
Hypothèse $\boldsymbol{(H)}$: _$g$ est une métrique dans l’espace de Sobolev
$H_{2}^{p}(M,T^{*}M\otimes T^{*}M)$ avec $p>n$. Il existe un point $P_{0}\in
M$ et $\delta>0$ tels que $g$ est $C^{\infty}$ sur la boule
$B_{P_{0}}(\delta)$._
Alors le problème que l’on résout est le suivant:
###### Problème 0.3.
Soit $g$ une métrique qui satisfait l’hypothèse $(H)$. Existe-t-il une
métrique $\tilde{g}$ conforme à $g$ pour laquelle la courbure scalaire
$R_{\tilde{g}}$ est constante (même aux points où $R_{g}$ n’est pas
régulière)?
Avant de résoudre ce problème, on commence par étudier plus généralement les
équations suivantes:
$\Delta_{g}\varphi+h\varphi=\tilde{h}\varphi^{\frac{n+2}{n-2}}$ (3)
où $h$ est une fonction qui est supposée seulement être dans $L^{p}(M)$ (c’est
là l’originalité de cette étude) et $\tilde{h}\in\mathbb{R}$. La métrique $g$
est supposée $C^{\infty}$ (la supposer $C^{2}$ donnerait les même résultats,
ce n’est pas un point important). On appellera ces équations les équations de
type Yamabe. Comme ces équations sont non linéaires et que $h$ est dans
$L^{p}$, les théorèmes de régularité standard ne s’appliquent pas directement.
On établit le résultat suivant (adaptation d’un théorème de N. Trudinger
[Trud] au cas où $h$ n’est que dans $L^{p}$)
###### Théorème 0.1.
Soit $(M,g)$ une variété riemannienne compacte $C^{\infty}$ de dimension
$n\geq 3$, $p$ et $\tilde{h}$ sont deux nombres réels, avec $p>n/2$. Si
$\varphi\in H_{1}(M)$ est une solution faible positive non triviale de
l’équation 3 alors $\varphi\in H_{2}^{p}(M)\subset C^{1-[n/p],\beta}(M)$ et
$\varphi$ est strictement positive.
La régularité donnée par ce théorème est optimale.
En ce qui concerne l’existence des solutions, on démontre que la fonctionnelle
$I_{g}$, définie pour tout $\psi\in H_{1}(M)-\\{0\\}$ par
$I_{g}(\psi)=\frac{\int_{M}|\nabla\psi|^{2}+h\psi^{2}\mathrm{d}v}{\|\psi\|^{2}_{\frac{2n}{n-2}}}$
atteint son minimum $\mu(g)$, si $\mu(g)<\frac{1}{4}n(n-2)\omega_{n}^{2/n}$
(où $\omega_{n}$ est le volume de la sphère standard $S_{n}$). On obtient
alors le résultat suivant:
###### Théorème 0.2.
Soit $(M,g)$ une variété riemannienne compacte $C^{\infty}$ de dimension
$n\geq 3$ et $p>n/2$. Si
$\mu(g)<\frac{1}{4}n(n-2)\omega_{n}^{2/n}$
alors l’équation (3) admet une solution strictement positive $\varphi\in
H^{p}_{2}(M)\subset C^{1-[n/p],\beta}(M)$, qui minimise la fonctionnelle
$I_{g}$, où $\beta\in]0,1[$.
Si $h$ est $G-$invariante, on définit
$\mu_{G}(g)=\inf_{\psi\in H_{1,G}(M)-\\{0\\}}I_{g}(\psi)$
où $H_{1,G}(M)$ est l’espace des fonctions dans $H_{1}(M)$, $G-$invariantes.
On note par $O_{G}(Q)$ l’orbite du point $Q\in M$. On obtient le résultat
suivant:
###### Théorème 0.3.
Si $0<\mu_{G}(g)<\frac{1}{4}n(n-2)\omega_{n}^{2/n}(\inf_{Q\in
M}\mathrm{card}O_{G}(Q))^{2/n}$ alors l’équation (3) admet une solution
$\varphi\in H_{2,G}^{p}(M)\subset C^{1-[n/p],\beta}(M)$ strictement positive,
$G-$invariante et minimisante pour la fonctionnelle $I_{g}$.
Ce théorème se démontre en utilisant la méthode variationnelle (comme dans les
cas classiques où $h$ est très régulière), les inclusions de Sobolev en
présence de symétries, trouvées par E. Hebey et M. Vaugon [HV2] et l’inégalité
de la meilleure constante en présence symétries calculée par Z. Faget [Fag].
Dans le chapitre 3, on étudie l’équation (3) lorsque
$h=\frac{n-2}{4(n-1)}R_{g}$ et $g$ est une métrique qui satisfait l’hypothèse
$(H)$. Ce cas a une signification géométrique, il permet de résoudre le
problème 0.3 (le problème de Yamabe avec singularités). La courbure scalaire
$R_{g}$ est dans $L^{p}(M)$ et l’équation (3) devient l’équation de Yamabe
(1). D’après le théorème 0.2, la résolution du problème 0.3 est ramenée à la
preuve de l’inégalité $\mu(g)<\frac{1}{4}n(n-2)\omega_{n}^{2/n}$ (cette
inégalité a déjà été démontrée lorsque $g$ est $C^{\infty}$). Dans le cas où
$g$ satisfait l’hypothèse $(H)$, on commence par démontrer certaines
propriétés (connues dans le cas $C^{\infty}$): l’invariance conforme de
$\mu(g)$, l’invariance conforme faible du Laplacien conforme
$L_{g}=\Delta_{g}+\frac{n-2}{4(n-1)}R_{g}$ et l’existence de la fonction de
Green pour cet opérateur. Ensuite, on démontre le résultat suivant:
###### Théorème 0.4.
Soit $M$ une variété compacte $C^{\infty}$ de dimension $n$, $g$ une métrique
riemannienne qui satisfait l’hypothèse $(H)$. Si $(M,g)$ n’est pas
conformément difféomorphe à la sphère $(S_{n},g_{can})$ alors
$\mu(g)<\frac{1}{4}n(n-2)\omega_{n}^{2/n}$.
Lorsque la métrique $g$ est $C^{\infty}$, ce théorème a résolu la conjecture
0.1. Les arguments utilisés pour le démontrer dans ce cas sont encore valables
lorsque $g$ satisfait l’hypothèse $(H)$. En effet, il suffit de construire une
certaine fonction test $\varphi$ qui vérifie
$I_{g}(\varphi)<\frac{1}{4}n(n-2)\omega_{n}^{2/n}$. Les fonctions test
construites par T. Aubin [Aub] et R. Schoen [Schoen], sont encore utilisables
dans ce cas singulier.
Dans le cas équivariant (en présence de symétries), le résultat obtenu est le
suivant:
###### Théorème 0.5.
Soit $M$ une variété compacte $C^{\infty}$ de dimension $n\geq 3$. $g$ une
métrique riemannienne qui appartient à $H_{2}^{p}(M,T^{*}M\otimes T^{*}M)$
avec $p>n/2$. Si
$\mu_{G}(g)<\frac{1}{4}n(n-2)\omega_{n}^{2/n}(\inf_{Q\in
M}\mathrm{card}O_{G}(Q))^{2/n}$ (4)
alors l’équation (1) admet une solution strictement positive $\varphi\in
H_{2,G}^{p}(M)\subset C^{1-[n/p],\beta}(M)$ $G-$invariante.
Les résultats sur l’unicité des solutions de l’équation de Yamabe (1), connus
lorsque la métrique est $C^{\infty}$, restent valables dans le cas singulier.
On obtient le résultat suivant:
###### Théorème 0.6.
Soit $g$ une métrique dans $H^{p}_{2}(M,T^{*}M\otimes T^{*}M)$, avec $p>n$. Si
$\mu(g)\leq 0$ alors les solutions de l’équation (1) sont uniques à une
constante multiplicative près.
Dans cette première partie, on a montré que la majorité des résultats connus
sur le problème de Yamabe et certains dans le cas équivariant, lorsque la
métrique est $C^{\infty}$, restent vrais lorsque la métrique satisfait
l’hypothèse $(H)$, définie ci-dessus. Une question naturelle que l’on peut se
poser est de savoir s’il est possible de supprimer certaines conditions dans
l’hypothèse $(H)$. Par exemple, peut on considérer des métriques dans
$H^{p}_{2}$, sans qu’elles soit $C^{\infty}$ dans une boule? La réponse semble
difficile et le sujet ne sera pas abordé dans cette thèse (mais sera traité
ultérieurement).
#### Deuxième partie
La deuxième partie de cette thèse est indépendante de la première (elles sont
mathématiquement liées, mais aucun résultat de la première partie n’est
utilisé dans la seconde partie).
On suppose que $(M,g)$ est une variété riemannienne compacte $C^{\infty}$ de
dimension $n\geq 3$. Le but principal des deux chapitres de cette partie est
d’étudier la conjecture de Hebey–Vaugon qui s’énonce comme suit:
###### Conjecture 0.3 (E. Hebey et M. Vaugon [HV]).
Soit $G$ un sous groupe d’isométries de $I(M,g)$. Si $(M,g)$ n’est pas
conformément difféomorphe à $(S_{n},g_{can})$ ou bien si $G$ n’a pas de point
fixe, alors l’inégalité stricte suivante a toujours lieu
$\inf_{g^{\prime}\in[g]^{G}}J(g^{\prime})<n(n-1)\omega_{n}^{2/n}(\inf_{Q\in
M}\mathrm{card}O_{G}(Q))^{2/n}$ (5)
Cette conjecture généralise la conjecture de T. Aubin 0.1 puisque:
$\inf_{g^{\prime}\in[g]^{G}}J(g^{\prime})=4\frac{n-1}{(n-2)}\mu_{G}(g)$
(si $G=\\{\mathrm{id}\\}$ les deux conjectures sont identiques).
On note par $W_{g}$ le tenseur de Weyl associé à $g$. Pour tout $P\in M$, on
définit $\omega(P)$ par
$\omega(P)=\inf\\{|\beta|\in\mathbb{N}/\|\nabla^{\beta}W_{g}(P)\|\neq
0\\},\;\omega(P)=+\infty\text{ si
}\forall\beta\;\;\|\nabla^{\beta}W_{g}(P)\|=0$
où $\beta$ est un multi-indice de longueur $|\beta|$.
Pour prouver la conjecture, on doit construire une fonction test
$G-$invariante $\phi$ telle que
$I_{g}(\phi)<n(n-1)\omega_{n}^{2/n}(\inf_{Q\in M}\mathrm{card}O_{G}(Q))^{2/n}$
Toute la difficulté est dans la construction d’une telle fonction. Dans
certains cas, on peut utiliser les fonctions test introduites par T. Aubin
[Aub] et R. Schoen [Schoen] pour démontrer la conjecture 0.1. De nombreux cas
ont été traités ainsi par E. Hebey et M. Vaugon [HV], par contre le cas numéro
3 présenté dans le théorème suivant utilise des fonctions test qui sont
différentes de celles de T. Aubin et R. Schoen.
###### Théorème 0.7 (E. Hebey et M. Vaugon).
Soit $(M,g)$ une variété riemannienne compacte de dimension $n$ et $G$ un sous
groupe d’isométries du groupe $I(M,g)$. On a toujours:
$\inf_{g^{\prime}\in[g]^{G}}J(g^{\prime})\leq
n(n-1)\omega_{n}^{2/n}(\inf_{Q\in M}\mathrm{card}O_{G}(Q))^{2/n}$
et l’ inégalité stricte (5) est au moins vérifiée dans chacun des cas
suivants:
1. 1.
$G$ opère librement sur $M$
2. 2.
$3\leq\dim M\leq 11$
3. 3.
Il existe un point $P$ d’orbite minimale (finie) sous $G$ pour lequel soit
$\omega(P)>(n-6)/2$, soit $\omega(P)\in\\{0,1,2\\}$.
Les cas restant pour démontrer complètement la conjecture sont les cas où
$n\geq 12$ et $\omega\in[[3,[(n-6)/2]]]$. Dans le chapitre 5, on démontre les
résultats suivants:
###### Théorème 0.8.
La conjecture 0.3 est vraie s’il existe un point $P$ d’orbite minimale (finie)
pour lequel $\omega(P)\leq 15$ ou si le degré de la partie principale de
$R_{g}$, au voisinage de $P$ est plus grand ou égal à $\omega(P)+1$.
###### Corollaire 0.1.
La conjecture 0.3 est vraie si $M$ est de dimension $n\in[[3,37]]$.
Ce théorème se démontre en effectuant des calculs longs et délicats
(introduits par T. Aubin [Aub5]). Les fonctions test $\varphi_{\varepsilon}$
choisies sont définies comme suit: pour un point $P$ quelconque de $M$, on
pose pour tout $Q\in M$
$\displaystyle\varphi_{\varepsilon}(Q)=(1-r^{\omega+2}f(\xi))u_{\varepsilon}(Q)$
(6) $\displaystyle\text{avec
}u_{\varepsilon}(Q)=\begin{cases}\biggl{(}\displaystyle\frac{\varepsilon}{r^{2}+\varepsilon^{2}}\biggr{)}^{\frac{n-2}{2}}-\biggl{(}\frac{\varepsilon}{\delta^{2}+\varepsilon^{2}}\biggr{)}^{\frac{n-2}{2}}&\mbox{
si }Q\in B_{P}(\delta)\\\ \hskip 56.9055pt0&\mbox{ si }Q\in
M-B_{P}(\delta)\end{cases}$ (7)
où $r=d(Q,P)$ est la distance entre $P$ et $Q$. $(r,\xi^{j})$ sont les
coordonnées géodésiques de $Q$ au voisinage de $P$ et $B_{P}(\delta)$ est une
boule géodésique de centre $P$, de rayon $\delta$, fixé suffisamment petit.
$f$ est une fonction qui dépend seulement de $\xi$ et telle que
$\int_{S_{n-1}}fd\sigma=0$. C’est la précision sur le choix de cette fonction
$f$ qui va permettre d’obtenir les résultats énoncés dans cette deuxième
partie.
On obtient d’abord le théorème suivant:
###### Théorème 0.9.
Soit $(M,g)$ une variété riemannienne compacte de dimension $n$. Pour tout
$P\in M$ tel que $\omega(P)\leq(n-6)/2$, il existe $f\in C^{\infty}(S_{n-1})$,
d’intégrale nulle, telle que
$\mu(g)\leq I_{g}(\varphi_{\varepsilon})<\frac{n(n-2)}{4}\omega_{n-1}^{2/n}$
(Ce résultat généralise donc le théorème de T. Aubin [Aub], qui correspond à
$\omega=0$ et qui démontre la conjecture 0.1, dans certains cas). La fonction
$f$ de ce théorème est définie par
$f=\sum_{k=1}^{q}c_{k}\nu_{k}\varphi_{k}$
où $\varphi_{k}$ sont des fonctions propres du Laplacien sphérique de la
sphère $S_{n-1}$, $\nu_{k}$ sont les valeurs propres associées et
$q\in[[1,[\frac{\omega}{2}]]]$, les constantes $c_{k}$ sont données
explicitement. Si $f$ était $G-$invariante, on pouvait construire, à l’aide
des $\varphi_{k}$, des fonctions test $G-$invariantes qui permettraient de
démontrer la conjecture 0.3 dans tous les cas. Malheureusement, $f$ n’est
$G-$invariante que pour un choix particulier des $c_{k}$, et ce choix
particulier ne permet de montrer la conjecture que dans les cas énoncés dans
le théorème 0.8.
## Introduction (English version)
In the first part of this thesis, we study a certain kind of nonlinear partial
differential equations on compact manifolds. Solutions of these PDEs have a
geometric meaning. The particularity here is that one of the coefficients of
this equations doesn’t have the usual regularity, which allow us to obtain a
Yamabe theorem with singularities.
The Second part is dedicated to the study of Hebey–Vaugon conjecture.
#### First part
Consider $(M,g)$ a compact Riemannian manifold of dimension $n\geq 3$. Denote
by $R_{g}$ the scalar curvature of $g$. The Yamabe problem is the following:
###### Problem 0.1.
Does there exists a constant scalar curvature metric conformal to $g$?
Let $\tilde{g}=\varphi^{\frac{4}{n-2}}g$ be a conformal metric, where
$\varphi$ is a smooth positive function. $\tilde{g}$ is a solution of the
Yamabe problem if and only if $\varphi$ satisfies the following equation:
$\frac{4(n-1)}{n-2}\Delta_{g}\varphi+R_{g}\varphi=R_{\tilde{g}}\varphi^{\frac{n+2}{n-2}}$
(8)
where $\Delta_{g}=-\nabla^{i}\nabla_{i}$ is the Laplacian of $g$ and
$R_{\tilde{g}}$ is a constant which plays the role of the scalar curvature of
$\tilde{g}$. T. Aubin showed that it is sufficient to prove the following
conjecture:
###### Conjecture 0.1 (T. Aubin [Aub]).
For every smooth compact Riemannian manifold $(M,g)$ of dimension $n\geq 3$,
non conformal to $(S_{n},g_{can})$,
$\mu(M,g)<\mu(S_{n},g_{can})$ (9)
where
$\mu(M,g)=\inf\biggl{\\{}\displaystyle\frac{\int_{M}|\nabla\psi|^{2}+\frac{n-2}{4(n-1)}R_{g}\psi^{2}\mathrm{d}v}{\|\psi\|^{2}_{\frac{2n}{n-2}}},\;\psi\in
H_{1}(M)-\\{0\\}\biggr{\\}}$
It is known that $\mu(S_{n},g_{can})=\frac{1}{4}n(n-2)\omega_{n}^{2/n}$. The
works of T. Aubin [Aub], R. Schoen [Schoen] and H. Yamabe [Yam] showed that
this conjecture is always true, and the Yamabe problem has a solution. Namely,
in each conformal class $[g]$, there exists a constant scalar curvature
metric.
Denote by $I(M,g)$ and $C(M,g)$ the isometry group and the conformal group
respectively. Let $G$ be a subgroup of $I(M,g)$. E. Hebey and M. Vaugon [HV]
studied the equivariant Yamabe problem, which generalizes the Yamabe problem,
and which can be formulated in the following way:
###### Problem 0.2.
Is there some $G-$invariant metric $g_{0}$ which minimizes the functional
$J(g^{\prime})=\frac{\int_{M}R_{g^{\prime}}\mathrm{d}v(g^{\prime})}{(\int_{M}\mathrm{d}v(g^{\prime}))^{\frac{n-2}{n}}}$
where $g^{\prime}$ belongs to the $G-$conformal class of $g$:
$[g]^{G}:=\\{\tilde{g}=e^{f}g/f\in
C^{\infty}(M),\;\sigma^{*}\tilde{g}=\tilde{g}\quad\forall\sigma\in G\\}$
E. Hebey and M. Vaugon proved that this problem has always solutions. The
positive answer would have two consequences. The first is that there exists a
$I(M,g)-$invariant metric $g_{0}$ conformal to $g$ such that the scalar
curvature $R_{g_{0}}$ is constant. The second is that the following conjecture
is true.
##### Lichnerowicz conjecture
_For every compact Riemannian manifold $(M,g)$ which is not conformal to the
unit sphere $S_{n}$ endowed with its standard metric, there exists a metric
$\tilde{g}$ conformal to $g$ for which $I(M,\tilde{g})=C(M,g)$, and the scalar
curvature $R_{\tilde{g}}$ is constant._
In this part, we study the Yamabe problem 0.1 (without and in presence of the
isometry group), when the metric $g$ is not necessarily smooth. We suppose
that the metric is in the Sobolev space $H^{p}_{2}$, where $p>n$. Riemann
curvature tensor, Ricci tensor and the scalar curvature are in $L^{p}$. More
precisely, we make the following assumption on $g$ :
Assumption $\boldsymbol{(H)}$: _$g$ is a metric which belongs to the Sobolev
space $H_{2}^{p}(M,T^{*}M\otimes T^{*}M)$ with $p>n$. There exists a point
$P_{0}\in M$ and $\delta>0$ such that $g$ is smooth in the ball
$B_{P_{0}}(\delta)$._
The problem that we solve is the following:
###### Problem 0.3.
Let $g$ be a metric satisfying the assumption $(H)$. Does there exists a
constant scalar curvature metric $\tilde{g}$ conformal to $g$?
Before solving this problem, we start by studying these equations:
$\Delta_{g}\varphi+h\varphi=\tilde{h}\varphi^{\frac{n+2}{n-2}}$ (10)
where $h$ is a function in $L^{p}(M)$ (which makes this work original) and
$\tilde{h}\in\mathbb{R}$. The metric $g$ is assumed to be smooth. The
smoothness of $g$ is not an important point. Indeed, if $g$ is $C^{2}$, we
will obtain the same results. This kind of equations are called "Yamabe type
equations". We can not apply for these equations the standard regularity
theorems because of the nonlinearity and the fact that $h\in L^{p}(M)$. Thus,
we establish the following result (it is an adaptation of Trudinger’s theorem
when $h$ is more regular).
###### Theorem 0.1.
Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n\geq 3$,
$p$ and $\tilde{h}$ are two reel numbers such that $p>n/2$. If $\varphi\in
H_{1}(M)$ is nontrivial, nonnegative, weak solution of (10), then $\varphi$ is
positive and belongs to $H_{2}^{p}(M)\subset C^{1-[n/p],\beta}(M)$.
For the existence of solutions of (10), we prove that the functional $I_{g}$,
defined for all $\psi\in H_{1}(M)-\\{0\\}$ by
$I_{g}(\psi)=\frac{\int_{M}|\nabla\psi|^{2}+h\psi^{2}\mathrm{d}v}{\|\psi\|^{2}_{\frac{2n}{n-2}}}$
has a minimum $\mu(g)$ if $\mu(g)<\frac{1}{4}n(n-2)\omega_{n}^{2/n}$ (where
$\omega_{n}$ is the volume of the unit sphere $S_{n}$). Therefore, we obtain
the following:
###### Theorem 0.2.
Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n\geq 3$ and
$p>n/2$. If
$\mu(g)<\frac{1}{4}n(n-2)\omega_{n}^{2/n}$
then equation (10) admits a positive solution $\varphi\in H^{p}_{2}(M)\subset
C^{1-[n/p],\beta}(M)$, which minimizes the functional $I_{g}$, where
$\beta\in(0,1)$.
If $h$ is $G-$invariant, we define
$\mu_{G}(g)=\inf_{\psi\in H_{1,G}(M)-\\{0\\}}I_{g}(\psi)$
where $H_{1,G}(M)$ is the space of $G-$invariant functions in $H_{1}(M)$. We
denote by $O_{G}(Q)$ the orbit of $Q\in M$. Then,
###### Theorem 0.3.
If $0<\mu_{G}(g)<\frac{1}{4}n(n-2)\omega_{n}^{2/n}(\inf_{Q\in
M}\mathrm{card}O_{G}(Q))^{2/n}$ then equation (10) admits a positive
$G-$invariant solution $\varphi\in H_{2,G}^{p}(M)\subset
C^{1-[n/p],\beta}(M)$, which minimizes the functional $I_{g}$.
We prove this theorem by using the variational method (known in the classical
case when $h$ is smooth), Sobolev embedding in the presence of symmetries,
proven by E. Hebey and M. Vaugon [HV2] and the best constant inequality,
computed by Z. Faget [Fag].
In chapter 3, we consider the particular case when $h=\frac{n-2}{4(n-1)}R_{g}$
and the metric $g$ satisfies the assumption $(H)$. This case has a geometric
meaning. It allows us to solve the problem 0.3 (Yamabe problem with
singularities). The scalar curvature $R_{g}$ is in $L^{p}(M)$ and equation
(10) becomes the Yamabe equation (8). Using theorem 0.2 to solve problem 0.3,
it is sufficient to prove the inequality
$\mu(g)<\frac{1}{4}n(n-2)\omega_{n}^{2/n}$ (this inequality has been proven
when $g$ is smooth). When $g$ satisfies the assumption $(H)$, we establish
some properties (known in the smooth case) : conformal invariance of $\mu(g)$,
weak conformal invariance of the conformal Laplacian
$L_{g}=\Delta_{g}+\frac{n-2}{4(n-1)}R_{g}$ and the existence of the Green
function for this operator. We show afterwards the following theorem :
###### Theorem 0.4.
Let $M$ be a smooth compact manifold of dimension $n\geq 3$ and $g$ be a
Riemannian metric satisfying the assumption $(H)$. If $(M,g)$ is not conformal
to $(S_{n},g_{can})$, then $\mu(g)<\frac{1}{4}n(n-2)\omega_{n}^{2/n}$.
When the metric $g$ is smooth, this theorem solves the conjecture 0.1. The
arguments used to prove it are still valid when the metric $g$ satisfies the
assumption $(H)$. In fact, it is sufficient to construct a test function
$\varphi$ which satisfies $I_{g}(\varphi)<\frac{1}{4}n(n-2)\omega_{n}^{2/n}$.
Test functions, constructed by T. Aubin [Aub] and R. Schoen [Schoen], are
still useful in the singular case.
In the equivariant case (in the presence of the isometry group), the result
obtained is the following:
###### Theorem 0.5.
Let $M$ be a smooth compact manifold of dimension $n\geq 3$ and $g\in
H_{2}^{p}(M,T^{*}M\otimes T^{*}M)$ be a Riemannian metric with $p>n/2$. If
$\mu_{G}(g)<\frac{1}{4}n(n-2)\omega_{n}^{2/n}(\inf_{Q\in
M}\mathrm{card}O_{G}(Q))^{2/n}$ (11)
then equation (8) has a positive $G-$invariant solution $\varphi\in
H_{2,G}^{p}(M)\subset C^{1-[n/p],\beta}(M)$.
The known result about uniqueness of solutions of the Yamabe equation (8), for
smooth metrics, is valid in the singular case. Therefore, we have the
following result:
###### Theorem 0.6.
Let $g$ be a metric in $H^{p}_{2}(M,T^{*}M\otimes T^{*}M)$, with $p>n$. If
$\mu(g)\leq 0$ then the solutions of (8) are proportional.
In this part, we showed that almost all of the results and properties known
about the Yamabe problem, and some properties in the equivariant case, holds
in the singular case (when the metric satisfies the assumption $(H)$ defined
above). A question that naturally arises is the possibility of deleting some
conditions in the assumption $(H)$. For example, can we consider metrics in
$H^{p}_{2}$ without the smoothness condition in a ball? The answer seems
difficult and this question will not be treated in this thesis.
#### Second part
The second part is independent from the first (they are mathematically linked,
but the results of the first part are not used in the second).
Suppose that $(M,g)$ is a smooth compact Riemannian manifold of dimension
$n\geq 3$. The principal goal of the two last chapters of this part is to
study Hebey–Vaugon conjecture that can be stated in the following way:
###### Conjecture 0.2 (E. Hebey and M. Vaugon [HV]).
Let $G$ be a subgroup of $I(M,g)$. If $(M,g)$ is not conformal to
$(S_{n},g_{can})$ or if the action of $G$ has no fixed point, then the
following inequality holds
$\inf_{g^{\prime}\in[g]^{G}}J(g^{\prime})<n(n-1)\omega_{n}^{2/n}(\inf_{Q\in
M}\mathrm{card}O_{G}(Q))^{2/n}$ (12)
This conjecture generalizes naturally T. Aubin’s conjecture 0.1. In fact
$\inf_{g^{\prime}\in[g]^{G}}J(g^{\prime})=4\frac{n-1}{(n-2)}\mu_{G}(g)$
(if $G=\\{\mathrm{id}\\}$ then the two conjectures are the same).
Denote by $W_{g}$ the Weyl tensor associated to $g$. For all $P\in M$, we
define $\omega(P)$ by
$\omega(P)=\inf\\{|\beta|\in\mathbb{N}/\|\nabla^{\beta}W_{g}(P)\|\neq
0\\},\;\omega(P)=+\infty\text{ if
}\forall\beta\;\;\|\nabla^{\beta}W_{g}(P)\|=0$
To prove the conjecture, we need to construct a $G-$invariant test function
$\phi$ such that
$I_{g}(\phi)<n(n-1)\omega_{n}^{2/n}(\inf_{Q\in M}\mathrm{card}O_{G}(Q))^{2/n}$
Thus, all of the difficulties are in the construction of a such function. For
some cases, we can use the test functions constructed by T. Aubin [Aub] and R.
Schoen [Schoen] to prove the conjecture 0.1. They have been already proven by
E. Hebey and M. Vaugon [HV]. But the item 3, presented in the following
theorem, uses test functions different than T. Aubin and R. Schoen ones.
###### Theorem 0.7 (E. Hebey and M. Vaugon).
Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n\geq 3$ and
$G$ be a subgroup of $I(M,g)$. We always have :
$\inf_{g^{\prime}\in[g]^{G}}J(g^{\prime})\leq
n(n-1)\omega_{n}^{2/n}(\inf_{Q\in M}\mathrm{card}O_{G}(Q))^{2/n}$
and inequality (12) holds if one of the following items is satisfied.
1. 1.
The action of $G$ on $M$ is free
2. 2.
$3\leq\dim M\leq 11$
3. 3.
There exists a point $P$ with minimal orbit (finite) under $G$ such that
$\omega(P)>(n-6)/2$ or $\omega(P)\in\\{0,1,2\\}$.
The remaining case of the conjecture, is the case when $n\geq 12$ and
$\omega\in[[3,[(n-6)/2]]]$. In chapter 5, we prove the following result:
###### Theorem 0.8.
The conjecture 0.2 holds if there exists a point $P\in M$ with minimal orbit
(finite) for which $\omega(P)\leq 15$ or if the degree of the leading part of
$R_{g}$ is greater or equal to $\omega(P)+1$, in the neighborhood of this
point $P$.
###### Corollary 0.1.
The conjecture 0.2 holds for every smooth compact Riemannian manifold $(M,g)$
of dimension $n\in[3,37]$.
We prove this theorem using long and subtle computations (introduced by T.
Aubin [Aub5]). We use the test function $\varphi_{\varepsilon}$, defined in
the following way: for an arbitrary fixed point $P$ in $M$, for any $Q\in M$
$\displaystyle\varphi_{\varepsilon}(Q)=(1-r^{\omega+2}f(\xi))u_{\varepsilon}(Q)$
(13) $\displaystyle\text{with
}u_{\varepsilon}(Q)=\begin{cases}\biggl{(}\displaystyle\frac{\varepsilon}{r^{2}+\varepsilon^{2}}\biggr{)}^{\frac{n-2}{2}}-\biggl{(}\frac{\varepsilon}{\delta^{2}+\varepsilon^{2}}\biggr{)}^{\frac{n-2}{2}}&\mbox{
if }Q\in B_{P}(\delta)\\\ \hskip 56.9055pt0&\mbox{ if }Q\in
M-B_{P}(\delta)\end{cases}$ (14)
where $r=d(Q,P)$ is the distance between $P$ and $Q$. $(r,\xi^{j})$ is a
geodesic coordinates system of $Q$, defined in the neighborhood of $P$, and
$B_{P}(\delta)$ is a geodesic ball of center $P$ and of radius $\delta$, fixed
sufficiently small, and $f$ is a function depending only on $\xi$ such that
$\int_{S_{n-1}}fd\sigma=0$. The choice of the this function $f$ allow us to
prove the results of this part.
We obtain also the following theorem:
###### Theorem 0.9.
Let $(M,g)$ be a compact Riemannian manifold of dimension $n\geq 3$. For any
$P\in M$ such that $\omega(P)\leq(n-6)/2$, there exists $f\in
C^{\infty}(S_{n-1})$ with vanishing mean integral, such that
$\mu(g)\leq I_{g}(\varphi_{\varepsilon})<\frac{n(n-2)}{4}\omega_{n-1}^{2/n}$
This result generalizes T. Aubin’s [Aub] theorem (which corresponds to
$\omega=0$ and proves conjecture 0.1). For the above theorem, the function $f$
is defined as
$f=\sum_{k=1}^{q}c_{k}\nu_{k}\varphi_{k}$
where $\varphi_{k}$ are the eigenfunctions of the Laplacian on the sphere
$S_{n-1}$, $\nu_{k}$ are the associated eigenvalues, and
$q\in[[1,[\frac{\omega}{2}]]]$. The constants $c_{k}$ are given explicitly. If
$f$ is $G-$invariant, then we would construct, using $\varphi_{k}$, a
$G-$invariant test function, which would prove the conjecture 0.2, in all the
remaining cases. Unfortunately, $f$ is only $G-$invariant for a special choice
of $c_{k}$, and this particular choice allows us to prove the conjecture only
in the cases stated in theorem 0.8.
## Chapter 1 Théorèmes de régularité et généralités
Tout au long de cette thèse, on utilise la convention d’Einstein pour les
indices. $M$ sera toujours une variété compacte, sans bord, $C^{\infty}$ de
dimension $n\geq 3$, sauf mention contraire. On commence par rappeler les
définitions des courbures de Riemann, Ricci, scalaire et de Weyl.
### 1.1 Les courbures
###### Définition 1.1.
Soient $(M,g)$ une variété riemannienne $C^{\infty}$ et $\nabla_{g}$ (ou
simplement $\nabla$) la connexion riemannienne associée (i.e. la connexion
sans torsion pour laquelle $g$ est à dérivée covariante nulle). On note par
$\Gamma(M)$ l’ensemble des champs de vecteurs $C^{\infty}$ définis sur $M$.
1. 1.
$X,\;Y$, $Z$ et $T$ étant quatre champs de vecteurs dans $\Gamma(M)$. La
courbure de Riemann $R$ est l’application bilinéaire antisymétrique de
$\Gamma(M)\times\Gamma(M)$ dans $Hom(\Gamma(M),\Gamma(M))$, définie par
$R(X,Y)Z=\nabla_{X}\nabla_{Y}Z-\nabla_{Y}\nabla_{X}Z-\nabla_{[X,Y]}Z$
On appelle tenseur de courbure de Riemann de $g$ le champ de tenseur
$C^{\infty}$ quatre fois covariants défini par
$R(X,Y,Z,T)=g(X,R(Z,T)Y)=R_{ijkl}X^{i}Y^{j}Z^{k}T^{l}$
dans une carte locale; $R_{ijkl}$ sont les composantes du tenseur de courbure.
2. 2.
La courbure de Ricci de $g$ est le champ de tenseur $C^{\infty}$, deux fois
covariants, obtenu en contractant par $g$ le tenseur de courbure de Riemann de
$g$ de la manière suivante
$Ric_{ij}=g^{kl}R_{kilj}$
où $g^{kl}$ sont les composantes de $g^{-1}$.
3. 3.
La courbure scalaire de $g$ est la trace du tenseur de Ricci, notée $R_{g}$.
Dans une carte locale $R_{g}=g^{ij}Ric_{ij}$
###### Propriétés 1.1.
Soient $X$ un champ de vecteurs et $\omega$ une $1-$forme. Dans un système de
coordonnées locales, $(\nabla_{\partial_{i}}X)^{k}$ est notée
$\nabla_{i}X^{k}$ et $(\nabla_{\partial_{i}}\omega)_{k}$ est notée
$\nabla_{i}\omega_{k}$. Rappelons les formules de permutation des dérivées
covariantes suivantes
$\nabla_{ij}X^{l}-\nabla_{ji}X^{l}=R^{l}_{kij}X^{k},\qquad\nabla_{ij}\omega_{l}-\nabla_{ji}\omega_{l}=-R^{k}_{lij}\omega_{k}$
où $R^{l}_{kij}=g^{lm}R_{mkij}$.
Pour tout champ de tenseur $C^{2}$ deux fois covariants $T$:
$\nabla_{ij}T_{kl}-\nabla_{ji}T_{kl}=-R^{m}_{kij}T_{ml}-R^{m}_{lij}T_{km}$
On aura l’occasion d’utiliser ces propriétés dans le chapitre 4 et
l’appendice.
###### Définition 1.2.
La courbure de Weyl $W$ de la variété riemannienne $(M,g)$, de dimension
$n\geq 3$ est définie par le champ de tenseurs quatre fois covariants dont les
composantes sont
$W_{ijkl}=R_{ijkl}-\frac{1}{n-2}(R_{ik}g_{jl}-R_{il}g_{jk}+R_{jl}g_{ik}-R_{jk}g_{il})+\frac{R_{g}}{(n-1)(n-2)}(g_{ik}g_{jl}-g_{il}g_{jk})$
Le tenseur de Weyl est obtenu à partir du tenseur de courbure de Riemann, en
recherchant un tenseur invariant par transformation conforme de la variété: si
$\tilde{g}=e^{f}g$ est une métrique conforme à $g$ alors
$W_{\tilde{g}}=e^{f}W_{g}$.
###### Définition 1.3.
Une variété riemannienne $(M,g)$ est dite conformément plate si pour tout
$Q\in M$, il existe un voisinage ouvert $\Omega$ de $Q$ et une métrique
$\tilde{g}$ conforme à $g$, tels que le tenseur de courbure de Riemann associé
à la métrique $\tilde{g}$ est identiquement nul sur $\Omega$.
Le tenseur de Weyl est identiquement nul si la variété est de dimension 3 ou
si elle est conformément plate.
### 1.2 Le Laplacien
###### Définition 1.4.
Sur $(M,g)$ une variété riemannienne $C^{\infty}$, le Laplacien $\Delta_{g}f$
d’une fonction $f\in C^{2}(M)$ est l’opposé de la trace de la hessienne de
$f$, donné par
$\Delta_{g}f=-\nabla_{i}\nabla^{i}f=-g^{ij}\nabla_{i}\nabla_{j}f=-g^{ij}(\partial_{ij}f-\Gamma^{k}_{ij}\partial_{k}f)$
Dans un système de coordonnées polaires $(r,\xi^{i})$ ($i.e.\;g_{rr}=1$,
$g_{r\xi^{i}}=0$) si $f(r)$ est une fonction radiale alors le Laplacien de $f$
s’écrit
$\Delta_{g}f(r)=-f^{\prime\prime}(r)-\frac{n-1}{r}f^{\prime}(r)-f^{\prime}(r)\partial_{r}\log\sqrt{\det
g}$
##### Remarques.
Tout au long de cette thèse, on utilise le Laplacien géométrique défini ci-
dessus, avec des valeurs propres positives.
On définit le Laplacien $\Delta_{g}f$ d’une fonction $f\in H_{1}(M)$ (voir
plus bas pour la définition de $H_{1}(M)$) par : pour tout $\psi\in H_{1}(M)$
$(\Delta_{g}f,\psi)_{g,L^{2}}=(\nabla f,\nabla\psi)_{g,L^{2}}$
où $(\cdot,\cdot)_{g,L^{2}}$ est le produit scalaire standard dans $L^{2}(M)$
muni de la métrique $g$, dont on omettra la lettre $g$ lorsque il n y a pas
d’ambiguïté.
### 1.3 Les espaces de Sobolev
###### Définition 1.5.
Soit $(M,g)$ une variété riemannienne $C^{\infty}$ de dimension $n$, $p\geq 1$
un nombre réel, $k$ et $r$ sont deux entiers naturels
1. 1.
L’espace de Sobolev $H_{k}^{p}(M)$ est le complété de l’espace $\\{f\in
C^{\infty}(M),\;|\nabla^{l}f|\in L^{p}(M)\quad\forall\;0\leq l\leq k\\}$ pour
la norme
$\|f\|_{p,k}=\sum_{l=0}^{k}\|\nabla^{l}f\|_{p}$
2. 2.
$C^{r,\beta}(M)$ est l’espace de Hölder des fonctions $C^{r}$ dont la r-ème
dérivée appartient à
$C^{\beta}(M)=\\{f\in C^{0}(M),\;\|f\|_{C^{\beta}}:=\|f\|_{\infty}+\sup_{P\neq
Q}\frac{|f(P)-f(Q)|}{d(P,Q)^{\beta}}<+\infty\\}$
avec $\beta\in[0,1[$.
$C^{0,1}(M)$ est l’ensemble des fonctions lipschitzienne.
L’espace $H^{2}_{k}(M)$ est un espace de Hilbert pour le produit scalaire
suivant
$(f,h)_{H_{k}}=\sum_{l=0}^{k}(\nabla^{l}f,\nabla^{l}h)_{L^{2}}$
Dans la suite, $H^{2}_{k}(M)$ est noté $H_{k}(M)$.
La norme correspondante au produit scalaire sur $H_{k}(M)$ est équivalente à
la norme $\|\cdot\|_{2,k}$.
###### Définition 1.6.
Soit $(M,g_{0})$ une variété riemannienne compacte de dimension $n$. On note
par $T^{*}(M)$ le fibré cotangent de $M$. L’espace $H^{p}_{k}(M,T^{*}M\otimes
T^{*}M)$ est l’ensemble des sections $g$ (des tenseurs 2 fois covariants)
telles que dans toute carte exponentielle, les composantes $g_{ij}$ de $g$
sont dans $H^{p}_{k}$.
L’espace $H^{p}_{k}(M,T^{*}M\otimes T^{*}M)$ ne dépend pas de la métrique
$g_{0}$. On peut aussi définir cet espace, en utilisant le théorème du
plongement isométrique de Nash. Les deux théorèmes qui suivent sont encore
valables pour cet espace $H^{p}_{k}(M,T^{*}M\otimes T^{*}M)$.
###### Théorème 1.1 (Théorème d’inclusions de Sobolev).
Soit $(M,g)$ une variété riemannienne compacte de dimension $n$.
$(i)$
Si $k$ et $l$ deux entiers ($k>l\geq 0$), $p$ et $q$ deux réels ($p>q\geq 1$)
qui vérifient $1/p=1/q-(k-l)/n$ alors $H_{k}^{q}(M)$ est inclus dans
$H_{l}^{p}(M)$ et l’inclusion $H_{k}^{q}(M)\subset H_{l}^{p}(M)$ est continue.
$(ii)$
Si $r\in\mathbb{N}$ et $(k-r)/n>1/q$ alors l’inclusion $H_{k}^{q}(M)\subset
C^{r}(M)$ est continue
$(iii)$
Si $(k-r-\beta)/n\geq 1/q$ alors l’inclusion $H_{k}^{q}(M)\subset
C^{r,\beta}(M)$ est continue avec $\beta\in]0,1[$
dans tous les cas $H_{k}^{q}(M)$ ne dépend pas de la métrique $g$
Une preuve détaillée du théorème est donnée dans le livre de T. Aubin [Aubin],
chapitre 2, celui de Adams [Ada] ou de E. Hebey [Heb]. On utilisera souvent
l’espace de Hilbert $H_{1}(M)$ muni de la norme
$\|\varphi\|_{H_{1}}^{2}=\|\varphi\|^{2}_{2}+\|\nabla\varphi\|^{2}_{2}$
pour minimiser des fonctionnelles. Cet espace est inclus continûment dans
$L^{q}(M)$, pour tout $q\in[1,2n/(n-2)]$.
Kondrakov a montré que les inclusions de Sobolev sont compactes dans les cas
suivants:
###### Théorème 1.2 (Kondrakov).
Soit $(M_{n},g)$ une variété riemannienne compacte. $k$ un entier naturel, $p$
et $q$ deux nombres réels qui vérifient $1\geq 1/p>1/q-k/n>0$ alors
(i)
l’inclusion $H_{k}^{q}(M)\subset L^{p}(M)$ est compacte
(ii)
l’inclusion $H_{k}^{q}(M)\subset C^{\alpha}(M)$ est compacte si $k-\alpha>n/q$
avec $0\leq\alpha<1$
Grâce aux inclusions de Sobolev, on montre le résultat suivant:
###### Proposition 1.1.
Soit $(M,g)$ une variété riemannienne compacte de dimension $n$. Si $p>n/2$
alors $H^{p}_{2}(M)$ est une algèbre.
###### Preuve.
Il suffit de montrer que si $\varphi$ et $\psi$ sont dans $H^{p}_{2}(M)$ alors
$\psi\varphi\in H^{p}_{2}(M)$. Par les inclusions de Sobolev (théorème 1.1),
$H^{p}_{2}(M)\subset C^{\beta}(M)$ donc $\varphi$ et $\psi$ sont continues.
Par la compacité de $M$ et la continuité de $\varphi$ et $\psi$:
$\nabla(\psi\varphi)=\psi\nabla\varphi+\varphi\nabla\psi\in L^{p}(M)$
D’autre part
$\nabla^{2}(\psi\varphi)=\psi\nabla^{2}\varphi+\varphi\nabla^{2}\psi+\nabla\varphi\otimes\nabla\psi+\nabla\psi\otimes\nabla\varphi\in
L^{p}(M)$
En effet, $|\psi\nabla^{2}\varphi|+|\varphi\nabla^{2}\psi|\in L^{p}(M)$ par le
même argument que précédemment, et comme
$\||\nabla\varphi||\nabla\psi|\|_{p}\leq\|\nabla\varphi\|_{2p}\|\nabla\psi\|_{2p}$
est borné (cf. théorème 1.1) alors $|\nabla\varphi||\nabla\psi|\in L^{p}(M)$.
D’où $\varphi\psi\in H^{p}_{2}(M)$ ∎
#### 1.3.1 Théorèmes des espaces de Banach
###### Théorème 1.3.
Un espace de Banach $\mathcal{B}$ est réflexif si et seulement si sa boule
unité fermée est faiblement compact.
Puisque les espaces de Sobolev sont réflexifs, on utilisera ce théorème comme
suit: si on a une certaine suite de fonctions $(\varphi_{i})_{i\in\mathbb{N}}$
bornée dans $H_{k}(M)$ alors il existe une sous-suite
$(\varphi_{q_{i}})_{i\in\mathbb{N}}$ qui converge vers $\varphi\in H_{k}(M)$
et $\liminf_{i\to+\infty}\|\varphi_{q_{i}}\|_{H_{k}}\geq\|\varphi\|_{H_{k}}$
###### Théorème 1.4.
Soit $p\in]1,+\infty[$ et $(\varphi_{i})_{i\in\mathbb{N}}$ une suite bornée
dans $L^{p}(\mathcal{B})$, qui converge presque partout vers $\varphi$, alors
$\varphi\in L^{p}(\mathcal{B})$ et $(\varphi_{i})$ converge faiblement vers
$\varphi$ dans $L^{p}(\mathcal{B})$.
### 1.4 Inégalité de la meilleure constante
###### Théorème 1.5 (Aubin–Talenti).
Soit $(M,g)$ une variété riemannienne compacte de dimension $n$. Pour tout
$\varepsilon>0$ il existe $A(\varepsilon)>0$ tel que
$\forall\varphi\in
H_{1}^{p}(M)\quad\|\varphi\|_{p^{*}}\leq(K(n,p)+\varepsilon)\|\nabla\varphi\|_{p}+A(\varepsilon)\|\varphi\|_{p}$
$p^{*}=\frac{np}{n-p}\mbox{ et
}K(n,p)=\frac{p-1}{n-p}\biggl{(}\frac{n-p}{n(p-1)}\biggr{)}^{1/p}\biggl{[}\frac{\Gamma(n+1)}{\Gamma(n/p)\Gamma(n+1-n/p)\omega_{n-1}}\biggr{]}^{1/n}$
$K(n,1)=\frac{1}{n}\biggl{[}\frac{n}{\omega_{n-1}}\biggr{]}^{1/n}$
$K(n,p)$ est la meilleure constante au sens où pour toute constante plus
petite qui remplace $K(n,p)$, l’inégalité ci-dessus devient fausse pour une
certaine fonction $\varphi\in H^{p}_{1}(M)$. La preuve détaillée du théorème
de T. Aubin est reprise dans le livre [Aubin]. Beaucoup de travaux ont été
faits depuis sur la validité de cette inégalité (sur les puissances dans cette
inégalité aussi) lorsque $\varepsilon=0$. Des résultats ont été obtenus par T.
Aubin et Y.Y. Li [AL], R.J. Biezuner [Bie], O. Druet [Dru, Dru2], E. Hebey et
M. Vaugon [HV3, HVI]…
Dans le chapitre suivant (cf. théorème 1.7), on généralisera cette inégalité
par l’inégalité de Hardy.
### 1.5 L’inégalité de Hardy sur une variété compacte
###### Définition 1.7.
Soit $P$ un point d’une variété riemannienne $(M,g)$. $\rho_{P}$ est la
fonction définie par:
$\rho_{P}(Q)=\begin{cases}&d(P,Q)\mbox{ si }d(P,Q)<\delta(M)\\\
&\delta(M)\mbox{ si }d(P,Q)\geq\delta(M)\end{cases}$ (1.1)
avec $\delta(M)$ le rayon d’injectivité de la variété $M$
La fonction $\rho$ dépend évidemment du point $P\in M$ que l’on omettra
parfois dans les notations.
###### Définition 1.8.
Sur une variété riemannienne $(M,g)$, on définit $L^{p}(M,\rho^{\gamma})$
comme étant l’espace des fonctions $u$ telles que $\rho^{\gamma}|u|^{p}$ soit
intégrable. On le munit de la norme
$\|u\|^{p}_{p,\rho^{\gamma}}:=\int_{M}\rho^{\gamma}|u|^{p}\mathrm{d}v$
où $p\geq 1$ et $\rho$ est la fonction introduite dans la définition
précédente.
###### Proposition 1.2.
Pour tout $p\geq 1$, $L^{p}(M,\rho^{\gamma})$ muni de la norme
$\|\cdot\|_{p,\rho^{\gamma}}$ est un espace de Banach
###### Preuve.
La complétude de l’espace $L^{p}(M,\rho^{\gamma})$ pour la norme
$\|\cdot\|_{p,\rho^{\gamma}}$ découle du fait que $L^{p}(M)$ est un espace
complet et que $\|u\|_{p,\rho^{\gamma}}=\|\rho^{\gamma/p}u\|_{p}$ pour tout
$u\in L^{p}(M,\rho^{\gamma})$ ∎
###### Théorème 1.6 (Inégalité de Hardy).
Pour toute fonction $u\in C_{o}^{\infty}(\mathbb{R}^{n})$, il existe une
constante $c>0$ telle que
$\||x|^{\gamma}u\|_{p}\leq c\||x|^{\beta}\nabla_{l}u\|_{q}$
où $1\leq q\leq p\leq qn/(n-lq)$, $\gamma=\beta-l+n(1/q-1/p)>-n/p$ et $n>lq$
Ce type d’inégalité à une dimension a été introduite par Hardy, puis
généralisée pour toute dimension, le livre de V.G. Maz’ja [Maz] est une bonne
référence où on trouvera la preuve de ce théorème. Dans notre étude, on
s’intéresse à cette inégalité dans le cas où $\beta=0$ et $l=1$. Dans ce cas
précis, la constante $c=K(n,q,\gamma)$ est la meilleure constante dans
l’inégalité ci-dessus. Si $p\gamma>-q$, cette constante est atteinte pour la
fonction
$x\mapsto(1+|x|^{(q+p\gamma)/(q-1)})^{(q-n)/(q+p\gamma)}$
et $K(n,q,-q)=q/(n-q)$. (cf. [Chu], [Lieb])
###### Théorème 1.7.
Soit $(M,g)$ une variété riemannienne compacte de dimension $n$ et
$p,\,q\mbox{ et }\gamma$ des nombres réels qui satisfont
$(\gamma+n)/p=-1+n/q>0$ et $1\leq q\leq p\leq qn/(n-q)$. Pour tout
$\varepsilon>0$, il existe $A(\varepsilon,q,\gamma)$ tel que
$\forall u\in
H^{q}_{1}(M)\quad\|u\|_{p,\rho^{\gamma}}\leq(K(n,q,\gamma)+\varepsilon)\|\nabla
u\|_{q}+A(\varepsilon,q,\gamma)\|u\|_{q}$ (1.2)
en particulier $K(n,q,0)=K(n,q)$ la meilleure constante dans l’inégalité de
Sobolev
###### Preuve.
La preuve de ce théorème est quasiment identique à celle de T. Aubin (voir
[Aubin], chapitre 2) dans le cas des inclusions de Sobolev sur les variétés
riemanniennes complètes à courbure bornée.
On commence par montrer le lemme suivant:
###### Lemme 1.1.
Pour tout $f\in H^{q}_{1}(M)$ à support dans $B_{P}(\delta)$
$\|f\|_{p,\rho^{\gamma}}\leq K_{\delta}(n,q,\gamma)\|\nabla f\|_{q}$
avec $B_{P}(\delta)$ une boule de centre $P$ et de rayon $\delta<\delta(M)$.
Lorsque $\delta\to 0$, $K_{\delta}(n,q,\gamma)\to K(n,q,\gamma)$
##### _Preuve du lemme._
On se place dans un système de coordonnées géodésiques $\\{r,\theta^{i}\\}$,
centré en $P$. Soit $\varepsilon>0$ donné, si $\delta$ est choisi suffisamment
petit, on a les estimées de la métrique suivantes (Aubin [Aubin], p. 20) :
$1-\varepsilon\leq\sqrt{g_{\theta^{i}\theta^{i}}(r,\theta)}\leq
1+\varepsilon\mbox{ et }(1-\varepsilon)^{n-1}\leq\sqrt{\det
g(r,\theta)}\leq(1+\varepsilon)^{n-1}$
où $g=dr^{2}+r^{2}g_{\theta^{i}\theta^{j}}d\theta^{i}d\theta^{j}$.
Si on pose $\tilde{f}(x)=f(\exp_{P}x)$, on obtient une fonction bien définie
sur $\mathbb{R}^{n}$ à support dans $\\{x\in\mathbb{R}^{n};\;|x|<1\\}$ qui
vérifie, d’après le théorème 1.6:
$\biggl{(}\int_{\mathbb{R}^{n}}|x|^{\gamma}|\tilde{f}|^{p}dx\biggr{)}^{1/p}\leq
K(n,q,\gamma)\biggl{(}\int_{\mathbb{R}^{n}}|\nabla\tilde{f}|^{q}dx\biggr{)}^{1/q}$
de plus si $Q=\exp_{P}x\in B_{P}(\delta)$ alors
$|x|=d(P,Q)=\rho(Q)\mbox{ et
}(1-\varepsilon)|\nabla\tilde{f}|_{\mathcal{E}}(x)\leq|\nabla
f|_{g}(\exp_{P}x)$
On déduit que
$\|f\|_{p,\gamma}\leq(1+\varepsilon)^{(n-1)/p}\|\tilde{f}\|_{p,\gamma}\mbox{
et }\|\nabla f\|_{q}\geq(1-\varepsilon)^{1+(n-1)/q}\|\nabla\tilde{f}\|_{q}$
Finalement
$\|f\|_{p,\rho^{\gamma}}\leq K_{\delta}(n,q,\gamma)\|\nabla f\|_{q}$
avec
$K_{\delta}(n,q,\gamma)=(1-\varepsilon)^{-1+(1-n)/q}(1+\varepsilon)^{(n-1)/p}K(n,q,\gamma)$.
Ce qui achève la preuve du lemme.
Pour terminer la preuve du théorème, on considère un recouvrement fini
$\\{B_{P_{i}}(\delta)\\}_{1\leq i\leq m}$ de $M$ qui existe puisque la variété
est compacte. Soit $\\{h_{i}\\}_{1\leq i\leq m}$ une partition de l’unité
associée à ce recouvrement. On pose
$\eta_{i}=\frac{h_{i}^{[q]+1}}{\sum_{k=1}^{m}h_{k}^{[q]+1}}$
où $[q]$ est la partie entière de $q$.
$\\{B_{P_{i}}(\delta),\eta_{i}\\}_{1\leq i\leq m}$ est aussi une partition de
l’unité de $M$ et $\eta_{i}^{1/q}\in C^{1}(M)$, donc il existe $H>0$ tel que,
pour tout $i\leq m$: $|\nabla\eta_{i}^{1/q}|\leq H$
Pour tout $u\in H_{1}^{q}(M)$, on a
$\|u\|_{p,\rho^{\gamma}}^{q}=\|u^{q}\|_{p/q,\rho^{\gamma}}=\|\sum_{i=1}^{m}\eta_{i}u^{q}\|_{p/q,\rho^{\gamma}}\leq\sum_{i=1}^{m}\|\eta_{i}u^{q}\|_{p/q,\rho^{\gamma}}\leq\sum_{i=1}^{m}\|\eta^{1/q}_{i}u\|^{q}_{p,\rho^{\gamma}}$
Or d’après le lemme 1.1, on a pour tout $i\leq m$
$\|\eta^{1/q}_{i}u\|^{q}_{p,\rho^{\gamma}}\leq
K^{q}_{\delta}(n,q,\gamma)\|\nabla(\eta^{1/q}_{i}u)\|^{q}_{q}$
donc
$\displaystyle\|u\|_{p,\rho^{\gamma}}^{q}$ $\displaystyle\leq
K^{q}_{\delta}(n,q,\gamma)\sum_{i=1}^{m}\int_{M}(|\nabla\eta^{1/q}_{i}||u|+\eta^{1/q}_{i}|\nabla
u|)^{q}\mathrm{d}v$ $\displaystyle\leq
K^{q}_{\delta}(n,q,\gamma)\sum_{i=1}^{m}\int_{M}\eta_{i}|\nabla
u|^{q}+\mu|\nabla
u|^{q-1}|\nabla\eta_{i}^{1/q}|\eta_{i}^{(q-1)/q}|u|+\nu|\nabla\eta^{1/q}_{i}|^{q}|u|^{q}\mathrm{d}v$
$\displaystyle\leq K^{q}_{\delta}(n,q,\gamma)(\|\nabla u\|_{q}^{q}+\mu
mH\|\nabla u\|_{q}^{q-1}\|u\|_{q}+\nu mH^{q}\|u\|_{q}^{q})$
car il existe $\mu,\;\nu\in\mathbb{R}_{+}$ tel que pour tout $t\geq
0,\quad(1+t)^{q}\leq 1+\mu t+\nu t^{q}$
On a aussi pour tout $z,\;y,\;\lambda\in\mathbb{R}_{+}^{*}\qquad
qz^{q-1}y\leq\lambda(q-1)z^{q}+\lambda^{1-q}y^{q}$
Si on pose $z=\|\nabla u\|_{q}$, $y=\|u\|_{q}$ et
$\lambda=q\varepsilon_{0}/(\mu mH(q-1))$ avec $\varepsilon_{0}>0$ petit, on
obtient
$\|u\|_{p,\rho^{\gamma}}^{q}\leq
K^{q}_{\delta}(n,q,\gamma)[(1+\varepsilon_{0})\|\nabla
u\|_{q}^{q}+A(\varepsilon_{0})\|u\|_{q}^{q}]$
On peut choisir $\delta$ et $\varepsilon_{0}$ suffisamment petits de sorte que
$K_{\delta}(n,q,\gamma)(1+\varepsilon_{0})^{1/q}\leq
K(n,q,\gamma)+\varepsilon$ et si on pose
$A(\varepsilon,q,\gamma)=(K(n,q,\gamma)+\varepsilon)A(\varepsilon_{0})^{1/q}$
alors l’inégalité (1.2) est établie ∎
###### Théorème 1.8.
Soit $(M,g)$ une variété riemannienne compacte de dimension $n$.
1. 1.
Si $(\gamma+n)/p=-1+n/q>0$ et $1\leq q\leq p$ alors l’inclusion
$H^{q}_{1}(M)\subset L^{p}(M,\rho^{\gamma})$ est continue.
2. 2.
Si $(\gamma+n)/p>-1+n/q>0$, $\gamma\leq 0$ et $q\leq p$ alors cette inclusion
est compacte.
###### Preuve.
La preuve de la première partie de ce théorème est évidente compte tenu de
l’inégalité démontrée dans le théorème 1.7. La seconde partie du théorème est
établie si on montre que $H^{q}_{1}(M)\subset L^{r}(M)\subset
L^{p}(M,\rho^{\gamma})$ continûment, où la première inclusion est compacte
pour un certain $r\geq 1$ que l’on déterminera.
D’après l’inégalité de Hölder, on a pour tout $u\in H^{q}_{1}(M)$
$\|u\|_{p,\rho^{\gamma}}^{p}=\int_{M}\rho^{\gamma}|u|^{p}\mathrm{d}v\leq\biggl{(}\int_{M}\rho^{\gamma
r^{\prime}}\mathrm{d}v\biggr{)}^{1/r^{\prime}}\|u\|^{p}_{r}$
où $r^{\prime}=r/(r-p)$. Pour que le second membre de cette inégalité soit
fini, il suffit que $\gamma r/(r-p)>-n$, pour le premier facteur, et
$1/r>1/q-1/n$, pour le second facteur. De plus le théorème de Kondrakov 1.2
assure que si $r\geq 1$ satisfait la deuxième inégalité alors l’inclusion
$H^{q}_{1}(M)\subset L^{r}(M)$ est compacte. On en déduit que l’on doit avoir
$\frac{n}{r}<\frac{\gamma+n}{p}\mbox{ et }\frac{n}{r}>-1+\frac{n}{q}$
Puisque $(\gamma+n)/p>-1+n/q$ par hypothèse alors, pour que $u\in
L^{p}(M,\rho^{\gamma})$ et que l’inclusion soit compacte, il suffit de poser
$\frac{n}{r}=\frac{1}{2}(\frac{\gamma+n}{p}-1+\frac{n}{q})$
Comme $\gamma\leq 0$ on a $n/r<(\gamma+n)/p\leq n/p$ donc $r>p\geq 1$. ∎
##### Remarque.
Puisque la fonction $\rho$ (cf. définition 1.7) dépend de $P\in M$, l’espace
$L^{p}(M,\rho_{P}^{\gamma})$ dépend aussi du point $P$ choisi, et si $P\neq
P^{\prime}$, il n’y a pas en général d’inclusions entre
$L^{p}(M,\rho_{P}^{\gamma})$ et $L^{p}(M,\rho_{P^{\prime}}^{\gamma})$.
Cependant les inclusions et les inégalités qu’on a déjà montrées dans les
théorèmes 1.7 et 1.8 sont valables pour tout point $P\in M$.
### 1.6 La régularité des solutions de l’équation de type Yamabe
Lorsque on cherche des solutions d’équations aux dérivées partielles, la
première étape donne fréquemment des solutions faibles (dans notre cas, elles
seront dans $H_{1}(M)$). Dans la plupart des cas on trouve la régularité des
solutions en appliquant le théorème de régularité pour les opérateurs
elliptiques à coefficients continus suivant:
###### Théorème 1.9.
Soient $\Omega$ un ouvert de $\mathbb{R}^{n}$ et $L$ un opérateur linéaire
d’ordre 2 uniformément elliptique qui s’écrit sous la forme
$L(u)=a^{ij}\partial_{ij}u+b^{i}\partial_{i}u+hu$ (1.3)
où $a^{ij},\;b^{i}\mbox{ et }h$ sont des fonctions bornées dans $C^{k}$,
$k\in\mathbb{N}$.
Soit $u$ une solution de l’équation $Lu=f$ au sens des distributions.
1. (i)
Si $f\in C^{k,\alpha}(\Omega)$ alors $u\in C^{k+2,\alpha}(\Omega)$
2. (ii)
Si $f\in H_{k}^{p}(\Omega)$ alors $u\in H_{k+2}^{p}(\Omega)$
Ce théorème est standard, on peut en trouver une preuve dans le livre de D.
Gilbarg et N. Trudinger [GT].
Les deux théorèmes suivants permettent de trouver la meilleure régularité des
solutions d’un certains type d’équations. Ils sont fondamentaux pour la suite,
associés aux théorèmes de régularité habituels pour les opérateurs elliptiques
ci-dessus. N. Trudinger [Trud] avait montré que les solutions faibles de
l’équation de Yamabe (3.1) (voir chapitre 3) sont toujours $C^{\infty}$ grâce
à ces deux théorèmes. Le premier théorème a été utilisé implicitement par H.
Yamabe [Yam] et on peut en trouver une preuve dans l’article de J. Serrin
[Ser]. Le deuxième théorème est plus spécifique car il s’applique à des
équations de type Yamabe qu’on étudiera dans le prochain chapitre.
###### Théorème 1.10.
Sur une variété riemannienne compacte $(M,g)$, si $u\geq 0$ est une solution
faible dans $H_{1}(M)$, non triviale, de l’équation $\Delta u+hu=0$, c’est à
dire si
$\forall v\in H_{1}(M)\qquad(\nabla u,\nabla v)_{L^{2}}+(hu,v)_{L^{2}}=0$
avec $h\in L^{p}(M)$ et $p>n/2$, alors $u\in C^{1-[n/p],\beta}(M)$ et est
strictement positive bornée.
$[n/p]$ est la partie entière de $n/p$ et $\beta\in]0,1[$.
Observons que si $u$ est une fonction qui satisfait les hypothèses de ce
théorème alors $\Delta u\in L^{p}(M)$. Par le théorème de régularité 1.9,
$u\in H^{p}_{2}(M)$ et par les inclusions de Sobolev, $u\in
C^{1-[n/p],\beta}(M)$
Le théorème 1.10 permet de montrer le théorème suivant:
###### Théorème 1.11.
Soit $(M,g)$ une variété riemannienne compacte $C^{\infty}$ de dimension $n$.
$p$ et $\tilde{h}$ sont deux nombres réels, avec $p>n/2$. Si $\varphi\in
H_{1}(M)$ une solution faible positive non triviale de l’équation
$\Delta_{g}\psi+h\psi=\tilde{h}\psi^{\frac{n+2}{n-2}}$ (1.4)
alors $\varphi\in H_{2}^{p}(M)\subset C^{1-[n/p],\beta}(M)$ et $\varphi$ est
strictement positive.
###### Preuve.
Pour montrer ce théorème, il suffit de montrer qu’il existe $\varepsilon>0$
tel que $\varphi\in L^{(\varepsilon+2n)/(n-2)}(M)$. En effet si $\varphi$
satisfait aux hypothèses du théorème et qu’elle est dans
$L^{(\varepsilon+2n)/(n-2)}(M)$, alors elle est solution de l’équation
$\Delta_{g}u+(h-\tilde{h}\varphi^{\frac{4}{n-2}})u=0$
avec $h-\tilde{h}\varphi^{\frac{4}{n-2}}\in L^{r}(M)$ et
$r=\min(p,\frac{2n+\varepsilon}{4})>n/2$. Par le théorème 1.10, on en déduit
que $\varphi$ est strictement positive bornée. Par le théorème de régularité
1.9 et les inclusions de Sobolev, on montre que $\varphi$ appartient à
$H^{p}_{2}(M)$ avec $p>n/2$.
Soient $l$ un nombre réel strictement positif et $H$, $F$ deux fonctions
réelles continues sur $\mathbb{R}_{+}$ définies par:
$\displaystyle H(t)$ $\displaystyle=\begin{cases}t^{\gamma}&\text{ si }0\leq
t\leq l\\\ l^{q-1}(ql^{q-1}t-(q-1)l^{q})&\text{ si }t>l\end{cases}$
$\displaystyle F(t)$ $\displaystyle=\begin{cases}t^{q}&\text{ si }0\leq t\leq
l\\\ ql^{q-1}t-(q-1)l^{q}\qquad&\text{ si }t>l\end{cases}$
$\text{o\\`{u} }\gamma=2q-1,\text{ et }1<q<\frac{n(p-1)}{p(n-2)}$
Comme $\varphi$ est une fonction positive appartenant à $H_{1}(M)$,
$H\circ\varphi$ et $F\circ\varphi$ sont également dans $H_{1}(M)$. Notons que
pour tout $t\in\mathbb{R}_{+}-\\{l\\}$
$qH(t)=F(t)F^{\prime}(t),\;(F^{\prime}(t))^{2}\leq qH^{\prime}(t)\text{ et
}F^{2}(t)\geq tH(t)$ (1.5)
Si $\varphi$ est une solution faible de l’équation (1.4) alors
$\forall\psi\in
H_{1}(M)\quad\int_{M}\nabla\varphi\cdot\nabla\psi\mathrm{d}v+\int_{M}h\varphi\psi\mathrm{d}v=\tilde{h}\int_{M}\varphi^{N-1}\psi\mathrm{d}v$
(1.6)
où $N=2n/(n-2)$.
On choisit $\psi=\eta^{2}H\circ\varphi$, où $\eta$ est une fonction de classe
$C^{1}$ à support dans la boule $B_{P}(2\delta)$ de rayon $2\delta$
suffisamment petit telle que $\eta=1$ sur $B_{P}(\delta)$. Si on substitue
dans (1.6), on obtient
$\int_{M}\eta^{2}H^{\prime}\circ\varphi|\nabla\varphi|^{2}\mathrm{d}v+2\int_{M}\eta
H\circ\varphi\nabla\varphi\cdot\nabla\eta\mathrm{d}v=\tilde{h}\int_{M}\varphi^{N-1}\eta^{2}H\circ\varphi\mathrm{d}v-\int_{M}h\varphi\eta^{2}H\circ\varphi\mathrm{d}v$
(1.7)
On pose $f=F\circ\varphi$. On estimera les quatre intégrales ci-dessus, en
utilisant la fonction $f$ et les relations (1.5). On a $\nabla
f=F^{\prime}\circ\varphi\nabla\varphi$ donc, en utilisant la deuxième relation
de (1.5)
$|\nabla f|^{2}=(F^{\prime}\circ\varphi)^{2}|\nabla\varphi|^{2}\leq
qH^{\prime}\circ\varphi|\nabla\varphi|^{2}$
On en déduit que la première intégrale de l’égalité (1.7) est minorée par
$\frac{1}{q}\|\eta\nabla
f\|_{2}^{2}\leq\int_{M}\eta^{2}H^{\prime}\circ\varphi|\nabla\varphi|^{2}\mathrm{d}v$
La première relation de (1.5) et l’inégalité de Cauchy–Schwarz impliquent que
la deuxième intégrale de (1.7) est minorée par:
$2\int_{M}\eta
H\circ\varphi\nabla\varphi\cdot\nabla\eta\mathrm{d}v=\frac{2}{q}\int_{M}\eta
f\nabla f\nabla\eta\mathrm{d}v\geq\frac{-2}{q}\|f\nabla\eta\|_{2}\|\eta\nabla
f\|_{2}$
Grâce à la dernière relation de (1.5), on a $\varphi H\circ\varphi\leq f^{2}$.
Les deux intégrales de droite dans (1.7) sont donc majorées par:
$\biggl{|}\tilde{h}\int_{M}\varphi^{N-1}\eta^{2}H\circ\varphi\mathrm{d}v-\int_{M}h\varphi\eta^{2}H\circ\varphi\mathrm{d}v\biggr{|}\leq|\tilde{h}|\|\varphi\|^{4/(n-2)}_{N,2\delta}\|\eta
f\|^{2}_{N}+\|h\|_{p}\|\eta f\|_{2p/(p-1)}^{2}$
où $\|\varphi\|^{N}_{N,r}=\int_{B_{P}(r)}\varphi^{N}\mathrm{d}v$. Si on
regroupe ces estimées, l’égalité (1.7) devient:
$\|\eta\nabla f\|_{2}^{2}-2\|f\nabla\eta\|_{2}\|\eta\nabla f\|_{2}\leq
q(|\tilde{h}|\|\varphi\|^{4/(n-2)}_{N,2\delta}\|\eta
f\|^{2}_{N}+\|h\|_{p}\|\eta f\|_{2p/(p-1)}^{2})$ (1.8)
Remarquons que pour tout nombre réel positif $a,\;b,\;c\text{ et }d$, si
$a^{2}-2ab\leq c^{2}+d^{2}$ alors $a\leq c+d+2b$. En utilisant cette remarque,
l’inégalité (1.8) devient:
$\|\eta\nabla
f\|_{2}\leq\sqrt{q|\tilde{h}|}\|\varphi\|^{2/(n-2)}_{N,2\delta}\|\eta
f\|_{N}+\sqrt{q\|h\|_{p}}\|\eta f\|_{2p/(p-1)}+2\|f\nabla\eta\|_{2}$ (1.9)
Par les inclusions de Sobolev (cf. théoème 1.1) on sait qu’il existe une
constante $c>0$ qui dépend seulement de $n$ telle que
$\|\eta f\|_{N}\leq c(\|\eta\nabla f\|_{2}+\|f\nabla\eta\|_{2}+\|\eta
f\|_{2})$
Le choix de $q$ ($q<N$) et l’inégalité (1.9) permettent d’écrire
$(1-c\sqrt{N|\tilde{h}|}\|\varphi\|^{2/(n-2)}_{N,2\delta})\|\eta f\|_{N}\leq
c\bigl{(}\sqrt{N\|h\|_{p}}\|\eta f\|_{2p/(p-1)}+3\|f\nabla\eta\|_{2}+\|\eta
f\|_{2}\bigr{)}$
On choisit $\delta$ suffisamment petit pour que
$\|\varphi\|^{2/(n-2)}_{N,2\delta}\leq 1/(2c\sqrt{N|\tilde{h}|})$
ensuite on fait tendre $l$ vers $+\infty$, on en déduit qu’il existe une
constante $C>0$ qui dépend de
$n,\;\delta,\|\eta\|_{\infty},\;\|\nabla\eta\|_{\infty},\;\|h\|_{p}$ et
$|\tilde{h}|$ telle que
$\|\varphi^{q}\|_{N,2\delta}\leq
C(\|\varphi^{q}\|_{2}+\|\varphi^{q}\|_{2p/(p-1)})$
Comme $\frac{2p}{p-1}q<N$ et que $\varphi$ est bornée dans $L^{N}$ on a
$\|\varphi\|_{qN,2\delta}\leq C$
Si $(\eta_{i})_{i\in I}$ est une partition de l’unité subordonnée au
recouvrement $\\{B_{P_{i}}(\delta)\\}_{i\in J}$ de la variété $M$ alors
$\|\varphi\|^{qN}_{qN}=\sum_{i\in
I}\|\eta_{i}\varphi\|^{qN}_{qN,\delta_{i}}\leq C$
on en déduit que $\varphi\in L^{qN}$ avec $qN>N$. En tenant compte de ce qui a
été dit au début de la preuve, le théorème est démontré. ∎
###### Proposition 1.3.
Soit $(M,g)$ une variété riemannienne compacte, si $u$ est une solution faible
dans $H_{1}(M)$ de l’équation $\Delta u+hu=f$, où $h$ et $f$ sont deux
fonctions telles que $h\in L^{p}(M)$ et $f\in L^{q}(M)$, $p>n/2$ et $q\geq 1$,
alors $u\in H^{\min(p,q)}_{2}(M)$
###### Preuve.
Distinguons les deux cas $q\geq p$ et $q<p$.
1. $(i)$
Si $q\geq p$ . Supposons que $u\in L^{s_{i}}(M)$ et satisfait les hypothèses
de la proposition. Alors $hu\in L^{\frac{ps_{i}}{p+s_{i}}}(M)$, donc $\Delta
u\in L^{\frac{ps_{i}}{p+s_{i}}}(M)$ car $ps_{i}/(p+s_{i})<q$. Le théorème de
régularité 1.9 assure que $u\in H^{\frac{ps_{i}}{p+s_{i}}}_{2}(M)$. Ensuite,
les inclusions de Sobolev $H_{2}^{r}(M)\subset L^{s}(M)$ si $r\leq n/2$ avec
$s=nr/(n-2r)$ et $H^{r}_{2}(M)\subset C^{1-[n/r],\beta}(M)$ si $r>n/2$
permettent d’écrire
$\begin{cases}s_{0}=N\\\ u\in L^{s_{i+1}}(M)\text{ o\\`{u}
}s_{i+1}=\frac{nps_{i}}{np-(p-2n)s_{i}}&\mbox{ si }s_{i}\leq\frac{np}{2p-n}\\\
u\in H^{p}_{2}(M)&\mbox{ si }s_{i}>\frac{np}{2p-n}\end{cases}$
S’il existe $i\in\mathbb{N}$ tel que $s_{i}>\frac{np}{2p-n}$ ce qui est
équivalent à $\frac{ps_{i}}{p+s_{i}}>n/2$ alors $u\in C^{0,\beta}(M)$, ce qui
implique que $\Delta u\in L^{p}(M)$, donc $u\in H^{p}_{2}(M)$ et la
proposition est démontrée. S’il existe $i\in\mathbb{N}$ tel que
$s_{i}=\frac{np}{2p-n}$ alors $u\in L^{\infty}(M)$ et on conclut par le
théorème de régularité que $u\in H^{p}_{2}(M)$. Supposons que pour tout
$i\in\mathbb{N}$, $s_{i}<\frac{np}{2p-n}$ alors la suite
$(s_{i})_{i\in\mathbb{N}}$ est croissante majorée, donc elle converge vers
$s=0$ ce qui est impossible.
2. $(ii)$
Supposons que $q<p$ alors on doit montrer que $u\in H^{q}_{2}(M)$. Supposons
que $u\in L^{s_{i}}(M)$ et satisfait les hypothèses de la proposition. Ceci
implique que $hu\in L^{\frac{ps_{i}}{p+s_{i}}}(M)$ donc $\Delta u\in
L^{r_{i}}(M)$ avec $r_{i}=\min(q,\frac{ps_{i}}{p+s_{i}})$. Par le théorème de
régularité 1.9, $u\in H^{r_{i}}_{2}(M)$. Donc
$\begin{cases}s_{0}=N\\\ u\in L^{s_{i+1}}(M)\text{ o\\`{u}
}s_{i+1}=\frac{nr_{i}}{n-2r_{i}}&\mbox{ si }r_{i}\leq n/2\\\ u\in
H^{q}_{2}(M)&\mbox{ si }r_{i}>n/2\end{cases}$
En effet, comme $u\in H^{r_{i}}_{2}(M)$, s’il existe $i\in\mathbb{N}$ tel que
$r_{i}>n/2$ alors $u$ est continue, donc $\Delta u=hu-f\in L^{q}(M)$ d’où
$u\in H^{q}_{2}(M)$. Si $r_{i}=n/2$ alors $u\in L^{\infty}(M)$ donc $hu-f\in
L^{q}(M)$, d’où $u\in H^{q}_{2}(M)$.
Le seul cas qui reste à étudier est bien le cas où $r_{i}<n/2$ pour tout
$i\in\mathbb{N}$. Dans ce cas, s’il existe $i\in\mathbb{N}$ tel que
$q\leq\frac{ps_{i}}{p+s_{i}}$ alors $r_{i}=q$ et $u\in H^{q}_{2}(M)$. Sinon
pour tout $i\in\mathbb{N}$, $r_{i}=\frac{ps_{i}}{p+s_{i}}<n/2$ et on retrouve
le cas $(i)$ où la suite $(s_{i})$ est croissante majorée et converge vers 0,
ce qui est absurde.
∎
###### Proposition 1.4.
Soit $(M,g)$ une variété riemannienne compacte de dimension $n$ et soit
$L:=\Delta+h$ un opérateur linéaire avec $h\in L^{p}(M)$ et $p>n/2$. Si la
plus petite valeur propre $\lambda$ de $L$ est strictement positive alors
* i.
$L$ est coercif, autrement dit il existe $c>0$ tel que
$\forall\psi\in H_{1}(M)\quad(L\psi,\psi)_{L^{2}}\geq
c(\|\nabla\psi\|^{2}_{2}+\|\psi\|^{2}_{2})$
* ii.
pour tout $q>2n/(n+2)$, $L:H_{2}^{\min(p,q)}(M)\longrightarrow L^{q}(M)$ est
inversible
###### Preuve.
$L$ admet une plus petite valeur propre, car si $\lambda$ est une valeur
propre de fonction propre $\psi$ alors il existe $C>0$ tel que
$\lambda\|\psi\|_{2}^{2}=(L\psi,\psi)_{L^{2}}=\|\nabla\psi\|_{2}^{2}+\int_{M}h\psi^{2}\mathrm{d}v\geq-\|h\|_{p}\|\psi\|_{2p/(p-1)}^{2}\geq-C\|h\|_{p}\|\psi\|_{2}^{2}$
Donc $\lambda\geq-C\|h\|_{p}$. Si $\lambda$ est la plus petite valeur propre
de $L$ alors
$\lambda=\inf_{\varphi\in
H_{1}(M)-\\{0\\}}\frac{E(\varphi)}{\|\varphi\|_{2}^{2}}$
où
$E(\varphi)=(L\varphi,\varphi)_{L^{2}}=\int_{M}|\nabla\varphi|^{2}+h\varphi^{2}\mathrm{d}v$
Alors pour tout $\varphi\in H_{1}(M)$
$E(\varphi)\geq\lambda\|\varphi\|_{2}^{2}$ (1.10)
Supposons que $L$ ne soit pas coercif, alors il existe une suite
$(\psi_{i})_{i\in\mathbb{N}}$ dans $H_{1}(M)$ qui satisfait
$E(\psi_{i})<\frac{1}{i}(\|\nabla\psi_{i}\|^{2}_{2}+vol(M)^{2/n})\mbox{ et
}\|\psi_{i}\|_{N}=1$
ce qui entraîne
$(1-\frac{1}{i})E(\psi_{i})<\frac{vol(M)^{2/n}}{i}-\frac{1}{i}\int_{M}h\psi_{i}^{2}\mathrm{d}v$
Puisque $|\int_{M}h\psi_{i}^{2}\mathrm{d}v|\leq\|h\|_{n/2}$,
$\lim_{i\to+\infty}E(\psi_{i})\leq 0$. D’autre part
$E(\psi_{i})\geq\lambda\|\psi_{i}\|_{2}^{2}$ avec $\lambda>0$. Ce qui est
impossible.
Il est clair que $L$ est injective car si $L\psi=0$ alors par l’inégalité
(1.10) $,\varphi=0$.
Soit $f\in L^{q}(M)$ avec $q>2n/(n+2)$. Montrons que l’équation
$\Delta\varphi+h\varphi=f$ (1.11)
admet une solution $\psi\in H^{\min(p,q)}_{2}(M)$. On minimise la
fonctionnelle $E$ définie au début de la preuve, pour cela on pose
$\mu=\inf\\{E(\varphi)/\varphi\in H_{1}(M),\;\int_{M}f\varphi\mathrm{d}v=1\\}$
Soit $(\psi_{i})_{i\in\mathbb{N}}$ une suite dans $H_{1}(M)$ qui minimise $E$,
alors
$\lim_{i\to+\infty}E(\psi_{i})=\mu\text{ et }\int_{M}f\psi_{i}\mathrm{d}v=1$
Sans perte de généralité, on peut supposer que pour tout entier naturel $i$,
$E(\psi_{i})\leq\mu+1$. Ce qui implique
$c(\|\nabla\psi_{i}\|^{2}_{2}+\|\psi_{i}\|^{2}_{2})\leq E(\psi_{i})\leq\mu+1$
car $L$ est coercif. On en conclut que la suite $(\psi_{i})_{i\in\mathbb{N}}$
est bornée dans $H_{1}(M)$. Par le théorème de Banach (voir section 1.3.1) et
le théorème de compacité de Kondrakov 1.2, on en déduit qu’il existe une sous-
suite $(\psi_{j})_{j\in\mathbb{N}}$ telle que
$*$
$\psi_{j}\rightharpoonup\psi$ faiblement dans $H_{1}(M)$
$*$
$\psi_{j}\rightarrow\psi$ fortement dans $L^{s}(M)$ pour tout $1\leq s<N$
$*$
$\psi_{j}\rightarrow\psi$ presque partout.
En particulier la suite $(\psi_{j})$ converge fortement dans $L^{q/(q-1)}(M)$
et $L^{2p/(p-1)}(M)$ car $q/(q-1)<N$ et $2p/(p-1)<N$. Par conséquent
$\int_{M}f\psi\mathrm{d}v=1\text{ et
}\int_{M}h\psi_{j}^{2}\mathrm{d}v\rightarrow\int_{M}h\psi^{2}\mathrm{d}v$
La convergence faible dans $H_{1}(M)$ et forte dans $L^{2}(M)$ entraînent que
$\lim_{j\to+\infty}\|\nabla\psi_{j}\|_{2}\geq\|\nabla\psi\|_{2}$
On en conclut que $E(\psi)\leq\mu$ et donc nécessairement que $E(\psi)=\mu$.
En écrivant l’équation d’Euler–Lagrange pour $\psi$, on trouve qu’elle est
solution faible dans $H_{1}(M)$ de l’équation (1.11). Par la proposition 1.3,
on déduit que $\psi\in H^{\min(p,q)}_{2}(M)$. ∎
## Chapter 2 Étude d’équations de type Yamabe
### 2.1 Existence de solutions sans présence de symétries
Soit $(M,g)$ une variété riemannienne compacte $C^{\infty}$ de dimension
$n\geq 3$. On considère l’équation suivante :
$\Delta_{g}\psi+h\psi=\tilde{h}\psi^{\frac{n+2}{n-2}}$ (2.1)
Où $\psi\in H_{1}(M)$, $h\in L^{p}(M)$ avec $p>n/2$ et $\tilde{h}$ une
constante. Dorénavant, ce type d’équation s’appellera équation de type Yamabe.
Dans le cas particulier $h=\frac{n-2}{4(n-1)}R_{g}$, l’équation (2.1) est
celle de Yamabe qu’on verra plus en détail dans la section 3.1. Ce type
d’équation a été déjà considéré par Z. Faget [Fagt], lorsque $h$ est continue
sur $M$ et invariante par un sous groupe d’isométries.
Pour résoudre ce type d’équations, on utilisera la méthode variationnelle, qui
consiste à trouver une fonctionnelle à minimiser sur un espace bien choisi.
Dans notre cas l’espace est $H_{1}(M)$. On montrera ensuite que le minimum de
cette fonctionnelle est atteint pour une certaine fonction qui sera solution
de l’équation d’Euler–Lagrange. On aura l’occasion d’appliquer cette méthode
plusieurs fois.
On se place dans l’espace $H_{1}(M)$, on définit l’énergie $E$ de $\psi\in
H_{1}(M)$ par:
$E(\psi)=\int_{M}|\nabla\psi|^{2}+h\psi^{2}\mathrm{d}v$
Et on considère la fonctionnelle $I_{g}$ définie, pour tout $\psi\in
H_{1}(M)-\\{0\\}$, par
$I_{g}(\psi)=\frac{E(\psi)}{\|\psi\|^{2}_{N}}$
On note
$\mu(g)=\inf_{\psi\in H_{1}(M)-\\{0\\},\psi\geq
0}I_{g}(\psi)=\inf_{\|\psi\|_{N}=1,\psi\geq 0}E(\psi)$
avec $N=\frac{2n}{n-2}$. On note $[p]$ la partie entière d’un nombre réel $p$.
Dans le cas du problème de Yamabe (i.e. $h=\frac{n-2}{4(n-1)}R_{g}$), $I_{g}$
est appelée la fonctionnelle de Yamabe, et $\mu(g)$ l’invariant conforme de
Yamabe (voir section 3.1). L’un des résultats important de ce chapitre est le
suivant:
###### Théorème 2.1.
Soit $(M,g)$ une variété riemannienne compacte $C^{\infty}$ de dimension
$n\geq 3$ et $p>n/2$. Si
$\mu(g)<K^{-2}(n,2)$
alors l’équation (2.1) admet une solution strictement positive $\varphi\in
H^{p}_{2}(M)\subset C^{1-[n/p],\beta}(M)$, qui minimise la fonctionnelle
$I_{g}$ (i.e. $E(\varphi)=\mu(g)=\tilde{h}$ et $\|\varphi\|_{N}=1$). où
$\beta\in]0,1[$.
Dans la preuve de ce théorème, on aura besoin du lemme suivant dû à H. Brezis
et E.H. Lieb [BL]
###### Lemme 2.1.
Soit $(f_{i})_{i\in\mathbb{N}}$ une suite de fonctions dans un espace mesuré
$(\Omega,\Sigma,\mu)$. Si $(f_{i})_{i\in\mathbb{N}}$ est uniformément bornée
dans $L^{p}$ avec $0<p<+\infty$ et $f_{i}\rightarrow f$ p.p, alors
$\lim_{i\to+\infty}[\|f_{i}\|_{p}^{p}-\|f_{i}-f\|_{p}^{p}]=\|f\|_{p}^{p}$
###### Preuve du théorème 2.1.
On commence par vérifier que $\mu(g)$ est fini. En effet, d’après l’inégalité
de Hölder, on a
$E(\psi)\geq-\|h\|_{n/2}\|\psi\|^{2}_{N}$
on en déduit que $\mu(g)\geq-\|h\|_{n/2}>-\infty$.
Soit $(\varphi_{i})_{i\in\mathbb{N}}$ une suite minimisante:
$E(\varphi_{i})=\mu(g)+o(1),\;\|\varphi_{i}\|_{N}=1\mbox{ et }\varphi_{i}\geq
0$ (2.2)
En utilisant l’inégalité de Hölder encore une fois dans l’équation ci-dessus,
on obtient
$\displaystyle\|\nabla\varphi_{i}\|_{2}^{2}\leq\|h\|_{n/2}+\mu(g)+o(1)$
$\displaystyle\|\varphi_{i}\|_{2}^{2}\leq(vol(M))^{2/n}$
On en déduit que $(\varphi_{i})_{i\in\mathbb{N}}$ est bornée dans $H_{1}(M)$.
Quitte à extraire une sous-suite, on peut supposer qu’il existe $\varphi\in
H_{1}(M)$ tel que
$*$
$\varphi_{i}\rightharpoonup\varphi$ faiblement dans $H_{1}(M)$ par le théorème
de Banach (cf. section 1.3.1).
$*$
$\varphi_{i}\rightarrow\varphi$ fortement dans $L^{s}(M)$, pour tout
$s\in[1,N[$, par l’inclusion compacte de Kondrakov (cf. théorème 1.2).
$*$
$\varphi_{i}\rightarrow\varphi$ presque partout.
On en conclut que:
$\int_{M}|h||\varphi_{i}-\varphi|^{2}\mathrm{d}v\leq\|h\|_{p}\|\varphi_{i}-\varphi\|_{2p/(p-1)}^{2}\rightarrow
0\text{ fortement car }2p/(p-1)<N$
On pose $\psi_{i}=\varphi_{i}-\varphi$, alors $\psi_{i}\rightarrow 0$
faiblement dans $H_{1}(M)$, fortement dans $L^{q}(M)$ pour tout $q<N$.
On a
$\|\nabla\varphi_{i}\|_{2}^{2}=\|\nabla\psi_{i}\|_{2}^{2}+\|\nabla\varphi\|_{2}^{2}+2\int_{M}\nabla\psi_{i}\cdot\nabla\varphi\mathrm{d}v$.
On en déduit que
$E(\varphi_{i})=E(\varphi)+\|\nabla\psi_{i}\|_{2}^{2}+o(1)$
Puisque $E(\varphi)\geq\mu(g)\|\varphi\|_{N}^{2}$ par définition de $\mu(g)$
et $E(\varphi_{i})=\mu(g)+o(1)$ par définition de la suite
$(\varphi_{i})_{i\in\mathbb{N}}$, on en déduit que
$\mu(g)\|\varphi\|_{N}^{2}+\|\nabla\psi_{i}\|_{2}^{2}\leq\mu(g)+o(1)$ (2.3)
On applique le lemme 2.1 à la suite $(\varphi_{i})_{i\in\mathbb{N}}$, on
trouve
$\displaystyle\|\psi_{i}\|_{N}^{N}$ $\displaystyle+\|\varphi\|_{N}^{N}+o(1)=1$
(2.4) $\displaystyle\|\psi_{i}\|_{N}^{2}$
$\displaystyle+\|\varphi\|_{N}^{2}+o(1)\geq 1$ (2.5)
Par le théorème 1.5
$\|\psi_{i}\|_{N}^{2}\leq(K^{2}(n,2)+\varepsilon)\|\nabla\psi_{i}\|_{2}^{2}+o(1)$
l’inégalité (2.5) devient donc
$(K^{2}(n,2)+\varepsilon)\|\nabla\psi_{i}\|_{2}^{2}+\|\varphi\|_{N}^{2}+o(1)\geq
1$
Si on utilise cette dernière inégalité dans (2.3), on trouve
$\mu(g)\|\varphi\|_{N}^{2}+\|\nabla\psi_{i}\|_{2}^{2}\leq\mu(g)[(K^{2}(n,2)+\varepsilon)\|\nabla\psi_{i}\|_{2}^{2}+\|\varphi\|_{N}^{2}]+o(1)$
Finalement
$[1-\mu(g)(K^{2}(n,2)+\varepsilon)]\|\nabla\psi_{i}\|_{2}^{2}\leq o(1)$
Si $\mu(g)<K^{-2}(n,2)$, on peut choisir $\varepsilon$ de sorte que le premier
facteur de cette inégalité soit strictement positif. On en déduit que
$(\psi_{i})_{i\in\mathbb{N}}$ converge fortement vers 0 dans $H_{1}(M)$,
$\varphi_{i}\rightarrow\varphi$ fortement dans $H_{1}(M)$ et $L^{N}(M)$ d’où
$I_{g}(\varphi)=\mu(g)$.
On vient de mettre en évidence une solution non triviale de l’équation de type
Yamabe
$\Delta\psi+h\psi=\mu(g)\psi^{N-1}$
qui satisfait $\|\varphi\|_{N}=1$ et $\varphi\geq 0$. Par le théorème 1.11,
$\varphi\in H^{p}_{2}(M)\subset C^{1-[n/p],\beta}(M)$ et $\varphi>0$. ∎
###### Proposition 2.1.
Soit $(M,g)$ une variété riemannienne compacte $C^{\infty}$. On a toujours:
$\mu(g)\leq K^{-2}(n,2)$
###### Preuve.
Soient $P$ un point fixé de $M$ et $u_{\varepsilon}$ une fonction radiale
définie sur $M$ par
$\displaystyle
u_{\varepsilon}(Q)=\begin{cases}\biggl{(}\displaystyle\frac{\varepsilon}{r^{2}+\varepsilon^{2}}\biggr{)}^{\frac{n-2}{2}}-\biggl{(}\frac{\varepsilon}{\delta^{2}+\varepsilon^{2}}\biggr{)}^{\frac{n-2}{2}}&\mbox{
if }Q\in B_{P}(\delta)\\\ 0&\mbox{ if }Q\in M-B_{P}(\delta)\end{cases}$
où $r=d(P,Q)$ et $B_{P}(\delta)$ est la boule géodésique de centre $P$ et de
rayon $\delta$. Montrons que $\lim_{\varepsilon\to
0}I_{g}(u_{\varepsilon})=K^{-2}(n,2)$, ce qui entraînera l’inégalité de la
proposition car $\mu(g)$ est bien le minimum de $I_{g}$.
Puisque $u_{\varepsilon}$ est radiale
$\nabla
u_{\varepsilon}=\partial_{r}u_{\varepsilon}=-(n-2)\varepsilon^{(n-2)/2}\frac{r}{(r^{2}+\varepsilon^{2})^{n/2}}$
En intégrant le carré de ce gradient sur $M$, on obtient:
$\int_{M}|\nabla
u_{\varepsilon}|^{2}\mathrm{d}v=(n-2)^{2}\omega_{n-1}\varepsilon^{n-2}\int_{0}^{\delta}\frac{r^{n+1}}{(r^{2}+\varepsilon^{2})^{n}}\mathrm{d}r$
En effectuant le changement de variable $t=r/\varepsilon$ on trouve
$\int_{M}|\nabla
u_{\varepsilon}|^{2}\mathrm{d}v=(n-2)^{2}\omega_{n-1}\int_{0}^{\delta/\varepsilon}\frac{t^{n+1}}{(t^{2}+1)^{n}}\mathrm{d}t$
(2.6)
D’autre part $h\in L^{p}(M)$ avec $p>n/2$ donc
$\int_{M}hu_{\varepsilon}^{2}\mathrm{d}v\leq\|h\|_{p}\|u_{\varepsilon}\|^{2}_{2p/(p-1)}$
Par le même changement de variable $t=r/\varepsilon$, on a
$\|u_{\varepsilon}\|^{2p/(p-1)}_{2p/(p-1)}\leq\int_{0}^{\delta}\biggl{(}\frac{\varepsilon}{r^{2}+\varepsilon^{2}}\biggr{)}^{\frac{p(n-2)}{p-1}}r^{n-1}\mathrm{d}r\leq\varepsilon^{\frac{2p-n}{p-1}}\int_{0}^{\delta/\varepsilon}\biggl{(}\frac{1}{t^{2}+1}\biggr{)}^{\frac{p(n-2)}{p-1}}t^{n-1}\mathrm{d}t$
donc $\|u_{\varepsilon}\|^{2}_{2p/(p-1)}=O(\varepsilon^{2-\frac{n}{p}})$.
Puisque $p>n/2$, on en déduit que
$\lim_{\varepsilon\to 0}\int_{M}hu_{\varepsilon}^{2}\mathrm{d}v=0$ (2.7)
Il nous reste à calculer $\|u_{\varepsilon}\|_{N}^{-2}$. Lorsque on prend
l’intégrale des puissances de $u_{\varepsilon}$ on peut négliger le terme
constant dans l’expression de $u_{\varepsilon}$ (des détails sur les
puissances de $u_{\varepsilon}$ sont donnés dans l’appendice A, équation
(A.20)). D’où
$\|u_{\varepsilon}\|^{N}_{N}=\omega_{n-1}\int_{0}^{\delta/\varepsilon}\frac{t^{n-1}}{(t^{2}+1)^{n}}\mathrm{d}t+O(\varepsilon^{n-2})$
(2.8)
Il est bien connu que la fonction
$v_{\varepsilon}:x\longmapsto\biggl{(}\frac{\varepsilon}{|x|^{2}+\varepsilon^{2}}\biggr{)}^{\frac{n-2}{2}}$
est solution de l’équation $\Delta_{\mathcal{E}}u=n(n-2)u^{N-1}$ sur
$\mathbb{R}^{n}$, où $\Delta_{\mathcal{E}}$ est le Laplacien euclidien sur
$\mathbb{R}^{n}$. C’est aussi la fonction qui réalise la meilleure constante
de l’inégalité du théorème 1.5 (page 1.5) sur $\mathbb{R}^{n}$. On a donc
$K^{2}(n,2)\|\nabla v_{\varepsilon}\|^{2}_{2}=\|v_{\varepsilon}\|_{N}^{2}$.
Autrement dit, si on calcule $\|\nabla v_{\varepsilon}\|^{2}_{2}$ et
$\|v_{\varepsilon}\|^{2}_{N}$, en passant aux coordonnées polaires, on trouve:
$\biggl{(}(n-2)^{2}\omega_{n-1}\int_{0}^{+\infty}\frac{t^{n+1}}{(t^{2}+1)^{n}}\mathrm{d}t\biggr{)}\biggl{(}\omega_{n-1}\int_{0}^{+\infty}\frac{t^{n-1}}{(t^{2}+1)^{n}}\mathrm{d}t\biggr{)}^{-\frac{n-2}{n}}=K^{-2}(n,2)$
(2.9)
En combinant (2.6), (2.7), (2.8) et (2.9) on conclut que
$\lim_{\varepsilon\to 0}I_{g}(u_{\varepsilon})=\lim_{\varepsilon\to
0}(\int_{M}|\nabla
u_{\varepsilon}|^{2}\mathrm{d}v+\int_{M}hu_{\varepsilon}^{2}\mathrm{d}v)\|u_{\varepsilon}\|^{-2/N}_{N}=K^{-2}(n,2)$
Ce qui entraîne que $\mu(M,g)\leq\lim_{\varepsilon\to
0}I_{g}(u_{\varepsilon})=K^{-2}(n,2)$. ∎
#### 2.1.1 Application
On considère l’équation suivante:
$\Delta\psi+\frac{R}{\rho^{\alpha}}\psi=\tilde{R}\psi^{\frac{n+2}{n-2}}$
(2.10)
où $R\in C^{0}(M)$, $\alpha,\;\tilde{R}$ sont deux nombres réels et $\rho$ la
fonction distance (cf définition 1.7). On pose
$\displaystyle
E_{\alpha}(\varphi)=\int_{M}|\nabla\varphi|^{2}+\frac{R}{\rho^{\alpha}}\varphi^{2}\mathrm{d}v$
$\displaystyle
I_{g,\alpha}(\varphi)=\frac{E_{\alpha}(\varphi)}{\|\varphi\|^{2}_{N}}$
$\displaystyle\mu_{\alpha}(g)=\inf_{\varphi\in H_{1}(M)-\\{0\\},\varphi\geq
0}I_{g,\alpha}(\varphi)=\inf_{\|\varphi\|_{N}=1,\varphi\geq
0}E_{\alpha}(\varphi)$
###### Proposition 2.2.
Si $0<\alpha<2$ et $\mu_{\alpha}(g)<K^{-2}(n,2)$ alors l’équation (2.10) admet
une solution $\varphi_{\alpha}\in C^{1-[\alpha],\beta}(M)$ strictement
positive qui satisfait
$E_{\alpha}(\varphi_{\alpha})=\mu_{\alpha}(g)=\tilde{R}$ et
$\|\varphi_{\alpha}\|_{N}=1$.
###### Preuve.
Si on pose $h:=R/\rho^{\alpha}\in L^{p}(M)$ avec $2>n/p>\alpha$, alors cette
proposition est un corollaire immédiat du théorème 2.1 ∎
#### Le cas critique $\boldsymbol{\alpha=2}$
Ce cas correspond à l’équation non linéaire de Schrödinger avec le potentiel
de Hardy et l’exposant critique. Il a été déjà étudié sur $\mathbb{R}^{n}$ par
S. Terracini [Ter] et D. Smets [Sme] qui ont montré l’existence et non
existence de solutions de l’équation ci-dessous pour $\alpha=2$ et $\rho=|x|$
sous certaines conditions. Le théorème obtenu ici est le suivant:
###### Théorème 2.2.
Si $\mu_{2}(g)<[1+\min(R(P),0)K^{2}(n,2,-2)]K^{-2}(n,2)$ et
$1+R(P)K^{2}(n,2,-2)>0$ alors il existe $\varphi_{2}\in H_{1}(M)$ solution non
triviale de l’équation (2.10) pour $\alpha=2$.
###### Preuve.
$(a).$ On montre que $\mu_{2}(g)$ est fini et $\lim_{\alpha\rightarrow
2^{-}}\mu_{\alpha}(g)=\mu_{2}(g)$. Pour tout $\varepsilon>0$ il existe
$\delta>0$ tel que si $Q\in B_{\delta}(P)$ alors $|R(Q)-R(P)|<\varepsilon$, de
plus si $\psi\in H_{1}(M)$ et $\|\psi\|_{N}=1$ alors
$E_{2}(\psi)\geq\|\nabla\psi\|_{2}^{2}-\frac{\|R\|_{\infty}}{\delta^{2}}\|\psi\|_{2}^{2}+(R(P)-\varepsilon)\int_{B_{\delta}(P)}\rho^{-2}\psi^{2}\mathrm{d}v$
Par le lemme 1.1 et l’inégalité de Hölder:
$E_{2}(\psi)\geq[1+(\min(R(P),0)-\varepsilon)K_{\delta}^{2}(n,2,-2)]\|\nabla\psi\|_{2}^{2}-\|R\|_{\infty}\delta^{-2}vol(M)^{2/n}$
Si $1+R(P)K^{2}(n,2,-2)>0$ alors il existe $\varepsilon$ et $\delta$ tels que
$E_{2}(\psi)>-\|R\|_{\infty}\delta^{-2}vol(M)^{2/n}$
Le théorème de la convergence dominée de Lebesgue, nous permet d’écrire que
pour tout $\psi\in H_{1}(M)-\\{0\\}$: $\lim_{\alpha\rightarrow
2^{-}}I_{g,\alpha}(\psi)=I_{g,2}(\psi)$. On en déduit que
$\lim_{\alpha\rightarrow 2^{-}}\mu_{\alpha}(g)=\mu_{2}(g)$. Il existe alors
$\alpha_{0}$ tel que pour tout $\alpha\in[\alpha_{0},2]$:
$\mu_{\alpha}(g)<K^{-2}(n,2)$
$(b).$ On montre que la famille
$\\{\varphi_{\alpha}\\}_{\alpha\in[\alpha_{0},2[}$ est uniformément bornée
dans $H_{1}(M)$. Cette famille satisfait les résultats de la proposition 2.2
donc pour tout $\alpha\in[\alpha_{0},2[$
$\|\varphi_{\alpha}\|_{2}\leq vol(M)^{1/n}\mbox{ et
}\|\nabla\varphi_{\alpha}\|_{2}^{2}+\int_{B_{\delta}(P)}\frac{R}{\rho^{\alpha}}\varphi_{\alpha}^{2}\mathrm{d}v\leq
K^{-2}(n,2)+\delta^{-2}\|R\|_{\infty}\|\varphi_{\alpha}\|_{2}^{2}$
Mais
$\int_{B_{\delta}(P)}\frac{R}{\rho^{\alpha}}\varphi_{\alpha}^{2}\mathrm{d}v\geq(\min(R(P),0)-\varepsilon)K_{\delta}^{2}(n,2,-2)\|\nabla\varphi_{\alpha}\|_{2}^{2}$
d’où
$[1+(\min(R(P),0)-\varepsilon)K_{\delta}^{2}(n,2,-2)]\|\nabla\varphi_{\alpha}\|_{2}^{2}\leq
K^{-2}(n,2)+\delta^{-2}\|R\|_{\infty}vol(M)^{2/n}$
Compte tenu de l’hypothèse sur $R(P)$, on peut choisir $\varepsilon$
suffisamment petit pour que le premier facteur de cette inégalité soit
strictement positif.
$(c).$ Il existe une suite $(\alpha_{i})_{i\in\mathbb{N}}$ à valeur dans
$[\alpha_{0},2[$ qui converge vers 2, telle que la suite de fonctions
$(\varphi_{\alpha_{i}})_{i\in\mathbb{N}}$ converge faiblement dans $H_{1}(M)$,
$L^{2}(M,\rho^{-2})$, $L^{N}(M)$ et fortement dans $L^{q}(M)$ vers une
fonction $\varphi_{2}\geq 0$, avec $q<N$ (voir la section 1.5 pour la
définition de $L^{2}(M,\rho^{\gamma})$ et le théorème 1.7).
Pour tout $\psi\in H_{1}(M)$
$\int_{M}\nabla\varphi_{\alpha_{i}}\nabla\psi\mathrm{d}v+\int_{M}\frac{R}{\rho^{\alpha_{i}}}\varphi_{\alpha_{i}}\psi\mathrm{d}v=\mu_{\alpha_{i}}(g)\int_{M}\varphi_{\alpha_{i}}^{N-1}\psi\mathrm{d}v$
On veut passer à la limite dans cette égalité. C’est immédiat pour la première
intégrale, d’après la convergence faible dans $H_{1}(M)$. Pour la seconde
intégrale:
$\biggl{|}\int_{M}\frac{R}{\rho^{\alpha_{i}}}\varphi_{\alpha_{i}}\psi-\frac{R}{\rho^{2}}\varphi_{2}\psi\mathrm{d}v\biggr{|}\leq\biggl{|}\int_{M}\frac{R\psi}{\rho^{2}}(\varphi_{\alpha_{i}}-\varphi_{2})\mathrm{d}v\biggr{|}+\int_{M}|R\psi\varphi_{\alpha_{i}}||\frac{1}{\rho^{\alpha_{i}}}-\frac{1}{\rho^{2}}|\mathrm{d}v$
La convergence faible dans $L^{2}(M,\rho^{-2})$ et le théorème de la
convergence dominée de Lebesgue impliquent que le second membre converge vers
0.
Comme $(\varphi_{\alpha_{i}})_{i\in\mathbb{N}}$ est uniformément bornée dans
$L^{N}(M)$, $(\varphi_{\alpha_{i}}^{N-1})_{i\in\mathbb{N}}$ est uniformément
bornée dans $L^{N/(N-1)}$. Alors
$\mu_{\alpha_{i}}(g)\int_{M}\varphi_{\alpha_{i}}^{N-1}\psi\mathrm{d}v\rightarrow\mu_{2}(g)\int_{M}\varphi_{2}^{N-1}\psi\mathrm{d}v$
On en conclut que $\varphi_{2}$ est une solution faible de l’équation (2.10)
pour $\alpha=2$. Il nous reste à montrer que $\varphi_{2}$ n’est pas
identiquement nulle. Le théorème 1.5 montre que
$1=\|\varphi_{\alpha_{i}}\|_{N}^{2}\leq(K^{2}(n,2)+\varepsilon)(\mu_{\alpha_{i}}(g)-\int_{M}\frac{R}{\rho^{\alpha_{i}}}\varphi_{\alpha_{i}}^{2}\mathrm{d}v)+A\|\varphi_{\alpha_{i}}\|_{2}^{2}$
(2.11)
Ce même théorème implique encore une fois
$\displaystyle\int_{M}\frac{R}{\rho^{\alpha_{i}}}\varphi_{\alpha_{i}}^{2}\mathrm{d}v$
$\displaystyle=\int_{B_{\delta}(P)}\frac{R}{\rho^{\alpha_{i}}}\varphi_{\alpha_{i}}^{2}\mathrm{d}v+\int_{M-B_{\delta}(P)}\frac{R}{\rho^{\alpha_{i}}}\varphi_{\alpha_{i}}^{2}\mathrm{d}v$
$\displaystyle\geq(\min(R(P),0)-\varepsilon)(K^{2}(n,2,-2)+\varepsilon^{\prime})(\mu_{\alpha_{i}}(g)-\int_{M}\frac{R}{\rho^{\alpha_{i}}}\varphi_{\alpha_{i}}^{2}\mathrm{d}v)-A\|\varphi_{\alpha_{i}}\|_{2}^{2}$
d’où
$\int_{M}\frac{R}{\rho^{\alpha_{i}}}\varphi_{\alpha_{i}}^{2}\mathrm{d}v\geq\frac{(\min(R(P),0)-\varepsilon)(K^{2}(n,2,-2)+\varepsilon^{\prime})}{[1+(\min(R(P),0)-\varepsilon)(K^{2}(n,2,-2)+\varepsilon^{\prime})]}\mu_{\alpha_{i}}(g)-A^{\prime}\|\varphi_{\alpha_{i}}\|_{2}^{2}$
(2.12)
Le dénominateur ci-dessus est strictement positif, si $\varepsilon$ et
$\varepsilon^{\prime}$ sont suffisamment petit. Les constantes $A$ et
$A^{\prime}$ ne dépendent pas de $\alpha_{i}$. Des inégalités (2.11) et
(2.12), on tire
$A^{\prime\prime}\|\varphi_{\alpha_{i}}\|_{2}^{2}\geq\frac{1+(\min(R(P),0)-\varepsilon)(K^{2}(n,2,-2)+\varepsilon^{\prime})-(K^{2}(n,2)+\varepsilon)\mu_{\alpha_{i}}(g)}{1+(\min(R(P),0)-\varepsilon)(K^{2}(n,2,-2)+\varepsilon^{\prime})}$
Le second membre de cette expression reste strictement positif lorsque
$i\to+\infty$, alors il existe $c>0$ tel que $\|\varphi_{2}\|_{2}^{2}>c$ ∎
### 2.2 Existence de solutions en présence de symétries
#### 2.2.1 Le groupe d’isométries et le groupe conforme
###### Définition 2.1.
Soit $(M,g)$ une variété riemannienne $C^{\infty}$. le groupe d’isométries
$I(M,g)$ et le groupe conforme $C(M,g)$ de $(M,g)$ sont définis par
$\displaystyle I(M,g)=\\{f\in C^{\infty}(M,M)/f^{*}g=g\\}$ $\displaystyle
C(M,g)=\\{f\in C^{\infty}(M,M)/f^{*}g=e^{h}g,\;h\in C^{\infty}(M)\\}$
###### Définition 2.2.
Soit $G$ un sous groupe du groupe $I(M,g)$.
1. 1.
On dit qu’une fonction $f$ dans $H^{q}_{k}(M)$ est $G-$invariante si et
seulement si pour tout $\sigma\in G$, $\sigma^{*}f=f$ presque partout, où
$k\in\mathbb{N}$ et $q\geq 1$. L’ensemble de ces fonctions est noté
$H^{q}_{k,G}(M)$ si $k\geq 1$, $L_{G}^{q}(M)$ si $k=0$, et $H_{k,G}(M)$ si
$q=2$.
2. 2.
Une métrique $g^{\prime}$ est dite $G-$invariante si et seulement si $G\subset
I(M,g^{\prime})$
3. 3.
$[g]^{G}$ est la classe des métriques $G-$invariantes conforment à $g$ définie
par:
$[g]^{G}=\\{\tilde{g}=e^{f}g/f\in C^{\infty}(M),\;G\subset I(M,\tilde{g})\\}$
Résoudre l’équation de type Yamabe (2.1) en présence de symétries revient à
chercher une solution $G-$invariante, strictement positive de l’équation
(2.1), où $h$ est fonction $G-$invariante presque partout. E. Hebey et M.
Vaugon [HV] ont introduit cette équation lorsque $h$ est proportionnelle à la
courbure scalaire $R_{g}$, qui est évidemment $G-$invariante. Dans ce cas le
problème a une signification géométrique que l’on précisera dans le chapitre
3. Afin de trouver des solutions à ce problème, E. Hebey et M. Vaugon ont
utilisé la technique des points de concentration, sans utiliser l’analogue de
l’inégalité de la meilleure constante pour l’espace $H_{1,G}(M)$. Cette
inégalité s’avérera fondamentale pour trouver la condition suffisante dans la
résolution de l’équation de type Yamabe sans présence de symétries (2.1) (cf.
théorème 2.1), elle a été obtenue par E. Hebey et M. Vaugon [HV2], après leurs
travaux sur le problème de Yamabe équivariant, lorsqu’ils ont étudié les
inclusions de Sobolev pour les espaces $G-$invariants. Ils ont obtenu les
résultats suivants:
#### 2.2.2 Inégalité de la meilleure constante en présence de symétries
###### Théorème 2.3 (Hebey–Vaugon).
Soit $(M,g)$ une variété riemannienne compacte de dimension $n$, $G$ un sous
groupe compact du groupe $I(M,g)$. Soit $k$ la plus petite dimension des
orbites de $M$ sous $G$. On pose $p^{*}=\frac{(n-k)p}{n-k-p}$ si $n-k-p\neq
0$.
1. 1.
Si $p$ est un réel tel que $1\leq p<n-k$ alors pour tout $q\in[1,p^{*}]$,
l’inclusion $H_{1,G}^{p}(M)\subset L_{G}^{q}(M)$ est continue. De plus si
$q\in[1,p^{*}[$ elle est compacte.
2. 2.
Si $p\geq n-k$ alors pour tout $q\geq 1$, l’inclusion $H_{1,G}^{p}(M)\subset
L_{G}^{q}(M)$ est continue et compacte
(T. Parker [Par] avait aussi travaillé sur les inclusions de Sobolev pour les
espaces $G-$invariants).
On note par $O_{G}(P)$ l’orbite du point $P$ sous l’action de $G$. La
meilleure constante dans ces inclusions a été calculée par Z. Faget [Fag].
###### Théorème 2.4 (Z. Faget).
Sous les hypothèses du théorème précédent, si on pose
$A=\min\\{vol(O_{G}(Q))/Q\in M\text{ et }\dim O_{G}(Q)=k\\}$
(si $G$ a des orbites finies alors $k=0$ et $A=\min_{Q\in M}cardO_{G}(Q)$) et
$1\leq p<n-k$ alors pour tout $\varepsilon>0$, il existe $B(\varepsilon)$ tel
que
$\forall\varphi\in
H_{1,G}^{p}(M)\quad\|\varphi\|^{p}_{p^{*}}\leq(\frac{K^{p}(n-k,p)}{A^{p/(n-k)}}+\varepsilon)\|\nabla\varphi\|^{p}_{p}+B(\varepsilon)\|\varphi\|^{p}_{p}$
$K(n-k,p)A^{-1/(n-k)}$ est la meilleure constante.
Soit $h$ une fonction dans $L_{G}^{p}(M)$ avec $p>n/2$ et
$q\in[2,\frac{2n}{n-2}]$. On considère l’équation de type Yamabe (avec un
exposant $q$) suivante:
$\Delta_{g}\psi+h\psi=\tilde{h}\psi^{q-1}$ (2.13)
où $\tilde{h}$ est une constante. Le but de cette section est de chercher des
solutions $\psi>0$ et $G-$invariante dans $H^{p}_{2,G}$. On attachera plus
d’attention au cas $q=N=\frac{2n}{n-2}$. Posons pour tout $\varphi\in
H_{1,G}(M)$.
$I_{q,g}(\varphi)=\frac{E(\varphi)}{\|\varphi\|^{2}_{q}},\qquad\mu_{q,G}(g)=\inf_{\varphi\in
H_{1,G}(M)-\\{0\\}}I_{q,g}(\varphi)$
où $E(\varphi)$ a été défini au début de la section 2.1.
Notons que si $q=N$, l’équation (2.13) et la fonctionnelle $I_{q,g}$
s’identifient à l’équation (2.1) et à la fonctionnelle $I_{g}$ respectivement.
Par contre $\mu(g)\leq\mu_{N,G}(g)$ car $\mu_{N,G}(g)$ est obtenu en prenant
des fonctions tests dans $H_{1,G}(M)\subset H^{2}_{1}(M)$ (voir la section
2.1, pour les définitions de $I_{g}$ et $\mu(g)$).
###### Proposition 2.3.
Si $q\in[\frac{2p}{p-1},\frac{2n}{n-2}[$ et $\mu_{q,G}(g)>0$ alors l’équation
(2.13) admet une solution $\varphi_{q}\in H_{2,G}^{p}(M)$, $G-$invariante,
strictement positive et qui minimise $I_{q,g}$, pour $\tilde{h}=\mu_{q,G}(g)$.
###### Preuve.
* $*$
Soit $(\varphi_{i})_{i\in\mathbb{N}}$ une suite minimisante dans $H_{1,G}(M)$
telle que $\|\varphi_{i}\|_{q}=1$ et $\varphi_{i}\geq 0$ alors
$(\varphi_{i})_{i\in\mathbb{N}}$ est bornée dans $H_{1,G}(M)$, en effet
$(E(\varphi_{i}))_{i\in\mathbb{N}}$ est une suite convergente dans
$\mathbb{R}$, on peut donc supposer qu’elle est majorée par $\mu_{q,G}(g)+1$,
d’où
$\begin{split}\|\varphi_{i}\|_{2}^{2}&\leq
vol(M)^{(q-2)/q}\|\varphi_{i}\|_{q}^{2}\leq vol(M)^{(q-2)/q}\\\
\|\nabla\varphi_{i}\|_{2}^{2}&=E(\varphi_{i})-\int_{M}h\varphi_{i}^{2}\mathrm{d}v\\\
&\leq\mu_{q,G}(g)+1+C\|h\|_{p}\end{split}$
* $*$
Le théorème de Banach (voir section 1.3.1) assure l’existence d’une sous-suite
$(\varphi_{j})_{j\in\mathbb{N}}$ de $(\varphi_{i})_{i\in\mathbb{N}}$, qui
converge faiblement dans $H_{1,G}(M)$ vers une fonction $\varphi_{q}$, et que
$\displaystyle\liminf_{j\to+\infty}\|\nabla\varphi_{j}\|_{2}+\|\varphi_{j}\|_{2}\geq\|\nabla\varphi_{q}\|_{2}+\|\varphi_{q}\|_{2}$
* $*$
Il existe une sous-suite $(\varphi_{k})_{k\in\mathbb{N}}$ de
$(\varphi_{j})_{j\in\mathbb{N}}$, qui converge fortement dans $L^{q}(M)$ vers
la fonction $\varphi_{q}$ si $q\in[\frac{2p}{p-1},\frac{2n}{n-2}[$. Il en
résulte que $\|\varphi_{q}\|_{q}=1$
il en résulte aussi que
$\mu_{q,G}(g)=\lim_{k\to+\infty}I_{q,g}(\varphi_{k})\geq I_{q,g}(\varphi_{q})$
on en déduit que $I_{q,g}(\varphi_{q})=\mu_{q,G}(g)$, $\varphi_{g}\geq 0$ et
que $\varphi_{q}$ est $G-$invariante presque partout. Donc $\varphi_{q}$
minimise la fonctionnelle $I_{q,g}$. On écrit l’équation d’Euler-Lagrange pour
la fonction $\varphi_{q}$, on trouve:
$\forall\psi\in
H_{1,G}(M)\qquad\int_{M}\nabla_{i}\varphi_{q}\nabla^{i}\psi+h\psi\varphi_{q}-\mu_{q,G}(g)\psi\varphi_{q}^{q-1}\mathrm{d}v=0$
(2.14)
On doit montrer que l’égalité (2.14) reste vraie pour tout $\psi\in H_{1}(M)$.
C’est là qu’on utilise l’hypothèse $\mu_{q,G}(g)>0$ qui montre que la plus
petite valeur propre $\lambda$ de l’opérateur $L:=\Delta_{g}+h$ est
strictement positive. En effet si $\lambda\leq 0$, il existe une fonction
propre $\psi\geq 0$ non identiquement nulle telle que
$E(\psi)=(L\psi,\psi)_{L^{2}}=\lambda\|\psi\|_{2}^{2}<0$
D’autre part $E(\psi)\geq\mu_{N,G}(g)\|\psi\|_{N}^{2}>0$, ce qui est absurde.
Maintenant la proposition 1.4 montre que $L$ est inversible. Comme
$\varphi_{q}\in L^{N/(q-1)}(M)$ et $N/(q-1)>2n/(n+2)$, il existe une unique
fonction $\tilde{\varphi}_{q}$ solution faible de l’équation
$L\tilde{\varphi}_{q}=\mu_{q,G}(g)\varphi_{q}^{q-1}$
$h$ est $G-$invariante, ainsi que $\Delta_{g}$, donc
$\sigma^{*}\tilde{\varphi}_{q}$ est solution de la même équation pour tout
$\sigma\in G$. Par unicité
$\sigma^{*}\tilde{\varphi}_{q}=\tilde{\varphi}_{q}$, $\tilde{\varphi}_{q}$ est
donc $G-$invariante. D’autre part
$\forall\psi\in
H_{1,G}(M)\qquad(L(\varphi_{q}-\tilde{\varphi}_{q}),\psi)_{L^{2}}=0$
Si on choisit $\psi=\varphi_{q}-\tilde{\varphi}_{q}$ alors
$\varphi_{q}=\tilde{\varphi}_{q}$, car $L$ est coercif, d’après la proposition
1.4. Finalement $\varphi_{q}$ est une solution faible, non triviale de
l’équation
$\Delta_{g}\varphi+(h-\mu_{q,G}(g)\varphi_{q}^{q-2})\varphi=0$
avec $(h-\mu_{q,G}(g)\varphi_{q}^{q-2})\in L^{s}(M)$, où
$s=\min(p,\frac{2n}{(q-2)(n-2)})>n/2$. Par le théorème 1.10, $\varphi_{q}$ est
bornée, strictement positive, donc $\Delta\varphi_{q}\in L^{p}(M)$. Par le
théorème de régularité, $\varphi_{q}\in H^{p}_{2,G}(M)$. ∎
On s’intéresse maintenant au cas où $q=N$ dans l’équation (2.13). On obtient
d’abord le résultat suivant:
###### Proposition 2.4.
Si $k:=\inf_{Q\in M}\dim O_{G}(Q)\geq 1$ et $\mu_{N,G}(g)>0$ alors l’équation
(2.13) admet une solution $\varphi_{N}\in H^{p}_{2}(M)$, qui minimise
$I_{N,g}$, $G-$invariante et strictement positive pour $q=N$ et
$\tilde{h}=\mu_{G}(M,g)$.
###### Preuve.
D’après le théorème 2.3, si $k\geq 1$, l’inclusion $H_{1,G}(M)\subset
L_{G}^{N}(M)$ est compacte. C’est ce qui manquait pour que la preuve de la
proposition 2.3 soit valable pour $q=N$. $\varphi_{N}$ est donc solution
faible dans $H_{1,G}(M)$ de (2.14). Pour montrer qu’elle est solution faible
pour tout $\psi\in H_{1}(M)$, il suffit d’utiliser l’argument déjà utilisé à
la fin de la preuve de la proposition 2.3, en utilisant le fait que
l’inclusion $H_{1,G}(M)\subset L_{G}^{2^{*}}(M)$ est continue, où
$2^{*}=2(n-k)/(n-k-2)$ (cf. théorème 2.3). Ceci entraîne qu’il existe
$s>2n/(n+2)$ tel que $\varphi_{N}^{N-1}\in L^{s}(M)$. Le résultat de la
proposition 2.3 s’étend donc à $q=N$ lorsque $k\geq 1$. ∎
###### Théorème 2.5.
Soit $(M,g)$ une variété riemannienne compacte. $G$ un sous groupe de
$I(M,g)$. Si
$0<\mu_{N,G}(g)<K^{-2}(n,2)(\inf_{Q\in M}\mathrm{card}O_{G}(Q))^{2/n}$
alors pour $q=N$, l’équation (2.13) admet une solution $\varphi\in
H_{2,G}^{p}(M)\subset C^{1-[n/p],\beta}(M)$ strictement positive,
$G-$invariante et minimisante pour la fonctionnelle $I_{N,g}$.
###### Preuve.
On fait tendre $q$ vers $N$ pour les solutions $\varphi_{q}$ de l’équation
(2.13), obtenues grâce à la proposition (2.3). En utilisant la proposition
2.4, le problème est résolu si $k=\inf_{Q\in M}\dim O_{G}(Q)\geq 1$.
Supposons que $k=\inf_{Q\in M}\dim O_{G}(Q)=0$. On pose
$\Phi=\\{\varphi_{q}\text{ solution de \eqref{eygi}
},\;\varphi_{q}>0,\;\|\varphi_{q}\|_{q}=1\text{ et
}\mu_{q,G}(g)=I_{q,g}(\varphi_{q})/q\in[q_{0},N[\\}$
l’ensemble des solutions données par la proposition 2.3, avec
$q_{0}\in]2p/(p-1),N[$ suffisamment proche de $N$ de sorte que $\mu_{q,G}(g)$
reste strictement positive pour tout $q\in[q_{0},N[$. Ce qui est possible car
$\forall
q\in[q_{0},N[\quad\mu_{q,G}(g)=I_{q,g}(\varphi_{q})=I_{N,g}(\varphi_{q})\|\varphi_{q}\|_{N}^{-2}\geq\mu_{N,G}(g)\|\varphi_{q}\|_{N}^{-2}>0$
D’autre part, pour tout $\varepsilon>0$, il existe $\varphi_{\varepsilon}\in
H^{p}_{2,G}(M)$ strictement positive telle que
$I_{N,g}(\varphi_{\varepsilon})<\mu_{N,G}(g)+\varepsilon$
Puisque
$\limsup_{q\to N}\mu_{q,G}(g)\leq\lim_{q\to
N}I_{q,g}(\varphi_{\varepsilon})=I_{N,g}(\varphi_{\varepsilon})$
on en déduit que
$\limsup_{q\to N}\mu_{q,G}(g)\leq\mu_{N,G}(g)$ (2.15)
L’ensemble $\Phi$ est borné dans $H_{1}^{2}(M)$, en effet:
$\begin{split}\|\varphi_{q}\|_{2}&\leq vol(M)^{1/2-1/q}\|\varphi_{q}\|_{q}\leq
1+vol(M)^{1/2-1/N}\\\
\|\nabla\varphi_{q}\|^{2}_{2}&=\mu_{q,G}(g)-\int_{M}h\varphi_{q}^{2}\mathrm{d}v\\\
&\leq I_{q,g}(1)+\|h\|_{p}\|\varphi_{q}\|_{2p/(p-1)}^{2}\\\
&\leq\|h\|_{1}vol(M)^{-2/q}+\|h\|_{p}\|\varphi_{q}\|_{2p/(p-1)}^{2}\\\ &\leq
C\|h\|_{p}\end{split}$
où $C$ est une constante strictement positive qui dépend seulement de $n$.
L’ensemble $\Phi$ est donc faiblement compact dans $H_{1}^{2}(M)$, on en
déduit qu’il existe une suite $(q_{i})_{i\in\mathbb{N}}$ qui converge vers $N$
telle que
$*$
$\varphi_{q_{i}}\rightharpoonup\varphi_{N}$ faiblement dans $H_{1}(M)$.
$*$
$\varphi_{q_{i}}\rightarrow\varphi_{N}$ fortement dans $L^{s}(M)$ pour tout
$1\leq s<N$.
$*$
$\varphi_{q_{i}}\rightarrow\varphi_{N}$ presque partout.
Donc $\varphi_{N}$ est nécessairement $G-$invariante presque partout.
Puisque $\varphi_{q_{i}}$ satisfait l’équation (2.13) pour
$\tilde{h}=\mu_{q_{i},G}(g)$ et $q=q_{i}$, alors pour tout $\psi\in H_{1}(M)$:
$\int_{M}\nabla^{j}\psi\nabla_{j}\varphi_{q_{i}}\mathrm{d}v+\int_{M}h\psi\varphi_{q_{i}}\mathrm{d}v=\mu_{q_{i},G}(g)\int_{M}\psi\varphi_{q_{i}}^{q_{i}-1}\mathrm{d}v$
(2.16)
D’autre part, l’inclusion de Sobolev $H_{1}(M)\subset L^{N}(M)$ et l’inégalité
de Hölder permettent d’écrire
$\|\varphi_{q_{i}}^{q_{i}-1}\|_{N/(N-1)}\leq
vol(M)^{\frac{N-q_{i}}{N-1}}\|\varphi_{q_{i}}\|^{q_{i}-1}_{N}\leq
c(\|\nabla\varphi_{q_{i}}\|_{2}+\|\varphi_{q_{i}}\|_{2})^{N-1}\leq C$
car $\Phi$ est bornée dans $H_{1}(M)$. Donc, à extraction de sous-suite près,
$\varphi_{q_{i}}^{q_{i}-1}$ converge faiblement vers $\varphi_{N}^{N-1}$ dans
$L^{N/(N-1)}(M)$ (voir les théorèmes des espaces de Banach dans la section
1.3.1) et par l’inégalité (2.15), on peut supposer que $\mu_{q_{i},G}(g)$
converge vers $\mu$. Par conséquent on peut passer à la limite dans (2.16), on
en déduit que $\varphi_{N}$ est solution faible de l’équation (2.13) pour
$q=N$ et $\tilde{h}=\mu$. Montrons que $\varphi_{N}$ n’est pas identiquement
nulle. Puisque $\varphi_{q}$ est $G-$invariante presque partout, on peut
appliquer l’inégalité de la meilleure constante en présence de symétrie du
théorème 2.4 :
$\forall\varepsilon>0\quad\|\varphi_{q_{i}}\|^{2}_{N}\leq(K^{2}(n,2)[\inf_{Q\in
M}\mathrm{card}O_{G}(Q)]^{-2/n}+\varepsilon)\|\nabla\varphi_{q_{i}}\|^{2}_{2}+B(\varepsilon)\|\varphi_{q_{i}}\|^{2}_{2}$
$\varphi_{q_{i}}\in\Phi$ et en utilisant l’inégalité de Hölder:
$\|\varphi_{q_{i}}\|^{2}_{N}\geq
vol(M)^{2/N-2/q_{i}}\|\varphi_{q_{i}}\|^{2}_{q_{i}}=vol(M)^{2/N-2/q_{i}}$
on peut donc écrire que
$vol(M)^{2/N-2/q_{i}}\leq(K^{2}(n,2)[\inf_{Q\in
M}\mathrm{card}O_{G}(Q)]^{-2/n}+\varepsilon)(\mu_{q_{i},G}(g)-\int_{M}h\varphi_{q_{i}}^{2}\mathrm{d}v)+B(\varepsilon)\|\varphi_{q_{i}}\|^{2}_{2}$
Quand $i\rightarrow+\infty$, $\mu_{q_{i},G}(g)\rightarrow\mu$ et
$vol(M)^{2/N-2/q_{i}}\rightarrow 1$ donc
$\displaystyle 1\leq(K^{2}(n,2)[\inf_{Q\in
M}\mathrm{card}O_{G}(Q)]^{-2/n}+\varepsilon)(\mu-\int_{M}h\varphi_{N}^{2}\mathrm{d}v)+B(\varepsilon)\|\varphi_{N}\|^{2}_{2}$
Comme $\mu<\mu_{N,G}(g)<K^{-2}(n,2)(\inf_{Q\in
M}\mathrm{card}O_{G}(Q))^{2/n}$, on peut même supposer qu’il existe
$\varepsilon_{0}>0$ tel que
$(K^{2}(n,2)[\inf_{Q\in
M}\mathrm{card}O_{G}(Q)]^{-2/n}+\varepsilon_{0})\mu<1-\varepsilon_{0}$
cela entraîne l’existence d’une constante $C(\varepsilon_{0})>0$ telle que
$B(\varepsilon_{0})\|\varphi_{N}\|^{2}_{2}+C(\varepsilon_{0})\|h\|_{p}\|\varphi_{N}\|^{2}_{\frac{2p}{p-1}}\geq\varepsilon_{0}$
alors $\varphi_{N}$ n’est pas identiquement nulle. On vient donc de montrer
que $\varphi_{N}$ est une solution faible positive, non identiquement nulle et
$G-$invariante presque partout de l’équation
$\Delta_{g}\varphi_{N}+h\varphi_{N}=\mu\varphi_{N}^{N-1}$ (2.17)
Par le théorème 1.11, $\varphi_{N}\in H^{p}_{2,G}(M)$ est strictement
positive. Il reste à montrer que $\varphi_{N}$ est minimisante pour la
fonctionnelle $I_{N,g}=I_{g}$ et que $\mu=\mu_{N}(g)$. On revient pour celà à
la suite $(\varphi_{q_{i}})$ qui converge fortement vers $\varphi_{N}$ dans
$L^{s}$ pour tout $1\leq s<N$. En utilisant l’inégalité de Hölder et le fait
que $\|\varphi_{q_{i}}\|_{q_{i}}=1$, on a l’inégalité suivante:
$\int_{M}\varphi_{q_{i}}^{N-1}\varphi_{N}\mathrm{d}v\leq\|\varphi_{N}\|_{q_{i}/(q_{i}-N+1)}$
En passant à la limite dans cette inégalité et grâce au fait que
$\varphi_{q_{i}}\rightarrow\varphi_{N}$ fortement dans $L^{N-1}$ et que
$\varphi_{N}$ est continue sur $M$ (i.e. $\varphi_{N}\in H^{p}_{2}(M)$), on en
déduit que $\|\varphi_{N}\|_{N}\leq 1$. D’autre part, si on multiplie
l’équation (2.17) par $\varphi_{N}$ et on intégre sur $M$, on trouve que
$\mu\|\varphi_{N}\|^{N-2}_{N}=I_{N,g}(\varphi_{N})\geq\mu_{N,G}(g)$
D’où $\mu\geq\mu_{N,G}(g)$. En combinant avec l’inégalité (2.15), on conclut
que $\mu=\mu_{N,G}(g)$ et $\|\varphi_{N}\|_{N}=1$. ∎
##### Remarque
La méthode que l’on vient d’utiliser dans la preuve de ce théorème n’est pas
valable dans le cas où $\mu_{q,G}(g)\leq 0$, car l’opérateur $L=\Delta_{g}+h$
n’est plus inversible. On verra dans la section 3.12 que si la fonction $h$
est proportionnelle à la courbure scalaire $R_{g}$ de $g$, alors on peut s’en
tirer grâce au théorème d’unicité des solutions 3.7.
Si on reprend la même démarche utilisée pour montrer le théorème 2.1 afin de
démontrer le théorème 2.5, on montre qu’il existe $\varphi_{N}$ solution
faible dans $H_{1,G}(M)$ de l’équation (2.13). Plus précisément $\varphi_{N}$
est solution de l’équation (2.14), pour tout $\psi\in H_{1,G}(M)$ et pour
$q=N$. Pour que $\varphi_{N}$ soit une solution de l’équation (2.14), pour
tout $\psi\in H^{2}_{1}(M)$ et pour $q=N$, il suffit de montrer que l’équation
$Lu=\mu_{N,G}(g)\varphi_{N}^{N-1}$ admet une unique solution faible
$u=\tilde{\varphi}_{N}\in H_{1,G}(M)$ puis utiliser le même argument que celui
de la fin de la preuve de la proposition 2.3 (voir page 2.2.2).
Malheureusement, on ne peut pas conclure qu’il existe une telle solution
$\tilde{\varphi}_{N}$, car la proposition 1.4 assure l’existence d’une telle
fonction, si $f\in L^{q}(M)$ avec $q>2n/(n+2)$, or $\varphi_{N}^{N-1}\in
L^{2n/(n+2)}(M)$.
Dans le cas positif (i.e. $\mu(g)>0$), le théorème 2.1 est une conséquence du
théorème 2.5, en prenant $G=\\{\mathrm{id}\\}$.
###### Proposition 2.5.
Soit $(M,g)$ une variété riemannienne compacte $C^{\infty}$. $G$ un sous
groupe de $I(M,g)$. On a toujours:
$\mu_{G}(g)\leq K^{-2}(n,2)(\inf_{Q\in M}\mathrm{card}O_{G}(Q))^{2/n}$
###### Preuve.
L’inégalité est triviale si $\inf_{Q\in M}\mathrm{card}O_{G}(Q)=+\infty$.
Supposons qu’il existe une orbite minimale finie et soit $P$ un point de cette
orbite. Autrement dit
$\inf_{Q\in M}\mathrm{card}O_{G}(Q)=\mathrm{card}O_{G}(P)<+\infty$
$O_{G}(P)=\\{P_{i}\\}_{1\leq i\leq k}$, $P=P_{1}$ et
$k=\mathrm{card}O_{G}(P)$. Soit $u_{\varepsilon}$ la fonction définie dans la
preuve de la proposition 2.1, que l’on note $u_{\varepsilon,P}$ car elle
dépend du point $P$ qu’on avait fixé arbitrairement. Soit donc
$u_{\varepsilon,P_{i}}$ les fonctions obtenues en remplaçant $P$ par $P_{i}$
dans l’expression qui définit $u_{\varepsilon,P}$. Enfin, on pose
$U_{\varepsilon}=\sum_{i=1}^{k}u_{\varepsilon,P_{i}}$
D’autre part on choisit $\delta$ suffisamment petit tel que pour tout
$\sigma\in G-\\{\mathrm{id}\\}$
$B_{P}(\delta)\cap B_{\sigma(P)}(\delta)=\emptyset$
Puisque $u_{\varepsilon,P_{i}}$ est radiale (i.e. pour tout $\sigma\in
I(M,g)$,
$\sigma^{*}u_{\varepsilon,P_{i}}=u_{\varepsilon,\sigma^{-1}(P_{i})}$), on en
déduit par cette construction que la fonction $U_{\varepsilon}$ est
$G-$invariante, à support compact et que pour tout $1\leq i\leq k$:
$E(U_{\varepsilon})=\sum_{i=1}^{k}E(u_{\varepsilon,P_{i}})=kE(u_{\varepsilon,P})\text{
et }\|U_{\varepsilon}\|^{N}_{N}=k\|u_{\varepsilon,P_{i}}\|^{N}_{N}$
Finalement
$I_{g}(U_{\varepsilon})=k^{2/n}I_{g}(u_{\varepsilon,P})$
La proposition 2.1 montre que $\lim_{\varepsilon\to
0}I_{g}(U_{\varepsilon})=k^{2/n}K^{-2}(n,2)$ ∎
## Chapter 3 Le problème de Yamabe avec singularités
Dans ce chapitre on interprétera géométriquement les résultats obtenus dans le
chapitre 2. On donnera une signification géométrique aux équations de type
Yamabe qu’on a déjà résolues. On commence par un rappel historique sur le
problème de Yamabe.
### 3.1 Le problème de Yamabe
Soit $(M,g)$ une variété riemannienne compacte $C^{\infty}$ de dimension
$n\geq 3$, $R_{g}$ désigne la courbure scalaire de $g$. Le problème de Yamabe
est le suivant:
###### Problème 3.1.
Parmi les métriques conformes à $g$, existe-t-il une métrique à courbure
scalaire constante?
Yamabe [Yam] avait posé ce problème dans le but de résoudre la conjecture de
Poincaré. Si on pose $\tilde{g}=\varphi^{4/(n-2)}g$ une métrique conforme à
$g$, où $\varphi>0$ est une fonction $C^{\infty}$, alors les courbures
scalaires $R_{g}$, $R_{\tilde{g}}$ sont reliées par l’équation suivante:
$\frac{4(n-1)}{n-2}\Delta_{g}\varphi+R_{g}\varphi=R_{\tilde{g}}\varphi^{N-1}$
(3.1)
avec $N=\frac{2n}{n-2}$.
Pour résoudre ce problème, il suffit de chercher une fonction $C^{\infty}$,
strictement positive $\varphi$ solution de l’équation aux dérivées partielles
non linéaire ci-dessus. L’équation (3.1) est appelée l’équation de Yamabe. On
utilise la méthode variationnelle pour résoudre cette équation. H. Yamabe a
posé la fonctionnelle suivante, définie pour tout $\psi\in H_{1}(M)-\\{0\\}$
par
$I_{g}(\psi)=\frac{E(\psi)}{\|\psi\|_{N}^{2}}=\frac{\displaystyle\int_{M}|\nabla\psi|^{2}+\frac{n-2}{4(n-1)}R_{g}\psi^{2}\mathrm{d}v}{\|\psi\|_{N}^{2}}$
(3.2)
ensuite, il a considéré le minimum de $I_{g}$ et a défini l’invariant conforme
suivant:
$\mu(g)=\inf_{\psi\in H_{1}(M)-\\{0\\}}I_{g}(\psi)$
La difficulté majeure dans la recherche des solutions est le fait que
l’inclusion de Sobolev $H_{1}(M)\subset L^{q}(M)$ est seulement continue pour
$q=N$. Par contre cette inclusion est compacte si $1\leq q<N$. Yamabe a donc
commencé par résoudre une "sous-équation":
$\frac{4(n-1)}{n-2}\Delta_{g}\varphi+R_{g}\varphi=\mu_{q}(g)\varphi^{q-1}$
(3.3)
où $q\in[2,N[,\;N=2n/(n-2)$ et $\mu_{q}(g)\in\mathbb{R}$, ensuite a fait
tendre $q$ vers $N$. H. Yamabe a affirmé que l’ensemble
$\\{\varphi_{q}>0\text{ solution de }\eqref{EY},q\in[2,2n/(n-2)[\\}$ est
uniformément borné dans $C^{0}(M)$. Or N. Trudinger [Trud] a montré que c’est
seulement vrai lorsque $\mu_{q}(g)\leq\leavevmode\nobreak\ 0$. Finalement, H.
Yamabe a seulement réussi à résoudre le problème dans le cas négatif et nul de
$\mu(g)$. Le cas positif est resté ouvert jusqu’à ce que T. Aubin [Aub] montre
qu’il suffit de prouver la conjecture suivante pour résoudre le problème dans
tout les cas.
###### Conjecture 3.1 (T. Aubin [Aub]).
Si $(M,g)$ est une variété riemannienne compacte $C^{\infty}$ de dimension $n$
et non conformément difféomorphe à $(S_{n},g_{can})$ alors
$\mu(M,g)<\mu(S_{n},g_{can})$ (3.4)
où $\mu(M,g)=\inf\\{I_{g}(\psi),\;\psi\in H_{1}(M)-\\{0\\}\\}$
Dans la suite, on écrira $\mu(g)$ en place de $\mu(M,g)$.
T. Aubin a montré que cette inégalité est vraie pour les variétés de dimension
$n\geq 6$, non conformément plates et pour les variétés conformément plates de
groupe fondamental fini, non trivial. Le cas des variétés conformément plates
et des dimensions 3,4 et 5 a été résolu par Schoen [Schoen], en admettant le
théorème de la masse positive. Finalement, la conjecture ci-dessus est
toujours vraie. Grâce essentiellement aux travaux de Yamabe [Yam], T. Aubin
[Aub] et Schoen [Schoen], le problème de Yamabe est complètement résolu dans
le cas des variétés riemanniennes compactes $C^{\infty}$ (voir aussi
[Bah],[BB], [BC] pour résolution avec une méthode topologique).
###### Théorème 3.1 (Aubin–Schoen).
Soit $M$ une variétés compacte $C^{\infty}$, de dimension
$n\geq\leavevmode\nobreak\ 3$. pour toute métrique riemannienne $g$ de classe
$C^{\infty}$, il existe une métrique conforme $\tilde{g}=\varphi^{4/(n-2)}g$
de courbure scalaire constante $R_{\tilde{g}}$, où $\varphi$ est une fonction
$C^{\infty}$, strictement positive, qui minimise la fonctionnelle de Yamabe
$I_{g}$.
On s’intéresse maintenant au problème de Yamabe avec singularités.
### 3.2 Choix de la métrique
Soit $M$ une variété compacte $C^{\infty}$ de dimension $n\geq 3$ et $g$ une
métrique riemannienne sur $M$.
Hypothèse $\boldsymbol{(H)}$: _$g$ est une métrique dans l’espace de Sobolev
$H_{2}^{p}(M,T^{*}M\otimes T^{*}M)$ avec $p>n$. Il existe un point $P_{0}\in
M$ et $\delta>0$ tels que $g$ est $C^{\infty}$ sur la boule
$B_{P_{0}}(\delta)$._
Les métriques que l’on considère sont dans l’espace $H_{2}^{p}(M,T^{*}M\otimes
T^{*}M)$, défini dans la section 1.3. On a choisi cet espace de métriques pour
donner un sens aux courbures, qui sont donc dans $L^{p}$. (On peut supposer
$g$ de classe $C^{2}$ dans la boule $B_{P_{0}}(\delta)$ au lieu de
$C^{\infty}$, mais ce n’est pas un point important).
En fait, l’objectif de cette partie est surtout d’étudier le problème de
Yamabe dans le cas où la métrique $g$ a un nombre fini de points de
singularités et est $C^{\infty}$ en dehors de ces points, l’hypothèse $(H)$
généralise ces conditions et précise la notion de "singularité".
Par les inclusions de Sobolev 1.1, $H_{2}^{p}(M,T^{*}M\otimes T^{*}M)\subset
C^{1,\beta}(M,T^{*}M\otimes T^{*}M)$, pour un certain $\beta\in]0,1[$. Donc
les métriques qui satisfont l’hypothèse $(H)$ sont de classe $C^{1,\beta}$.
Les Christoffels sont dans $C^{\beta}$ et les courbures de Riemann, Ricci et
scalaire sont dans $L^{p}$ car elles font appel à la dérivée seconde de la
métrique $g$ qui est seulement dans $L^{p}$. Comme exemple de métrique qui
satisfait l’hypothèse $(H)$, on peut considérer
$g=(1+\rho^{2-\alpha})^{m}g_{0}$, où $g_{0}$ est une métrique $C^{\infty}$,
$\alpha\in]0,1[$ et $\rho$ est définie dans 1.7. Les dérivées secondes de $g$
ont alors des singularités du type $\rho^{-\alpha}$.
Dans la suite, beaucoup de résultats seront vrais pour toute métrique dans
$H^{p}_{2}(M,T^{*}M\otimes T^{*}M)$, avec $p>n/2$ (c’est la valeur minimale de
$p$ qui donne un sens à la fonctionnelle de Yamabe. Le cas $p=n/2$ est un cas
critique, il est hors de considération). L’hypothèse $(H)$ impose en plus que
la métrique est $C^{\infty}$ dans une certaine boule et que $p>n$. On rajoute
la condition $p>n$ pour que les Christoffels de la métrique $g\in
H^{p}_{2}(M,T^{*}M\otimes T^{*}M)$ soient continus. L’hypothèse $(H)$ est
suffisante pour montrer la conjecture 3.1 (cf. théorème 3.5) et pour
construire la fonction de Green du Laplacien conforme (cf. section 3.5).
On considère le problème suivant:
###### Problème 3.2.
Soit $g$ une métrique qui satisfait l’hypothèse $(H)$. Existe-t-il une
métrique $\tilde{g}$ conforme à $g$ pour laquelle la courbure scalaire
$R_{\tilde{g}}$ est constante (même aux points où $R_{g}$ n’est pas
régulière)?
Il est clair que si la métrique initiale $g$ est de classe $C^{\infty}$, alors
le problème ci-dessus n’est autre que le problème de Yamabe 3.1 qui a été déjà
complètement résolu. On montrera plus loin que la réponse à ce problème est
positive. La proposition suivante, permet de préciser ce que l’on entend par
changement de métrique conforme lorsque les métriques sont dans $H^{p}_{2}$.
###### Proposition 3.1.
Soit $g$ une métrique dans $H^{p}_{2}$ et $\psi\in H^{p}_{2}(M)$, strictement
positive. Si $p>n/2$ alors la métrique $\tilde{g}=\psi^{\frac{4}{n-2}}g$ est
bien définie, et elle est dans le même espace que $g$.
###### Preuve.
Cette proposition découle du fait que $H^{p}_{2}(M)$ est une algèbre, pour
tout $p>n/2$ (cf. proposition 1.1, page 1.1). ∎
### 3.3 Le Laplacien conforme
###### Définition 3.1.
Le Laplacien conforme d’une variété riemannienne $(M,g)$ est l’opérateur
$L_{g}$ défini par :
$L_{g}=\Delta_{g}+\frac{n-2}{4(n-1)}R_{g}$
#### 3.3.1 L’invariance conforme faible
Il est bien connu que le Laplacien conforme lorsque $g$ est $C^{\infty}$, est
conformément invariant, c’est à dire qu’il vérifie (3.5) fortement. On montre
qu’on a toujours la même propriété lorsque la métrique est dans
$H^{p}_{2}(M,T^{*}M\otimes T^{*}M)$.
###### Proposition 3.2.
Soient $M$ une variété compacte $C^{\infty}$ de dimension $n\geq 3$ et $g\in
H_{2}^{p}(M,T^{*}M\otimes T^{*}M)$ est une métrique riemannienne sur $M$, avec
$p>n/2$. Si $\tilde{g}=\psi^{\frac{4}{n-2}}g$ est une métrique conforme à $g$,
avec $\psi\in H_{2}^{p}(M)$ et $\psi>0$, alors $L$ est faiblement conformément
invariant, autrement dit
$\forall u\in H_{1}(M)\qquad\psi^{\frac{n+2}{n-2}}L_{\tilde{g}}(u)=L_{g}(\psi
u)\quad faiblement$ (3.5)
De plus si $\mu(g)>0$ alors le Laplacien conforme
$L_{g}=\Delta_{g}+\frac{n-2}{4(n-1)}R_{g}$ est inversible et coercif.
###### Preuve.
Rappelons que $\mathrm{d}v_{\tilde{g}}=\psi^{\frac{2n}{n-2}}\mathrm{d}v$ et
que
$\forall u,w\in L^{2}(M)\quad(u,w)_{g,L^{2}}=\int_{M}uw\mathrm{d}v_{g}$
est le produit scalaire sur l’espace $L^{2}(M)$ muni de la métrique $g$.
Pour tout $u,w\in H_{1}(M)$:
$\begin{split}(\psi^{\frac{2n}{n-2}}L_{\tilde{g}}u,w)_{g,L^{2}}&=(L_{\tilde{g}}u,w)_{\tilde{g},L^{2}}\\\
&=\int_{M}\tilde{g}(\nabla u,\nabla
w)+\frac{n-2}{4(n-1)}R_{\tilde{g}}uw\mathrm{d}v_{\tilde{g}}\\\
&=\int_{M}\psi^{2}g(\nabla u,\nabla
w)+\frac{n-2}{4(n-1)}R_{\tilde{g}}\psi^{\frac{n+2}{n-2}}(uw\psi)\mathrm{d}v_{g}\end{split}$
D’autre part, on sait que les deux courbures scalaires $R_{g}$ et
$R_{\tilde{g}}$ sont reliées par l’équation de Yamabe (3.1), ce qui est
équivalent à
$L_{g}\psi=\frac{n-2}{4(n-1)}R_{\tilde{g}}\psi^{\frac{n+2}{n-2}}\quad
faiblement$
ce que l’on écrit
$(L_{g}\psi,uw\psi)_{g,L^{2}}=\frac{n-2}{4(n-1)}(R_{\tilde{g}}\psi^{\frac{n+2}{n-2}},uw\psi)_{g,L^{2}}$
où il y a un abus de notation car $uw\psi$ n’appartient pas forcément à
$L^{2}(M)$. Par contre $L_{g}\psi\in L^{p}(M)\subset L^{n/2}(M)$ et $uw\psi\in
L^{n/(n-2)}(M)$, le produit est donc bien défini. Par conséquent
$\begin{split}(\psi^{\frac{2n}{n-2}}L_{\tilde{g}}u,w)_{g,L^{2}}&=\int_{M}\psi^{2}g(\nabla
u,\nabla
w)+g(\nabla\psi,\nabla(uw\psi))+\frac{n-2}{4(n-1)}R_{g}\psi(uw\psi)\mathrm{d}v_{g}\\\
&=\int_{M}g(\nabla(\psi u),\nabla(w\psi))+\frac{n-2}{4(n-1)}R_{g}(\psi
u)(w\psi)\mathrm{d}v_{g}\\\ &=(\psi L_{g}(\psi u),w)_{g,L^{2}}\end{split}$
On a utilisé le fait que $u\psi$ et $w\psi$ appartiennent à $H_{1}(M)$, car on
a les inclusions
$H_{2}^{p}(M)\subset C^{1-[n/p],\beta}(M),\;H^{p}_{1}(M)\subset
L^{\frac{pn}{n-p}}(M)\text{ et }H_{1}(M)\subset L^{\frac{2n}{n-2}}(M)$
Maintenant, montrons que $L_{g}$ est inversible et coercif. Soit $\lambda$ la
plus petite valeur propre de $L_{g}$, de fonction propre $\varphi\in H_{1}(M)$
positive, non identiquement nulle, alors
$\lambda\|\varphi\|_{2}^{2}=(L_{g}\varphi,\varphi)_{g,L^{2}}=I_{g}(\varphi)\|\varphi\|_{N}^{2}\geq\mu(g)\|\varphi\|_{N}^{2}>0$
d’où $\lambda>0$. Il suffit donc d’appliquer la proposition 1.4.
∎
### 3.4 L’invariant conforme de Yamabe
Dans le cas des métriques de classe $C^{\infty}$, $\mu(g)$ est un invariant
conforme, ce qui signifie que si $g$ et $\tilde{g}$ sont deux métriques
conformes de classe $C^{\infty}$ alors
$\mu(g)=\mu(\tilde{g})$
(voir la section 3.1 pour la définition). La proposition suivante montre qu’on
peut étendre cette propriété à des métriques dans $H^{p}_{2}$. Elle nous
permettra aussi de prendre une métrique quelconque dans la classe conforme
$[g]$ comme métrique initiale, tout en gardant la valeur de $\mu(g)$
inchangée.
###### Proposition 3.3.
Soit $M$ une variété compacte $C^{\infty}$, de dimension $n$. Soit $g$ et
$\tilde{g}=\psi^{\frac{4}{n-2}}g$ deux métriques dans $H^{p}_{2}$, avec
$\psi\in H^{p}_{2}(M)$, strictement positive. Si $p>n/2$ alors
$\mu(g)=\mu(\tilde{g})$
###### Preuve.
Soient $u\in H_{1}(M)$ une fonction test et $I_{g}$ la fonctionnelle de Yamabe
(3.2). Remarquons que $E(u)=(L_{g}(u),u)_{g,L^{2}}$. Donc
$I_{\tilde{g}}(u)=(L_{\tilde{g}}(u),u)_{\tilde{g},L^{2}}\|u\psi\|_{N}^{-2}$
De la proposition 3.2, on en déduit que
$I_{\tilde{g}}(u)=(L_{g}(\psi u),\psi u)_{g,L^{2}}\|u\psi\|_{N}^{-2}$
Finalement
$I_{\tilde{g}}(u)=I_{g}(\psi u)$ (3.6)
ce qui implique que $\mu(g)=\mu(\tilde{g})$, et que cet invariant dépend
seulement de la classe conforme $[g]$ et de la variété $M$. ∎
### 3.5 La fonction de Green du Laplacien conforme
###### Définition 3.2.
Soit $(M,g)$ une variété riemannienne compacte et $P$ un point de $M$. On
appelle fonction de Green au point $P$ d’un opérateur linéaire $L$, la
fonction $G_{P}$ qui vérifie au sens des distributions
$LG_{P}=\delta_{P}(\Longleftrightarrow\forall f\in C^{\infty}(M)\quad\langle
G_{P},Lf\rangle=f(P))$
La fonction de Green peut être vue comme l’inverse de l’opérateur $L$, lorsque
ce dernier est inversible. La proposition 3.5 montre l’existence d’une telle
fonction pour un opérateur du type $L=\Delta+h$ avec $h>0$ continue.
Malheureusement, la méthode utilisée pour construire cette fonction de Green
n’est pas valable lorsque la fonction $h$ est dans $L^{p}(M)$. Ce cas se
présente pour le Laplacien conforme $L_{g}$, car $R_{g}\in L^{p}(M)$. Mais,
grâce à la proposition 3.6, on pourra s’en tirer, et obtenir le corollaire
3.7. Pour montrer son existence lorsque $h$ est continue, on aura besoin du
résultat suivant dû à G. Giraud [Gir] (On peut aussi consulter [Aubin], page
108).
###### Proposition 3.4.
Soit $\Omega$ un ouvert d’une variété riemannienne compacte $(M,g)$.
$\varphi$, $\psi$ deux fonctions continues sur
$\Omega\times\Omega-\\{(x,x)\in\Omega\times\Omega\\}$ qui vérifient:
$|\varphi(P,Q)|\leq c(d(P,Q))^{\alpha-n}\mbox{ et }|\psi(P,Q)|\leq
c(d(P,Q))^{\beta-n}$
pour tout $(P,Q)\in\Omega\times\Omega-\\{(x,x)\in\Omega\times\Omega\\}$, où
$\alpha,\;\beta\in]0,n[$.
alors la fonction $\chi$ définie par:
$\chi(P,Q)=\int_{\Omega}\varphi(P,R)\psi(R,Q)\mathrm{d}v(R)$
est continue sur $\Omega\times\Omega-\\{(x,x)\in\Omega\times\Omega\\}$ et est
vérifie:
$|\chi(P,Q)|\leq\begin{cases}c(d(P,Q))^{\alpha+\beta-n}&\mbox{ si
}\alpha+\beta<n\\\ c(1+\log d(P,Q))&\mbox{ si }\alpha+\beta=n\\\ c&\mbox{ si
}\alpha+\beta>n\end{cases}$
dans le dernier cas la fonction $\chi$ est continue sur $\Omega\times\Omega$.
###### Proposition 3.5.
Soit $M$ une variété compacte $C^{\infty}$ de dimension $n\geq 3$, $h$ une
fonction continue, strictement positive et $P$ un point de $M$. $g$ une
métrique qui satisfait l’hypothèse $(H)$ (cf. section 3.2). Il existe une
unique fonction de Green $G_{P}$ de l’opérateur $L=\Delta_{g}+h$ qui satisfait
au sens des distributions $LG_{P}=\delta_{P}$ et
* $(i)$
$G_{P}$ est $C^{\infty}$ sur $B_{P_{0}}(\delta)-\\{P\\}$
* $(ii)$
$G_{P}\in C^{2}(M-\\{P\\})$
* $(iii)$
Il existe $c>0$ tel que pour tout $Q\in M-\\{P\\}$, $|G_{P}(Q)|\leq
cd(P,Q)^{2-n}$
###### Preuve.
L’unicité de $G_{P}$ est due au fait que $L$ est inversible. En effet, si
$\lambda$ est une valeur propre de $L$ et $\varphi$ une fonction propre, non
identiquement nulle, associée à $\lambda$ alors
$\lambda\|\varphi\|^{2}_{2}=(L\varphi,\varphi)_{L^{2}}=E(\varphi)>0$
D’où $\lambda>0$. Pour conclure, il suffit d’appliquer la proposition 1.4. En
ce qui concerne l’existence de cette fonction, on reprend la construction de
T. Aubin [Aubin] pour le Laplacien, dans le cas des métriques $C^{\infty}$. On
choisit $f(r)$ une fonction radiale décroissante $C^{\infty}$ positive, égale
à $1$ pour $r<\delta/2$ et nulle pour $r\geq\delta(M)$, le rayon d’injectivité
de $M$. On définit les fonctions suivantes:
$\displaystyle H(P,Q)=\frac{f(r)}{(n-2)\omega_{n-1}}r^{2-n}\mbox{ avec
}r=d(P,Q)$ $\displaystyle\Gamma^{1}(P,Q)=-L_{Q}H(P,Q)$ $\displaystyle\forall
i\in\mathbb{N^{*}}\qquad\Gamma^{i+1}(P,Q)=\int_{M}\Gamma^{i}(P,S)\Gamma^{1}(S,Q)\mathrm{d}v(S)$
où $L_{Q}H(P,Q)$ signifie qu’on applique l’opérateur $L$ à la fonction
$H(P,Q)$ par rapport à $Q$.
On observe que $\Gamma^{1}$ est continue sur $M\times M-\\{(Q,Q)\in M\times
M\\}$, et il existe $c>0$ tel que pour tout $P,Q\in M$:
$|\Gamma^{1}(P,Q)|\leq cd(P,Q)^{2-n}$
En utilisant la proposition 3.4, on montre les inégalités suivantes:
$\forall i\geq 1\qquad|\Gamma^{i}(P,Q)|\leq\begin{cases}&cd(P,Q)^{2i-n}\hskip
78.24507pt\text{ si }2i<n\\\ &c(1+\log d(P,Q))\hskip 56.9055pt\text{ si
}2i=n\\\ &c\hskip 133.72786pt\text{ si }2i>n\end{cases}$
La fonction de Green de $L$ s’écrit
$G_{P}(Q)=H(P,Q)+\sum_{i=1}^{k}\int_{M}\Gamma^{i}(P,S)H(S,Q)\mathrm{d}v(S)+F_{P}(Q)$
(3.7)
où $F_{P}$ est une fonction que l’on détermine dans les lignes qui suivent. On
prend $k=[n/2]$ alors $\Gamma^{k+1}(P,\cdot)$ est continue (cf. proposition
3.4). On veut $L_{Q}G_{P}(Q)=0$ pour $Q\neq P$. On a l’identité
$\psi(Q)=\Delta_{g}\int_{M}H(P,Q)\psi(P)\mathrm{d}v(P)-\int_{M}\Delta_{Q}H(P,Q)\psi(P)\mathrm{d}v(P)$
(La preuve est donnée dans [Aubin], page 106). D’où
$\psi(Q)=L\int_{M}H(P,Q)\psi(P)\mathrm{d}v(P)-\int_{M}L_{Q}H(P,Q)\psi(P)\mathrm{d}v(P)$
En utilisant cette dernière identité, on trouve que
$L_{Q}G_{P}(Q)=-\Gamma^{k+1}(P,Q)+L_{Q}F_{P}(Q)$
Puisque $L$ est inversible, il suffit de poser $F_{P}$ comme l’unique solution
de l’équation
$LF_{P}=\Gamma^{k+1}(P,\cdot)$
Par le théorème de régularité 1.9, $F_{P}$ est de classe $C^{2}$.
$(i)$ Comme $L_{g}G_{P}=0$ sur $B_{P_{0}}(\delta)-\\{P\\}$ et que la métrique
est $C^{\infty}$ sur $B_{P_{0}}(\delta)$, le théorème de régularité affirme
que $G_{P}$ est $C^{\infty}$ sur $B_{P_{0}}(\delta)-\\{P\\}$, avec $P\in M$ et
$B_{P_{0}}(\delta)-\\{P\\}=B_{P_{0}}(\delta)$ si $P\notin B_{P_{0}}(\delta)$.
$(ii)$ On a aussi $LG_{P}=0$ sur $M-\\{P\\}$. On conclut par le théorème de
régularité que $G_{P}$ est $C^{2}$ sur $M-\\{P\\}$.
$(iii)$ En observant l’expression (3.7) qui définit $G_{P}$, on remarque que
le terme dominant, au voisinage de $P$, est bien $H(P,Q)$, donc pour tout
$P\neq Q$,
$|G_{P}(Q)|\leq cd(P,Q)^{2-n}$
∎
###### Proposition 3.6.
Soit $g$ une métrique dans $H^{p}_{2}(M,T^{*}M\otimes T^{*}M)$,
$\tilde{g}=\psi^{\frac{4}{n-2}}g$ une métrique conforme à $g$, avec $\psi\in
H^{p}_{2}(M)$, strictement positive et $p>n/2$. On suppose que le Laplacien
conforme $L_{\tilde{g}}$ admet une fonction de Green $\tilde{G}_{P}$, alors
$L_{g}$ admet aussi une fonction de Green notée $G_{P}$ et elle donnée par
$\forall Q\in M-\\{P\\}\qquad G_{P}(Q)=\psi(P)\psi(Q)\tilde{G}_{P}(Q)$
###### Preuve.
Pour toute fonction $\varphi\in C^{\infty}(M)$:
$\begin{split}\langle\psi(P)\psi\tilde{G}_{P},L_{g}\varphi\rangle_{g}&=\psi(P)\int_{M}\tilde{G}_{P}\psi
L_{g}[\psi(\frac{\varphi}{\psi})]\mathrm{d}v_{g}\\\
&=\psi(P)\int_{M}\tilde{G}_{P}L_{\tilde{g}}\frac{\varphi}{\psi}\mathrm{d}v_{\tilde{g}}\\\
&=\psi(P)\langle\tilde{G}_{P},L_{\tilde{g}}\frac{\varphi}{\psi}\rangle_{\tilde{g}}\\\
&=\varphi(P)\end{split}$
La deuxième égalité ci-dessus vient de l’invariance conforme faible du
Laplacien conforme (cf. proposition 3.2). La troisième inégalité est réalisée
car pour tout $Q\in M-\\{P\\}$
$|\tilde{G}_{P}(Q)|\leq cd(P,Q)^{2-n}$
donc $G_{P}\in L^{s}(M)$, pour tout $1\leq s<n/(n-2)$ et
$L_{\tilde{g}}\frac{\varphi}{\psi}\in L^{p}(M)$ avec $p>n/2$. On peut donc
choisir $s$ pour que
$\langle\tilde{G}_{P},L_{\tilde{g}}\frac{\varphi}{\psi}\rangle_{\tilde{g}}$
soit fini. ∎
###### Proposition 3.7.
Soit $M$ une variété compacte $C^{\infty}$ de dimension $n\geq 3$. $g$ une
métrique riemannienne qui satisfait l’hypothèse $(H)$. Si $\mu(g)>0$, alors le
Laplacien conforme $L_{g}$ admet une fonction de Green $G_{P_{0}}$, qui
satisfait au sens des distributions $LG_{P_{0}}=\delta_{P_{0}}$ et
* $(i)$
$G_{P_{0}}$ est $C^{\infty}$ sur $B_{P_{0}}(\delta)-\\{P_{0}\\}$
* $(ii)$
$G_{P_{0}}\in H^{p}_{2}(M-B_{P_{0}}(r))$ pour tout $r>0$.
* $(iii)$
Il existe $c>0$ tel que pour tout $Q\in B_{P_{0}}(\delta)-\\{P_{0}\\}$,
$|G_{P_{0}}(Q)|\leq cd(P_{0},Q)^{2-n}$
###### Preuve.
Puisque $\mu(g)>0$, $L_{g}$ est nécessairement inversible. On en déduit que si
$L_{g}$ admet une fonction de Green, celle-ci est unique. La proposition 2.3
permet de montrer que l’équation
$\Delta_{g}\psi+\frac{n-2}{4(n-1)}R_{g}\psi=\mu_{q,G}(g)\psi^{q-1}$ (3.8)
admet une solution $\psi\in H^{p}_{2}(M)$, strictement positive (pour $q<N$
suffisamment proche de $N$ et $G=\\{\mathrm{id}\\}$). De plus, puisque la
métrique $g$ est $C^{\infty}$ dans $B_{P_{0}}(\delta)$, les théorèmes de
régularité montrent que $\psi$ est également $C^{\infty}$ dans cette même
boule. La métrique $\tilde{g}:=\psi^{\frac{4}{n-2}}g$ satisfait donc
l’hypothèse $(H)$. D’après l’équation de Yamabe (3.1) (cf. page 3.1), la
courbure scalaire de la métrique $\tilde{g}$ est
$R_{\tilde{g}}=\frac{4(n-1)}{n-2}\mu_{q,G}(g)\psi^{q-N}$
Par conséquent, $R_{\tilde{g}}$ est continue et strictement positive car
$\mu_{q,G}(g)>0$. On est maintenant en mesure d’utiliser la proposition 3.5,
qui assure l’existence d’une fonction de Green $\tilde{G}_{P_{0}}$ du
Laplacien conforme $L_{\tilde{g}}$ pour la variété $M$ muni de la métrique
$\tilde{g}$. Par la proposition 3.6, on conclut que
$G_{P_{0}}=\psi(P_{0})\psi\tilde{G}_{P_{0}}$ est la fonction de Green du
Laplacien $L_{g}$. Comme les métriques $g$ et $\tilde{g}$ sont $C^{\infty}$
sur $B_{P_{0}}(\delta)$ et que $\tilde{G}_{P_{0}}$ satisfait les propriétés de
la proposition 3.5, les propriétés énoncées pour $G_{P_{0}}$ sont vérifiées. ∎
On dit la fonction de Green $G_{P}$ est normalisée si
$\lim_{r\to 0}r^{2-n}G_{P}(Q)=1$
Autrement dit, si $G_{P}$ est normalisée alors
$L_{g}G_{P}=(n-2)\omega_{n-1}\delta_{P}$
où $r=d(P,Q)$ et $\omega_{n-1}$ est le volume de la sphère $S_{n-1}$. Lorsque
il s’agit de la fonction de Green $G_{P_{0}}$ du Laplacien conforme $L_{g}$,
on peut toujours la normaliser car elle est d’ordre $r^{2-n}$. On gardera la
même notation pour la fonction de Green normalisée.
### 3.6 La métrique de Cao–Günther
Dans l’article [LP] sur le problème de Yamabe, J.M. Lee et T. Parker ont
montré que sur une variété riemannienne $(M,g)$, il existe un système de
coordonnées normale $\\{(U_{i},x_{i})\\}_{i\in I}$ et une métrique
$g^{\prime}$ conforme à $g$ tels que $\det g^{\prime}=1+O(|x|^{m})$ avec $m$
aussi grand que l’on veut. J. Cao [Cao] et M. Günther [Gun] ont montré
(indépendamment) qu’on peut avoir, en fait, $\det g^{\prime}=1$.
###### Définition 3.3.
Soit $(M,g)$ une variété riemannienne compacte. $\tilde{g}$ est une métrique
de Cao–Günther, si elle est conforme à $g$ et s’il existe un système de
coordonnées dans lequel $\det\tilde{g}=1$.
###### Théorème 3.2 (Cao–Günther).
Soient $M$ une variété de dimension $n$ et de classe $C^{a+2,\beta}$ avec
$a\in\mathbb{N}$, $\beta\in]0,1[$. $g$ une métrique riemannienne de classe
$C^{a+1,\beta}$, et $P$ un point de $M$. Alors il existe une fonction
$\varphi$ strictement positive, de classe $C^{a+1,\beta^{\prime}}$, avec
$\beta^{\prime}\in]0,\beta[$ telle que $\det(\varphi g)=1$, dans un système de
coordonnées normales pour la métrique $\varphi g$ d’origine $P$.
On remarque que si la métrique $g\in H_{2}^{p}(M,T^{*}M\otimes T^{*}M)$ avec
$p>n$, alors elle est de classe $C^{1,\beta}$, la variété $(M,g)$ admet une
métrique de Cao–Günther. Il n’est donc pas utile de supposer que la métrique
$g$ est $C^{\infty}$ dans une boule pour l’existence de telles coordonnées.
### 3.7 Le théorème de la masse positive
Dans cette section, on rappelle les résultats obtenus au sujet de la masse
positive.
###### Définition 3.4.
Une variété riemannienne $M$ muni d’une métrique $C^{\infty}$, $g$ est dite
asymptotiquement plate d’ordre $\tau>0$, s’il existe une décomposition
$M=M_{0}\cup M_{\infty}$ (avec $M_{0}$ compacte) et un difféomorphisme
$M_{\infty}\rightarrow\mathbb{R}^{n}-B_{R}(O)$ pour un certain $R>0$ tels que:
$g_{ij}=\delta_{ij}+O(\rho^{-\tau}),\quad\partial_{k}g_{ij}=O(\rho^{-\tau-1}),\quad\partial_{kl}g_{ij}=O(\rho^{-\tau-2})$
(3.9)
quand $\rho=|z|\to+\infty$ dans les coordonnées $\\{z^{i}\\}$ induites sur
$M_{\infty}$. Les $\\{z^{i}\\}$ sont appelés les coordonnées asymptotiques.
On écrit $g_{ij}=\delta_{ij}+O^{\prime\prime}(\rho^{-\tau})$ si $g_{ij}$
satisfait (3.9). D’une façon analogue, on peut définir $O^{\prime\prime}$ pour
tout fonction.
###### Définition 3.5.
Etant donné une variété riemannienne asymptotiquement plate $(M,g)$ avec des
coordonnées asymptotiques $\\{z^{i}\\}$, on définit la masse de la façon
suivante:
$m(g)=\lim_{\rho\to+\infty}\omega_{n-1}^{-1}\int_{\partial
B_{P}(\rho)}\partial_{\rho}(g_{\rho\rho}-g_{ii})+\rho^{-1}(ng_{\rho\rho}-g_{ii})d\sigma_{\rho}$
Cette définition de la masse dépend des coordonnées asymptotiques. R. Bartnik
[Bar] a montré que si $(M,g)$ asymptotiquement plate d’ordre $\tau>(n-2)/2$,
alors $m(g)$ est bien définie et dépend seulement de la métrique $g$.
Le théorème de la masse positive s’énonce comme suit:
###### Théorème 3.3.
Soit $(M,g)$ une variété riemannienne de dimension $n\geq 3$, asymptotiquement
plate d’ordre $\tau>(n-2)/2$, de courbure scalaire positive. La masse $m(g)$
est toujours positive ou nulle. De plus $m(g)=0$ si et seulement si $(M,g)$
est isométrique à l’espace euclidien $(\mathbb{R}^{n},\mathcal{E})$ muni de sa
métrique canonique.
Beaucoup de mathématiciens ont contribué à la preuve de ce théorème,
essentiellement T. Aubin [Aub2, Aub6] R. Schoen et S.T. Yau [SY, SY2, SY3], E.
Witten [Wit].
Récemment T. Aubin [Aub2] a montré que:
###### Théorème 3.4.
Si $g$ est une métrique de Cao–Günther, $L_{g}$ est inversible et si au
voisinage de $P_{0}\in M$ la fonction de Green normalisée $G_{P_{0}}$ de
$L_{g}$ s’écrit
$G_{P_{0}}(Q)=r^{2-n}+A+O(r)$
avec $r=d(P_{0},Q)$, alors $A>0$ sauf si $(M,g)$ est conformément difféomorphe
à la sphère $(S_{n},g_{can})$, auquel cas $A=0$.
On utilisera les deux théorèmes 3.3, 3.4, sous réserve de leur validité.
### 3.8 Théorème d’existence de solutions sans présence de symétries
###### Théorème 3.5.
Soit $M$ une variété compacte $C^{\infty}$ de dimension $n\geq 3$, $g$ une
métrique riemannienne qui satisfait l’hypothèse $(H)$. Si $(M,g)$ n’est pas
conformément difféomorphe à la sphère $(S_{n},g_{can})$, alors
$\mu(g)<K^{-2}(n,2)$.
On montre ce théorème sous réserve de la validité du théorème 3.4.
Ce théorème affirme que la conjecture de T. Aubin 3.1 reste vraie pour des
métriques qui satisfont l’hypothèse $(H)$ (pas nécessairement $C^{\infty}$
partout).
Pour montrer ce théorème, on se base sur les travaux de T. Aubin et R. Schoen
dans le cas où $g$ est $C^{\infty}$. La stratégie est la suivante: on
construit des fonctions test pour la fonctionnelle $I_{g}$, à support dans des
petites boules géodésiques. Puisque le problème est local et que la métrique
$g$ est $C^{\infty}$ sur la boule $B_{P_{0}}(\delta)$, alors la preuve du
théorème ci-dessus est identique à celle dont la métrique $g$ est $C^{\infty}$
sur $M$ (c’est pour cette raison qu’on a supposé que la métrique est
$C^{\infty}$ dans la boule $B_{P_{0}}(\delta)$). On prendra donc les fonctions
test de T. Aubin et R. Schoen à support dans $B_{P_{0}}(\delta)$.
###### Preuve du théorème 3.5.
Si $\mu(g)\leq 0$ alors l’inégalité est triviale. À partir de maintenant
jusqu’à la fin de la preuve, on suppose que $\mu(g)>0$. Quitte à considérer
une métrique conforme, on peut supposer que $g$ est la métrique de Cao–Günther
donnée par le théorème 3.2. En effet, $\mu(g)$ est un invariant conforme
d’après la proposition 3.3.
Deux cas se présentent:
$(a)$ Soit $(M,g)$ n’est pas conformément plate en $P_{0}$ et $n\geq 6$. Dans
ce cas, on pose $\varphi_{\varepsilon}=\eta v_{\varepsilon}$, $\eta$ une
fonction cut-off de support dans $B_{P_{0}}(2\varepsilon)$, $\eta=1$ sur
$B_{P_{0}}(\varepsilon)$, $2\varepsilon<\delta$ et
$v_{\varepsilon}(Q)=\biggl{(}\frac{\varepsilon}{r^{2}+\varepsilon^{2}}\biggr{)}^{\frac{n-2}{2}}\quad
r=d(P_{0},Q)$
Comme $supp\varphi\subset B_{P_{0}}(\delta)$ et que la métrique $g$ est de
classe $C^{\infty}$ sur cette boule, on obtient le lemme suivant (cf. T. Aubin
[Aub]):
###### Lemme 3.1.
$\mu(g)\leq
I_{g}(\varphi_{\varepsilon})\leq\begin{cases}&K^{-2}(n,2)-c|W(P_{0})|^{2}\varepsilon^{4}+o(\varepsilon^{4})\text{
si }n>6\\\
&K^{-2}(n,2)-c|W(P_{0})|^{2}\varepsilon^{4}\log\frac{1}{\varepsilon}+O(\varepsilon^{4})\text{
si }n=6\end{cases}$
où $|W(P_{0})|$ est la norme du tenseur de Weyl au point $P_{0}$.
J.M. Lee et T. Parker ont donné une preuve simple de ce lemme, en utilisant
les coordonnées géodésiques conformes en $P_{0}$ (cf. [LP]). Par hypothèse la
métrique n’est pas conformément plate au voisinage de $P_{0}$ et $n\geq 6$
donc $|W(P_{0})|\neq 0$ d’où $\mu(g)<K^{-2}(n,2)$.
$(b)$ Soit $(M,g)$ est conformément plate en $P_{0}$ ou $n=3,\;4\text{ ou }5$:
Puisque $\mu(g)$ est un invariant conforme, quitte à considérer une métrique
conforme à $g$, on peut supposer que la métrique est celle de Cao–Günther et
que la fonction de Green normalisée $G_{P_{0}}$, construite dans la
proposition 3.7, s’écrit:
$G_{P_{0}}(Q)=r^{2-n}+A+O(r)$
au voisinage de $P_{0}$, avec $r=d(P_{0},Q)$ (cf. l’article de J.M. Lee et T.
Parker [LP] pour la preuve de ce développement limité).
Si la métrique $g$ satisfait l’ hypothèse $(H)$ et que $(M,g)$ n’est pas
conformément difféomorphe à la sphère $(S_{n},g_{can})$, par le théorème 3.4,
nous savons que $A>0$. Considérons alors $\varphi_{\varepsilon}$, la fonction
test introduite par R. Schoen [Schoen], définie pour tout $Q\in M$ par:
$\varphi_{\varepsilon}(Q)=\begin{cases}&v_{\varepsilon}(Q)\text{ si }Q\in
B_{P_{0}}(\rho_{0})\\\
&\varepsilon_{0}[G_{P_{0}}-\eta(G_{P_{0}}-r^{2-n}-A)](Q)\text{ si }Q\in
B_{P_{0}}(2\rho_{0})-B_{P_{0}}(\rho_{0})\\\ &\varepsilon_{0}G_{P_{0}}(Q)\text{
si }Q\in M-B_{P_{0}}(2\rho_{0})\end{cases}$
avec $2\rho_{0}<\delta$,
$(\frac{\varepsilon}{\rho_{0}^{2}+\varepsilon^{2}})^{(n-2)/2}=\varepsilon_{0}(\rho_{0}^{2-n}+A)$
et $\eta$ une fonction réelle positive $C^{\infty}$, décroissante sur
$\mathbb{R}_{+}$, à support dans $]-2\rho_{0},2\rho_{0}[$, identiquement égale
à $1$ sur $[0,\rho_{0}]$, dont le gradient vérifie
$|\nabla\eta(r)|\leq\rho_{0}^{-1}$. Puisque la métrique $g$ est $C^{\infty}$
sur $B_{P_{0}}(2\rho_{0})\subset B_{P_{0}}(\delta)$ et que $G_{P_{0}}\in
H^{p}_{2}(M-B_{P_{0}}(\rho_{0}))$ (voir le corollaire 3.7), alors on a
l’estimée suivante de $\mu(g)$, obtenue par R. Schoen [Schoen]:
###### Lemme 3.2.
$\mu(g)\leq I_{g}(\varphi_{\varepsilon})\leq
K^{-2}(n,2)+c\varepsilon_{0}^{2}(c\rho_{0}-A)$
Comme $A>0$ alors on peut choisir $\rho_{0}$ suffisamment petit
($c\rho_{0}<A$) pour que $\mu(g)<K^{-2}(n,2)$.
∎
On est maintenant en mesure d’énoncer le théorème qui résout le problème 3.2
pour les métriques qui satisfont l’hypothèse $(H)$.
###### Théorème 3.6.
Soit $M$ une variété compacte $C^{\infty}$ de dimension $n\geq 3$, $g$ une
métrique riemannienne qui satisfait l’hypothèse $(H)$, alors il existe une
métrique $\tilde{g}$ conforme à $g$ ayant une courbure scalaire
$R_{\tilde{g}}$ constante, solution du problème 3.2.
Ce théorème affirme qu’il existe toujours des solutions pour l’équation de
type Yamabe (2.1) (page 2.1) et que l’hypothèse du théorème 2.1 est toujours
satisfaite avec $h=\frac{n-2}{4(n-1)}R_{g}$.
##### Remarque
Dans l’énoncé du théorème 2.1, la métrique $g$ est supposée être de classe
$C^{\infty}$. Ce théorème reste vrai si l’on suppose que la métrique est dans
$H^{p}_{2}$ avec $p>n$. Pour le voir, il suffit de remarquer que si $g\in
H^{p}_{2}$, il existe une solution faible pour l’équation (2.1) (preuve
identique). La seule chose qui peut changer est la régularité de la solution
faible. Dans ce cas, on aura la même régularité car les coefficients de
$\Delta_{g}$ sont continus.
###### Preuve.
Si $(M,g)$ est conformément difféomorphe à la sphère $S_{n}$, munie de la
métrique canonique $g_{can}$, alors il n’y a rien à montrer car
$(S_{n},g_{can})$ est à courbure scalaire constante. Sinon $(M_{n},g)$ n’est
pas conformément difféomorphe à $(S_{n},g_{can})$. Au quel cas, on a
l’inégalité
$\mu(g)<K^{-2}(n,2)$
par le théorème 3.5. Le théorème 2.1 nous fournit une solution $\psi\in
H^{p}_{2}(M)$, strictement positive, de l’équation (2.1), où
$h=\frac{n-2}{4(n-1)}R_{g}$ et $\tilde{h}=\mu(g)$. D’après l’équation (3.1),
la métrique $\tilde{g}=\psi^{\frac{4}{n-2}}g$ est à courbure scalaire
constante $R_{\tilde{g}}=\frac{4(n-1)}{n-2}\mu(g)$. ∎
### 3.9 Unicité des solutions
Pour le problème de Yamabe classique (i.e. la métrique $g$ est $C^{\infty}$),
on sait qu’on a unicité des solutions à une constante multiplicative près dans
le cas où l’invariant conforme de Yamabe $\mu(g)$ est négatif ou nul. Le
théorème suivant, montre qu’on a toujours les mêmes résultats lorsque la
métrique est seulement de classe $H^{p}_{2}$, avec $p>n$.
###### Théorème 3.7.
Soit $g$ une métrique dans $H^{p}_{2}(M,T^{*}M\otimes T^{*}M)$, avec $p>n$. Si
$\mu(g)\leq 0$, alors les solutions de l’équation (3.1) sont uniques à une
constante multiplicative près.
###### Preuve.
Soit $\varphi_{1}$ et $\varphi_{2}$ deux solutions strictement positives de
l’équation (3.1). Les métriques $g_{i}=\varphi_{i}^{\frac{4}{n-2}}g$ sont à
courbures scalaires constantes $R_{i}$, où $i=1$ ou $2$. On pose
$\psi=\frac{\varphi_{1}}{\varphi_{2}}$, donc
$g_{1}=\psi^{\frac{4}{n-2}}g_{2}$. Ce qui entraîne que $\psi$ satisfait
$\Delta_{g_{2}}\psi+\frac{n-2}{4(n-1)}R_{2}\psi=\frac{n-2}{4(n-1)}R_{1}\psi^{\frac{n+2}{n-2}}$
(3.10)
Par le théorème de régularité 1.9, on en déduit que $\psi$ est de classe
$C^{2,\beta}$ car les coefficients du Laplacien sont $C^{0}$. En effet, dans
une carte locale:
$\Delta_{g}\psi=-\nabla_{i}\nabla^{i}\psi=-g^{ij}(\partial_{ij}\psi-\Gamma^{k}_{ij}\partial_{k}\psi)$
et les Christoffels sont donnés par
$\Gamma^{k}_{ij}=g^{kl}(\partial_{i}g_{lj}+\partial_{j}g_{il}-\partial_{l}g_{ij})$
Ils sont dans $H^{p}_{1}$, et continus si $p>n$. D’autre part, remarquons que
$R_{1}$ et $R_{2}$ sont forcément de même signe. Pour le voir, il suffit
d’intégrer l’équation (3.10) sur $M$, avec l’élément de volume de $g_{2}$, et
utiliser le fait que l’intégrale du Laplacien d’une fonction $C^{2}$ est
toujours nulle.
Si $\mu(g)<0$, alors $R_{i}<0$ pour $i=1$ et 2. Supposons que $\psi$ atteint
son maximum en $Q_{1}\in M$ et son minimum en $Q_{2}\in M$ alors
$\Delta_{g_{2}}\psi(Q_{1})\geq 0$ et $\Delta_{g_{2}}\psi(Q_{2})\leq 0$. Par
conséquent, si on évalue l’équation (3.10) au point $Q_{1}$ et $Q_{2}$, on
obtient les deux inégalités suivantes:
$\psi^{\frac{4}{n-2}}(Q_{1})\leq\frac{R_{2}}{R_{1}}\mbox{ et
}\psi^{\frac{4}{n-2}}(Q_{2})\geq\frac{R_{2}}{R_{1}}$
de là on tire que $\psi=\frac{R_{2}}{R_{1}}$ et que $\varphi_{1}$ et
$\varphi_{2}$ sont proportionnelles.
Si $\mu(g)=0$ alors $R_{1}=R_{2}=0$ et l’équation (3.10) est réduite à
$\Delta_{g_{2}}\psi=0$, d’où $\psi$ est constante. ∎
### 3.10 Application
Prenons le cas particulier d’une métrique
$g_{\alpha}=(1+\rho_{P_{0}}^{2-\alpha})^{m}g_{0}$
où $g_{0}$ est une métrique riemannienne $C^{\infty}$, $\alpha\in]0,1[$ et
$\rho_{P_{0}}$ la fonction distance donnée par la définition 1.7 (page 1.7).
Les dérivées secondes de $g_{\alpha}$ ont des singularités du type
$\rho^{-\alpha}$, ce qui entraîne qu’il existe une fonction continue $R_{0}$
telle que la courbure scalaire de $g$ soit de la forme
$R_{g_{\alpha}}=\frac{R_{0}}{\rho^{\alpha}}$. Cette fonction est dans
$L^{p}(M)$, si $p<\frac{n}{\alpha}$. Comme $\alpha\in]0,1[$ alors on peut
trouver un $p>n$ car $\frac{n}{\alpha}>n$. Pour cette valeur de $p$, on
conclut que la métrique $g_{\alpha}$ est dans $H^{p}_{2}(M,T^{*}M\otimes
T^{*}M)$ et elle satisfait l’hypothèse $(H)$ car la fonction $\rho_{P_{0}}$
est $C^{\infty}$ sur $B_{P_{0}}(\delta(M))-\\{P_{0}\\}$, avec $\delta(M)$ le
rayon d’injectivité de $M$.
Soit $\varphi$ une fonction strictement positive dans $H^{p}_{2}(M)$ et
$\tilde{g}=\varphi^{\frac{4}{n-2}}g_{\alpha}$ une métrique conforme à
$g_{\alpha}$. Si on veut que $\tilde{g}$ soit une métrique qui résout le
problème 3.2 (cf. page 3.2) alors il suffit que $\varphi$ soit solution de
l’équation de type Yamabe (2.10). D’après le théorème 3.6 et la proposition
2.2, une telle solution existe toujours.
### 3.11 Le problème de Yamabe équivariant
#### 3.11.1 Le problème de Hebey–Vaugon
Soit $(M,g)$ une variété riemannienne compacte $C^{\infty}$ de dimension $n$.
$G$ un sous groupe du groupe d’isométries $I(M,g)$. E. Hebey et M. Vaugon [HV]
ont considéré le problème suivant:
###### Problème 3.3.
Existe-t-il une métrique $g_{0}$, $G-$invariante qui minimise la fonctionnelle
$J(g^{\prime})=\frac{\int_{M}R_{g^{\prime}}\mathrm{d}v(g^{\prime})}{(\int_{M}\mathrm{d}v(g^{\prime}))^{\frac{n-2}{n}}}$
où $g^{\prime}$ appartient à la classe $G-$conforme de $g$:
$[g]^{G}:=\\{\tilde{g}=e^{f}g/f\in
C^{\infty}(M),\;\sigma^{*}\tilde{g}=\tilde{g}\quad\forall\sigma\in G\\}$
(Les définitions sont données dans la section 2.2).
Ils ont démontré que ce problème à toujours des solutions, sous réserve de la
validité du théorème de la masse positive 3.3. La résolution de ce problème a
deux conséquences. La première est l’existence d’une métrique $g_{0}$,
$I(M,g)-$invariante et conforme à $g$, telle que la courbure scalaire de
$g_{0}$ est constante. En effet, si $g_{0}=\varphi^{\frac{4}{n-2}}g$ est une
métrique qui minimise $J$, alors $\varphi$ est $I(M,g)-$invariante, solution
de l’équation d’Euler–Lagrange de $J$. Cette équation est bien celle de Yamabe
(3.1), avec $R_{\tilde{g}}=R_{g_{0}}$ une constante qui joue le rôle de la
courbure scalaire de $g_{0}$. La deuxième conséquence est que la conjecture de
A. Lichnerowicz [Lic] ci-dessous est vraie. Par les travaux de J. Lelong-
Ferrand [Lel] et M. Obata [Oba], on sait que si $(M,g)$ n’est pas conformément
difféomorphe à $(S_{n},g_{can})$ alors le groupe conforme $C(M,g)$ est compact
et il existe une métrique $g^{\prime}$ conforme à $g$ telle que
$I(M,g^{\prime})=C(M,g)$.
###### Conjecture 3.2 (A. Lichnerowicz [Lic]).
Pour tout variété riemannienne $(M,g)$, compacte $C^{\infty}$, de dimension
$n$ et qui n’est pas conformément difféomorphe à $(S_{n},g_{can})$, il existe
une métrique $\tilde{g}$ conforme à $g$ de courbure scalaire $R_{\tilde{g}}$
constante et pour laquelle $I(M,\tilde{g})=C(M,g)$.
On a déjà remarqué que les métriques qui résolvent le problème de Hebey–Vaugon
3.3 sont nécessairement solutions de l’équation de Yamabe (3.1). Par
conséquent, le problème de Yamabe classique, décrit à la section 3.1,
correspond au cas particulier $G=\\{\mathrm{id}\\}$ du problème 3.3.
Au début de ce chapitre, on a rappelé le problème de Yamabe, ensuite on a
montré que les équations de type Yamabe (2.1) admettent toujours des
solutions, si la fonction $h$ est proportionnelle à la courbure scalaire
$R_{g}$ (cf. théorème 3.6). On essaye de faire le même travail lorsque un sous
groupe $G$ du groupe d’isométries agit sur $M$. Les métriques ne seront pas
nécessairement $C^{\infty}$, mais elles vérifient l’hypothèse $(H)$ (cf.
section 3.2).
###### Problème 3.4.
Supposons que la métrique $g\in H^{p}_{2}(M,T^{*}M\otimes T^{*}M)$. Existe-t-
il une métrique $\tilde{g}$ dans la classe conforme $G-$invariante de $g$ qui
minimise la fonctionnelle $J$ et pour laquelle la courbure scalaire
$R_{\tilde{g}}$ est constante partout?
Si la métrique $g$ est $C^{\infty}$ alors ce problème est exactement le
problème de Hebey–Vaugon 3.3. Si la métrique $\tilde{g}$ minimise la
fonctionnelle $J$ définie au début de la section 3.11, alors la courbure
scalaire de $\tilde{g}$ est automatiquement constante. Plus précisément, si
$\tilde{g}=\psi^{4/(n-2)}g$, avec $\psi\in H_{2}^{p}(M)$, strictement positive
et $G-$invariante, alors $\psi$ est solution de l’équation de Yamabe (3.1).
#### 3.11.2 L’invariant de Yamabe $\boldsymbol{G-}$conforme
Soit $I_{g}$ la fonctionnelle de Yamabe définie par (3.2) (page 3.2). Pour ce
problème, on considère seulement des fonctions test dans $H_{1,G}(M)$,
l’espace des fonctions dans $H_{1}(M)$, $G-$invariante.
###### Définition 3.6.
L’invariant $G-$conforme de Yamabe $\mu_{G}(g)$ est défini par:
$\mu_{G}(g)=\inf_{\psi\in H_{1,G}(M)-\\{0\\}}I_{g}(\psi)$
La proposition suivante justifie la terminologie employée.
###### Proposition 3.8.
Soit $M$ une variété compacte $C^{\infty}$. $g\in H^{p}_{2}(M,T^{*}M\otimes
T^{*}M)$ une métrique riemannienne, avec $p>n/2$. Alors
1. 1.
$\mu_{G}(g)=\frac{n-2}{4(n-1)}\inf_{g^{\prime}\in[g]^{G}}J(g^{\prime})$
2. 2.
Si $\tilde{g}\in[g]^{G}$ alors $\mu_{G}(\tilde{g})=\mu_{G}(g)$.
###### Preuve.
Pour tout $g^{\prime}\in[g]^{G}$, il existe $\psi\in H^{p}_{2,G}(M)$,
strictement positive telle que $g^{\prime}=\psi^{\frac{4}{n-2}}g$. Par
l’équation de Yamabe (3.1):
$R_{g^{\prime}}=\psi^{-\frac{n+2}{n-2}}(4\frac{n-1}{n-2}\Delta_{g}\psi+R_{g}\psi)$
En intégrant cette équation sur $M$ par rapport à l’élément de volume
$\mathrm{d}v_{g^{\prime}}$, on obtient
$\int_{M}R_{g^{\prime}}\mathrm{d}v_{g^{\prime}}=\int_{M}\psi(4\frac{n-1}{n-2}\Delta_{g}\psi+R_{g}\psi)\mathrm{d}v_{g}=4\frac{n-1}{n-2}E(\psi)$
D’autre part
$\int_{M}\mathrm{d}v_{g^{\prime}}=\|\psi\|_{N}^{N}$
On en déduit que
$J(g^{\prime})=4\frac{n-1}{n-2}I_{g}(\psi)$ (3.11)
En prenant la borne inférieure, on obtient la première propriété. Pour la
seconde propriété, il suffit de reprendre la preuve de la proposition 3.3. En
effet, si $g^{\prime}$ est la métrique considérée ci-dessus alors, d’après
l’équation (3.6)
$\forall\varphi\in H_{1,G}(M)\qquad
I_{g^{\prime}}(\varphi)=I_{g}(\psi\varphi)$
∎
### 3.12 Théorème d’existence de solutions en présence de symétries
###### Théorème 3.8.
Soit $M$ une variété compacte $C^{\infty}$ de dimension $n\geq 3$. $g$ une
métrique riemannienne qui appartient à $H_{2}^{p}(M,T^{*}M\otimes T^{*}M)$,
avec $p>n$. Si
$\mu_{G}(g)<\frac{1}{4}n(n-2)\omega_{n}^{2/n}(\mathrm{card}O_{G}(P))^{2/n}$
alors l’équation (3.1) admet une solution strictement positive $\varphi\in
H_{2,G}^{p}(M)\subset C^{1-[n/p],\beta}(M)$, $G-$invariante. De plus la
métrique $\tilde{g}=\varphi^{\frac{4}{n-2}}g$ est solution du problème 3.4 et
de courbure scalaire constante $R_{\tilde{g}}=\frac{4(n-1)}{n-2}\mu_{G}(g)$.
###### Preuve.
Si $\mu_{G}(g)\leq 0$, d’après le théorème 3.7, les solutions de l’équation de
Yamabe sont proportionnelles. Si $\varphi$ est une solution de (3.1) alors
pour tout $\sigma\in G$, $\sigma^{*}\varphi$ est également une solution. Il
existe donc une constante $c>0$ telle que $\sigma^{*}\varphi=c\varphi$.
D’autre part, $\|\sigma^{*}\varphi\|_{2}=\|\varphi\|_{2}$. On en déduit que
$c=1$ et que $\varphi$ est $G-$invariante.
Supposons que $\mu_{G}(g)>0$. Notons que
$K^{-2}(n,2)=\frac{1}{4}n(n-2)\omega_{n}^{2/n}$
l’expression de $K(n,q)$ est donnée dans le théorème 1.5 (page (1.5)). Il
suffit d’appliquer le théorème 2.5 pour $h=\frac{n-2}{4(n-1)}R_{g}$, qui
entraîne que l’équation (3.1) admet une solution $\varphi\in H^{p}_{2,G}(M)$,
strictement positive et minimisante pour la fonctionnelle $I_{g}$. D’après la
relation (3.11), la métrique $\varphi^{\frac{4}{n-2}}g$ minimise la
fonctionnelle $J^{\prime}$. ∎
##### Remarque
D’après le théorème 3.8, la condition suffisante, pour trouver une solution
$G-$invariante de l’équation de Yamabe (3.1) est que l’inégalité
$\mu_{G}(g)<n(n-1)\omega_{n-1}^{2/n}(\mathrm{card}O_{G}(P))^{2/n}$
soit toujours vraie.
On a vu que dans le cas particulier où $G=\\{\mathrm{id}\\}$, cette inégalité
est vraie pour toute variété compacte $(M,g)$, non conformément difféomorphe à
$(S_{n},g_{can})$, munie d’une métrique $g$ qui satisfait l’hypothèse $(H)$
(cf. théorème 3.5).
Dans le cas où $G$ est un sous groupe quelconque de $I(M,g)$ lorsque $(M,g)$
est une variété riemannienne compacte $C^{\infty}$, E. Hebey et M. Vaugon [HV]
ont annoncé cette inégalité sous forme de conjecture (cf. conjecture 5.1). Ils
l’ont démontrée dans certains cas (cf. théorème 5.2). Dans le chapitre 5, on
démontre qu’elle est vraie dans de nouveaux cas (par contre, vu la complexité
de la preuve et des arguments utilisés, on n’est pas encore en mesure
d’adapter la preuve, dans le cas où la métrique est seulement dans
$H^{p}_{2}$).
## Chapter 4 Calculs techniques sur la courbure scalaire
Dans tout ce chapitre, on suppose que $M$ est une variété compacte
$C^{\infty}$, de dimension $n\geq 3$, $g$ est une métrique riemannienne
$C^{\infty}$, munie de sa connexion riemannienne, notée $\nabla_{g}$. On note
par $\nabla^{\beta}$ la dérivée covariante
$\nabla^{\beta_{1}}\cdots\nabla^{\beta_{i}}$, où $\beta\in[[1,n]]^{i}$ sont
des multi-indices, et $|\beta|=i$. On note par $[[1,n]]$ l’ensemble des
entiers naturels entre $1$ et $n$.
###### Définition 4.1.
Soit $(M,g)$ une variété riemannienne et $W_{g}$ le tenseur de Weyl associé à
$g$. On définit l’entier $\omega$ au point $P$ par
$\omega(P)=\inf\\{|\beta|\in\mathbb{N}/\|\nabla^{\beta}W_{g}(P)\|\neq 0\\}$
(et si $\|\nabla^{\beta}W_{g}(P)\|=0$ pour tout multi-indices $\beta$, alors
$\omega(P)=+\infty$).
Pour des raisons de simplicité, on omet $P$ dans $\omega(P)$. On a les
propriétés suivantes:
###### Propriétés 4.1.
Soit $\tilde{g}$ une métrique conforme à $g$. On note $\tilde{\omega}$
l’entier défini ci-dessus associé à la métrique $\tilde{g}$. Alors
$\omega=\tilde{\omega}$
$\omega$ est conformément invariant.
###### Preuve.
Si $\tilde{g}=\varphi^{\frac{4}{n-2}}g$, alors
$W_{\tilde{g}}=\varphi^{\frac{4}{n-2}}W_{g}$ (cf. remarque après la définition
1.2), avec $\varphi$ une fonction $C^{\infty}$, strictement positive. Par
conséquent
$\forall i<\omega\quad\nabla^{i}W_{g}(P)=0\Longleftrightarrow\forall
i<\tilde{\omega}\quad\nabla^{i}W_{\tilde{g}}(P)=0$
∎
### 4.1 Calculs sur l’intégrale de la courbure scalaire
Cette section est consacrée au calcul de l’intégrale de la courbure scalaire
sur une sphère de rayon $r$ assez petit. Ces calculs ont été effectués par T.
Aubin [Aub5, Aub4], que l’on reprendra, avec des preuves détaillées. Notons
par $S(r)$ la sphère de dimension $n-1$ et de rayon $r$, et par
$\mathrm{d}\sigma_{r}$ l’élément de volume sur $S(r)$. On note par
$\bar{\int}$ la valeur moyenne
$\bar{\int}_{M}\varphi\mathrm{d}v=\frac{1}{vol(M)}\int_{M}\varphi\mathrm{d}v$
L’ intégrale de la courbure scalaire que l’on calculera joue un rôle important
dans la fonctionnelle de Yamabe (4.10). On verra que si elle est négative
alors la conjecture de Hebey–Vaugon 5.1 est démontrée. Mais dans certains cas,
elle est positive, ce qui complique la preuve de la conjecture 5.1. Notons
qu’on a déjà démontré que l’inégalité large suivante est toujours vraie
$\mu_{G}(g)\leq\frac{n(n-2)}{4}\omega_{n}^{2/n}(\mathrm{card}O_{G}(P))^{2/n}$
pour tout variété compacte $(M,g)$, de dimension $n\geq 3$ (cf. proposition
2.5, page 2.5 ), même dans le cas où on met $h$ une fonction quelconque à la
place de $R_{g}$. On constate qu’il y a certaines informations contenues dans
$R_{g}$ qu’il faut absolument utiliser pour démontrer la conjecture.
###### Définition 4.2.
Soit $P$ un point fixé de $M$. On note $\mu(P)$ l’entier naturel, défini comme
suit: $|\nabla_{\beta}R_{g}(P)|=0$ pour tout $|\beta|<\mu(P)$ et il existe
$\beta\in[[1,n]]^{\mu(P)}$ tel que $|\nabla_{\beta}R_{g}(P)|\neq 0$. Dans un
système de coordonnées normales $\\{x^{i}\\}$ d’origine $P$
$R_{g}(Q)=\bar{R}+O(r^{\mu(P)+1})$
où $\bar{R}=r^{\mu(P)}\sum_{|\beta|=\mu(P)}\nabla_{\beta}R_{g}(P)\xi^{\beta}$
est un polynôme homogène de degré $\mu(P)$, qui représente la partie
principale de $R_{g}$, $r=d(P,Q)=|x|$ et $\xi^{i}=\frac{x^{i}}{r}$.
Pour des raisons de simplicité, on omet $P$ dans $\mu(P)$.
Le lemme 5.2, énoncé dans le chapitre suivant, et le développement limité de
la métrique donnent:
###### Lemme 4.1.
On a toujours $\mu\geq\omega$, $g_{ij}=\delta_{ij}+O(r^{\omega+2})$ et
$\bar{\int}_{S(r)}R_{g}\mathrm{d}\sigma_{r}=O(r^{2\omega+2})$ ce qui entraîne
que $\int_{S(r)}\bar{R}\mathrm{d}\sigma_{r}=0$ lorsque $\mu<2\omega+2$.
###### Preuve.
Par le développement limité (5.3) (voir le chapitre suivant),
$g_{ij}=\delta_{ij}+O(r^{\omega+2})$. Puisque la courbure scalaire $R_{g}$ est
obtenue, en dérivant deux fois les composantes de la métrique, alors
$R_{g}=O(r^{\omega})$. Ce qui veut dire que la partie principale $\bar{R}$ est
d’ordre $\mu\geq\omega$.
Intéressons nous à $\bar{\int}_{S(r)}R_{g}\mathrm{d}\sigma_{r}$, elle est
d’ordre $2\omega+2$. En effet, on a le développement suivant
$R_{g}(Q)=\sum_{m=\mu}^{2\omega+1}(\sum_{|\beta|=m}\nabla_{\beta}R_{g}(P)\xi^{\beta})r^{m}+O(r^{2\omega+2})$
avec $r=d(P,Q)$ et $(r,\xi^{j})$ un système de coordonnées géodésiques. En
intégrant cette égalité sur la sphère $S(r)$, sachant que l’intégrale d’un
polynôme homogène de degré impair sur la sphère est nulle, on obtient
$\bar{\int}_{S(r)}R_{g}\mathrm{d}\sigma_{r}=\sum_{m=\mu}^{\omega}C(m,n)\Delta_{g}^{m}R_{g}(P)r^{m}+O(r^{2\omega+2})$
pour une certaine constante $C(m,n)$, qui dépend seulement de $n$ et $m$.
Comme les courbures de la métrique $g$ satisfont le lemme 5.2, pour tout
$m\leq\omega$, $\Delta_{g}^{m}R_{g}(P)=0$. Donc
$\bar{\int}_{S(r)}R_{g}\mathrm{d}\sigma_{r}=O(r^{2\omega+2})$
∎
Soit $\\{x^{\alpha}\\}$ un système de coordonnées normal en $P$ et
$\\{r,\xi^{i}\\}$ un système de coordonnées géodésiques. Le lemme 4.1 entraîne
qu’il existe un tenseur symétrique $h$ tel que
$g=\mathcal{E}+h$
avec $h=O(r^{\omega+2})$, alors
$g=\mathcal{E}+h=(\delta_{\alpha\beta}+h_{\alpha\beta})dx^{\alpha}\otimes
dx^{\beta}=dr^{2}+(s_{ij}+h_{ij})(rd\xi^{i})\otimes(rd\xi^{j})$
où $(s_{ij})$ sont les composantes de la métrique standard sur la sphère
$S_{n-1}$ et
$h_{ij}=\frac{\partial x^{\alpha}}{r\partial\xi^{i}}\frac{\partial
x^{\beta}}{r\partial\xi^{j}}h_{\alpha\beta},\quad h_{ir}=h_{rr}=0$
Remarquons que $h_{ij}=O(r^{\omega+2})$. On peut donc décomposer $(h_{ij})$ de
la façon suivante:
$h_{ij}=r^{\omega+2}\bar{g}_{ij}+r^{2(\omega+2)}\hat{g}_{ij}+\tilde{h}_{ij}$
(4.1)
où $\bar{g}$, $\hat{g}$ et $\tilde{h}$ sont des 2-tenseurs symétriques définis
sur la sphère $S_{n-1}$. On choisit $\\{\frac{\partial}{\partial
r},\frac{\partial}{r\partial\xi^{i}}\\}_{1\leq i\leq n-1}$ et
$\\{dr,rd\xi^{i}\\}_{1\leq i\leq n-1}$ comme bases locales de l’espace tangent
$TM$ et cotangent $T^{*}M$ respectivement. Notre but dans le choix de ces
bases est d’avoir
$g_{ij}=s_{ij}+h_{ij},\;g_{rr}=1\text{ et }g_{ir}=0$
et d’éliminer une fois pour toutes le $r^{2}$ qui apparaît, en passant aux
coordonnées géodésiques. Les composantes $g^{ij}$ de l’inverse de la métrique
sont
$g^{ij}=s^{ij}-h^{ij}+O(r^{2\omega+4})$
où $h^{ij}=s^{ik}s^{jl}h_{lk}$. On fait monter et baisser les indices, en
utilisant la métrique $(s_{ij})$, sauf pour la métrique $g$. On note par
$\nabla$ la connexion riemannienne sur la sphère, associée à $s$. Par des
calculs directs, T. Aubin [Aub5] a montré que:
###### Théorème 4.1.
$\bar{R}=\nabla^{ij}\bar{g}_{ij}r^{\omega}\quad\text{et}$
$\bar{\int}_{S(r)}R\mathrm{d}\sigma_{r}=[B/2-C/4-(1+\omega/2)^{2}Q]r^{2(\omega+1)}+o(r^{2(\omega+1)})$
où
$B=\bar{\int}_{S_{n-1}}\nabla^{i}\bar{g}^{jk}\nabla_{j}\bar{g}_{ik}\mathrm{d}\sigma$,
$C=\bar{\int}_{S_{n-1}}\nabla^{i}\bar{g}^{jk}\nabla_{i}\bar{g}_{jk}\mathrm{d}\sigma$
et $Q=\bar{\int}_{S_{n-1}}\bar{g}_{ij}\bar{g}^{ij}\mathrm{d}\sigma$
Une preuve détaillée de ce lemme est donnée dans l’appendice A.
De plus T. Aubin [Aub3] a montré que
###### Théorème 4.2.
Si $\mu\geq\omega+1$ alors il existe une constante $C(n,\omega)>0$ telle que
$\bar{\int}_{S(r)}R\mathrm{d}\sigma_{r}=C(n,\omega)(-\Delta_{g})^{\omega+1}R(P)r^{2\omega+2}+o(r^{2\omega+2})$
$(-\Delta_{g})^{\omega+1}R(P)$ est strictement négative et
$I_{g}(u_{\varepsilon})<\frac{n(n-2)}{4}\omega_{n-1}^{2/n}$.
La fonction $u_{\varepsilon}$ est définie plus bas (voir équation (4.8)).
On rappellera le schéma de la preuve et on donnera des détails sur ce théorème
dans l’appendice A.
On considèrera à partir de maintenant et jusqu’à la fin de cette section que
$\boldsymbol{\mu=\omega}$.
On sait que $\bar{R}$ est un polynôme homogène de degré $\omega$,
$\Delta_{\mathcal{E}}\bar{R}$ est donc homogène de degré $\omega-2$ et
$\Delta_{\mathcal{E}}\bar{R}=r^{-2}(\Delta_{s}\bar{R}-\omega(n+\omega-2)\bar{R})$
où $\Delta_{\mathcal{E}}$ est le Laplacien euclidien et $\Delta_{s}$ est le
Laplacien de la sphère $S_{n-1}$, muni de la métrique $s$.
$\Delta_{\mathcal{E}}^{k-1}\bar{R}$ est homogène de degré $\omega-2k+2$ et
$\Delta^{k}_{\mathcal{E}}\bar{R}=r^{-2}(\Delta_{s}-\nu_{k}\mathrm{id})\Delta^{k-1}_{\mathcal{E}}\bar{R}=r^{-2k}\prod_{p=1}^{k}(\Delta_{S}-\nu_{p}\mathrm{id})\bar{R}$
avec
$\nu_{k}=(\omega-2k+2)(n+\omega-2k)$ (4.2)
Cette suite d’entiers naturels $(\nu_{k})_{\\{1\leq k\leq[\omega/2]\\}}$ est
décroissante. Elle est formée de valeurs propres du Laplacien sur la sphère
$S_{n-1}$ (il est bien connu que les valeurs propres du Laplacien géométrique
sont positives et qu’elles forment une suite croissante. Nos valeurs $\nu_{k}$
sont prises dans l’ordre opposé).
Puisque $\bar{R}$ est homogène de degré $\omega$, deux cas se présentent. Soit
$\omega$ est pair, alors $\Delta^{[\omega/2]}_{\mathcal{E}}\bar{R}$ est une
constante, mais d’après le 4ème point du lemme 5.2,
$\Delta^{[\omega/2]}_{\mathcal{E}}\bar{R}(P)=0$, d’où
$\Delta^{[\omega/2]}_{\mathcal{E}}\bar{R}=0$
Soit $\omega$ est impair, alors $\Delta^{[\omega/2]}_{\mathcal{E}}\bar{R}$ est
une forme linéaire. D’après le 4ème point du lemme 5.2
$\Delta^{[\omega/2]}_{\mathcal{E}}\bar{R}(P)=0\text{ et
}\nabla\Delta^{[\omega/2]}_{\mathcal{E}}\bar{R}(P)=0$
Finalement $\Delta^{[\omega/2]}_{\mathcal{E}}\bar{R}=0$ dans tous les cas.
On a $r^{-\omega}\bar{R}\in\bigoplus_{k=1}^{q}E_{k}$, où $E_{k}$ l’espace
propre associé à la valeur propre $\nu_{k}$, du Laplacien $\Delta_{s}$, sur la
sphère $S_{n-1}$, et où on a noté
$q=\min\\{k\in\mathbb{N}/\Delta_{\mathcal{E}}^{k}\bar{R}=0\\}$
Si $j\neq k$, $E_{k}$ est bien orthogonal à $E_{j}$, pour le produit scalaire
dans $L^{2}(S_{n-1})$ et le produit scalaire sur $H_{1}(S_{n-1})$ définis ci-
dessous, puisque si $j\neq k$ et $\varphi_{k}\in E_{k}$
$\nu_{k}(\varphi_{k},\varphi_{j})_{L^{2}}=(\Delta_{s}\varphi_{k},\varphi_{j})_{L^{2}}=(\varphi_{k},\Delta_{s}\varphi_{j})_{L^{2}}=\nu_{j}(\varphi_{k},\varphi_{j})_{L^{2}}$
(4.3)
Le produit suivant est bien un produit scalaire sur l’ensemble des fonctions
dans $H_{1}(S_{n-1})$, d’intégrales nulles
$(\varphi_{k},\varphi_{j})_{H_{1}}=(\nabla\varphi_{k},\nabla\varphi_{j})_{L^{2}}=\nu_{k}(\varphi_{k},\varphi_{j})_{L^{2}}=\nu_{j}(\varphi_{k},\varphi_{j})_{L^{2}}$
(4.4)
De plus, puisque $\int_{S(r)}\bar{R}d\sigma_{r}=0$ (d’après le lemme 4.1), il
existe des $\varphi_{k}\in E_{k}$ (fonctions propres de $\Delta_{s}$) telles
que
$\bar{R}=r^{\omega}\Delta_{s}\sum_{k=1}^{q}\varphi_{k}=r^{\omega}\sum_{k=1}^{q}\nu_{k}\varphi_{k}$
(4.5)
On pose
$b_{ij}=\sum_{k=1}^{q}\frac{1}{(n-2)(\nu_{k}+1-n)}[(n-1)\nabla_{ij}\varphi_{k}+\nu_{k}\varphi_{k}s_{ij}]$
et $a_{ij}=\bar{g}_{ij}-b_{ij}$.
On note $\bar{R}_{a}=\bar{R}$ lorsque $\bar{g}_{ij}=a_{ij}$ et
$\bar{R}_{b}=\bar{R}$ lorsque $\bar{g}_{ij}=b_{ij}$. En tenant compte de
l’expression (4.5) de $\bar{R}$, on établit les relations suivantes:
###### Lemme 4.2.
$\nabla^{i}b_{ij}=-\sum_{k=1}^{q}\nabla_{j}\varphi_{k},\;\bar{R}=\bar{R}_{b}=\nabla^{ij}b_{ij}r^{\omega},\;\bar{R}_{a}=\nabla^{ij}a_{ij}r^{\omega}=0\text{
et }s^{ij}b_{ij}=s^{ij}a_{ij}=0$
La preuve détaillée est donnée dans l’appendice A.
Regardons les deux cas particuliers suivants:
Si $\bar{g}_{ij}=a_{ij}$. Alors $\bar{R}=\bar{R}_{a}=0$, ce qui entraîne que
la partie principale de $R_{g}$, est de degré $\mu\geq\omega+1$. Par le
théorème 4.2
$\bar{\int}_{S(r)}R\mathrm{d}\sigma_{r}=\bar{\int}_{S(r)}R_{a}\mathrm{d}\sigma_{r}<0$
Si $\bar{g}_{ij}=b_{ij}$. D’après le théorème 4.1, on a
$\bar{\int}_{S(r)}R\mathrm{d}\sigma_{r}=\bar{\int}_{S(r)}R_{b}\mathrm{d}\sigma_{r}=[B_{b}/2-C_{b}/4-(1+\omega/2)^{2}Q_{b}]r^{2(\omega+1)}+o(r^{2(\omega+1)})$
où l’on note par $B_{b}$, $C_{b}$ et $Q_{b}$ les intégrales $B$, $C$ et $Q$
respectivement, définies dans le théorème 4.1 lorsque $\bar{g}_{ij}=b_{ij}$.
On peut les calculer en fonction des fonctions propres $\varphi_{k}$, on
trouve:
$\displaystyle
Q_{b}=\bar{\int}_{S_{n-1}}b_{ij}b^{ij}\mathrm{d}\sigma=\frac{n-1}{n-2}\sum_{k=1}^{q}\frac{\nu_{k}}{\nu_{k}-n+1}\bar{\int}_{S_{n-1}}\varphi_{k}^{2}\mathrm{d}\sigma$
$\displaystyle
B_{b}=-(n-1)Q_{b}+\sum_{k=1}^{q}\nu_{k}\bar{\int}_{S_{n-1}}\varphi_{k}^{2}\mathrm{d}\sigma$
$\displaystyle
C_{b}=-(n-1)Q_{b}+\frac{n-1}{n-2}\sum_{k=1}^{q}\nu_{k}\bar{\int}_{S_{n-1}}\varphi_{k}^{2}\mathrm{d}\sigma$
Dans le calcul de ces expressions on a utilisé plusieurs fois l’identité
$\nabla^{i}b_{ij}=-\sum_{k=1}^{q}\nabla_{j}\varphi_{k}$ et la formule de
Stokes (les calculs détaillés sont donnés dans l’appendice A).
Dans le cas général (i.e. $\bar{g}_{ij}=a_{ij}+b_{ij}$), on obtient le lemme
suivant:
###### Lemme 4.3.
Si $\mu=\omega$ et $\bar{g}_{ij}=a_{ij}+b_{ij}$, alors
$\bar{\int}_{S(r)}R\mathrm{d}\sigma_{r}=\bar{\int}_{S(r)}R_{a}+R_{b}\mathrm{d}\sigma_{r}+o(r^{2(\omega+1)})\leq[B_{b}/2-C_{b}/4-(1+\omega/2)^{2}Q_{b}]r^{2(\omega+1)}+o(r^{2(\omega+1)})$
$B_{b}/2-C_{b}/4-(1+\omega/2)^{2}Q_{b}=\sum_{k=1}^{q}u_{k}\bar{\int}_{S_{n-1}}\varphi_{k}^{2}\mathrm{d}\sigma$
avec
$u_{k}=\biggl{(}\frac{n-3}{4(n-2)}-\frac{(n-1)^{2}+(n-1)(\omega+2)^{2}}{4(n-2)(\nu_{k}-n+1)}\biggr{)}\nu_{k}$
(4.6)
les nombres réels $u_{k}$ sont obtenus à partir des expressions $Q_{b}$,
$B_{b}$ et $C_{b}$ ci-dessus (voir l’appendice A pour une preuve détaillée de
ce lemme).
### 4.2 Généralisation d’un théorème de T. Aubin
Dans son article sur le problème de Yamabe, T. Aubin [Aub] a démontré que s’il
existe un point $P_{0}\in M$ tel que $\omega(P_{0})=0$ (voir la définition 4.1
), alors il existe une fonction $\varphi_{\varepsilon}$ telle que
$I_{g}(\varphi_{\varepsilon})<\frac{n(n-2)}{4}\omega_{n-1}^{2/n}$
(cf lemme 3.1).
Le but de cette section est de généraliser ce résultat pour tout
$\omega\leq(n-6)/2$.
Soit $u_{\varepsilon}$ et $\varphi_{\varepsilon}$ deux fonctions définies par:
$\displaystyle\varphi_{\varepsilon}(Q)=(1-r^{\omega+2}f(\xi))u_{\varepsilon}(Q)$
(4.7) $\displaystyle
u_{\varepsilon}(Q)=\begin{cases}\biggl{(}\displaystyle\frac{\varepsilon}{r^{2}+\varepsilon^{2}}\biggr{)}^{\frac{n-2}{2}}-\biggl{(}\frac{\varepsilon}{\delta^{2}+\varepsilon^{2}}\biggr{)}^{\frac{n-2}{2}}&\mbox{
si }Q\in B_{P}(\delta)\\\ \hskip 56.9055pt0&\mbox{ si }Q\in
M-B_{P}(\delta)\end{cases}$ (4.8)
pour tout $Q\in M$, où $r=d(Q,P)$ est la distance entre $P$ et $Q$.
$(r,\xi^{j})$ sont les coordonnées géodésiques de $Q$ au voisinage de $P$ et
$B_{P}(\delta)$ est une boule géodésique de centre $P$, de rayon $\delta$,
fixé suffisamment petit. $f$ est une fonction qui dépend seulement de $\xi$
telle que $\int_{S_{n-1}}fd\sigma=0$ et le choix précis sera décidé plus tard.
Soit
$I_{a}^{b}(\varepsilon)=\int_{0}^{\delta/\varepsilon}\frac{t^{b}}{(1+t^{2})^{a}}dt\text{
et }I_{a}^{b}=\lim_{\varepsilon\to 0}I_{a}^{b}(\varepsilon)$
alors $I_{a}^{2a-1}(\varepsilon)=\log\varepsilon^{-1}+O(1)$. Si $2a-b>1$ alors
$I_{a}^{b}(\varepsilon)=I_{a}^{b}+O(\varepsilon^{2a-b-1})$ et par intégration
par parties, on établit les relations suivantes :
$I_{a}^{b}=\frac{b-1}{2a-b-1}I_{a}^{b-2}=\frac{b-1}{2a-2}I_{a-1}^{b-2}=\frac{2a-b-3}{2a-2}I_{a-1}^{b},\quad\frac{4(n-2)I_{n}^{n+1}}{(I^{n-2}_{n})^{(n-2)/n}}=n$
(4.9)
Rappelons que la fonctionnelle de Yamabe $I_{g}$ (cf. (3.2) page 3.2) est
définie, pour tout $\psi\in H_{1}(M)$, par
$I_{g}(\psi)=\biggr{(}\int_{M}|\nabla_{g}\psi|^{2}\mathrm{d}v+\frac{(n-2)}{4(n-1)}\int_{M}R_{g}\psi^{2}\mathrm{d}v\biggl{)}\|\psi\|_{N}^{-2}$
(4.10)
où $N=2n/(n-2)$ et $\nabla_{g}$ est le gradient de la métrique $g$.
Voici donc le résultat principal de ce chapitre:
###### Théorème 4.3.
Soit $(M,g)$ une variété riemannienne compacte de dimension $n$. Pour tout
$P\in M$, si $\omega(P)\leq(n-6)/2$, alors il existe $f\in
C^{\infty}(S_{n-1})$, d’intégrale moyenne nulle et $\varepsilon>0$ telles que
$\mu(g)\leq I_{g}(\varphi_{\varepsilon})<\frac{n(n-2)}{4}\omega_{n-1}^{2/n}$
où $\varphi_{\varepsilon}$ est définie par (4.7).
##### Remarque
L’hypothèse "$\omega$ est fini" affirme que la variété $(M,g)$ n’est pas
conformément difféomorphe à $(S_{n},g_{can})$.
###### Preuve.
Soit $P\in M$. On écrit $\omega$ au lieu de $\omega(P)$. Si $\mu\geq\omega+1$
alors l’inégalité est vraie par le théorème 4.2. On peut donc supposer que
$\mu=\omega$ jusqu’à la fin de la preuve. On commence par calculer la première
intégrale dans la fonctionnelle (4.10), avec $\psi=\varphi_{\varepsilon}$ et
$f$ inconnue pour l’instant, en utilisant la formule
$|\nabla_{g}\varphi_{\varepsilon}|^{2}=(\partial_{r}\varphi_{\varepsilon})^{2}+r^{-2}|\nabla_{s}\varphi_{\varepsilon}|^{2}$
On trouve :
$\int_{M}|\nabla_{g}\varphi_{\varepsilon}|^{2}\mathrm{d}v=\int_{M}|\nabla_{g}u_{\varepsilon}|^{2}\mathrm{d}v+\int_{0}^{\delta}[\partial_{r}(r^{(\omega+2)}u_{\varepsilon})]^{2}r^{n-1}\mathrm{d}r\int_{S_{n-1}}f^{2}\mathrm{d}\sigma+\\\
\int_{0}^{\delta}u^{2}_{\varepsilon}r^{n+2\omega+1}\mathrm{d}r\int_{S_{n-1}}|\nabla
f|^{2}\mathrm{d}\sigma$ (4.11)
Le changement de variable $t=r/\varepsilon$ donne
$\int_{M}|\nabla_{g}\varphi_{\varepsilon}|^{2}\mathrm{d}v=(n-2)^{2}\omega_{n-1}I_{n}^{n+1}(\varepsilon)+\varepsilon^{2\omega+4}\biggl{\\{}\int_{S_{n-1}}|\nabla
f|^{2}\mathrm{d}\sigma I_{n-2}^{2\omega+n+1}(\varepsilon)+\\\
\int_{S_{n-1}}f^{2}\mathrm{d}\sigma[(\omega-n+4)^{2}I_{n}^{2\omega+n+5}(\varepsilon)+2(\omega+2)(\omega-n+4)I_{n}^{2\omega+n+3}(\varepsilon)+(\omega+2)^{2}I_{n}^{2\omega+n+1}(\varepsilon)]\biggr{\\}}$
(4.12)
Pour la seconde intégrale qui contient la courbure scalaire $R_{g}$, on a
$\begin{split}\int_{M}R_{g}\varphi_{\varepsilon}^{2}\mathrm{d}v&=\int_{M}R_{g}u_{\varepsilon}^{2}\mathrm{d}v-2\int_{M}fu_{\varepsilon}^{2}R_{g}r^{\omega+2}\mathrm{d}v+\int_{M}f^{2}u_{\varepsilon}^{2}R_{g}r^{2\omega+4}\mathrm{d}v\\\
&=\varepsilon^{2\omega+4}\omega_{n-1}\bar{\int}_{S(r)}r^{-2\omega-2}R_{g}\mathrm{d}\sigma_{r}I_{n-2}^{n+2\omega+1}(\varepsilon)-\\\
&2\varepsilon^{2\omega+4}I_{n-2}^{2\omega+n+1}(\varepsilon)\omega_{n-1}\bar{\int}_{S(r)}r^{-\omega}f\bar{R}\mathrm{d}\sigma_{r}+O(\varepsilon^{n-2})\\\
\end{split}$ (4.13)
où $\omega$ est l’ordre de la partie principale $\bar{R}$ (voir définition
4.2). La fonction $f$ est définie sur $S_{n-1}$. Sans aucune difficulté, on
peut la redéfinir sur $S(r)$, pour tout $r>0$, en posant $f(\xi/r)$, où
$\xi\in S(r)$. On garde la même notation pour cette redéfinition de $f$.
On calcule d’abord le développement limité de
$\|\varphi_{\varepsilon}\|_{N}^{-2}$, on a:
$\varphi_{\varepsilon}^{N}(Q)=\bigl{[}1-Nr^{\omega+2}f(\xi)+\frac{N(N-1)}{2}r^{2\omega+4}f^{2}(\xi)+O(r^{3\omega+6})\bigr{]}u_{\varepsilon}^{N}$
En utilisant le fait que $\int_{S_{n-1}}f\mathrm{d}\sigma=0$, on conclut que
$\begin{split}\|\varphi_{\varepsilon}\|_{N}^{N}&=\int_{0}^{\delta}\int_{S_{n-1}}[1+\frac{N(N-1)}{2}r^{2(\omega+2)}f^{2}(\xi)+O(r^{3(\omega+2)})]r^{n-1}u^{N}_{\varepsilon}\mathrm{d}r\mathrm{d}\sigma(\xi)\\\
&=\omega_{n-1}I^{n-1}_{n}+\frac{N(N-1)}{2}\varepsilon^{2(\omega+2)}\int_{S_{n-1}}f^{2}\mathrm{d}\sigma
I_{n}^{2\omega+n+3}+o(\varepsilon^{2\omega+4})\end{split}$
alors
$\|\varphi_{\varepsilon}\|_{N}^{-2}=(\omega_{n-1}I^{n-1}_{n})^{-2/N}\bigl{\\{}1+\\\
-(N-1)\varepsilon^{2(\omega+2)}\int_{S_{n-1}}f^{2}\mathrm{d}\sigma
I_{n}^{2\omega+n+3}/(\omega_{n-1}I^{n-1}_{n})\bigr{\\}}+o(\varepsilon^{2\omega+4})$
(4.14)
Par (4.12), (4.13), (4.14) et les relations (4.9), on trouve que (les détails
de ces calculs sont dans l’appendice A):
Si $n>2\omega+6$ alors :
$I_{g}(\varphi_{\varepsilon})=\frac{n(n-2)}{4}\omega_{n-1}^{2/n}+(\omega_{n-1}I^{n-1}_{n})^{-2/N}I_{n-2}^{n+2\omega+1}\varepsilon^{2\omega+4}\times\\\
\biggl{\\{}\frac{(n-2)\omega_{n-1}}{4(n-1)}\bar{\int}_{S(r)}r^{-2\omega-2}R_{g}\mathrm{d}\sigma_{r}-\frac{n-2}{2(n-1)}\int_{S_{n-1}}f\bar{R}\mathrm{d}\sigma+\int_{S_{n-1}}|\nabla
f|^{2}\mathrm{d}\sigma+\\\
-\frac{n(n-2)^{2}-(\omega+2)^{2}(n^{2}+n+2)}{(n-1)(n-2)}\int_{S_{n-1}}f^{2}\mathrm{d}\sigma\biggr{\\}}+o(\varepsilon^{2\omega+4})$
Si $n=2\omega+6$ alors
$I_{g}(\varphi_{\varepsilon})=\frac{n(n-2)}{4}\omega_{n-1}^{2/n}+(\omega_{n-1}I^{n-1}_{n})^{-2/N}\varepsilon^{2\omega+4}\log\varepsilon^{-1}\times\\\
\biggl{\\{}\frac{(n-2)\omega_{n-1}}{4(n-1)}\bar{\int}_{S(r)}r^{-2\omega-2}R_{g}\mathrm{d}\sigma_{r}-\frac{n-2}{2(n-1)}\int_{S_{n-1}}f\bar{R}\mathrm{d}\sigma+\\\
\int_{S_{n-1}}|\nabla
f|^{2}\mathrm{d}\sigma+(\omega+2)^{2}\int_{S_{n-1}}f^{2}\mathrm{d}\sigma\biggr{\\}}+O(\varepsilon^{2\omega+4})$
On considère maintenant la fonctionnelle $I_{S}$, définie sur la sphère
$S_{n-1}$, pour les fonctions dans $H_{1}(S_{n-1})$, d’intégrale moyenne
nulle, par
$I_{S}(f)=\bar{\int}_{S_{n-1}}4(n-1)(n-2)|\nabla
f|^{2}-[4n(n-2)^{2}-4(\omega+2)^{2}(n^{2}+n+2)]f^{2}+\\\
-2(n-2)^{2}f\bar{R}\mathrm{d}\sigma$
Alors si $n>2\omega+6$
$I_{g}(\varphi_{\varepsilon})=\frac{n(n-2)}{4}\omega_{n-1}^{2/n}+\frac{\omega_{n-1}^{2/n}I_{n-2}^{n+2\omega+1}\varepsilon^{2\omega+4}}{4(n-1)(n-2)(I^{n-1}_{n})^{2/N}}\times\\\
\\{(n-2)^{2}\bar{\int}_{S(r)}r^{-2\omega-2}R_{g}\mathrm{d}\sigma_{r}+I_{S}(f)\\}+o(\varepsilon^{2\omega+4})$
(4.15)
et si $n=2\omega+6$
$I_{g}(\varphi_{\varepsilon})=\frac{n(n-2)}{4}\omega_{n-1}^{2/n}+\frac{\omega_{n-1}^{2/n}I_{n-2}^{n+2\omega+1}\varepsilon^{2\omega+4}\log\varepsilon^{-1}}{4(n-1)(n-2)(I^{n-1}_{n})^{2/N}}\times\\\
\\{(n-2)^{2}\bar{\int}_{S(r)}r^{-2\omega-2}R_{g}\mathrm{d}\sigma_{r}+I_{S}(f)\\}+O(\varepsilon^{2\omega+4})$
(4.16)
Remarquons que si $k\neq j$ alors
$I_{S}(\varphi_{k}+\varphi_{j})=I_{S}(\varphi_{k})+I_{S}(\varphi_{j})$. En
effet, $\varphi_{k}$ et $\varphi_{j}$ sont orthogonales pour le produit
scalaire sur $H_{1}(S_{n-1})$. D’où
$\begin{split}I_{S}(c_{k}\nu_{k}\varphi_{k})&=\bigl{\\{}d_{k}c_{k}^{2}-2(n-2)^{2}c_{k}\bigr{\\}}\nu_{k}^{2}\bar{\int}_{S_{n-1}}\varphi_{k}^{2}\mathrm{d}\sigma\\\
&=-\frac{(n-2)^{4}}{d_{k}}\nu_{k}^{2}\bar{\int}_{S_{n-1}}\varphi_{k}^{2}\mathrm{d}\sigma\end{split}$
avec
$\displaystyle
d_{k}=4[(n-1)(n-2)\nu_{k}-n(n-2)^{2}+(\omega+2)^{2}(n^{2}+n+2)]$ (4.17)
$\displaystyle\text{et }c_{k}=\frac{(n-2)^{2}}{d_{k}}$ (4.18)
Ici on choisit les $c_{k}$ de sorte que $I_{S}(c_{k}\nu_{k}\varphi_{k})$ soit
minimal. En utilisant (4.2), on peut vérifier aisément que les $d_{k}$ sont
strictement positifs pour tout $1\leq k\leq\omega/2$.
Maintenant, On pose
$f=\sum_{1}^{q}c_{k}\nu_{k}\varphi_{k}$ (4.19)
Il est clair que $f$ ainsi définie est d’intégrale nulle sur $S_{n-1}$. C’est
bien la définition de $f$ qu’on utilisera dans la suite de la preuve. Par
l’orthogonalité des fonctions $\varphi_{k}$, on trouve que
$I_{S}(f)=-\sum_{1}^{q}\frac{(n-2)^{4}}{d_{k}}\nu_{k}^{2}\bar{\int}_{S_{n-1}}\varphi_{k}^{2}\mathrm{d}\sigma$
et par le lemme 4.3, on trouve l’inégalité suivante:
$(n-2)^{2}\bar{\int}_{S(r)}r^{-2\omega-2}R_{g}\mathrm{d}\sigma_{r}+I_{S}(f)\leq\sum_{1}^{q}(u_{k}(n-2)^{2}-\frac{(n-2)^{4}}{d_{k}}\nu_{k}^{2})\bar{\int}_{S_{n-1}}\varphi_{k}^{2}\mathrm{d}\sigma+o(1)$
Le lemme ci-dessous énoncé, assure que le membre de droite de cette dernière
inégalité est strictement négatif. En utilisant les inégalités (4.15), (4.16),
on en déduit que
$I_{g}(\varphi_{\varepsilon})<\frac{n(n-2)}{4}\omega_{n-1}^{2/n}$ ∎
###### Lemme 4.4.
Pour tout $k\leq q\leq[\omega/2]$, l’inégalité suivante est toujours vraie
$u_{k}-\frac{(n-2)^{2}}{d_{k}}\nu_{k}^{2}<0$
###### Preuve.
On rappelle l’expression des $\nu_{k}$ donnée dans (4.2):
$\nu_{k}=(\omega-2k+2)(n+\omega-2k)$
Pour tout $k\in[[1,\omega/2]]$, on définit les nombres $(U_{k})$ par
$U_{k}:=(\nu_{k}-n+1)d_{k}\\{(n-2)\frac{u_{k}}{\nu_{k}}-\frac{(n-2)^{3}}{d_{k}}\nu_{k}\\}$
On remarque que l’expression de $U_{k}$ est polynomiale, décroissante en
$\nu_{k}$ quand $\nu_{k}\geq 0$. $U_{k}=P(\nu_{k})$, où $P$ est le polynôme
défini par
$P(x)=[(n-1)(n-2)x-n(n-2)^{2}+(\omega+2)^{2}(n^{2}+n+2)]\times\\\
[(n-3)(x-n+1)-(n-1)^{2}-(n-1)(\omega+2)^{2}]-(n-2)^{3}(x^{2}-(n-1)x)$
Le polynôme dérivé est
$P^{\prime}(x)=-2(n-2)x-2n(n-2)^{3}+2(n^{2}-3n-2)(\omega+2)^{2}$
Par hypothèse $\omega+2\leq(n-2)/2$, donc $P$ est décroissant sur
$\mathbb{R}_{+}$. Ce qui entraîne que
$U_{k}=P(\nu_{k})\leq P(\nu_{\omega/2})=U_{\omega/2}$
pour tout $1\leq k\leq\omega/2$. Il est facile de vérifier que $u_{\omega/2}$
est strictement négatif et donc $U_{k}\leq U_{\omega/2}<0$. ∎
## Chapter 5 Autour de la conjecture de Hebey–Vaugon
Dans la section 3.11, on a étudié le problème de Yamabe équivariant, considéré
par E. Hebey et M. Vaugon [HV], lorsque la métrique n’est pas nécessairement
$C^{\infty}$. On a démontré que la condition suffisante pour résoudre ce
problème est que la conjecture 5.1 soit vraie (cf. théorème 3.8).
Malheureusement, on ne peut pas donner une preuve de cette conjecture, lorsque
$g\in H^{p}_{2}(M,T^{*}M\otimes T^{*}M)$. En effet, la courbure scalaire
appartient à $L^{p}$, et plusieurs arguments utilisés dans le cas $C^{\infty}$
ne sont plus valables dans ce cas.
Dans tout ce chapitre, on suppose que $M$ est une variété compacte
$C^{\infty}$, de dimension $n\geq 3$, $g$ est une métrique riemannienne
$C^{\infty}$, munie de sa connexion riemannienne, notée $\nabla_{g}$. On note
par $I(M,g)$, $C(M,g)$ le groupe d’isométries et le groupe des transformations
conformes respectivement (voir la définition dans la section 2.2.1). Soit $G$
un sous groupe du groupe d’isométries $I(M,g)$.
Ce chapitre utilise beaucoup de résultats déjà démontrés dans le chapitre
précédent.
### 5.1 La conjecture de Hebey–Vaugon
###### Conjecture 5.1 (E. Hebey et M. Vaugon [HV]).
Soit $G$ un sous groupe d’isométries de $I(M,g)$. Si $(M,g)$ n’est pas
conformément difféomorphe à $(S_{n},g_{can})$ ou bien si $G$ n’a pas de point
fixe, alors l’inégalité stricte suivante a toujours lieu
$\inf_{g^{\prime}\in[g]^{G}}J(g^{\prime})<n(n-1)\omega_{n}^{2/n}(\inf_{Q\in
M}\mathrm{card}O_{G}(Q))^{2/n}$ (5.1)
##### Remarques
* •
Cette conjecture est la généralisation de la conjecture de T. Aubin 3.1 pour
le problème de Yamabe, qui correspond à $G=\\{\mathrm{id}\\}$. Dans ce cas, la
conjecture est complètement prouvée. Elle est prouvée aussi dans le cas où la
métrique satisfait l’hypothèse $(H)$, définie dans la section 3.2 (voir
théorème 3.5).
* •
Cette inégalité est triviale si $\inf_{g^{\prime}\in[g]^{G}}J(g^{\prime})$ est
négatif.
* •
Si pour tout $Q\in M$, $\mathrm{card}O_{G}(Q)=+\infty$, alors la conjecture
est vérifiée trivialement.
Rappelons que la partie principale de la courbure scalaire $\bar{R}$ est
définie dans la section 4.1 (voir définition 4.2).
Les résultats principaux de ce chapitre sont
###### Théorème 5.1.
La conjecture 5.1 est vraie, s’il existe un point $P$ d’orbite minimale
(finie) pour lequel $\omega(P)\leq 15$, ou si au voisinage de $P$,
$\mathrm{deg}\bar{R}\geq\omega(P)+1$
###### Corollaire 5.1.
La conjecture 5.1 est vraie si $M$ est de dimension $n\in[[3,37]]$.
###### Preuve.
Supposons que $P$ est un point d’orbite minimale (finie) sous $G$ (sinon la
conjecture est trivialement vérifiée).
Si $\omega(P)>(n-6)/2$, on conclut par le troisième point du théorème 5.2 ci-
dessous.
Si $\omega(P)\leq[(n-6)/2]\leq 15$, on conclut par le théorème 5.1. ∎
### 5.2 Les travaux de Hebey–Vaugon
E. Hebey et M. Vaugon [HV] ont prouvé la conjecture 5.1 dans les cas suivants:
###### Théorème 5.2.
Soit $(M,g)$ une variété riemannienne compacte, de dimension $n\geq 3$ et $G$
un sous groupe d’isométries du groupe $I(M,g)$. On a toujours:
$\inf_{g^{\prime}\in[g]^{G}}J(g^{\prime})\leq
n(n-1)\omega_{n}^{2/n}(\inf_{Q\in M}\mathrm{card}O_{G}(Q))^{2/n}$
et l’ inégalité stricte (5.1) est au moins vérifiée dans chacun des cas
suivants:
1. 1.
$G$ opère librement su $M$
2. 2.
$3\leq\dim M\leq 11$
3. 3.
Il existe un point $P$ d’orbite minimale (finie) sous $G$, pour lequel soit
$\omega(P)>(n-6)/2$ soit $\omega(P)\in\\{0,1,2\\}$.
###### Idées de la preuve.
On s’intéresse à la démonstration du point 3 du théorème ci-dessus (c’est le
cas qui manque dans le théorème 4.3). Les hypothèses sont:
1. 1.
$\mathrm{card}O_{G}(P)<+\infty$.
2. 2.
Il existe $P\in M$ tel que $\mathrm{card}O_{G}(P)=\inf_{Q\in
M}\mathrm{card}O_{G}(Q)$.
3. 3.
$\omega>[\frac{n-6}{2}]\Longleftrightarrow\forall\beta\in[[1,n]]^{i}/i\leq[(n-6)/2],\quad\nabla^{\beta}W_{g}(P)=0$.
Notons $k=\mathrm{card}O_{G}(P)$, le cardinal de l’orbite
$O_{G}(P)=\\{P_{i},1\leq i\leq k\\}$, où l’on a posé $P_{1}=P$. La troisième
hypothèse implique que pour tout $1\leq i\leq k$,
$\omega(P_{i})>[\frac{n-6}{2}]$, puisque le tenseur de Weyl est invariant sous
l’action du groupe d’isométries $I(M,g)$.
Par les travaux de J.M. Lee et T. Parker [LP], on sait qu’on peut trouver un
système de coordonnées et une métrique conforme $g^{\prime}$ tels que
$g^{\prime}$ satisfait:
$\det(g^{\prime})=1+O(r^{m})\quad\text{ pour tout }m\gg 1$ (5.2)
(cf. section 3.6 ou [LP] pour l’existence). Dans le cas équivariant, on ne
peut pas considérer n’importe quelle métrique dans la classe conforme $[g]$,
cependant E. Hebey et M. Vaugon ont démontré que dans chaque classe $[g]^{G}$
on peut trouver au moins une métrique qui satisfait (5.2). En utilisant les
champs de Jacobi, ils ont obtenu le développement limité de la métrique $g$
suivant:
###### Lemme 5.1.
$g_{ij}(Q)=\delta_{ij}+\sum_{\omega+4\leq m\leq
2\omega+5}C_{m}\nabla_{p_{3}\cdots p_{m-2}}R_{ip_{1}p_{2}j}(P)x^{p_{1}}\cdots
x^{p_{m-2}}\\\ +C_{\omega}\sum_{pj}\nabla_{p_{3}\cdots
p_{2\omega+4}}R_{ip_{1}p_{2}j}(P)x^{p_{1}}\cdots x^{p_{2\omega+4}}\\\
+C^{\prime}_{\omega}\sum_{q=1}^{n}\sum_{pj}(\nabla_{p_{3}\cdots
p_{\omega+2}}R_{ip_{1}p_{2}q}(P))(\nabla_{p_{\omega+5}\cdots
p_{2\omega+4}}R_{jp_{\omega+3}p_{\omega+4}q}(P))x^{p_{1}}\cdots
x^{p_{2\omega+4}}+O(r^{2\omega+5})$ (5.3)
pour tout $Q$ au voisinage de $P$, où $\\{x^{l}\\}$ sont les coordonnées
locales de $Q$. $C_{\omega}$, $C^{\prime}_{\omega}$ et $C_{m}$ sont des
nombres réels, qui dépendent de $\omega$ et $m$ respectivement. Ces nombres
sont donnés explicitement dans [HV].
Ce développement est le point crucial dans la preuve du lemme suivant:
###### Lemme 5.2.
Dans chaque classe $[g]^{G}$, des métriques conformes $G-$invariantes, on peut
trouver une métrique $g^{\prime}$ qui satisfait
1. 1.
$\det(g^{\prime})=1+O(r^{m})$, $m\gg 1$
2. 2.
$\forall i<\omega,\;\nabla^{i}R^{\prime}_{jklm}(P)=0$
3. 3.
pour tout $\beta\in[[1,n]]^{i}$ tel que $i\leq 2\omega+1$
$\nabla_{\beta}R^{\prime}(P)=\partial_{\beta}R^{\prime}(P),\;\nabla_{\beta}Ric^{\prime}(P)=\partial_{\beta}Ric^{\prime}(P),\;\nabla_{\beta}R_{g^{\prime}}(P)=\partial_{\beta}R_{g^{\prime}}(P)$
4. 4.
$\forall j\leq\omega\quad\Delta_{g}^{j}R_{g^{\prime}}(P)=0\text{ et
}\nabla\Delta_{g^{\prime}}^{\omega^{\prime}}R_{g^{\prime}}(P)=0$
où $R^{\prime}$, $Ric^{\prime}$ et $R_{g^{\prime}}$ sont le tenseur de
courbure de Riemann, le tenseur de Ricci et la courbure scalaire de
$g^{\prime}$ respectivement.
##### Remarque
Dans leur article [HV], E. Hebey et M. Vaugon ont noté par
$Sym_{\beta}T_{\beta}$ le symetrisé du tenseur $T$, et par $C(2,2)$
l’application de contraction des indices deux à deux pour les tenseurs
symétrique. A titre d’exemple $C(2,2)T_{ij}=\sum_{i}T_{ii}$,
$(C(2,2)T_{ijk})_{l}=\sum_{i}T_{iil}$ et $C(2,2)T_{ijkl}=\sum_{i,j}T_{iijj}$.
Ils ont montré que pour tout $\beta\in[[1,n]]^{i}$ tel que $i\leq 2\omega+1$
$C(2,2)(Sym_{\beta}\nabla_{\beta}R_{g}(P))=0$
ce qui est équivalent au point 4 du lemme ci-dessus.
L’invariance $G-$conforme de $\mu_{G}(g)$ et de $\omega$ (cf. propriétés 4.1,
3.8) nous permettent de considérer n’importe quelle métrique, $G-$invariante
dans la classe $[g]^{G}$ (cf. définition 2.2, page 2.2). Sans perte de
généralités, on suppose que la métrique $g$ et les courbures associées à $g$,
satisfont le lemme 5.2. Soit $G_{P_{i}}$ la fonction de Green du Laplacien
conforme $L_{g}$ au point $P_{i}$ (voir la section 3.5 pour l’existence). En
utilisant les points 1 et 4 du lemme 5.2, on montre que le développement
limité de la fonction $G_{P_{i}}$ au voisinage de $P_{i}$ est
$G_{P_{i}}(x)=\frac{1}{(n-2)\omega_{n-1}r_{i}^{n-2}}(1+\sum_{p=1}^{n}\psi_{p}(x))+O^{\prime\prime}(1)$
où $r_{i}=d(P_{i},x)$ et les $\psi_{p}$ sont des polynômes homogènes de degré
$p$ qui s’annulent si $1\leq p\leq[(n-2)/2]$.
Considérons la métrique $\tilde{g}=G_{P}^{\frac{4}{n-2}}g$. $G_{P}$ est
$C^{\infty}$ sur $M-\\{P\\}$ et la variété $(M-\\{P\\},\tilde{g})$ est
asymptotiquement plate d’ordre $\frac{n}{2}$. Les coordonnées asymptotiques
sont $z^{i}=\frac{x^{i}}{|x|^{2}}$ et $\rho=|z|$, où $\\{x^{i}\\}$ est un
système de coordonnées normal en $P$. La masse $m(\tilde{g})$ est bien définie
positive car $\tau=\frac{n}{2}>\frac{n-2}{2}$. Soit
$\mathcal{G}=\sum_{i=1}^{k}G_{P_{i}}$ une fonction $C^{\infty}$,
$G-$invariante, définie sur $M-O_{G}(P)$. La fonction test utilisée par E.
Hebey et M. Vaugon pour démontrer la conjecture est $w_{\varepsilon}$, définie
comme suit:
$w_{i,\varepsilon}=\begin{cases}\mathcal{G}r_{i}^{n-2}\biggl{(}\displaystyle\frac{\varepsilon}{r_{i}^{2}+\varepsilon^{2}}\biggr{)}^{\frac{n-2}{2}}&\text{
si }r_{i}\leq\delta\\\
\mathcal{G}\delta^{n-2}\biggl{(}\displaystyle\frac{\varepsilon}{\delta^{2}+\varepsilon^{2}}\biggr{)}^{\frac{n-2}{2}}&\text{
si }r_{i}\geq\delta\end{cases}$
$w_{\varepsilon}=\sum_{i=1}^{k}w_{i,\varepsilon}$
Si $\delta$ est suffisamment petit, alors les fonctions $w_{i,\varepsilon}$ et
$w_{\varepsilon}$ sont bien définies sur $M$. Il est clair que la fonction
$w_{\varepsilon}$ est $G-$invariante. Après calculs, E. Hebey et M. Vaugon
obtiennent l’inégalité suivante:
$E(w_{\varepsilon})\leq\frac{n(n-1)}{4}\omega_{n}^{2/n}k^{2/n}\|w_{\varepsilon}\|_{N}^{-2}-C_{1}(m(\tilde{g})+(n-2)K)\varepsilon^{n-2}+\varepsilon^{n-2}O(\delta)+O(\varepsilon^{n-1})$
où $C_{1}$ et $K$ deux constantes positives. Alors $m(\tilde{g})+(n-2)K>0$, et
on peut choisir $\delta$ et $\varepsilon$ suffisamment petits tels que
$I_{g}(w_{\varepsilon})<\frac{n(n-1)}{4}\omega_{n}^{2/n}k^{2/n}$. Par
conséquent
$\mu_{G}(g)<\frac{n(n-2)}{4}\omega_{n}^{2/n}(\mathrm{card}O_{G}(P))^{2/n}$
∎
### 5.3 Preuve du théorème principal
En tenant compte des remarques de la section 5.1 (cf. page 5.1) et du théorème
5.2, on considère seulement le cas où $\inf_{Q\in M}\mathrm{card}O_{G}(Q)$ est
fini, strictement positif (i.e. $\mu_{G}(g)>0$) et $\omega\leq(n-6)/2$ . Alors
il existe $P\in M$ tel que
$O_{G}(P)=\\{P_{i}\\}_{1\leq i\leq m},\;\;m=\mathrm{card}O_{G}(P)=\inf_{Q\in
M}\mathrm{card}O_{G}(Q)\text{ et }P_{1}=P$
Un élément très important, dans la démonstration du théorème principal 5.1,
est le choix des fonctions test dans la fonctionnelle $I_{g}$. Les fonctions
test précédemment utilisées par T. Aubin et R. Schoen (voir la preuve du
théorème 3.5) ne fonctionnent pas ici, comme cela avait été remarqué par E.
Hebey et M. Vaugon [HV]. Les "bonnes" fonctions test seront construites de la
manière suivante, en modifiant les fonctions test de T. Aubin: on construit
une fonction test $G-$invariante, à partir des fonctions
$\tilde{\varphi}_{\varepsilon,i}$, définie de la même façon que
$\varphi_{\varepsilon}$ (voir section 4.2), dont on rappelle la définition.
$P$ est un point d’orbite minimale. Pour tout $Q\in M$
$\displaystyle\tilde{\varphi}_{\varepsilon,i}(Q)=(1-r_{i}^{\omega+2}\tilde{f}_{i}(Q))u_{\varepsilon,i}(Q)$
(5.4) $\displaystyle
u_{\varepsilon,i}(Q)=\begin{cases}\biggl{(}\displaystyle\frac{\varepsilon}{r_{i}^{2}+\varepsilon^{2}}\biggr{)}^{\frac{n-2}{2}}-\biggl{(}\frac{\varepsilon}{\delta^{2}+\varepsilon^{2}}\biggr{)}^{\frac{n-2}{2}}&\mbox{
si }Q\in B_{P_{i}}(\delta)\\\ \hskip 56.9055pt0&\mbox{ si }Q\in
M-B_{P_{i}}(\delta)\end{cases}$ (5.5)
où $r_{i}=d(Q,P_{i})$ est la distance entre $P_{i}$ et $Q$. Pour la
simplicité: $P=P_{1}$, $r=r_{1}$,
$\tilde{\varphi}_{\varepsilon}=\tilde{\varphi}_{\varepsilon,1}$,
$\tilde{f}=\tilde{f}_{1}$ et $u_{\varepsilon,1}=u_{\varepsilon}$.
$B_{P}(\delta)$ est une boule géodésique de centre $P$, de rayon $\delta$,
fixé suffisamment petit. Les $\tilde{f}_{i}$ sont définies de la façon
suivante : Soit $\exp_{P_{i}}$ l’application exponentielle, définie de
$B(\delta)$, la boule euclidienne centrée en 0 et de rayon $\delta$, dans
$B_{P_{i}}(\delta)$. Pour tout $Q\in B_{P_{i}}(\delta)$, on pose
$\tilde{f}_{i}(Q)=cr_{i}^{-\omega}\nabla_{g}^{\omega}R_{(P_{i})}(\exp_{P_{i}}^{-1}Q,\cdots,\exp_{P_{i}}^{-1}Q)$
(5.6)
où $\omega=\omega(P)$ et $\nabla^{\omega}_{g}R(P)$ est la $\omega-$ème dérivée
covariante de $R_{g}$ au point $P$, c’est un tenseur $\omega$ fois covariant.
Dans le système de coordonnées géodésiques $\\{r,\xi^{j}\\}$, centré en $P$,
induit par l’application $\exp_{P}$, $\tilde{f}$ s’écrit:
$\tilde{f}=cr^{-\omega}\bar{R}=c\sum_{k=1}^{q}\nu_{k}\varphi_{k}$
où $\bar{R}$, $\varphi_{k}$ et $\nu_{k}$ sont définis dans la section 4.1
(page 4.1). La fonction $\tilde{f}$ est définie sur la sphère $S_{n-1}$. Le
choix de la constante $c$ est très important dans le lemme suivant.
###### Lemme 5.3.
Supposons que $\omega\leq(n-6)/2$. Si $\omega\in[[3,15]]$ ou si
$\mathrm{deg}\bar{R}\geq\omega+1$ alors il existe $c\in\mathbb{R}$ telle que,
pour la fonction $\tilde{\varphi}_{\varepsilon}$ correspondante, on a:
$I_{g}(\tilde{\varphi}_{\varepsilon})<\frac{1}{4}n(n-2)\omega_{n}^{2/n}$ (5.7)
##### Remarque
1. 1.
Dans le chapitre précédent, on a démontré que l’inégalité de ce lemme est
vérifiée, pour tout $\omega\leq(n-6)/2$, pour une fonction test
$\varphi_{\varepsilon}$ (voir théorème 4.3). On remarque que la seule
différence entre les définitions de $\varphi_{\varepsilon}$ et
$\tilde{\varphi}_{\varepsilon}$ est dans la construction des fonctions $f$ et
$\tilde{f}$. En effet, $\tilde{f}$ est définie à l’aide d’une constante
globale $c$ et $f$ à l’aide des constantes $c_{k}$ qui changent avec les
fonctions propres $\varphi_{k}$. On verra dans la preuve du théorème 5.1, qu’à
partir de $\tilde{\varphi}_{\varepsilon}$, on peut construire une fonction
$G-$invariante qui possède les "bonnes" propriétés, cette chose n’est pas
possible avec les fonctions $\varphi_{\varepsilon}$.
2. 2.
Pour $\omega=16$ et $n$ suffisamment grand, on peut vérifier qu’il n’existe
pas une valeur de $c$ pour laquelle l’inégalité (5.7) est vraie.
###### Preuve.
1\. Si $\mathrm{deg}\bar{R}\geq\omega+1$, alors d’après le théorème 4.2
$I_{g}(u_{\varepsilon,1})<\frac{n(n-2)}{4}\omega_{n}^{2/n}$
où $u_{\varepsilon,1}=u_{\varepsilon}$ est définie par (5.5). Il suffit donc
de prendre $c=0$ et $\tilde{\varphi}_{\varepsilon}=u_{\varepsilon}$.
2\. Si $\mathrm{deg}\bar{R}=\omega$. D’après les estimées données dans la
preuve du théorème 4.3 (voir page 4.15), il suffit de montrer qu’il existe
$c\in\mathbb{R}$ telle que
$I_{S}(\tilde{f})+(n-2)^{2}\bar{\int}_{S(r)}r^{-2\omega-2}R_{g}\mathrm{d}\sigma_{r}<0$
(5.8)
Cherchons donc cette constante $c$. On garde les notations de la preuve du
théorème 4.3. On a
$I_{S}(\tilde{f})=\sum_{k=1}^{q}I_{S}(c\nu_{k}\varphi_{k})=\bigl{\\{}d_{k}c^{2}-2(n-2)^{2}c\bigr{\\}}\nu_{k}^{2}\bar{\int}_{S_{n-1}}\varphi_{k}^{2}\mathrm{d}\sigma$
$\text{et
}\bar{\int}_{S(r)}r^{-2\omega-2}R_{g}\mathrm{d}\sigma_{r}=\sum_{k=1}^{q}u_{k}\bar{\int}_{S_{n-1}}\varphi_{k}^{2}\mathrm{d}\sigma$
Pour montrer l’inégalité 5.8, il suffit de montrer que
$\forall k\leq
q\leq[\omega/2]\quad\frac{d_{k}}{2(n-2)}c^{2}-(n-2)c+(n-2)\frac{u_{k}}{2\nu_{k}^{2}}<0$
(5.9)
On a donc un trinôme du second degré en $c$, son discriminant est
$\Delta_{k}=(n-2)^{2}-\frac{d_{k}u_{k}}{\nu_{k}^{2}}$
D’après le lemme 4.4, $\Delta_{k}>0$ pour tout $k\leq q\leq[\omega/2]$. Par
conséquent, le trinôme ci-dessus admet deux racines, notées $x_{k}<y_{k}$ et
données par
$x_{k}=\frac{(n-2)^{2}-(n-2)\sqrt{\Delta_{k}}}{d_{k}},\qquad
y_{k}=\frac{(n-2)^{2}+(n-2)\sqrt{\Delta_{k}}}{d_{k}}$
L’inégalité (5.9) est vérifiée si et seulement si
$\bigcap_{k=1}^{q}]x_{k},y_{k}[\neq\varnothing$ (5.10)
Le lemme est donc démontré, si l’intersection ci-dessus n’est pas vide dans
les cas énoncés.
Puisque $(d_{k})_{k}$ est décroissante, il est facile de vérifier que
$\forall k<j\leq[\frac{\omega}{2}]\qquad x_{k}<y_{j}$ (5.11)
(voir équations (4.2), (4.17), pour la définition de $\nu_{k}$ et $d_{k}$). On
vérifie aussi que $u_{\omega/2}<0$ (voir équation (4.6)), cela entraîne que si
$\omega$ est pair alors $x_{\omega/2}<0$.
* $i.$
Si $\omega=3$ alors $k=q=1$, l’intersection ci-dessus est donc non vide. Il
suffit de prendre $c=(x_{1}+y_{2})/2$.
* $ii.$
Si $\omega=4$ alors $k\in\\{1,2\\}$, $x_{2}<0$ (car $u_{2}<0$) et
$0<x_{1}<y_{2}$. L’intersection
$]x_{1},y_{1}[\cap]x_{2},y_{2}[\neq\varnothing$. Ce qui entraîne l’inégalité
(5.7).
* $iii.$
Si $\omega=5$ alors $k\in\\{1,2\\}$. Par des calculs directs, on montre que
$x_{2}<y_{1}$ (voir les détails dans l’appendice B). Puisque $y_{2}>x_{1}$,
l’intersection des deux intervalles n’est pas vide.
* $iv.$
Si $\omega=6$ alors $k\in\\{1,2,3\\}$ et il est immédiat de voir que $x_{3}<0$
(car $u_{3}<0$), $y_{3}>x_{2}>0$ et $y_{3}>x_{1}>0$. Par des calculs directs,
on montre que $x_{2}<y_{1}$ (voir les détails dans l’appendice B). Ce qui
entraîne que l’intersection
$\bigcap_{k=1}^{3}]x_{k},y_{k}[$ (5.12)
est non vide.
* $iv.$
Si $\omega=7$ alors $k\in\\{1,2,3\\}$. Il y a trois intervalles. Par des
calculs directs, on montre que pour tout $3\geq j>k\geq 1$, $y_{k}>x_{j}$
(voir appendice B). Puisque $y_{j}>x_{k}$ pour tout $3\geq j>k\geq 1$ (voir
inégalité (5.11)), l’intersection des trois intervalles n’est donc pas vide.
* •
En se servant du logiciel "Maple", on montre que le lemme reste vrai jusqu’à
$\omega=15$ (voir appendice B pour plus de détails).
∎
###### Fin de la preuve du théorème 5.1.
Sans perte de généralités, on suppose que $3\leq\omega\leq(n-6)/2$, car si
$\omega>(n-6)/2$ ou si $\omega\leq 2$, il suffit d’appliquer le théorème 5.2.
L’orbite de $P$ sous l’action de $G$ est supposée être de cardinal fini et
minimal (i.e. $\mathrm{card}O_{G}(P)=\inf_{Q\in M}\mathrm{card}O_{G}(Q)$). À
partir de la fonction $\tilde{\varphi}_{\varepsilon}$, définie au début de la
section 5.3, on définit la fonction $\phi_{\varepsilon}$ comme suit:
$\phi_{\varepsilon}=\sum_{k=1}^{m}\tilde{\varphi}_{\varepsilon,i}$
$\phi_{\varepsilon}$ est $G-$invariante. En effet, pour tout $\sigma\in G$, si
$\sigma(P_{i})=P_{j}$ alors
$u_{\varepsilon,i}=u_{\varepsilon,j}\circ\sigma$
d’après la définition de $\tilde{f}_{i}$, donnée par (5.6),
$\tilde{f}_{i}=\tilde{f}_{j}\circ\sigma$ et donc
$\tilde{\varphi}_{\varepsilon,i}=\tilde{\varphi}_{\varepsilon,j}\circ\sigma$
Le support de la fonction $\tilde{\varphi}_{\varepsilon}$ est inclus dans la
boule $B_{P}(\delta)$. On choisit $\delta$ suffisamment petit tel que pour
tout $i\in[[2,m]]$, l’intersection $B_{P}(\delta)\cap
B_{P_{i}}(\delta)=\varnothing$. Donc
$E(\phi_{\varepsilon})=(\mathrm{card}O_{G}(P))E(\varphi_{\varepsilon})\text{
et
}\|\phi_{\varepsilon}\|_{N}^{N}=(\mathrm{card}O_{G}(P))\|\varphi_{\varepsilon}\|_{N}^{N}$
alors
$I_{g}(\phi_{\varepsilon})=(\mathrm{card}O_{G}(P))^{2/n}I_{g}(\varphi_{\varepsilon})$
Par le lemme 5.3, on en déduit que
$I_{g}(\phi_{\varepsilon})<\frac{n(n-2)}{4}\omega_{n-1}^{2/n}(\mathrm{card}O_{G}(P))^{2/n}$
Il nous reste à remarquer que si $\tilde{g}=\phi_{\varepsilon}^{4/(n-2)}g$
alors
$J(\tilde{g})=4\frac{n-1}{n-2}I_{g}(\phi_{\varepsilon})$ (5.13)
cette relation est déjà établie dans la preuve des propriétés 3.8 ( voir page
3.8). On conclut que
$J(\tilde{g})<n(n-1)\omega_{n-1}^{2/n}(\mathrm{card}O_{G}(P))^{2/n}$
où $\varepsilon$ est choisi suffisamment petit par rapport à $\delta$. ∎
## Appendix A Détails des calculs (Chapitre 4)
### Preuve du théorème 4.1
On reprend les notations et les définitions de la section 4.1. Voici l’énoncé
du théorème que l’on démontre dans cette section:
###### Théorème A.1.
$\bar{R}=\nabla^{ij}\bar{g}_{ij}r^{\omega}\quad\text{et}$
$\bar{\int}_{S(r)}R\mathrm{d}\sigma_{r}=[B/2-C/4-(1+\omega/2)^{2}Q]r^{2(\omega+1)}+o(r^{2(\omega+1)})$
où
$B=\bar{\int}_{S_{n-1}}\nabla^{i}\bar{g}^{jk}\nabla_{j}\bar{g}_{ik}\mathrm{d}\sigma$,
$C=\bar{\int}_{S_{n-1}}\nabla^{i}\bar{g}^{jk}\nabla_{i}\bar{g}_{jk}\mathrm{d}\sigma$
et $Q=\bar{\int}_{S_{n-1}}\bar{g}_{ij}\bar{g}^{ij}\mathrm{d}\sigma$
###### Preuve.
Soit donc $\\{x^{\alpha}\\}$ un système de coordonnées normal en $P$.
$\\{r,\xi^{i}\\}$ un système de coordonnées géodésiques. On a vu que la
métrique se décompose de la façon suivante:
$g=\mathcal{E}+h=(\delta_{\alpha\beta}+h_{\alpha\beta})dx^{\alpha}\otimes
dx^{\beta}=dr^{2}+(s_{ij}+h_{ij})(rd\xi^{i})\otimes(rd\xi^{j})$
où $(s_{ij})$ sont les composantes de la métrique standard sur la sphère
$S_{n-1}$ et
$h_{ij}=\frac{\partial x^{\alpha}}{r\partial\xi^{i}}\frac{\partial
x^{\beta}}{r\partial\xi^{j}}h_{\alpha\beta},\;\text{ and }h_{ir}=h_{rr}=0$
et que $h_{ij}=O(r^{\omega+2})$. On a aussi décomposé $h$ de la façon suivante
$h_{ij}=r^{\omega+2}\bar{g}_{ij}+r^{2(\omega+2)}\hat{g}_{ij}+\tilde{h}_{ij}$
(A.1)
où $\bar{g}$, $\hat{g}$ et $\tilde{h}$ sont des 2-tenseurs symétriques définis
sur la sphère $S_{n-1}$. On choisit $\\{\frac{\partial}{\partial
r},\frac{\partial}{r\partial\xi^{i}}\\}_{1\leq i\leq n-1}$ et
$\\{dr,rd\xi^{i}\\}_{1\leq i\leq n-1}$ comme bases locales de l’espace tangent
$TM$ et cotangent $T^{*}M$ respectivement. Alors
$g_{ij}=s_{ij}+h_{ij},\;g_{rr}=1\text{ et }g_{ir}=0$
Les composantes $g^{ij}$ de l’inverse de la métrique sont
$g^{ij}=s^{ij}-h^{ij}+O(r^{2\omega+4}),\;{g^{rr}=1}\text{ et }g^{ir}=0$
où $h^{ij}=s^{ik}s^{jl}h_{lk}$. On fait monter et baisser les indices, en
utilisant la métrique $(s_{ij})$, sauf pour la métrique $g$. À partir de
maintenant, on omet $O(r^{2\omega+4})$ qui apparaît dans l’expression de
$g^{ij}$ ci-dessus, car nos calculs sont à $o(r^{2\omega+2})$ près. On note
par $\nabla$ la connexion riemannienne sur la sphère, associée à $s$.
$\tilde{\nabla}$ la connexion associée à la métrique euclidienne $\mathcal{E}$
dans le corepère $\\{dr,rd\xi^{i}\\}$, alors
$\tilde{\nabla}_{i}=\frac{1}{r}\nabla_{i}\text{ et
}\tilde{\nabla}_{r}=\partial_{r}$
et $\tilde{\partial}_{i}=\frac{1}{r}\partial_{i}$. Dans le système de
coordonnées $\\{x^{\alpha}\\}$, $\det g=1+O(r^{m})$, et dans le système
$\\{r,\xi^{i}\\}$, $\det g=r^{2(n-1)}\det s+O(r^{m})$, avec $m$ suffisamment
grand. D’où $tr\log((\delta^{k}_{i}+s^{jk}h_{ij}))=1$. Par le développement
limité
$(\log((\delta^{k}_{i}+s^{jk}h_{ij})))^{k}_{i}=s^{jk}h_{ij}-\frac{1}{2}s^{mk}s^{jl}h_{mj}h_{il}+o(r^{2\omega+4})$
en tenant compte de la décomposition (A.1), on trouve que $\bar{g}$, $\hat{g}$
et $\tilde{h}$ doivent satisfaire les relations suivantes
$s^{ij}\bar{g}_{ij}=0,\;\bar{g}^{ij}\bar{g}_{ij}=2s^{ij}\hat{g}_{ij}\text{ et
}\bar{\int}_{S(r)}s^{ij}\tilde{h}_{ij}\mathrm{d}\sigma_{r}=o(r^{2\omega+2})$
La première relation vient du fait que le terme d’ordre $\omega+2$ dans le
développement de $tr\log((\delta^{k}_{i}+s^{jk}h_{ij}))$ est
$s^{ij}\bar{g}_{ij}r^{\omega+2}$ qui doit être nul. Le terme d’ordre
$2\omega+4$ est
$(s^{ij}\hat{g}_{ij}-1/2\bar{g}^{ij}\bar{g}_{ij})r^{2\omega+4}$ qui doit être
également nul. Dans $s^{ij}\hat{h}_{ij}$, il y a des termes d’ordre entre
$\omega+3$ et $2\omega+3$ qui doivent être nuls, les termes d’ordre supérieur
à $2\omega+5$ sont négligeables.
Soient $\tilde{\Gamma}^{k}_{ij}$ et $\Gamma^{k}_{ij}$ les Christoffels de la
métrique $g$ et de la métrique euclidienne
$\mathcal{E}=dr^{2}+r^{2}s_{ij}d\xi^{i}d\xi^{j}$ respectivement. On sait que
les
$C^{m}_{jl}=\tilde{\Gamma}^{m}_{lj}-\Gamma^{m}_{lj}$
sont les composantes d’un certain tenseur $C$, défini sur la sphère $S_{n-1}$,
données par
$C^{m}_{jl}=\frac{1}{2}g^{mp}(\tilde{\nabla}_{j}h_{pl}+\tilde{\nabla}_{l}h_{pj}-\tilde{\nabla}_{p}h_{jl}),\;C^{r}_{jl}=-\frac{1}{2}\partial_{r}h_{jl}\text{
et }C^{m}_{rj}=\frac{1}{2}g^{mp}\partial_{r}h_{pj}$ (A.2)
et $C^{i}_{rr}=C^{r}_{ri}=0$. Ici les indices latins varient entre 1 et $n-1$
et les indices grecs varient entre 1 et $n$. Dans le système de coordonnées
$\\{x^{\alpha}\\}$, $g_{\alpha\beta}=\delta_{\alpha\beta}+h_{\alpha\beta}$,
les composantes du tenseur de Ricci de la métrique $g$ sont
$R_{\alpha\beta}=\partial_{\gamma}\tilde{\Gamma}^{\gamma}_{\alpha\beta}-\partial_{\beta}\tilde{\Gamma}^{\gamma}_{\gamma\alpha}+\tilde{\Gamma}^{\gamma}_{\gamma\mu}\tilde{\Gamma}^{\mu}_{\alpha\beta}-\tilde{\Gamma}^{\gamma}_{\beta\mu}\tilde{\Gamma}^{\mu}_{\gamma\alpha}$
D’après la définition du tenseur $C$ et le fait que les Christoffels de la
métrique euclidienne $\Gamma^{\alpha}_{\beta\gamma}$ sont identiquement nuls,
on obtient l’expression suivante :
$R_{\alpha\beta}=\tilde{\nabla}_{\gamma}C^{\gamma}_{\alpha\beta}-\tilde{\nabla}_{\beta}C^{\gamma}_{\gamma\alpha}+C^{\gamma}_{\gamma\mu}C^{\mu}_{\alpha\beta}-C^{\gamma}_{\beta\mu}C^{\mu}_{\gamma\alpha}$
T. Aubin [Aub2] montre que cette expression de Ricci est encore valable si
$g=g_{0}+h$, où $g_{0}$ est une métrique riemannienne quelconque (pas
nécessairement la métrique euclidienne $\mathcal{E}$).
Dans le système de coordonnées $\\{r,\xi^{i}\\}$, l’expression du tenseur $C$
ci-dessus devient :
$R_{jl}=\partial_{r}C^{r}_{jl}+\tilde{\nabla}_{m}C^{m}_{jl}-\tilde{\nabla}_{j}C^{m}_{ml}+C^{m}_{mr}C^{r}_{jl}-C^{m}_{jr}C^{r}_{ml}-C^{r}_{jp}C^{p}_{rl}+C^{m}_{mp}C^{p}_{jl}-C^{m}_{jp}C^{p}_{ml}$
En utilisant la définition du tenseur $C$, on en déduit l’expression suivante
des composantes du tenseur de Ricci:
$\displaystyle
R_{jl}=-\frac{1}{2}\partial^{2}_{r}h_{jl}+\tilde{\nabla}_{m}C^{m}_{jl}-\frac{1}{4}g^{mp}\partial_{r}h_{mp}\partial_{r}h_{jl}+\frac{1}{2}\partial_{r}h_{ij}\partial_{r}h_{kl}g^{ik}+C^{i}_{ik}C^{k}_{jl}-C^{i}_{jk}C^{k}_{il}$
(A.3) $\displaystyle R_{rr}=-\partial_{r}C^{m}_{mr}-C_{rp}^{m}C_{mr}^{p}$
(A.4)
Si $h=O(r^{\omega+2})$ alors $R_{g}=O(r^{\omega})$. De plus, on peut calculer
$\bar{R}$ la partie principale de $R_{g}$. Pour cela, on doit se focaliser
uniquement sur les termes d’ordre $\omega$ dans l’expression de
$R_{g}=R_{r}r+g^{jl}R_{jl}$. Tous les termes de $R_{g}$ sont négligeables par
rapport à $r^{\omega}$, sauf à priori les deux termes suivants:
$-\frac{1}{2}g^{jl}(\partial_{rr}h_{jl}+\frac{n-1}{r}\partial_{r}h_{jl})=-\frac{1}{2}(\omega+2)(\omega+n)s^{jl}\bar{g}_{jl}r^{\omega}+o(r^{\omega})=o(r^{\omega})$
Finalement ce terme également est négligeable par rapport à $r^{\omega}$. Il
ne fera pas partie des termes de $\bar{R}$. Le second candidat est
$g^{jl}\tilde{\nabla}_{m}C^{m}_{jl}=(s^{jl}-h^{jl})\tilde{\nabla}_{m}[(s^{mp}-h^{mp})\tilde{\nabla}_{l}h_{jp}]+o(r^{\omega})$
car $g^{jl}\tilde{\nabla}_{p}h_{jl}=o(r^{\omega})$. Donc
$g^{jl}\tilde{\nabla}_{m}C^{m}_{jl}=s^{jl}s^{mp}\nabla_{ml}\bar{g}_{jp}r^{\omega}+o(r^{\omega})$.
On conclut que
$\bar{R}=\nabla^{jp}\bar{g}_{jp}r^{\omega}$ (A.5)
La première formule du théorème A.1 est démontrée. Par le lemme 4.1, on sait
que
$\bar{\int}_{S(r)}R_{g}\mathrm{d}\sigma_{r}=O(r^{2\omega+2})$
On cherche les termes d’ordre $2\omega+2$ de cette intégrale. En utilisant
l’expression (A.3) des composantes de $R_{jl}$, on a:
$\bar{\int}_{S(r)}R_{g}\mathrm{d}\sigma_{r}=\bar{\int}_{S(r)}R_{rr}+g^{jl}R_{jl}\mathrm{d}\sigma_{r}$
Ici encore, on doit se focaliser uniquement sur les termes d’ordre
$2\omega+2$, d’intégrales non nulles. On doit examiner sept intégrales, six
correspondent aux termes de $R_{jl}$ et une à $R_{rr}$. Les calculs suivants
sont à $o(r^{2\omega+2})$ près. On a $g^{ij}=s^{ij}-h^{ij}$ à
$o(r^{2\omega+2})$ près. Comme
$\int_{S_{n-1}}s^{ij}\hat{h}_{ij}\mathrm{d}\sigma=o(r^{2\omega+2})$, on n’aura
pas à se soucier des termes qui proviennent de $\tilde{h}_{ij}$. En se servant
des relations $s^{ij}\bar{g}_{ij}=0$ et
$\bar{g}^{jl}\bar{g}_{jl}=2s^{jl}\hat{g}_{jl}$, on trouve que l’intégrale
correspondant aux premiers termes de $R_{jl}$ donne:
$-\frac{1}{2}\bar{\int}_{S(r)}(s^{jl}-h^{jl})\partial^{2}_{r}h_{jl}\mathrm{d}\sigma_{r}=-\frac{(\omega+2)^{2}}{2}Qr^{2\omega+2}+o(r^{2\omega+2})$
où $Q=\bar{\int}_{S_{n-1}}\bar{g}^{jl}\bar{g}_{jl}\mathrm{d}\sigma$
et que l’intégrale correspondant au troisième terme de $R_{jl}$ est
$-\frac{1}{4}\bar{\int}_{S(r)}(s^{mp}-h^{mp})(s^{jl}-h^{jl})\partial_{r}h_{mp}\partial_{r}h_{jl}\mathrm{d}\sigma_{r}=o(r^{2\omega+2})$
L’intégrale correspondant au quatrième terme de $R_{jl}$ devient
$\frac{1}{2}\bar{\int}_{S(r)}s^{ik}s^{jl}\partial_{r}h_{ij}\partial_{r}h_{kl}\mathrm{d}\sigma_{r}=\frac{(\omega+2)^{2}}{2}Qr^{2\omega+2}+o(r^{2\omega+2})$
La dernière intégrale qui donne des termes du type $Qr^{2\omega+2}$ est
$\bar{\int}_{S(r)}R_{rr}\mathrm{d}\sigma_{r}=-\bar{\int}_{S(r)}\partial_{r}C^{m}_{mr}-C_{rp}^{m}C_{mr}^{p}\mathrm{d}\sigma_{r}=-\frac{(\omega+2)^{2}}{4}Qr^{2\omega+2}+o(r^{2\omega+2})$
où $C^{p}_{mr}$ et $C_{mp}^{r}$ sont définis par l’expression (A.2).
En utilisant la formule de Stokes (intégration par parties) et le fait que les
intégrales de type
$\bar{\int}_{S(r)}s^{jl}s^{mp}\nabla_{mj}h_{pl}\mathrm{d}\sigma_{r}=\bar{\int}_{S(r)}\nabla_{mj}h^{mj}\mathrm{d}\sigma_{r}=0$
sont nulles (ce sont des intégrales de la divergence d’un champ de vecteur),
l’intégrale correspondant au second terme de $R_{jl}$, se calcule de la façon
suivante:
$\begin{split}\bar{\int}_{S(r)}g^{jl}\tilde{\nabla}_{m}C^{m}_{jl}\mathrm{d}\sigma_{r}&=\frac{1}{2r^{2}}\bar{\int}_{S(r)}g^{jl}g^{mp}(\nabla_{mj}h_{pl}+\nabla_{ml}h_{pj}-\nabla_{mp}h_{jl})\\\
&+g^{jl}(\nabla_{m}g^{mp})(\nabla_{j}h_{pl}+\nabla_{l}h_{pj}-\nabla_{p}h_{jl})\mathrm{d}\sigma_{r}\\\
&=r^{2\omega+2}\bar{\int}_{S(r)}-s^{jl}\bar{g}^{mp}\nabla_{mj}\bar{g}_{pl}-\bar{g}^{jl}s^{mp}\nabla_{mj}\bar{g}_{pl}+\frac{1}{2}\bar{g}^{jl}s^{mp}\nabla_{mp}\bar{g}_{jl}\\\
&-s^{jl}\nabla_{m}\bar{g}^{mp}\nabla_{j}\bar{g}_{pl}\mathrm{d}\sigma_{r}+o(r^{2\omega+2})\\\
&=(B-\frac{C}{2})r^{2\omega+2}+o(r^{2\omega+2})\end{split}$
où l’on a posé
$B=\bar{\int}_{S_{n-1}}\nabla^{i}\bar{g}^{jk}\nabla_{j}\bar{g}_{ik}\mathrm{d}\sigma\text{
et
}C=\bar{\int}_{S_{n-1}}\nabla^{i}\bar{g}^{jk}\nabla_{i}\bar{g}_{jk}\mathrm{d}\sigma$
(A.6)
En utilisant $s^{ij}\bar{g}_{ij}=0$, et la définition des $C^{i}_{jk}$, on a
$C^{i}_{ik}=\frac{r^{\omega+2}}{2}g^{ip}(\nabla_{i}\bar{g}_{kp}+\nabla_{k}\bar{g}_{pi}-\nabla_{p}\bar{g}_{ik})+o(r^{\omega+2})=o(r^{\omega+2})$
L’intégrale correspondant au cinquième terme $R_{jl}$ vérifie donc
$\bar{\int}_{S(r)}g^{jl}C^{i}_{ik}C^{k}_{jl}\mathrm{d}\sigma_{r}=o(r^{2\omega+2})$
et est négligeable devant $r^{2\omega+2}$. Il nous reste à calculer
l’intégrale correspondant au sixième terme $R_{jl}$.
$\begin{split}-\bar{\int}_{S(r)}g^{jl}C^{m}_{jp}C^{p}_{ml}\mathrm{d}\sigma_{r}&=-\frac{r^{2\omega+2}}{4}\bar{\int}_{S(r)}(\nabla^{l}\bar{g}^{mk}+\nabla^{k}\bar{g}^{lm}-\nabla^{m}\bar{g}^{kl})\\\
&\hskip
56.9055pt\times(\nabla_{m}\bar{g}_{lk}+\nabla_{l}\bar{g}_{mk}-\nabla_{k}\bar{g}_{ml})\mathrm{d}\sigma_{r}+o(r^{2\omega+2})\\\
&=(\frac{C}{4}-\frac{B}{2})r^{2\omega+2}+o(r^{2\omega+2})\end{split}$
Finalement
$\bar{\int}_{S(r)}R_{g}\mathrm{d}\sigma_{r}=(B/2-C/4-(1+\omega/2)^{2}\bar{Q})r^{2\omega+2}+o(r^{2\omega+2})$
(A.7)
∎
### Preuve du lemme 4.2
Rappelons l’énoncé de ce lemme:
###### Lemme A.1.
$\nabla^{i}b_{ij}=-\sum_{k=1}^{q}\nabla_{j}\varphi_{k},\;\bar{R}=\bar{R}_{b}=\nabla^{ij}b_{ij}r^{\omega},\;\bar{R}_{a}=\nabla^{ij}a_{ij}r^{\omega}=0\text{
et }s^{ij}b_{ij}=s^{ij}a_{ij}=0$
###### Preuve.
Les $b_{ij}$ sont définis comme suit:
$b_{ij}=\sum_{k=1}^{q}\frac{1}{(n-2)(\lambda_{k}+1-n)}[(n-1)\nabla_{ij}\varphi_{k}+\lambda_{k}\varphi_{k}s_{ij}]$
En contractant par $\nabla^{i}$
$\nabla^{i}b_{ij}=\sum_{k=1}^{q}\frac{1}{(n-2)(\lambda_{k}+1-n)}[(n-1)s^{im}\nabla_{mj}\nabla_{i}\varphi_{k}+\lambda_{k}\nabla_{j}\varphi_{k}]$
(A.8)
D’après la définition du tenseur de courbure de Riemann (voir section 1.1), on
a
$\nabla_{mj}\nabla_{i}\varphi_{k}=\nabla_{jm}\nabla_{i}\varphi_{k}-R^{l}_{imj}\nabla_{l}\varphi_{k}$
avec
$R_{lijm}=s_{lj}s_{im}-s_{lm}s_{ij},\quad
R^{l}_{imj}=\delta^{l}_{m}s_{ij}-\delta_{j}^{l}s_{mi}$ (A.9)
qui sont les composantes du tenseur de courbure de Riemann de la sphère
$S_{n-1}$ muni de la métrique standard $s$. Ici les $\varphi_{k}$ sont des
fonctions propres du Laplacien sur la sphère (il ne faut pas les confondre
avec les composantes d’un tenseur une fois covariant).
D’après les propriétés 1.1, on en déduit que
$\nabla_{mj}\nabla_{i}\varphi_{k}-\nabla_{jm}\nabla_{i}\varphi_{k}=R_{imj}^{l}\nabla_{l}\varphi_{k}$
Puisque $\Delta\varphi_{k}=-s^{im}\nabla_{mi}\varphi_{k}$,
$s^{im}\nabla_{mj}\nabla_{i}\varphi_{k}=-\nabla_{j}\Delta\varphi_{k}+(n-2)\nabla_{j}\varphi_{k}=-(\lambda_{k}-n+2)\nabla_{j}\varphi_{k}$
qu’on substitue dans l’équation (A.8). On trouve
$\nabla^{i}b_{ij}=-\sum_{k=1}^{q}\nabla_{j}\varphi_{k}$ (A.10)
La première formule est démontrée. Pour la seconde, il suffit de calculer
$\nabla^{ij}b_{ij}=-\sum_{k=1}^{q}\nabla^{j}_{j}\varphi_{k}=\sum_{k=1}^{q}\Delta_{s}\varphi_{k}=r^{-\omega}\bar{R}$
d’après l’expression 4.5 qui définit $\bar{R}$. D’autre part, d’après le
théorème 4.1
$\bar{R}=\nabla^{ij}\bar{g}_{ij}r^{\omega}=\nabla^{ij}a_{ij}r^{\omega}+\nabla^{ij}b_{ij}r^{\omega}$
On en conclut que $\nabla^{ij}a_{ij}=0$.
Les deux dernières identités se déduisent aisément de la relation
$s^{ij}\bar{g}_{ij}=0$ et de la définition des $b_{ij}$. ∎
### Preuve du lemme 4.3
Rappelons d’abord la définition des intégrales $Q_{b}$, $B_{b}$ et $C_{b}$:
$Q_{b}=\bar{\int}_{S_{n-1}}b_{ij}b^{ij}\mathrm{d}\sigma,\;B_{b}=\bar{\int}_{S_{n-1}}\nabla^{i}b^{jk}\nabla_{j}b_{ik}\mathrm{d}\sigma\text{
et
}C_{b}=\bar{\int}_{S_{n-1}}\nabla^{i}b^{jk}\nabla_{i}b_{jk}\mathrm{d}\sigma$
On commence par démontrer les formules suivantes (voir équations (4.1), (4.1)
et (4.1)) :
$\displaystyle
Q_{b}=\bar{\int}_{S_{n-1}}b_{ij}b^{ij}\mathrm{d}\sigma=\frac{n-1}{n-2}\sum_{k=1}^{q}\frac{\lambda_{k}}{\lambda_{k}-n+1}\bar{\int}_{S_{n-1}}\varphi_{k}^{2}\mathrm{d}\sigma$
$\displaystyle
B_{b}=-(n-1)Q_{b}+\sum_{k=1}^{q}\lambda_{k}\bar{\int}_{S_{n-1}}\varphi_{k}^{2}\mathrm{d}\sigma$
$\displaystyle
C_{b}=-(n-1)Q_{b}+\frac{n-1}{n-2}\sum_{k=1}^{q}\lambda_{k}\bar{\int}_{S_{n-1}}\varphi_{k}^{2}\mathrm{d}\sigma$
où les $b_{ij}$ sont donnés par
$b_{ij}=\sum_{k=1}^{q}\frac{1}{(n-2)(\lambda_{k}+1-n)}[(n-1)\nabla_{ij}\varphi_{k}+\lambda_{k}\varphi_{k}s_{ij}]$
(A.11)
Concernant l’intégrale $Q_{b}$, par une intégration par parties et le fait que
$s^{ij}b_{ij}=0$ (voir lemme A.1), on obtient
$\bar{\int}_{S_{n-1}}b^{ij}b_{ij}\mathrm{d}\sigma=\sum_{k=1}^{q}\frac{1}{(n-2)(\lambda_{k}+1-n)}\bar{\int}_{S_{n-1}}-(n-1)\nabla^{j}\varphi_{k}\nabla^{i}b_{ij}\mathrm{d}\sigma$
D’après (A.10) (rappelons que $\Delta\varphi_{k}=\lambda_{k}\varphi_{k}$),
l’égalité (A) est démontrée.
Montrons la formule (A). Par définition
$B_{b}=\bar{\int}_{S_{n-1}}\nabla^{i}b^{jk}\nabla_{j}b_{ik}\mathrm{d}\sigma=-\bar{\int}_{S_{n-1}}b^{jk}s^{li}\nabla_{lj}b_{ik}\mathrm{d}\sigma$
On permute les dérivées covariantes dans $\nabla_{lj}b_{ik}$, ensuite on
utilise (A.10), pour avoir
$s^{li}\nabla_{lj}b_{ik}=s^{li}(\nabla_{jl}b_{ik}-R_{ilj}^{m}b_{mk}-R_{klj}^{m}b_{im})=-\sum_{l=1}^{q}\nabla_{jk}\varphi_{l}+(n-1)b_{jk}$
(A.12)
on a utilisé (A.9) et le fait que $s^{ij}b_{ij}=0$. En reprenant la dernière
expression de $B_{b}$, on en déduit que
$B_{b}=-(n-1)Q_{b}-\sum_{l=1}^{q}\bar{\int}_{S_{n-1}}\nabla_{j}b^{jk}\nabla_{k}\varphi_{l}\mathrm{d}\sigma=-(n-1)Q_{b}+\sum_{l=1}^{q}\sum_{p=1}^{q}\bar{\int}_{S_{n-1}}\nabla^{k}\varphi_{p}\nabla_{k}\varphi_{l}\mathrm{d}\sigma$
Sachant que $\\{\varphi_{l}\\}_{1\leq l\leq q}$ est une famille de fonctions
orthogonales pour le produit scalaire dans $L^{2}$ et celui de
$H_{1}(S_{n-1})$ (voir équations (4.3),(4.4), page 4.3), l’égalité (A) est
démontrée. Pour montrer l’égalité (A), on établit d’abord l’identité suivante:
$\nabla_{i}b_{jk}=\nabla_{j}b_{ik}+\frac{1}{n-2}(\nabla^{m}b_{jm}s_{ik}-\nabla^{m}b_{im}s_{jk})$
(A.13)
En effet, en utilisant (A.11), (A.10) et (A.9), on obtient
$\begin{split}\nabla_{i}b_{jk}&=\sum_{l=1}^{q}\frac{1}{(n-2)(\lambda_{l}+1-n)}[(n-1)\nabla_{ij}\nabla_{k}\varphi_{l}+\lambda_{l}\nabla_{i}\varphi_{l}s_{jk}]\\\
&=\sum_{l=1}^{q}\frac{1}{(n-2)(\lambda_{l}+1-n)}[(n-1)\nabla_{ji}\nabla_{k}\varphi_{l}-(n-1)R_{kij}^{m}\nabla_{m}\varphi_{l}+\lambda_{l}\nabla_{i}\varphi_{l}s_{jk}]\\\
&=\nabla_{j}b_{ik}+\sum_{l=1}^{q}\frac{1}{n-2}[\nabla_{i}\varphi_{l}s_{jk}-\nabla_{j}\varphi_{l}s_{ik}]\end{split}$
Alors
$C_{b}=\bar{\int}_{S_{n-1}}\nabla^{i}b^{jk}\nabla_{i}b_{jk}\mathrm{d}\sigma=B_{b}+\frac{1}{n-2}\bar{\int}_{S_{n-1}}\nabla_{i}b^{ji}\nabla^{m}b_{jm}\mathrm{d}\sigma$
Si on substitue (A.10) et (A) dans la dernière égalité, on trouve l’expression
(A).
Rappelons l’énoncé du lemme 4.3:
###### Lemme A.2.
Si $\mu=\omega$ et $\bar{g}_{ij}=a_{ij}+b_{ij}$, alors
$\bar{\int}_{S(r)}R\mathrm{d}\sigma_{r}=\bar{\int}_{S(r)}R_{a}+R_{b}\mathrm{d}\sigma_{r}+o(r^{2(\omega+1)})\leq[B_{b}/2-C_{b}/4-(1+\omega/2)^{2}Q_{b}]r^{2(\omega+1)}+o(r^{2(\omega+1)})$
$B_{b}/2-C_{b}/4-(1+\omega/2)^{2}Q_{b}=\sum_{k=1}^{q}u_{k}\bar{\int}_{S_{n-1}}\varphi_{k}^{2}\mathrm{d}\sigma$
avec
$u_{k}=\biggl{(}\frac{n-3}{4(n-2)}-\frac{(n-1)^{2}+(n-1)(\omega+2)^{2}}{4(n-2)(\nu_{k}-n+1)}\biggr{)}\nu_{k}$
###### Preuve.
D’après le lemme 4.2 (démontré ci-dessus):
$r^{-\omega}\bar{R}=\nabla^{ij}\bar{g}_{ij}=\nabla^{ij}b_{ij}\text{ et
}\nabla^{ij}a_{ij}=s^{ij}a_{ij}=s^{ij}b_{ij}=0$ (A.14)
Montrons que $Q=Q_{a}+Q_{b}$, $B=B_{a}+B_{b}$ et $C=C_{a}+C_{b}$.
$Q=\bar{\int}_{S_{n-1}}(a^{ij}+b^{ij})(a_{ij}+b_{ij})\mathrm{d}\sigma=Q_{a}+Q_{b}+2\bar{\int}_{S_{n-1}}a^{ij}b_{ij}\mathrm{d}\sigma$
On a
$a^{ij}b_{ij}=\sum_{k=1}^{q}\frac{1}{(n-2)(\lambda_{k}+1-n)}a^{ij}[(n-1)\nabla_{ij}\varphi_{k}+\lambda_{k}\varphi_{k}s_{ij}]$
En intégrant sur $S_{n-1}$ l’expression ci-dessus et en utilisant les
relations A.14, on en déduit que
$\bar{\int}_{S_{n-1}}a^{ij}b_{ij}\mathrm{d}\sigma=0\quad\text{et que
}Q=Q_{a}+Q_{b}$ (A.15)
Par un raisonnement analogue au précédent, montrons que $B=B_{a}+B_{b}$.
D’après la définition de $B$ (voir A.6), on a
$B=\bar{\int}_{S_{n-1}}\nabla^{i}(a^{jk}+b^{jk})\nabla_{j}(a_{ik}+b_{ik})\mathrm{d}\sigma=B_{a}+B_{b}+2\bar{\int}_{S_{n-1}}\nabla^{i}a^{jk}\nabla_{j}b_{ik}\mathrm{d}\sigma$
Par une intégration par parties, on obtient
$B=B_{a}+B_{b}-2\bar{\int}_{S_{n-1}}a^{jk}\nabla^{i}\nabla_{j}b_{ik}\mathrm{d}\sigma$
En utilisant l’identité (A.12), écrite sous la forme suivante
$\nabla^{i}\nabla_{j}b_{ik}=-\sum_{l=1}^{q}\nabla_{jk}\varphi_{l}+(n-1)b_{jk}$
et les relations (A.14), (A.15), on en conclut que
$\bar{\int}_{S_{n-1}}a^{jk}\nabla^{i}\nabla_{j}b_{ik}\mathrm{d}\sigma=0\text{
et }B=B_{a}+B_{b}$ (A.16)
La dernière formule à établir est $C=C_{a}+C_{b}$. Or, d’après la définition
de $C$ (voir (A.6)),
$C=\bar{\int}_{S_{n-1}}\nabla^{i}(a^{jk}+b^{jk})\nabla_{i}(a_{jk}+b_{jk})\mathrm{d}\sigma=C_{a}+C_{b}-2\bar{\int}_{S_{n-1}}a^{jk}\nabla^{i}\nabla_{i}b_{jk}\mathrm{d}\sigma$
D’après l’identité (A.13)
$\begin{split}\nabla^{i}\nabla_{i}b_{jk}&=\nabla^{i}\nabla_{j}b_{ik}+\frac{1}{n-2}(\nabla^{i}\nabla^{m}b_{jm}s_{ik}-\nabla^{i}\nabla^{m}b_{im}s_{jk})\\\
&=\nabla^{i}\nabla_{j}b_{ik}-\frac{1}{n-2}\sum_{l=1}^{q}(\nabla_{kj}\varphi_{l}+\lambda_{l}\varphi_{l}s_{jk})\end{split}$
Ici on a juste utilisé l’expression (A.10) et le fait que
$\Delta\varphi_{l}=\lambda\varphi_{l}$. En contractant cette expression de
$\nabla^{i}\nabla_{i}b_{jk}$ avec $a^{jk}$, en utilisant (A.16) et les
relations (A.14), on en conclut que
$\bar{\int}_{S_{n-1}}a^{jk}\nabla^{i}\nabla_{i}b_{jk}\mathrm{d}\sigma=0$ et
$C=C_{a}+C_{b}$.
D’après le théorème 4.1
$\bar{\int}_{S(r)}R\mathrm{d}\sigma_{r}=[B/2-C/4-(1+\omega/2)^{2}Q]r^{2(\omega+1)}+o(r^{2(\omega+1)})$
et par ce qu’on vient de prouver, on en déduit que
$\bar{\int}_{S(r)}R\mathrm{d}\sigma_{r}=\bar{\int}_{S(r)}R_{a}+R_{b}\mathrm{d}\sigma_{r}+o(r^{2\omega+2})$
Comme $\nabla^{ij}a_{ij}=0$, $\bar{R}_{a}=0$, l’ordre de la partie $R_{a}$ est
donc supérieur à $\omega+1$. D’après le théorème 4.2,
$\bar{\int}_{S(r)}R_{a}\mathrm{d}\sigma_{r}\leq 0$. D’où l’inégalité du lemme.
∎
### Détails des calculs du théorème 4.3
On commence par rappeller les définitions données dans la section 4.1.
$I_{a}^{b}(\varepsilon)=\int_{0}^{\delta/\varepsilon}\frac{t^{b}}{(1+t^{2})^{a}}\mathrm{d}t\text{
et }I_{a}^{b}=\lim_{\varepsilon\to 0}I_{a}^{b}(\varepsilon)$ (A.17)
alors
$I_{a}^{b}(\varepsilon)=\begin{cases}I_{a}^{b}+O(\varepsilon^{2a-b-1})\text{
si }2a-b>1\\\ \log\varepsilon^{-1}+O(1)\text{ si }b=2a-1\end{cases}$ (A.18)
En effet, si $2a-b>1$,
$I_{a}^{b}-I_{a}^{b}(\varepsilon)=\int_{\delta/\varepsilon}^{+\infty}\frac{t^{b}}{(1+t^{2})^{a}}\mathrm{d}t\leq\int_{\delta/\varepsilon}^{+\infty}t^{b-2a}\mathrm{d}t\leq\frac{\varepsilon^{2a-1-b}}{(2a-1-b)\delta^{2a-b-1}}$
Si $b=2a-1$ alors pour $\varepsilon$ suffisamment petit
$I_{a}^{2a-1}(\varepsilon)\leq\int_{0}^{1}\frac{t^{2a-1}}{(1+t^{2})^{a}}\mathrm{d}t+\int_{1}^{\delta/\varepsilon}\frac{1}{t}\mathrm{d}t$
Par des intégrations par parties, on établit les relations suivantes :
$I_{a}^{b}=\frac{b-1}{2a-b-1}I_{a}^{b-2}=\frac{b-1}{2a-2}I_{a-1}^{b-2}=\frac{2a-b-3}{2a-2}I_{a-1}^{b},\quad\frac{4(n-2)I_{n}^{n+1}}{(I^{n-2}_{n})^{(n-2)/n}}=n$
(A.19)
Soit $\varphi_{\varepsilon}$ une fonction test définie dans (4.7) (voir page
4.7). On calcule $I_{g}(\varphi_{\varepsilon})$. En utilisant l’inégalité
$(a-b)^{\beta}\geq a^{\beta}-\beta a^{\beta-1}b$ pour $0<b<a$, on a $\beta\geq
2$, $0\leq\alpha<(n-2)(\beta-1)-n$
$\int_{M}r^{\alpha}u_{\varepsilon}^{\beta}\mathrm{d}v=\omega_{n-1}\int_{0}^{\delta}r^{\alpha+n-1}u_{\varepsilon}^{\beta}(r)\mathrm{d}r=\omega_{n-1}I_{(n-2)\beta/2}^{\alpha+n-1}\varepsilon^{\alpha+n-\beta(n-2)/2}+O(\varepsilon^{n-2})$
(A.20)
Ce type d’intégrales apparait plusieurs fois dans les calculs suivants, il
permet de négliger le terme constant dans l’expression de $u_{\varepsilon}$,
définie dans (4.7), lorsque l’on choisit $\delta$ suffisamment petit et
$\varepsilon$ plus petit que $\delta$. On commence par calculer
$\|\nabla\varphi_{\varepsilon}\|^{2}$ (la définition de
$\varphi_{\varepsilon}$ est donnée dans la section 5.3). D’après la formule
$|\nabla_{g}\varphi_{\varepsilon}|^{2}=(\partial_{r}\varphi_{\varepsilon})^{2}+r^{-2}|\nabla_{s}\varphi_{\varepsilon}|^{2}$
on a l’équation (4.11) suivante:
$\int_{M}|\nabla_{g}\varphi_{\varepsilon}|^{2}\mathrm{d}v=\int_{M}|\nabla_{g}u_{\varepsilon}|^{2}\mathrm{d}v+\int_{0}^{\delta}[\partial_{r}(r^{(\omega+2)}u_{\varepsilon})]^{2}r^{n-1}\mathrm{d}r\int_{S_{n-1}}f^{2}\mathrm{d}\sigma+\\\
\int_{0}^{\delta}u^{2}_{\varepsilon}r^{n+2\omega+1}\mathrm{d}r\int_{S_{n-1}}|\nabla
f|^{2}\mathrm{d}\sigma$
On exprime les intégrales ci-dessus, en utilisant les intégrales $I_{b}^{a}$,
définies plus haut. On effectue le changement de variable $t=r/\varepsilon$.
Ce qui donne les expressions suivantes
$\displaystyle\int_{M}|\nabla_{g}u_{\varepsilon}|^{2}\mathrm{d}v=(n-2)^{2}\omega_{n-1}I_{n}^{n+1}+O(\varepsilon^{n-2})\text{
et }$
$\displaystyle\int_{0}^{\delta}u^{2}_{\varepsilon}r^{n+2\omega+1}\mathrm{d}r\int_{S_{n-1}}|\nabla
f|^{2}\mathrm{d}\sigma=I^{n+2\omega+1}_{n-2}\|\nabla_{s}f\|^{2}$
$\begin{split}\int_{0}^{\delta}[\partial_{r}(r^{\omega+2}u_{\varepsilon})]^{2}r^{n-1}\mathrm{d}r\int_{S_{n-1}}f^{2}\mathrm{d}\sigma&=\|f\|^{2}\int_{0}^{\delta}\varepsilon^{n-2}\biggl{(}\frac{(\omega-n+4)r^{\omega+3}+\varepsilon^{2}(\omega+2)r^{\omega+1}}{(\varepsilon^{2}+r^{2})^{n/2}}\biggr{)}^{2}r^{n-1}\mathrm{d}r\\\
&=[(\omega-n+4)^{2}I_{n}^{2\omega+n+5}(\varepsilon)+2(\omega+2)(\omega-n+4)I_{n}^{2\omega+n+3}(\varepsilon)\\\
&+(\omega+2)^{2}I_{n}^{2\omega+n+1}(\varepsilon)]\|f\|^{2}\varepsilon^{2\omega+4}+o(\varepsilon^{2\omega+4})\end{split}$
Si on regroupe ensemble ces trois intégrales, on obtient (4.12):
$\int_{M}|\nabla_{g}\varphi_{\varepsilon}|^{2}\mathrm{d}v=(n-2)^{2}\omega_{n-1}I_{n}^{n+1}(\varepsilon)+\varepsilon^{2\omega+4}\biggl{\\{}\int_{S_{n-1}}|\nabla
f|^{2}\mathrm{d}\sigma I_{n-2}^{2\omega+n+1}(\varepsilon)+\\\
\int_{S_{n-1}}f^{2}\mathrm{d}\sigma[(\omega-n+4)^{2}I_{n}^{2\omega+n+5}(\varepsilon)\\\
+2(\omega+2)(\omega-n+4)I_{n}^{2\omega+n+3}(\varepsilon)+(\omega+2)^{2}I_{n}^{2\omega+n+1}(\varepsilon)]\biggr{\\}}$
(A.21)
Pour avoir (4.14) (page 4.14), il suffit d’écrire le développement limité de
$\varphi_{\varepsilon}^{N}$ et ensuite utiliser l’égalité (A.20).
$\|\varphi_{\varepsilon}\|_{N}^{-2}=(\omega_{n-1}I^{n-1}_{n})^{-2/N}\bigl{\\{}1+\\\
-(N-1)\varepsilon^{2(\omega+2)}\int_{S_{n-1}}f^{2}\mathrm{d}\sigma
I_{n}^{2\omega+n+3}/(\omega_{n-1}I^{n-1}_{n})\bigr{\\}}+O(\varepsilon^{\min(3\omega+6,n-2)})$
(A.22)
Il nous reste seulement à calculer
$\int_{M}R_{g}\varphi_{\varepsilon}^{2}\mathrm{d}v$. La fonction $f$ est
définie sur la sphère $S_{n-1}$. On sait qu’on peut la définir sur $S(r)$ pour
tout $r>0$ en posant $f(\xi/r)$ si $\xi\in S(r)$. On garde la même notation
pour la fonction ainsi redéfinie. D’après le lemme 4.1, on sait que
$\bar{\int}_{S(r)}r^{-2\omega-2}R_{g}\mathrm{d}\sigma=O(1)$, on en déduit, en
effectuant le changement de variable $t=r/\varepsilon$, que
$\begin{split}\int_{M}R_{g}u_{\varepsilon}^{2}\mathrm{d}v&=\varepsilon^{2\omega+4}\omega_{n-1}\bar{\int}_{S(r)}r^{-2\omega-2}R_{g}\mathrm{d}\sigma
I_{n-2}^{n+2\omega+1}(\varepsilon)\\\
&=\begin{cases}\varepsilon^{2\omega+4}\omega_{n-1}\bar{\int}_{S(r)}r^{-2\omega-2}R_{g}\mathrm{d}\sigma
I_{n-2}^{n+2\omega+1}+o(\varepsilon^{2\omega+4})\text{ si }n>2\omega+6\\\
\varepsilon^{2\omega+4}\log\varepsilon^{-1}\omega_{n-1}\bar{\int}_{S(r)}r^{-2\omega-2}R_{g}\mathrm{d}\sigma+O(\varepsilon^{2\omega+4})\text{
si }n=2\omega+6\end{cases}\end{split}$
D’autre part $R=\bar{R}+o(r^{\mu})$ avec $\mu\geq\omega$ (cf. lemme 4.1), d’où
$\begin{split}\int_{M}fu_{\varepsilon}^{2}R_{g}r^{\omega+2}\mathrm{d}v&=\varepsilon^{\omega+\mu+4}I_{n-2}^{\omega+\mu+n+1}(\varepsilon)\omega_{n-1}\bar{\int}_{S(r)}r^{-\mu}f(\xi)\bar{R}\mathrm{d}\sigma+o(\varepsilon^{\omega+\mu+4})\\\
&=\begin{cases}\varepsilon^{\omega+\mu+4}I_{n-2}^{\omega+\mu+n+1}\omega_{n-1}\bar{\int}_{S(r)}r^{-\mu}f(\xi)\bar{R}\mathrm{d}\sigma+o(\varepsilon^{\omega+\mu+4})\text{
si }n-6>\omega+\mu\\\
\varepsilon^{\omega+\mu+4}\log\varepsilon^{-1}\omega_{n-1}\bar{\int}_{S(r)}r^{-\mu}f(\xi)\bar{R}\mathrm{d}\sigma+O(\varepsilon^{\omega+\mu+4})\text{
si }n-6=\omega+\mu\end{cases}\end{split}$
Si $n>\omega+\mu+6$ alors
$\begin{split}\int_{M}R_{g}\varphi_{\varepsilon}^{2}\mathrm{d}v&=\int_{M}R_{g}u_{\varepsilon}^{2}\mathrm{d}v-2\int_{M}fu_{\varepsilon}^{2}R_{g}r^{\omega+2}\mathrm{d}v+\int_{M}f^{2}u_{\varepsilon}^{2}R_{g}r^{2\omega+4}\mathrm{d}v\\\
&=\varepsilon^{2\omega+4}\omega_{n-1}\bar{\int}_{S(r)}r^{-2\omega-2}R_{g}\mathrm{d}\sigma
I_{n-2}^{n+2\omega+1}-\\\
&2\varepsilon^{\omega+\mu+4}I_{n-2}^{\omega+\mu+n+1}\omega_{n-1}\bar{\int}_{S(r)}r^{-\mu}f(\xi)\bar{R}\mathrm{d}\sigma(\xi)+o(\varepsilon^{2\omega+4})\\\
\end{split}$ (A.23)
Si $n=2\omega+6$ et $\mu=\omega$ alors
$\int_{M}R_{g}\varphi_{\varepsilon}^{2}\mathrm{d}v=\varepsilon^{2\omega+4}\log\varepsilon^{-1}\omega_{n-1}\\{\bar{\int}_{S(r)}r^{-2\omega-2}R_{g}\mathrm{d}\sigma-2\bar{\int}_{S(r)}r^{-\mu}f(\xi)\bar{R}\mathrm{d}\sigma(\xi)\\}+O(\varepsilon^{2\omega+4})$
(A.24)
Rappelons que
$I_{g}(\varphi_{\varepsilon})=\biggl{(}\int_{M}|\nabla\varphi_{\varepsilon}|^{2}\mathrm{d}v+\frac{n-2}{4(n-1)}\int_{M}R_{g}\varphi_{\varepsilon}^{2}\mathrm{d}v\biggr{)}\|\varphi_{\varepsilon}\|_{N}^{-2}$
Maintenant, on a tout les ingrédients nécessaires pour donner l’expression
détaillée de $I_{g}(\varphi_{\varepsilon})$. On l’obtient, en combinant
(A.21), (A.22), (A.23) et (A.24) et le lemme A.3 ci-dessous. On en conclut que
si $n>2\omega+6$ alors
$I_{g}(\varphi_{\varepsilon})=\frac{n(n-2)}{4}\omega_{n-1}^{2/n}+(\omega_{n-1}I^{n-1}_{n})^{-2/N}I_{n-2}^{n+2\omega+1}\varepsilon^{2\omega+4}\times\\\
\biggl{\\{}\frac{(n-2)\omega_{n-1}}{4(n-1)}\bar{\int}_{S(r)}r^{-2\omega-2}R_{g}\mathrm{d}\sigma-\frac{n-2}{2(n-1)}\int_{S_{n-1}}f(\xi)\bar{R}\mathrm{d}\sigma+\int_{S_{n-1}}|\nabla
f|^{2}\mathrm{d}\sigma+\\\
-\frac{n(n-2)^{2}-(\omega+2)^{2}(n^{2}+n+2)}{(n-1)(n-2)}\int_{S_{n-1}}f^{2}\mathrm{d}\sigma\biggr{\\}}+o(\varepsilon^{2\omega+4)})$
si $n=2\omega+6$ alors
$I_{g}(\varphi_{\varepsilon})=\frac{n(n-2)}{4}\omega_{n-1}^{2/n}+(\omega_{n-1}I^{n-1}_{n})^{-2/N}\varepsilon^{2\omega+4}\log\varepsilon^{-1}\times\\\
\biggl{\\{}\frac{(n-2)\omega_{n-1}}{4(n-1)}\bar{\int}_{S(r)}r^{-2\omega-2}R_{g}\mathrm{d}\sigma-\frac{n-2}{2(n-1)}\int_{S_{n-1}}f(\xi)\bar{R}\mathrm{d}\sigma+\\\
\int_{S_{n-1}}|\nabla
f|^{2}\mathrm{d}\sigma+(\omega+2)^{2}\int_{S_{n-1}}f^{2}\mathrm{d}\sigma\biggr{\\}}+O(\varepsilon^{2\omega+4})$
###### Lemme A.3.
On a les relations suivantes pour tout $n>2\omega+6$:
$(\omega-n+4)^{2}I_{n}^{2\omega+n+5}+2(\omega+2)(\omega-n+4)I_{n}^{2\omega+n+3}+(\omega+2)^{2}I_{n}^{2\omega+n+1}\\\
-(N-1)(n-2)^{2}\frac{I_{n}^{2\omega+n+3}I_{n}^{n+1}}{I_{n}^{n-1}}=-\frac{n(n-2)^{2}-(\omega+2)^{2}(n^{2}+n+2)}{(n-1)(n-2)}I_{n-2}^{n+2\omega+1}$
Si $n=2\omega+6$ alors
$(\omega-n+4)^{2}I_{n}^{2\omega+n+5}(\varepsilon)+2(\omega+2)(\omega-n+4)I_{n}^{2\omega+n+3}(\varepsilon)+(\omega+2)^{2}I_{n}^{2\omega+n+1}(\varepsilon)\\\
-(N-1)(n-2)^{2}\frac{I_{n}^{2\omega+n+3}(\varepsilon)I_{n}^{n+1}}{I_{n}^{n-1}}=(\omega+2)^{2}\log\varepsilon^{-1}+O(1)$
Ces relations apparaissent dans l’expression de
$I_{g}(\varphi_{\varepsilon})$, comme étant le coefficient du terme
$\int_{S_{n-1}}f^{2}\mathrm{d}\sigma$.
###### Preuve.
Si $n=2\omega+6$ alors
$I_{n}^{2\omega+n+3}(\varepsilon)=I_{n}^{2\omega+n+3}+O(\varepsilon^{n-2})$,
$I_{n}^{2\omega+n+1}(\varepsilon)=I_{n}^{2\omega+n+1}+O(\varepsilon^{n-2})$ et
$I_{n}^{2\omega+n+5}(\varepsilon)=\log\varepsilon^{-1}+O(1)$ (cf. équation
(A.18)); la deuxième expression du lemme est démontrée.
Maintenant, on suppose que $n>2\omega+6$. En utilisant les relations (A.19),
on trouve
$\displaystyle I_{n}^{2\omega+n+5}$
$\displaystyle=\frac{(2\omega+n+4)(2\omega+n+2)}{4(n-1)(n-2)}I_{n-2}^{n+2\omega+1}\qquad
I_{n}^{2\omega+n+3}$
$\displaystyle=\frac{(2\omega+n+2)(n-2\omega-6)}{4(n-1)(n-2)}I_{n-2}^{n+2\omega+1}$
$\displaystyle I_{n}^{2\omega+n+1}$
$\displaystyle=\frac{(n-2\omega-4)(n-2\omega-6)}{4(n-1)(n-2)}I_{n-2}^{n+2\omega+1}\qquad
I_{n}^{n+1}$ $\displaystyle=\frac{n}{n-2}I^{n-1}_{n}$
Il suffit de montrer que le polynôme $P_{2}$, défini pour tout
$\omega\in\mathbb{N}$ par
$P_{2}(\omega+2)=(\omega-n+4)^{2}(2\omega+n+4)(2\omega+n+2)+2(\omega+2)(\omega-n+4)(2\omega+n+2)(n-2\omega-6)\\\
+(\omega+2)^{2}(n-2\omega-4)(n-2\omega-6)-n(n+2)(2\omega+n+2)(n-2\omega-6)$
est de degré 2 et est égal à
$P_{2}(\omega+2)=4(\omega+2)^{2}(n^{2}+n+2)-4n(n-2)^{2}$
En effet, on vérifie aisément que les termes de degré 4 se simplifient et que
$P_{2}(-X)=P_{2}(X)$, alors $P_{2}$ est pair de degré 2. On en déduit que
$P_{2}(X)=a_{n}X^{2}+b_{n}$, où $b_{n}=P_{2}(0)=-4n(n-2)^{2}$ et
$a_{n}=P_{2}^{\prime\prime}(0)/2=4(n^{2}+n+2)$ ∎
### Théorème 4.2
Dans son article [Aub3], T. Aubin démontre le résultat suivant:
###### Théorème A.2.
Si $\mu\geq\omega+1$ alors il existe une constante $C(n,\omega)>0$ telle que
$\bar{\int}_{S(r)}R\mathrm{d}\sigma_{r}=C(n,\omega)(-\Delta_{g})^{\omega+1}R(P)r^{2\omega+2}+o(r^{2\omega+2})$
$(-\Delta_{g})^{\omega+1}R(P)$ est strictement négative et
$I_{g}(u_{\varepsilon})<\frac{n(n-2)}{4}\omega_{n-1}^{2/n}$.
où $u_{\varepsilon}$ est définie dans la section 5.3 (voir équation (4.8)).
Tout d’abord, remarquons que si $\bar{\int}_{S(r)}R\mathrm{d}\sigma_{r}<0$,
d’après ce qui a été fait à la section 5.3, il suffit de prendre $f=0$ pour
que $\varphi_{\varepsilon}=u_{\varepsilon}$. L’inégalité
$I_{g}(u_{\varepsilon})<\frac{n(n-2)}{4}\omega_{n-1}^{2/n}$
est une conséquence immédiate des inégalités (4.15), (4.16).
Il suffit de montrer que $(-\Delta_{g})^{\omega+1}R(P)<0$. Pour cela, T. Aubin
donne un schéma assez détaillé de la preuve. Le cas $\omega=1$ ou $2$ sont des
conséquences des travaux de E. Hebey et M. Vaugon [HV]. Le cas $\omega=3$ est
fait par L. Zhang (communication privée). La méthode de T. Aubin marche pour
$\omega$ quelconque. Notons par $SymT$ le symétrisé du tenseur $T$ par rapport
à tout ses indices, et par $C(2,2)$ l’application de contraction des indices
deux à deux (voir la remarque de la section 5.2 pour des exemples). On pose
$\displaystyle A$
$\displaystyle=C(2,2)Sym\nabla_{\alpha}R_{pijq}\nabla_{\beta}R_{pq}$
$\displaystyle
B=C(2,2)Sym\nabla_{\alpha}R_{pijq}\nabla_{\tilde{\beta}l}\nabla_{p}R_{qk}$
$\displaystyle\tilde{C}$
$\displaystyle=C(2,2)Sym\nabla_{\alpha}R_{ip}\nabla_{\beta}R_{jp}$
$\displaystyle Z=C(2,2)Sym\nabla_{\alpha}R_{pklq}$
$R_{ijkl}$, $R_{ij}$ sont les composantes du tenseur de courbure de Riemann et
de Ricci. Tout les calculs sont faits au point $P$, qu’on omettra dans les
expressions pour des raisons de simplicité. Les indices grecs sont des multi-
indices de longueur $\omega$ (i.e. $|\beta|=|\alpha|=\omega$), si ils
contiennent un tilde, alors ils deviennent de longueur $\omega-2$ (i.e.
$|\tilde{\beta}|=|\tilde{\alpha}|=\omega-2$). Les indices latins sont de
longueur 1. Un indice ou multi-indice noté deux fois, il y a sommation sur cet
indice. sur les autres indices on considère toutes les permutations, afin
d’avoir le symétrisé. Par des calculs combinatoires et les identités de
Bianchi, on a le résultat suivant:
$2(\omega+2)^{2}C(2,2)Sym\nabla_{\alpha\beta kl}R+C(\omega)I=0$
avec
$I=Z+2(\omega+3)^{2}(A+\tilde{C})+2\omega(\omega+3)B\text{ et
}C(\omega)=\frac{(\omega+1)^{2}(\omega+2)^{2}(2\omega+2)!}{[(\omega+3)!]^{2}}$
On sait qu’il existe une constante $K>0$ telle que
$(-\Delta)^{\omega+1}R=KC(2,2)Sym\nabla_{\alpha\beta kl}R$. Pour démontrer le
théorème, il suffit de montrer que $I>0$. Pour cela T. Aubin considère de
nouveaux termes et de types de contractions qui lui permettent d’écrire $I$
comme somme de ces termes qui vérifient certaines relations et inégalités
entre eux (ces relations sont obtenues par des contractions, en utilisant les
identités de Bianchi). Grâce à ces nouvelles relations, il en déduit la
positivité de $I$.
## Appendix B Détails des calculs (Chapitre 5)
### Lemme 5.3
On a vu que la preuve du lemme est ramenée à prouver que
$\bigcap_{k=1}^{q}]x_{k},y_{k}[\neq\varnothing$ (B.1)
où
$x_{k}=\frac{(n-2)^{2}-(n-2)\sqrt{\Delta_{k}}}{d_{k}},\;y_{k}=\frac{(n-2)^{2}+(n-2)\sqrt{\Delta_{k}}}{d_{k}}\text{
et }\Delta_{k}=\\{(n-2)^{2}-\frac{d_{k}u_{k}}{\nu_{k}^{2}}\\}$
D’après le lemme 4.4, $\Delta_{k}>0$ pour tout $k\leq q\leq[\omega/2]$.
Puisque $(d_{k})_{k}$ est décroissante, il est facile de vérifier que
$\forall k<j\leq[\frac{\omega}{2}]\qquad x_{k}<y_{j}$ (B.2)
(voir équations (4.2), (4.17) pour la définition de $\nu_{k}$ et $d_{k}$). On
vérifie aussi que $u_{\omega/2}<0$ (voir équations (4.6)), cela entraîne que
si $\omega$ est pair alors $x_{\omega/2}<0$.
#### Le cas $\boldsymbol{\omega=5}$
D’après les remarques ci-dessus, il suffit de montrer que $x_{2}<y_{1}$. Ce
qui revient à montrer que
$(n-2)(d_{2}-d_{1})+d_{1}\sqrt{\Delta_{2}}+d_{2}\sqrt{\Delta_{1}}>0$
Dans ce cas
$\displaystyle\nu_{1}=5(n+3),\quad\nu_{2}=3(n+1)$ $\displaystyle
d_{1}=4(4n^{3}+53n^{2}+10n+128),\quad d_{2}=4(2n^{3}+47n^{2}+42n+104)$
$\displaystyle\frac{u_{2}}{\nu_{2}}=\frac{n^{2}-49n+36}{8(n-2)(n+2)}$
Après une décomposition en éléments simples de la fraction rationnelle
$\displaystyle\frac{d_{2}}{\nu_{2}}\frac{u_{2}}{\nu_{2}}$ par rapport à $n$,
on établit que
$\begin{split}\Delta_{2}=(n-2)^{2}-\frac{d_{2}}{\nu_{2}}\frac{u_{2}}{\nu_{2}}&=\frac{2}{3}n^{2}+\frac{29}{6}n+\frac{1076}{3}+\frac{2842}{9(n-2)}-\frac{1104}{n+2}+\frac{4601}{9(n+1)}\\\
&>\frac{2}{3}(n+\frac{29}{8})^{2}\end{split}$
D’où
$(n-2)(d_{2}-d_{1})+d_{1}\sqrt{\Delta_{2}}>-8(n-2)(n^{3}+3n^{2}-16n+12)\\\
+4(4n^{3}+53n^{2}+10n+128)\sqrt{\frac{2}{3}}(n+\frac{29}{8})>0$
#### Le cas $\boldsymbol{\omega=6}$
On doit encore montrer que $x_{2}<y_{1}$. En effet l’intersection avec
l’intervalle $]x_{3},y_{3}[$ n’est pas vide car $x_{3}<0$, $y_{3}>x_{2}$ et
$y_{3}>x_{1}$. Il suffit donc de montrer que
$(n-2)(d_{2}-d_{1})+d_{1}\sqrt{\Delta_{2}}+d_{2}\sqrt{\Delta_{1}}>0$
Dans ce cas
$\displaystyle\nu_{1}=6(n+4),\quad\nu_{2}=4(n+2)$ $\displaystyle
d_{1}=4(5n^{3}+74n^{2}+176),\quad d_{2}=4(3n^{3}+64n^{2}+44n+144)$
$\displaystyle\frac{u_{2}}{\nu_{2}}=\frac{n^{2}-31n+18}{6(n-2)(n+3)}$
On répète les mêmes calculs que dans le cas précédent. On établit que
$\begin{split}\Delta_{2}=(n-2)^{2}-\frac{d_{2}}{\nu_{2}}\frac{u_{2}}{\nu_{2}}&=\frac{1}{2}n^{2}+\frac{7}{3}n+\frac{892}{3}+\frac{512}{3(n-2)}+\frac{1008}{n+2}-\frac{2028}{n+3}\\\
&>\frac{1}{2}(n+\frac{7}{3})^{2}\end{split}$
D’où
$(n-2)(d_{2}-d_{1})+d_{1}\sqrt{\Delta_{2}}>-8(n-2)(n^{3}+5n^{2}-22n+16)\\\
+2\sqrt{2}(5n^{3}+74n^{2}+176)(n+\frac{7}{3})>0$
#### Le cas $\boldsymbol{\omega=7}$
Par contre dans ce cas, on doit vérifier que l’intersection
$\bigcap_{k=1}^{3}]x_{k},y_{k}[$
est non vide. On a déjà les inégalités suivantes: $y_{3}>x_{2}>0$,
$y_{3}>x_{1}>0$ et $y_{2}>x_{1}$. Il suffit de montrer que $y_{1}>x_{3}$,
$y_{1}>x_{2}$ et $y_{2}>x_{3}$, ce qui est équivalent à montrer que
$\forall 1\leq i<j\leq
3\quad(n-2)(d_{j}-d_{i})+d_{i}\sqrt{\Delta_{j}}+d_{j}\sqrt{\Delta_{i}}>0$
On reprend les mêmes calculs.
$\displaystyle\nu_{1}=7(n+5),\quad\nu_{2}=5(n+3),\quad\nu_{3}=3(n+1)$
$\displaystyle d_{1}=4(6n^{3}+99n^{2}-14n+232),\quad
d_{2}=4(4n^{3}+85n^{2}+42n+192)$ $\displaystyle
d_{3}=4(2n^{3}+79n^{2}+74n+168)$
$\displaystyle\frac{u_{2}}{\nu_{2}}=\frac{3n^{2}-75n+32}{16(n-2)(n+4)},\quad\frac{u_{3}}{\nu_{3}}=\frac{n^{2}-81n+68}{8(n-2)(n+2)}$
$\begin{split}\Delta_{2}=(n-2)^{2}-\frac{d_{2}}{\nu_{2}}\frac{u_{2}}{\nu_{2}}&=\frac{2}{5}n^{2}+\frac{5}{4}n+\frac{1413}{5}-\frac{3572}{n+4}+\frac{51333}{25(n+3)}+\frac{2862}{25(n-2)}\\\
&>\frac{2}{5}(n+\frac{25}{16})^{2}\end{split}$
$\begin{split}\Delta_{3}=(n-2)^{2}-\frac{d_{3}}{\nu_{3}}\frac{u_{3}}{\nu_{3}}&=\frac{2}{7}n^{2}-\frac{9}{14}n+\frac{2708}{21}-\frac{11951}{3(n+6)}+\frac{135809}{49(n+5)}+\frac{1755}{49(n-2)}\\\
&>\frac{2}{7}(n-\frac{9}{8})^{2}\end{split}$
On montre que les inégalités suivantes sont strictes.
$(n-2)(d_{2}-d_{1})+d_{1}\sqrt{\Delta_{2}}>-8(n-2)(n^{3}+7n^{2}-28n+20)\\\
+4\sqrt{\frac{2}{5}}(6n^{3}+99n^{2}-14n+232)(n+\frac{25}{16})>0$
$(n-2)(d_{3}-d_{1})+d_{1}\sqrt{\Delta_{3}}>-8(n-2)(2n^{3}-10n^{2}-44n+32)\\\
+4\sqrt{\frac{2}{7}}(6n^{3}+99n^{2}-14n+232)(n-\frac{9}{8})>0$
$(n-2)(d_{3}-d_{2})+d_{2}\sqrt{\Delta_{3}}>-8(n-2)(n^{3}+3n^{2}-16n+12)\\\
+4\sqrt{\frac{2}{7}}(4n^{3}+85n^{2}+42n+192)(n-\frac{9}{8})>0$
#### Le cas $\boldsymbol{8\leq\omega\leq 15}$
Á partir de 8 jusqu’à 15, on utilise le logiciel Maple pour faire la
décomposition en éléments simples de la fraction rationnelle $\Delta_{k}$. On
obtient la forme suivante:
$\Delta_{k}=a_{k}n^{2}+b_{k}n+d_{k}+\frac{e_{k}}{n-2}+\frac{f_{k}}{\nu_{k}-n+1}$
En utilisant encore ce logiciel, on montre que
$\sqrt{\Delta_{k}}>\sqrt{a_{k}}(n+\frac{b_{k}}{2a_{k}})$
où les coefficients $a_{k}$, $b_{k}$, $d_{k}$ et $f_{k}$ sont donnés
explicitement en fonction de $\omega$, $n$ et $k$. Ensuite, on vérifie que
pour tout $i<j$
$(n-2)(d_{j}-d_{i})+d_{i}\sqrt{\Delta_{j}}+d_{j}\sqrt{\Delta_{i}}>(n-2)(d_{j}-d_{i})\\\
+d_{i}\sqrt{a_{j}}(n+\frac{b_{j}}{2a_{j}})+d_{j}\sqrt{a_{i}}(n+\frac{b_{i}}{2a_{i}})>0$
D’après ce qui a été dit dans le cas 7 ci-dessus, l’inégalité du lemme est
démontrée.
|
arxiv-papers
| 2009-10-03T18:24:06 |
2024-09-04T02:49:05.616929
|
{
"license": "Public Domain",
"authors": "Farid Madani",
"submitter": "Farid Madani",
"url": "https://arxiv.org/abs/0910.0562"
}
|
0910.0564
|
# Adjustable Microchip Ringtrap for Cold Atoms and Molecules
Paul M. Baker AFRL.RVB.PA@hanscom.af.mil Air Force Research Laboratory,
Hanscom AFB, MA 01731, USA Physics Department, Tufts University. James A.
Stickney Space Dynamics Laboratory, Bedford, MA 01730, USA Matthew B.
Squires Air Force Research Laboratory, Hanscom AFB, MA 01731, USA James A.
Scoville Air Force Research Laboratory, Hanscom AFB, MA 01731, USA Evan J.
Carlson Air Force Research Laboratory, Hanscom AFB, MA 01731, USA Walter R.
Buchwald Air Force Research Laboratory, Hanscom AFB, MA 01731, USA Steven M.
Miller Air Force Research Laboratory, Hanscom AFB, MA 01731, USA
###### Abstract
We describe the design and function of a circular magnetic waveguide produced
from wires on a microchip for atom interferometry using deBroglie waves. The
guide is a two-dimensional magnetic minimum for trapping weak-field seeking
states of atoms or molecules with a magnetic dipole moment. The design
consists of seven circular wires sharing a common radius. We describe the
design, the time-dependent currents of the wires and show that it is possible
to form a circular waveguide with adjustable height and gradient while
minimizing perturbation resulting from leads or wire crossings. This maximal
area geometry is suited for rotation sensing with atom interferometry via the
Sagnac effect using either cold atoms, molecules and Bose-condensed systems.
###### pacs:
03.75.Dg, 37.25.+k
## I Introduction
In recent years atom interferometry has been used to make precision
measurements of various phenomena such as rotations, acceleration, gravity
gradients and frequency separation of the hyperfine splitting for precision
timekeeping Gustavson et al. (1997); Lenef et al. (1997). The sensitivity of
these measurements are directly proportional to the interaction time or, in
the case of rotation measurements, the area enclosed by the two separate
interferometer paths. State of the art atom interferometers Gustavson et al.
(1997); Lenef et al. (1997); Wu et al. (2007); Shin et al. (2004); Wang et al.
(2005); Garcia et al. (2006) use unconfined launched atom clouds with minimal
external potentials during the interferometer cycle. Despite the success of
unconfined atom interferometers there are limitations on the ultimate
sensitivity which include: gravity accelerating the atoms Gustavson et al.
(2000), increasing the enclosed area requires increasing the size of the
required magnetic shielding, longer interaction times leads to lower signal to
noise resulting from lower densities because of ballistic expansion of the
atomic cloud, limited dynamic range due to the atom clouds impacting the
rotating vacuum enclosure during high dynamic rotations. One method of
addressing these difficulties is to place the atoms in a confining potential.
Several methods for building potentials suitable for use in confined atom
interferometers have been developed Wu et al. (2007); Shin et al. (2004); Wang
et al. (2005); Garcia et al. (2006); Horikoshi and Nakagawa (2006).
One method for producing a potential suitable for confined atom interferometry
involves fabricating small micrometer scale current carrying wires on an
insulating substrate, commonly referred to as an atom chip. Current carrying
wires on atom chips produce magnetic fields that can be used to trap atomic
samples when prepared in a low-field seeking state. However, to be effective
for trapped atom interferometry, the magnetic potential must be sufficiently
uniform to avoid decoherence. The requirements on the smoothness of the
potential are reduced if the separate atomic clouds propagate through
reciprocal paths, canceling common mode noise Wu et al. (2007) and when the
energy associated with the cloud is higher than the potential roughness
Stickney et al. (2009).
Atom interferometry for rotation sensing via the Sagnac effect is one of the
most promising applications of trapped atom interferometers. To maximize the
enclosed area, and thus sensitivity, the atomic clouds used in the
interferometer should propagate in a circle. In this paper, we propose a
method for fabricating an atom chip ring trap, specifically for use in atom
interferometry.
A challenge of the ring traps using atom chips is the elimination of potential
imperfections resulting from the input leads. One method of avoiding the input
leads is to use several turns in an effort to make the input lead perturbation
small in comparison to the ring field Gupta et al. (2005). More recently
magnetic induction has been proposed as means to avoid input leads Griffin et
al. (2008). In our previous paper Crookston et al. (2005) we proposed two sets
of wires that provide two overlapping ring traps about a common radius and the
ability to switch between the two in order to avoid the input leads. This
method also provided a means of loading the atoms directly into the waveguide
via a U-trap wire located adjacent to the ring to avoid atom losses Arnold et
al. (2006). An experimental limitation of this design was a fixed trapping
height based upon the wire spacing and current ratios of the wires. Often the
desired working distance is unknown and is not cost or time effective to
redsesign and replace chips often. For this reason a chip design with
adjustable trapping distance is strongly desired.
There are several reasons why a ring trap with an adjustable trapping height
is experimentally useful. First, the lifetime of the atomic cloud trapped near
the surface of an atom chip is limited by trap loss caused by Johnson noise
photon induced spin flips Lin et al. (2004). The number of Johnson noise
photons produced is dependent on the temperature of the chip, which depends
upon the current density of the wires and the thermal properties of the
microchip. Because producing the same magnetic confinement further from a wire
requires more current and therefore a higher chip temperature, optimizing the
distance of the atoms from the chip is experimentally important. Also, the
atom interferometer requires some form of splitting and re-combining of the
atoms. A common method used is to apply an optical standing wave. Once again
the distance of the atoms from the microchip is important because the Bragg
scattering efficiency is reduced by the scattering of the laser beams off the
chip surface. Finally, increasing the distance from the waveguide to the chip
surface also averages small imperfection in the potential resulting from
current fluctuations in the wires. For all of these reasons it is desirable to
have a chip design that allows for an adjustable trapping distance by changing
the wire currents.
## II The 7-wire microchip ring trap design
Previous work has shown that a waveguide with a magnetic field minimum can be
generated utilizing either 3 or 4 straight current carrying wires Cassettari
et al. (2000); Thywissen et al. (1999). Specific currents in the wires can be
chosen, so as to produce a trapping potential some distance away from the
wires. In this paper it will be demonstrated that it is possible to use 7
concentric circular wires to produce a uniform ring trap waveguide that avoids
perturbations resulting from the input leads. A schematic of our 7-wire ring
trap is shown in Fig. 1. Since there are a total of seven concentric current
rings used to form this ring trap, we will refer to this geometry as a 7-wire
ring trap for the remainder of this paper. The primary advantage of using the
7-wire ring trap is that the distance of the ring trap from the atom chip can
be varied simply by changing the currents in the wires.
The operation of the ring trap is similar to our previous ring trap work
Crookston et al. (2005). Initially, the atoms are cooled below the recoil
temperature and are loaded into the 3-wire waveguide at the position indicated
by $0~{}\mbox{rad}$ in Fig. 1. The atoms are coherently split using a standing
wave laser pulse Wang et al. (2005); Wu et al. (2007), half of the atomic
cloud is given a $2\hbar k_{l}$ momentum kick clockwise and the other half is
given a momentum kick counter-clockwise, where $k_{l}$ is the wave number of
the lasers beams used to produce the standing wave. Since the atomic cloud is
cooled below the recoil temperature the two clouds of atoms will spatially
separate Garcia et al. (2006). Since the atoms are confined in the ring trap,
the two atomic clouds will propagate in circular paths. When the clouds have
entered the regions located near $\pm\pi/2~{}\mbox{rad}$, (shown as shaded
boxes in Fig. 1), currents in the 3-wire ring trap are slowly turned off,
while the currents in the 4-wire ring trap are turned on. This switching
prevents the atoms in the ring trap from experiencing perturbations in the
potential due to the currents in the input leads. When the two clouds have
entered the regions near $\pm\pi/2~{}\mbox{rad}$ for a second time, the
currents are switched back into the 3-wire ring trap. When the clouds return
to their initial position, the are illuminated with a second standing wave
pulse. By counting the number of atoms in the $0,\pm\hbar k_{l}$ momentum
states, the Sagnac phase shift can be determined Sagnac (1913); Post (1967);
Lenef et al. (1997).
Figure 1: (color online) 7-wire ringtrap layout including the appropriate
current labels. The location of the wires used to form the 3-wire ring trap is
shown as blue (dashed) lines with the input leads entering from the left and
the wires used to produce the 4-wire ring trap are shown as red (solid) lines,
with the input leads entering from the right. (Inset) The wire spacing is
given by the parameter a and the currents assigned to each wire are labeled
$I_{n}$ accordingly.
In addition to the waveguide, a bias field can be applied to lift the minimum
of the waveguide minimum from zero. The bias field can be created by either a
central orthogonal current carrying wire or a Time-Orbiting potential (TOP)
Petrich et al. (1995). Since the atoms are cooled below the recoil
temperature, a non-zero waveguide minimum is essential to reduce atom loss
through Majorana spin flips. Although often experimentally necessary, the
inclusion of a bias field is simple and its effects will be neglected for the
remainder of this paper.
## III Theoretical development
Below we will introduce a simple theoretical model for a ring trap using $N$
(odd) concentric current carrying rings on the surface of an atom chip. The
center ring has radius $R$ and the center to center distance between the rings
is $a$. For simplicity, only the case where the radii of the rings, is much
larger than the distance between them $R\gg a$, and much larger than the
distance of the ring trap from the chip will be considered. Thus, we will
neglect the effects due to the curvature of the wires. We will also treat the
wires as thin and neglect any effects due to finite wire size. The lowest
order effects due to wire curvature have been analyzed, but the resulting
formula’s provide little new insight into the operation of our ring trap.
The vector potential due to a current carrying ring points in the azimuthal
direction. In the limit of large radius $R$ the vector potential for $N$ (odd)
equally spaced current carrying concentric rings is,
$A_{\phi}=-\frac{\mu_{0}}{4\pi}\sum_{n=-(N-1)/2}^{(N-1)/2}I_{n}\ln\left[(\delta
r-na)^{2}+z^{2}\right],$ (1)
where $I_{n}$ is the current in the $n$-th ring, $a$ is the spacing between
the wires, $\delta r=r-R$ is the radial distance from the center ring to the
field point, and $z$ is the height of the field point from the rings.
Expanding Eq. (1) about the point $\delta r=0$ and $z=z_{0}$ yields
$\displaystyle A_{\phi}$ $\displaystyle=$
$\displaystyle-\frac{\mu_{0}}{4\pi}\sum_{n}I_{n}\left[-\frac{2na}{(na)^{2}+z_{0}^{2}}\delta
r+\frac{2z_{0}}{(na)^{2}+z_{0}^{2}}\delta z\right.$ (2) $\displaystyle+$
$\displaystyle\left.\frac{\left(z_{0}\delta r+na\delta
z\right)^{2}-\left(z_{0}\delta z-na\delta
r\right)^{2}}{\left((na)^{2}+z_{0}^{2}\right)^{2}}\right],$
where $\delta z=z-z_{0}$ and the constant terms have been dropped. From Eq.
(2) it is clear that the magnetic field is zero at the point $\delta r=0$ and
$z=z_{0}$ when the two linear terms in Eq.( 2) each vanish. The first term is
zero when
$I_{n}=I_{-n},$ (3)
and the second term is zero when the currents are such that
$0=\sum_{n}\frac{I_{n}}{(na)^{2}+z_{0}^{2}}.$ (4)
When both Eqns. (3) and (4) are fulfilled the vector potential Eq. (2) becomes
$\displaystyle A_{\phi}$ $\displaystyle=$
$\displaystyle\frac{\mu_{0}}{4\pi}\sum_{n}I_{n}\frac{z_{0}^{2}-(na)^{2}}{\left(z_{0}^{2}+(na)^{2}\right)^{2}}\left(\delta
r^{2}-\delta z^{2}\right).$ (5)
Taking the curl of Eq. (5) yields the magnetic field components
$\displaystyle
B_{r}=\frac{\mu_{0}}{2\pi}\sum_{n}I_{n}\frac{z_{0}^{2}-(na)^{2}}{\left(z_{0}^{2}+(na)^{2}\right)^{2}}\delta
z$ $\displaystyle
B_{z}=\frac{\mu_{0}}{2\pi}\sum_{n}I_{n}\frac{z_{0}^{2}-(na)^{2}}{\left(z_{0}^{2}+(na)^{2}\right)^{2}}\delta
r.$ (6)
Equations (6) show that the magnetic field near the minima is of the same form
as a simple single wire wave guide Thywissen et al. (1999); Fortagh and
Zimmermann (2007), with field gradient given by
$B^{\prime}=\frac{\mu_{0}}{2\pi}\sum_{n}I_{n}\frac{z_{0}^{2}-(na)^{2}}{\left(z_{0}^{2}+(na)^{2}\right)^{2}}.$
(7)
Note that the sum still runs from $-(N-1)/2$ to $(N-1)/2$.
We will now limit our discussion to the case of seven concentric rings, with
seven independent currents. Equation (3) eliminates three of the currents,
leaving us with four currents independent currents $I_{0}$, $I_{1}$, $I_{2}$
and $I_{3}$ (As shown in Fig. 1). Initially, the atoms are loaded into a ring
trap, i.e. located at $0~{}\mbox{rad}$ below the center wire in Fig. 1. To
avoid the leads the currents in the wires of the 4-wire trap at this location
must be zero, $I_{1}=I_{3}=0$. To satisfy Eq. (4), the relation between the
remaining to currents must be
$I_{2}=-I_{0}\frac{4a^{2}+z_{0}^{2}}{2z_{0}^{2}},$ (8)
and the magnetic field gradient is
$B^{\prime}=\frac{\mu_{0}I_{0}}{2\pi}\frac{8a^{2}}{z_{0}^{2}(z_{0}^{2}+4a^{2})}.$
(9)
When the atomic clouds have propagated half way around the ring trap, they are
near the $\pi~{}\mbox{rad}$ in Fig. 1. To avoid the perturbations due to the
leads of the 3-wire trap the currents at that position must vanish,
$I_{0}=I_{2}=0$. At this point the trap is formed only by the currents in the
wires with current $I_{1}$ and $I_{3}$. To satisfy Eq. (4), the relation
between the nonzero currents must be
$I_{3}=-I_{1}\frac{9a^{2}+z_{0}^{2}}{(a^{2}+z_{0}^{2})},$ (10)
and the magnetic field gradient is
$B^{\prime}=\frac{\mu_{0}I_{1}}{2\pi}\frac{32a^{2}z^{2}_{0}}{(z_{0}^{2}+a^{2})^{2}(z_{0}^{2}+9a^{2})}.$
(11)
When the atoms are in the region near $\pm\pi/2~{}\mbox{rad}$ as shown in Fig.
1, there are no input leads and all seven wires can have nonzero current. In
this situation Eq. (4) has many solutions, but the simplest solution is to
assume that both Eq. (8) and (10) are fulfilled. There are now two free
parameters to specify the magnetic field and the field gradient and can be
expressed as
$B^{\prime}=\frac{\mu_{0}}{2\pi}\left(\frac{8a^{2}I_{0}}{z_{0}^{2}(z_{0}^{2}+4a^{2})}+\frac{32a^{2}z_{0}^{2}I_{1}}{(z_{0}^{2}+a^{2})^{2}(z_{0}^{2}+9a^{2})}\right).$
(12)
To avoid heating of the atomic gas as it moves around the ring, the magnetic
field gradient $B^{\prime}$ should be held constant. At time $t=0$, $I_{1}=0$,
and $I_{0}=I_{0}(0)$. To hold the gradient constant, the time dependence of
the current $I_{1}$ should be
$I_{1}(t)=\frac{(z_{0}^{2}+a^{2})^{2}(z_{0}^{2}+9a^{2})}{4z_{0}^{4}(z_{0}^{2}+4a^{2})}\left(I_{0}(0)-I_{0}(t)\right).$
(13)
## IV Switching between the 3-wire and 4-wire ring trap
To demonstrate the uniformity of the trapping potential while transferring
from the 3-wire guide to the 4-wire guide, plots are given in Figs. 2 and 3 of
the magnitude of magnetic field strength in steps of t. In Figs. 2 and 3 ,
$a=50~{}\mu\mbox{m}$, $z_{0}=100~{}\mu\mbox{m}$, and the current
$I_{ref}=0.5~{}\mbox{A}$ was chosen to give a gradient
$B^{\prime}=1000~{}\frac{\mbox{G}}{\mbox{cm}}$. Time dependent currents are
given as follows:
$I_{0}(t)=I_{ref}\frac{(t_{max}-t)}{t_{max}}$ (14)
$I_{1}(t)=I_{ref}\frac{(z_{0}^{2}+a^{2})^{2}(z_{0}^{2}+9a^{2})}{4z_{0}^{4}(z_{0}^{2}+4a^{2})}\frac{t}{t_{max}},$
(15)
where $t_{max}=1.0$ and t was chosen to allow each waveguide to have values of
$I_{ref}$ between 0 and 1. This procedure serves to switch between the 3-wire
and the 4-wire waveguide in a linear manner. Notice that both the location of
the minimum and the shape of the potential near the minimum remain constant as
seen in Figs. 2 and 3.
Figure 2: (color online) Trapping potential in Gauss along $\delta r$-axis for
$t=0-1$ in equal steps. Stepping through increments of $t$ is equivalent to
turning off the current in the 3-wire waveguide and turning on the 4-wire
waveguide. Notice that both the position of the minimum and the trapping shape
near the minimum remain constant. $\delta r=0$ is the location of the center
wire with current label $I_{0}$ and is perpendicular to the wires. Figure 3:
(color online) Trapping potential below the wires is given in Gauss along
$\delta z$-axis for $t=0-1$ in equal steps. Stepping through increments of $t$
is equivalent to turning off the current in the 3-wire waveguide and turning
on the 4-wire waveguide. Notice that both the position of the minimum and the
trapping shape near the minimum remain constant.
To characterize the effects of the inputs lead and curvature of the wires, we
numerically calculated the magnetic field strength along a constant radius as
shown in Fig. 4. In this calculation the current carrying wires are assumed to
be thin, which is valid when the distance of the ring trap from the wires is
larger than the size of the wires. Since the wires cannot be larger than the
spacing between them, the thin wire approximation is always valid when
$z_{0}\gg a$.
We have performed numerical and analytic studies of the effects on the ring
trap due to the finite curvature of the wires used to form the ring trap. Our
results show that the curvature of the wires causes a small shift in the
location of the ring trap towards the center; the ring trap is no longer
located directly above the center wire. This shift can be corrected by making
a correction to Eq. (3) on the order of $na/R$. A more complete discussion of
the curvature effects will be presented in future work. There are also
corrections to Eqns. (8), (10), and (12) on the order of $na/R$. The leads add
yet another perturbation that shifts the location of the minimum and alters
the shape of the potential. This effect is small in the region of interest and
the approach of this paper is to switch the input leads before this
perturbation is significant. None of these corrections have an effect on the
ability to smoothly guide atom clouds around a ring.
In Fig. 4 the magnitude of the magnetic field is plotted along $\theta$ at a
fixed radius at the field minimum for the same values given above. As
previously mentioned the minimum will deviate slightly from $r_{min}$ as
$\theta$ approaches the leads however the current in the leads is being turned
down as the atoms approach reducing this perturbation. When the atomic clouds
are located between $\theta=-\pi/4$ and $\theta=\pi/4$, the magnetic field has
large perturbations due to the four wire ring trap’s leads. Similarly, when
the atomic cloud is located between $\theta=3\pi/4$ and $\theta=-3\pi/4$ the
magnetic field has large perturbations due to the leads of the three wire ring
trap. However, the atomic clouds are between $\theta=\pm\pi/4$ and $\theta=\pm
3\pi/4$, there are no perturbations due to either the three or four wire
leads. This numerical solution demonstrates that there is a large region where
the current can be switched between the two sets of wires.
Figure 4: Maintaining a constant waveguide minimum during the transfer from
3-wire to 4-wire waveguide is represented by the uniform flatness of the
potential in the switching areas. Solutions are given for arcs connected to
wires representing the input leads. The blue (dashed) curve is the magnitude
of the magnetic field when only the three-wire ring trap has nonzero current
and the red (solid) curve is the field when only the four-wire ring trap has
nonzero current. There are two solutions depending upon the symmetry of the
current direction, only the symmetric current configuration is discussed in
this paper, however the anti-symmetric case is shown above with thicker lines
for completeness and to illustrate experimental flexibility. A small constant
bias field is applied to lift the minimum from zero as would be required
experimentally to reduce Majorana losses.
## V Conclusions
We have designed and developed a 7-wire ring trap with adjustable height that
encloses area and avoids perturbation from input leads. We have introduced a
1-D theoretical model demonstrating it is possible to fabricate a chip where
the currents can be switched between the three and four wire ring trap while
holding the minimums location and gradient constant. We have numerically
analyzed the effects of the input leads and shown that there is a large
switching region where the perturbations due to the input leads of both the
three and four wire rings can be avoided. Finally, we have briefly discussed
our preliminary results of the effects of the curvature of the wires on the
ring trap.
The choice of atomic cloud temperature plays a pivotal role in the ring trap
operation. Bose Einstein Condensates in microchip waveguides can suffer from
fragmentation and de-phasing which are undesirable in atom interferometers.
Recently, Bouchoule et. al., Trebbia et al. (2007); Bouchoule et al. (2008)
has demonstrated a possible solution to the fragmentation issue and it is
possible to operate the 7-wire ring trap in a manner that makes use of this
technique. A BEC can also have reduced coherence times resulting from
potential noise and mean field interactions. The short coherence times
resulting from mean field interactions are density dependent Horikoshi and
Nakagawa (2006); Garcia et al. (2006); therefore a tightly confined BEC would
have additional dispersion and dephasing. The 7-wire ring trap design has the
additional feature of an adjustable gradient and by adjusting the gradient and
utilizing dilute samples the 7-wire ring trap would be able to reduce the
atom-atom interactions. Furthermore a weaker transverse confinement allows for
more transverse oscillation which can be used for dispersion management Murch
et al. (2006). The ability to adjust the gradient thus affords more
experimental flexibility.
It is experimentally useful to adjust the ring trap radius, i.e for a choice
of interferometer interrogation time. De-coupling of temporal and spatial
sources of error is useful for systematically identifying and eliminating
sources of noise and atom loss. Also, the possibility of adjusting the radius
of the waveguide dynamically allows for the study of the coupling of
longitudinal and transverse modes that could be used to damp out transverse
oscillations if desired and help overlap the clouds at the recombination point
Gupta et al. (2005); Murch et al. (2006). This concept can be extended to an
N-wire ring trap where the radial location of the minimum can be adjusted.
Finally it should be remarked that care must be taken during loading of the
ring trap. Small shot-to-shot uncertainty in the initial momentum of the atom
cloud, resulting from poor loading or coupling is sufficient to mask the small
phase shifts resulting from rotation Sackett (2009). The 7-wire ring trap and
ring traps in general may require additional loading wires that allow the atom
cloud to come to equilibrium before optical splitting.
## VI Acknowledgments
The authors acknowledge support from the Air Force Office of Scientific
Research under program/task 2301DS/03VS02COR and DARPA gBECi program.
## References
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* Garcia et al. (2006) O. Garcia, B. Deissler, K. J. Hughes, J. M. Reeves, and C. A. Sackett, Phys. Rev. A 74, 031601(R) (2006).
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* Petrich et al. (1995) W. Petrich, M. H. Anderson, J. R. Ensher, and E. A. Cornell, Phys. Rev. Lett. 74, 3352 (1995).
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* Bouchoule et al. (2008) I. Bouchoule, J.-B. Trebbia, and C. L. Garrido Alzar, Phys. Rev. A 77, 023624 (2008).
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|
arxiv-papers
| 2009-10-05T14:48:47 |
2024-09-04T02:49:05.639632
|
{
"license": "Public Domain",
"authors": "Paul M. Baker, James A. Stickney, Matthew B. Squires, James A.\n Scoville, Evan J. Carlson, Walter R. Buchwald, Steven M. Miller",
"submitter": "Paul Baker",
"url": "https://arxiv.org/abs/0910.0564"
}
|
0910.0607
|
††thanks: e-mail: shufw@cqupt.edu.cn
# The Quantum Viscosity Bound In Lovelock Gravity
Fu-Wen Shu College of Mathematics and Physics,Chongqing University of Posts
and Telecommunications, Chongqing, 400065, China
###### Abstract
Based on the finite-temperature AdS/CFT correspondence, we calculate the ratio
of shear viscosity to entropy density in any Lovelock theories to any order.
Our result shows that any Lovelock correction terms except the Gauss-Bonnet
term have no contribution to the value of $\eta/s$. This result is consistent
with that of Brustein and Medved’s prediction.
PACS: number(s): 11.25.Tq; 04.50.-h; 04.70.Dy; 11.25.Hf
Keywords: AdS/CFT, KSS bound, Lovelock theory, AdS black brane.
Stimulated by the conjecture of AdS/CFT correspondence [1, 2, 3], string
theory has attracted a lot of attention, especially after the discovery that
some theoretical results of the dual theory are consistent with that of the
RHIC experiment, say, the ratio of the viscosity to the entropy density [4,
5]. Recently, it was conjectured, based on the AdS/CFT correspondence, that
for all possible nonrelativistic fluids, there may exist a universal lower
bound (the KSS bound) on the viscosity/entropy-density ratio (we set
$G=c=\hbar=k_{B}=1$)[6]
$\frac{\eta}{s}=\frac{1}{4\pi}.$ (1)
This bound received great supports from several kinds of field theories [7, 8,
9], as well as the case with chemical potential in the theory[10, 11].
However, more recent work on the higher derivative gravity theories (see [12,
13, 14, 15, 16, 17, 18]) showed that the KSS bound is violated when the dual
gravity is enlarged to include a stringy correction (see [19] for more about
the KSS bound in higher derivative gravity). This correction is frequently
referred to as the quantum correction, since in CFT side this is a correction
of the ’t Hooft coupling $\lambda=g_{YM}^{2}N_{c}$. It is of particular
significance to consider the $1/\lambda$ correction when we are dealing with
non-extremely strong coupling fluids.
Recently, the authors of [20] predicted that all Lovelock terms higher than
the second order(the Gauss-Bonnet term) do NOT contribute to the value of
$\eta/s$ at all, and this prediction was partially confirmed in [21] for the
third-order Lovelock gravity. In this paper we calculate the
viscosity/entropy-density ratio directly in the Lovelock theory to any order,
trying to make a complete verification of the prediction, and indeed, our
result provides a direct support of this prediction as will see below.
We start with the Lovelock theory of gravity. This is one of the most general
second order gravity theories in higher dimensional spacetimes and is free of
ghost when expanding on a flat space[22] and hence is of particular interest.
The Lagrangian density for general Lovelock gravity in $D$ dimensions is
${\mathcal{L}}=\sum_{m=0}^{[D/2]}c_{m}\,{\mathcal{L}}_{m},$ where
${\mathcal{L}}_{m}$ is given by [23]
${\mathcal{L}}_{m}=\frac{1}{2^{m}}\sqrt{-g}\delta^{\lambda_{1}\sigma_{1}\cdots\lambda_{m}\sigma_{m}}_{\rho_{1}\kappa_{1}\cdots\rho_{m}\kappa_{m}}R_{\lambda_{1}\sigma_{1}}{}^{\rho_{1}\kappa_{1}}\cdots
R_{\lambda_{m}\sigma_{m}}{}^{\rho_{m}\kappa_{m}}\,,$ (2)
$c_{m}$ is the $m$’th order coupling constant, $[D/2]$ denotes the integer
value of $D/2$ and the Greek indices $\lambda$, $\rho$, $\sigma$ and $\kappa$
go from $0$ to $D-1$. The symbol $R_{\lambda\sigma}{}^{\rho\kappa}$ is the
Riemann tensor in $D$-dimensions and
$\delta^{\lambda_{1}\sigma_{1}\cdots\lambda_{m}\sigma_{m}}_{\rho_{1}\kappa_{1}\cdots\rho_{m}\kappa_{m}}$
is the generalized totally antisymmetric Kronecker delta. The term
${\mathcal{L}}_{0}=\sqrt{-g}$ is the cosmological term, while
${\mathcal{L}}_{1}=\sqrt{-g}\delta_{\rho_{1}\kappa_{1}}^{\lambda_{1}\sigma_{1}}\,R_{\lambda_{1}\sigma_{1}}{}^{\rho_{1}\kappa_{1}}/2$
is the Einstein term. In general ${\mathcal{L}}_{m}$ is the Euler class of a
$2m$ dimensional manifold.
Variation of the Lagrangian with respect to the metric yields the Lovelock
equation of motion
$\displaystyle
0={\mathcal{G}}_{\mu}^{\nu}=-\sum_{m=0}^{[D/2]}\frac{c_{m}}{2^{(m+1)}}\delta^{\nu\lambda_{1}\sigma_{1}\cdots\lambda_{m}\sigma_{m}}_{\mu\rho_{1}\kappa_{1}\cdots\rho_{m}\kappa_{m}}R_{\lambda_{1}\sigma_{1}}{}^{\rho_{1}\kappa_{1}}\cdots
R_{\lambda_{m}\sigma_{m}}{}^{\rho_{m}\kappa_{m}}\ ,$ (3)
As is shown in [24], there exist static exact solutions of Lovelock equation.
Let us consider the following metric
$\displaystyle
ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}\sum_{i,j}^{D-2}\gamma_{ij}dx^{i}dx^{j},$
(4)
where $\gamma_{ij}dx^{i}dx^{j}$ represents the line element of a
$(D-2)$-dimensional Einstein space. With this ansatz, we have
$\mathcal{R}_{ijkl}=\kappa(\gamma_{ik}\gamma_{jl}-\gamma_{il}\gamma_{jk}),\ \
\mathcal{R}_{ij}=\kappa(D-3)\gamma_{ij},\ \ \mathcal{R}=\kappa(D-2)(D-3).$ (5)
where $\kappa$ is the curvature constant, whose value determines the geometry
of the horizon. Without loss of the generality, one may take $\kappa=1$, $0$,
or $-1$ representing sphere, flat and hyperbolic respectively.
Using this metric ansatz, we can calculate Riemann tensor components as
$\displaystyle R_{tr}{}^{tr}=-\frac{f^{{}^{\prime\prime}}}{2},\
R_{ti}{}^{tj}=R_{ri}{}^{rj}=-\frac{f^{{}^{\prime}}}{2r}\delta_{i}{}^{j},\
R_{ij}{}^{kl}=\left(\frac{\kappa-f}{r^{2}}\right)\left(\delta_{i}{}^{k}\delta_{j}{}^{l}-\delta_{i}{}^{l}\delta_{j}{}^{k}\right)\
.$ (6)
Substituting (6) into (3) derives a simple equation
$\displaystyle
W[\psi]\equiv\sum_{m=0}^{n}\tilde{c}_{m}\psi^{m}=\frac{\mu}{r^{D-1}},$ (7)
where $\psi=r^{-2}(\kappa-f)$, $\mu>0$ is a constant of integration which is
related to the ADM mass by
$\displaystyle M=\frac{\mu V_{D-2}}{16\pi G_{D}}\ ,$ (8)
where $V_{D-2}$ is the volume of the $(D-2)$-dimensional hypersurface and
$G_{D}$ is the Newton constant. In (7), we also defined
$\tilde{c}_{m}\equiv\frac{(D-3)!}{(D-2m-1)!}c_{m}$ and $n$ is an integer with
$0<n\leq[D/2]$. In this paper we are considering AdS black brane in Lovelock
gravity, so we have $c_{0}=-2\Lambda$ with the cosmological constant
$\Lambda=-(D-1)(D-2)/2l^{2}$ and $c_{1}=1$.
We would like to extract some information from the Lovelock black brane, such
as their thermodynamic properties. One quantity which is of particular
interest is the entropy $S$. Generally speaking, one can obtain the entropy of
a black hole in higher derivative theories by using the thermodynamic relation
$S=-\partial F/\partial T$ with $F$ the free energy and $T$ the Hawking
temperature. By doing so one finds that the entropy of the Lovelock black
brane is given by[25]
$\displaystyle
S=\frac{V_{D-2}r_{+}^{D-2}}{4G_{D}}\sum_{m=1}^{n}\frac{m(D-2)}{(D-2m)}\tilde{c}_{m}(\kappa
r_{+}^{-2})^{m-1},$ (9)
where $r_{+}$ is the event horizon of the black brane which is the positive
root of $f(r_{+})=0$. In the present paper, we mainly focus on the case where
$\kappa=0$. In this case we have a simple formula for the entropy density of
the Lovelock black brane
$\displaystyle s=\frac{r_{+}^{D-2}}{4G_{D}},$ (10)
and now, $r_{+}$ is a solution of $\psi(r_{+})=0$. The Hawking temperature of
this case is given by
$\displaystyle T=\frac{(D-1)\tilde{c}_{0}}{4\pi}r_{+}.$ (11)
In what following, we would like to see the waves generated by a metric
perturbation of the background. Generally speaking, there are scalar, vector
and tensor modes depending on the rotation symmetry. In this paper, we only
study tensor perturbations which is closely related to the shear viscosity as
will see below.
We now add a small tensor perturbations to the solution (4)
$\displaystyle\delta g_{ij}=r^{2}\phi(t,r)h_{ij}(x^{i})\ ,\ \ \ (others)=0$
(12)
where $\phi(t,r)$ represents the dynamical degrees of freedom. Here, $h_{ij}$
are defined by
$\displaystyle\nabla^{k}\nabla_{k}h_{ij}=k^{2}h_{ij}\
,\qquad\nabla^{i}h_{ij}=0\ ,\quad\gamma^{ij}h_{ij}=0.$ (13)
Here, $\nabla^{i}$ denotes a covariant derivative with respect to
$\gamma_{ij}$ and $k^{2}$ is the eigenvalue playing a role of momentum.
With these definition, one can obtain the first order perturbation equation of
the Lovelock equation (3)[26]
$\displaystyle
0=\delta{\mathcal{G}}_{\mu}^{\nu}=-\sum_{m=1}^{k}\frac{a_{m}}{2^{(m+1)}}\delta^{\nu\lambda_{1}\sigma_{1}\cdots\lambda_{m}\sigma_{m}}_{\mu\rho_{1}\kappa_{1}\cdots\rho_{m}\kappa_{m}}R_{\lambda_{1}\sigma_{1}}{}^{\rho_{1}\kappa_{1}}\cdots
R_{\lambda_{m-1}\sigma_{m-1}}{}^{\rho_{m-1}\kappa_{m-1}}\delta
R_{\lambda_{m}\sigma_{m}}{}^{\rho_{m}\kappa_{m}},$ (14)
where $\delta R_{ab}{}^{cd}$ represents the first order variation of the
Riemann tensor and we have introduced a new quantity $a_{m}=mc_{m}(m>0)$. As
shown in [26], it is straightforward once we know the expressions of
quantities $\delta R_{ti}{}^{tj}$, $\delta R_{ri}{}^{rj}$ and $\delta
R_{ij}{}^{kl}$. Then the calculation becomes a mathematical game and the
result is ready-made[26]
$\displaystyle
0=\delta{\mathcal{G}}_{i}{}^{j}=\frac{1}{r^{D-4}}\left[\frac{h}{2f}\left(\ddot{\phi}-f^{2}\phi^{{}^{\prime\prime}}\right)-\left(\frac{(r^{2}fh)^{\prime}}{2r^{2}}\right)\phi^{{}^{\prime}}+\frac{(k^{2}+2\kappa)h^{{}^{\prime}}}{2(D-4)r}\phi\right]h_{i}{}^{j},$
(15)
where
$\displaystyle h(r)$ $\displaystyle=$
$\displaystyle\frac{d}{dr}\left[\frac{r^{D-3}}{D-3}\frac{dW[\psi]}{d\psi}\right]$
(16) $\displaystyle=$ $\displaystyle
r^{D-4}-\sum_{m=2}^{n}\Biggl{[}\frac{m\tilde{c}_{m}r^{D-2m-2}(\kappa-f)^{m-2}}{D-3}\left\\{(m-1)rf^{{}^{\prime}}-(D-2m-1)(\kappa-f)\right\\}\Biggr{]}.$
Using the Fourier decomposition
$\displaystyle\phi(t,r)=\int\frac{d\omega}{2\pi}e^{-i\omega t}\phi(r),$ (17)
we obtain the linearized equation of motion for $\phi(r)$:
$\displaystyle\phi^{\prime\prime}(r)+\left(\frac{(r^{2}fh)^{{}^{\prime}}}{r^{2}fh}\right)\phi^{\prime}(r)+\frac{1}{f^{2}}\left(\omega^{2}-\frac{(k^{2}+2\kappa)fh^{{}^{\prime}}}{(D-4)rh}\right)\phi(r)=0\
.$ (18)
It is convenient to introduce a new dimensionless coordinate
$u=(r_{+}/r)^{(D-1)/2}$ with $r_{+}$ the event horizon of the black brane. In
this coordinate frame, $u=0$ corresponds to the boundary and $u=1$ the
horizon. The linearized equation of motion (18) then becomes (for $\kappa=0$)
$\displaystyle\phi^{\prime\prime}(u)+\frac{g^{\prime}(u)}{g(u)}\phi^{\prime}(u)+\frac{\bar{\omega}^{2}}{u^{\frac{2D-6}{D-1}}\psi^{2}(u)}\phi(u)-\frac{D-1}{2(D-4)}\cdot\frac{h^{\prime}\bar{k}^{2}}{u^{\frac{D-5}{D-1}}\psi(u)h}\phi(u)=0\,$
(19)
where
$\displaystyle g(u)=-r_{+}^{4-D}\psi(u)h(u)u^{\frac{D-7}{D-1}},$ (20)
$\displaystyle\bar{\omega}\equiv\frac{2}{(D-1)r_{+}}\omega,\ \ \ \
\bar{k}\equiv\frac{2}{(D-1)r_{+}}k,$ (21)
and the prime denotes the derivative with respect to $u$.
Now we shall calculate the shear viscosity in Lovelock gravity theories.
Generally speaking, the shear viscosity $\eta$ can be calculated via Kubo
formula,
$\eta=-\lim_{\omega\rightarrow 0}\frac{\mbox{Im}(G^{R}(\omega,0))}{\omega},$
(22)
where $G^{R}$ is the retarded Green’s function
$G^{R}(\omega,\vec{k})=-i\int
dtd\vec{x}e^{-i\vec{k}.\vec{x}}\theta(t)<[\hat{\mathcal{O}}(x)\hat{\mathcal{O}}(0)]>,$
(23)
with $\hat{\mathcal{O}}$ some boundary CFT operators. According to AdS/CFT
correspondence, the Green’s function can be calculated from the dual gravity
side via the Gubser-Klebanov-Polyakov/Witten relation [2, 3]
$\langle e^{\int_{\partial
M}\phi_{0}\hat{\mathcal{O}}}\rangle=e^{-S_{cl}[\phi_{0}]},$
where $\phi$ is the bulk field and $\phi_{0}$ is its value at the boundary,
i.e., $\phi_{0}=\lim_{u\rightarrow 0}\phi(u)$. Extracting the part of $S_{cl}$
that is quadratic in $\phi$ and inserting the solution of the linearized field
equation we may get a surface term in four dimensions by using the equation of
motion,
$S_{cl}[\phi_{0}]=\\!\int\\!\frac{d^{D-1}k}{(2\pi)^{D-1}}\phi_{0}(-k)G(k,u)\phi_{0}(k)\bigg{|}_{u=0}^{u=1},$
(24)
where $u=(r_{+}/r)^{(D-1)/2}$ as defined previously. In this way, we obtain
the following relation for the retarded Green’s function[27]
$G^{R}(k)=2G(k,u)\bigg{|}_{u=0},$ (25)
where the incoming boundary condition at the horizon is imposed. The shear
viscosity then can be calculated by using (22).
In the following we would like to calculate the shear viscosity, following the
procedures introduced above. The main task is to solve the equation of motion
(19) in hydrodynamic regime $\it{i.e.}$, small $\omega$ and $k$. To solve the
wave equation (19) we first examine the behavior around the horizon where
$u=1$. For this purpose it is convenient to impose a solution as
$\phi(u)=(1-u)^{\nu}F(u),$ (26)
with $F(u)$ regular at the horizon. Substituting (26) into the wave equation
(19) and leaving the most divergent terms, we can obtain
$\displaystyle\nu=\pm i\frac{\bar{\omega}}{\psi^{\prime}(1)}\,$ (27)
where we have used the relations
$\displaystyle g(u\rightarrow 1)$ $\displaystyle=$
$\displaystyle-g^{\prime}(1)(1-u)+\mathcal{O}((1-u)^{2}),$ (28)
$\displaystyle\psi(u\rightarrow 1)$ $\displaystyle=$
$\displaystyle-\psi^{\prime}(1)(1-u)+\mathcal{O}((1-u)^{2}).$ (29)
In present paper we choose $``-^{\prime\prime}$ sign in eq. (27) for
convenience.
To get the viscosity via Kubo formula (22), the standard procedure is to
consider series expansion of the solution in terms of frequencies up to the
linear order of $\omega$,
$F(u)=F_{0}(u)+\nu F_{1}(u)+{\mathcal{O}}(\nu^{2},k^{2}).$ (30)
Then the equation of motion (19) becomes the following form up to
${\mathcal{O}}(\nu)$,
$\left[g(u)F^{\prime}(u)\right]^{\prime}-\nu\left(\frac{1}{1-u}g(u)\right)^{\prime}F(u)-\frac{2\nu}{1-u}g(u)F^{\prime}(u)=0.$
(31)
After substituting the series expansion (30) into the equation (31), we obtain
the following equations of motion for $F_{0}(u)$ and $F_{1}(u)$
$\displaystyle\left[g(u)F^{\prime}_{0}(u)\right]^{\prime}=0,$ (32)
$\displaystyle\left[g(u)F^{\prime}_{1}(u)\right]^{\prime}-\left(\frac{1}{1-u}g(u)\right)^{\prime}F_{0}(u)=0.$
(33)
By requiring that the functions $F_{0}(u)$ and $F_{1}(u)$ are regular at the
horizon one gets the following results
$\displaystyle F_{0}(u)=C,$ (34) $\displaystyle
F_{1}^{\prime}(u)=\left(\frac{1}{1-u}+\frac{g^{\prime}(1)}{g(u)}\right)C,$
(35)
where again we have used the relation (28) and the constant $C$ can be
determined in terms of boundary value of the field, i.e.,
$C=\phi_{0}\Big{(}1+{\mathcal{O}}(\nu)\Big{)}.$
Now we shall calculate the retarded Green’s function. Using the equation of
motion, the action reduces to the surface terms. The relevant part is given by
$S_{cl}[\phi(u)]=-\frac{(D-1)r^{D-1}_{+}}{64\pi
G_{D}}\\!\int\\!\frac{d^{D-1}k}{(2\pi)^{D-1}}\Big{(}g(u)\phi(u)\phi^{\prime}(u)+\cdots\Big{)}\Bigg{|}_{u=0}^{u=1}.$
(36)
Near the boundary $u=\varepsilon$, using the perturbative solution of
$\phi(u)$, we get
$\displaystyle\phi^{\prime}(\varepsilon)$ $\displaystyle=$
$\displaystyle\nu\frac{g^{\prime}(1)}{g({\varepsilon})}\phi_{0}+{\mathcal{O}}(\nu^{2},k^{2})$
(37) $\displaystyle=$
$\displaystyle-i\frac{\bar{\omega}}{\psi^{\prime}(1)}\frac{g^{\prime}(1)}{g(\varepsilon)}\phi_{0}+{\mathcal{O}}(\omega^{2},k^{2}).$
Therefore we can read off the correlation function from the relation (25),
$G^{R}(\omega,k)=i\omega\frac{1}{16\pi
G_{D}}\left(\frac{r_{+}^{D-2}}{\psi^{\prime}(1)}\right)g^{\prime}(1)+{\mathcal{O}}(\omega^{2},k^{2}),$
(38)
where contact terms are subtracted. Then the shear viscosity can be obtained
by using Kubo formula (22),
$\eta=-\frac{1}{16\pi
G_{D}}\left(\frac{r^{D-2}_{+}}{\psi^{\prime}(1)}\right)g^{\prime}(1).$ (39)
The ratio of the shear viscosity to the entropy density is concluded as
$\frac{\eta}{s}=-\frac{1}{4\pi}\frac{g^{\prime}(1)}{\psi^{\prime}(1)}.$ (40)
From (20) we have a relation $g^{\prime}(1)=-r_{+}^{4-D}\psi^{\prime}(1)h(1)$
and $h(1)$ can be obtained from (16) by inserting $\kappa=0$
$h(1)=r_{+}^{D-4}\Big{(}1-(D-1)(D-4)\tilde{c}_{0}a_{2}\Big{)}.$
It is straightforward to show that
$\frac{\eta}{s}=\frac{1}{4\pi}\Big{(}1-(D-1)(D-4)\tilde{c}_{0}a_{2}\Big{)}=\frac{1}{4\pi}\Big{(}1-\frac{2(D-1)(D-4)\lambda}{l^{2}}\Big{)},$
(41)
where we have defined $\lambda=c_{2}$. This result is exactly the one
predicted in [20].
In summary, we have computed the ratio of shear viscosity to entropy density
for any Lovelock theories. Our result shows that any correction terms except
the Gauss-Bonnet term do not affect the value of $\eta/s$, and this confirms
the prediction made by [20]. During our calculation, we have chosen a
vanishing curvature constant $\kappa$. Actually, our result is still valid
(for leading term) for nonzero $\kappa$ if we focus on a large black brane. In
the large black brane limit, both the entropy and the viscosity have the same
leading terms as those of $\kappa=0$. This can be seen by noting the
expressions of entropy density and viscosity. From (9), the entropy density of
the Lovelock black brane with nonzero $\kappa$ can be expanded, in the large
black brane limit($\it{i.e.}$, $\frac{\kappa}{r_{+}^{2}}\ll 1$), to the first
order as
$s=\frac{r_{+}^{D-2}}{4G_{D}}\left[1+\frac{2(D-2)\tilde{c}_{2}}{D-4}\cdot\frac{\kappa}{r_{+}^{2}}\right]+\mathcal{O}\left(\frac{\kappa}{r_{+}^{2}}\right).$
(42)
With the same spirit one can also expand the shear viscosity to the first
order in the large black brane limit. This can be done by repeating the
previous procedures and noting that $h(1)=h(u=1)$ can be obtained from (16).
In this way, the shear viscosity for nonvanishing $\kappa$ can be expanded to
the first order as
$\displaystyle\eta$ $\displaystyle=$ $\displaystyle\frac{r_{+}^{D-2}}{16\pi
G_{D}}\left\\{1-\frac{2(D-1)(D-4)\lambda}{l^{2}}-\right.$ (43)
$\displaystyle\left.\frac{2(D-1)}{D-3}\left[\tilde{c}_{2}(1-2\tilde{c}_{2})+3\frac{\tilde{c}_{3}}{l^{2}}-(D-5)\tilde{c}_{2}\right]\cdot\frac{\kappa}{r_{+}^{2}}\right\\}+\mathcal{O}\left(\frac{\kappa}{r_{+}^{2}}\right).$
From (42) and (43) it is obvious that the leading terms of the entropy density
and the viscosity for $\kappa\neq 0$ are the same as those of $\kappa=0$. In
other words, in the large black brane limit, the curvature constant $\kappa$
has no contribution to the shear viscosity to entropy density ratio for the
leading term. The sub-leading terms, however, receive contributions from
$\kappa$.
So far we are confident with the violation of the KSS bound while we are not
sure the existence of a universal lower bound of $\eta/s$. A great progress
alone this line appeared several months ago when the authors of [28] gave a
proof for the existence of a universal bound of $\eta/s$ for any ghost-free
extension of Einstein theory. However, the work in[14] shows that the
causality violation of the dual gauge theory may put constraints on the
coefficients of higher derivative terms and this in turn will put constraints
on the value of $\eta/s$. Then it is natural to ask if the lower bound still
exists as these constraints are taken into account. Recent progress made by
Camanho and Edelstein in [29] provides us with an answer that the causality
violation, as expected, may impose a constraint on the bound of the $\eta/s$
at least for cubic Lovelock gravity. For completeness, we briefly catch some
important results from [29], so as to compare the lower limit on $\eta/s$ from
causality violation and the result in the present paper. Actually, the formula
of the ratio between $\eta$ and $s$ obtained in [29] is not different from our
result (41). What is new of their result is that by imposing a condition so
that the causality violation can be avoided, they found constraints on the
coefficient $\lambda$ (or $c_{2}$ as defined) in (41). For any order ($n\geq
2$) Lovelock gravity, the condition to be free of causality is that
$\sum_{m=1}^{n}mc_{m}\Lambda^{m-1}\left(1+\frac{\gamma(m-1)(D-1)}{D-3}\right)\geq
0,$ (44)
where $\gamma=-2,-1,2/(D-4)$ represent helicity zero, helicity one and
helicity two graviton, respectively. Therefore, though any correction terms
higher than the second order of Lovelock gravity do not manifestly contribute
to the ratio of viscosity to entropy density, it does not mean that they are
irrelevant to this ratio. Through (44) we see these terms impose a constraint
on the value of $c_{2}$ (or $\lambda$) thus in turn affecting the lower bound
for $\eta/s$.
ACKNOWLEDGEMENTS
The author would like to thank Profs. Y.-G. Gong and S.-J. Sin for their
valuable comments. This work was supported in part by Natural Science
Foundation Project of CQ CSTC under Grant No. 2009BB4084 and key project from
NNSFC (No. 10935013).
## References
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|
arxiv-papers
| 2009-10-04T14:01:37 |
2024-09-04T02:49:05.645617
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Fu-Wen Shu",
"submitter": "Fu-Wen Shu",
"url": "https://arxiv.org/abs/0910.0607"
}
|
0910.0632
|
# The Square Kilometre Array
Naval Research Laboratory and Square Kilometre Array Program Development
Office, 4555 Overlook Ave. SW, Washington, DC 20375-5351 USA
E-mail Basic research in radio astronomy at the NRL is supported by 6.1 Base
funding.
###### Abstract:
The Square Kilometre Array (SKA) is intended as the next-generation radio
telescope and will address fundamental questions in astrophysics, physics, and
astrobiology. The international science community has developed a set of Key
Science Programs: (1) Emerging from the Dark Ages and the Epoch of
Reionization, (2) Galaxy Evolution, Cosmology, and Dark Energy, (3) The Origin
and Evolution of Cosmic Magnetism, (4) Strong Field Tests of Gravity Using
Pulsars and Black Holes, and (5) The Cradle of Life/Astrobiology. In addition,
there is a design philosophy of “exploration of the unknown,” in which the
objective is to keep the design as flexible as possible to allow for future
discoveries. Both a significant challenge and opportunity for the SKA is to
obtain a significantly wider field of view than has been obtained with radio
telescopes traditionally. Given the breadth of coverage of cosmic magnetism
and galaxy evolution in this conference, I highlight some of the opportunities
that an expanded field of view will present for other Key Science Programs.
## 1 Introduction
In the $20^{\mathrm{th}}$ Century, we discovered our place in the Universe. We
learned that it was much bigger than we imagined and much more exotic. Beyond
our Milky Way, the Universe is filled with galaxies—each their own island
universe. They range in size from dwarf galaxies barely able to survive near
their larger neighbors to giant elliptical galaxies, orders of magnitudes
larger than the Milky Way. These galaxies of stars also contain a multitude of
other components including gas with a wide range of temperatures; compact
objects including white dwarfs, neutron stars, and black holes; and planets.
Over the course of the century, black holes moved from a theoretical curiosity
to a well-recognized endpoint of stellar evolution and a likely fundamental
component of the centers of galaxies, with the potential to power immense jets
of relativistic particles that affect their surroundings. By the end of the
century, we were beginning to unveil the basic structure and processes of the
Universe in which these objects are embedded, including evidence of its origin
and the still mysterious properties of dark matter and dark energy.
Our probes of the Universe have expanded dramatically as well. Electromagnetic
radiation has been detected from celestial objects at frequencies below 1 MHz
($\lambda\sim 300$ m) to energies exceeding 1 TeV. Moreover, the range of
possible signals has expanded beyond just electromagnetic radiation. Cosmic
rays rain down on the Earth, some with energies approaching those of
macroscopic objects. Gravitational radiation has been detected indirectly, and
numerous potential classes of sources have been suggested, with the
expectation that the Earth is awash in gravitational waves. Neutrinos have
been detected from both the Sun and supernova 1987A, and many of the processes
that generate high energy cosmic rays should also produce a spectrum of high-
energy neutrinos.
In the $21^{\mathrm{st}}$ Century, we seek to understand the Universe we
inhabit. To do so will require a suite of powerful new instruments, on the
ground and in space, operating across the entire electromagnetic spectrum and
for multiple decades. Observations at centimeter- to meter wavelengths have
provided deep insight to a wide range of phenomena ranging from the solar
system to the most distant observable celestial emission. This long and rich
record of important discoveries in the radio spectrum, including 3 Nobel
prizes, has been possible since many of the relevant physical phenomena can
only be observed, or understood best, at these wavelengths. These phenomena
include the cosmic microwave background (CMB), quasars, pulsars, gravitational
waves, astrophysical masers, magnetism from planets through galaxies, the
ubiquitous jets from black holes and other objects, and the spatial
distribution of hydrogen gas, the predominant baryonic constituent of the
Universe. Moreover, through the invention of aperture synthesis, also
recognized by the Nobel committee, radio astronomy has reached unprecedented
levels of imaging resolution and astrometric precision, providing the fuel for
further discovery.
With only a handful of exceptions, radio telescopes and arrays have been
limited to apertures of about $10^{4}$ m2, constraining, for instance, studies
of the 21-centimeter hydrogen emission to the nearby Universe ($z\sim 0.2$)
[4]. Contemporaneous with the astronomical discoveries in the latter half of
the $20^{\mathrm{th}}$ Century have been technological developments that offer
a path to substantial improvements in future radio astronomical measurements.
Among the improvements are mass production of centimeter-wavelength antennas
enabling apertures potentially 100 times larger than previously available,
fiber optics for the transmission of large volumes of data, high-speed digital
signal processing hardware for the acquisition and analysis of the signals,
and computational improvements leading to massive processing and storage.
These new technologies, combined with dramatically improved survey speeds and
the other advances, can open up an enormous expanded volume of discovery
space, providing access to many new celestial phenomena and structures,
including 3-dimensional mapping of the web of hydrogen gas through much of
cosmic history ($z\sim 2$).
The realization that radio astronomy was on the doorstep of a revolutionary
age of scientific breakthrough has led the international community to
investigate this opportunity in great detail over the last decade. That
coordinated effort, involving a significant fraction of the world’s radio
astronomers and engineers, has resulted in the Square Kilometre Array (SKA)
Program (Figure 1), an international roadmap for the future of radio astronomy
over the next two decades and one for which access to a wide field of view is
an integral part of the science.
Figure 1: An artist’s impression of the core of the SKA illustrating the
various technologies over the frequency range 70 MHz to 10 GHz. All of these
technologies would enable various levels of wide-field imaging.
From its inception, development of the SKA Program has been a global endeavor.
In the early 1990s, there were multiple, independent suggestions for a “large
hydrogen telescope.” It was recognized that probing the fundamental baryonic
component of the Universe much beyond the local Universe would require a
substantial increase in collecting area. The IAU established a working group
in 1993 to begin a worldwide study of the next generation radio observatory.
Since that time, the effort has grown to comprise 19 countries and more than
50 institutes, including about 200 scientists and engineers.
## 2 Key SKA Science
Over the past several years, there has been extensive activity related to
developing a detailed science case for the SKA, culminating in the SKA Science
Book [4]. Highlighting the SKA Science Case are Key Science Projects (KSPs),
which represent unanswered questions in fundamental physics, astrophysics, and
astrobiology. Furthermore, each of these projects has been selected using the
criterion that it represents science that is either unique to the SKA or in
which the SKA will provide essential data for a multi-wavelength analysis [6].
The KSPs are
Emerging from the Dark Ages and the Epoch of Reionization
The ionizing ultra-violet radiation from the first stars and galaxies produced
a fundamental change in the surrounding intergalactic medium, from a nearly
completely neutral state to the nearly completely ionized Universe in which we
live today. The most direct probe of this Epoch of Re-ionization (EoR), and of
the first large-scale structure formation, will be obtained by imaging neutral
hydrogen and tracking the transition of the intergalactic medium from a
neutral to ionized state. Moreover, as the first galaxies and AGN form, the
SKA will provide an unobscured view of their gas content and dynamics via
observations of highly redshifted, low-order molecular transitions (e.g., CO).
Galaxy Evolution, Cosmology, and Dark Energy
Hydrogen is the fundamental baryonic component of the Universe. The SKA will
have sufficient sensitivity to the 21-cm hyperfine transition of H i to detect
galaxies to redshifts $z>1$. One of the key questions for $21^{\mathrm{st}}$
Century astronomy is the assembly of galaxies; the SKA will probe how galaxies
convert their gas to stars over a significant fraction of cosmic time and how
the environment affects galactic properties. Simultaneously, baryon acoustic
oscillations (BAOs), remnants of early density fluctuations in the Universe,
serve as a tracer of the early expansion of the Universe. The SKA will
assemble a large enough sample of galaxies to measure BAOs as a function of
redshift to constrain the equation of state of dark energy.
The Origin and Evolution of Cosmic Magnetism
Magnetic fields likely play an important role throughout astrophysics,
including in particle acceleration, cosmic ray propagation, and star
formation. Unlike gravity, which has been present since the earliest times in
the Universe, magnetic fields may have been generated essentially ab initio in
galaxies and clusters of galaxies. By measuring the Faraday rotation toward
large numbers of background sources, the SKA will track the evolution of
magnetic fields in galaxies and clusters of galaxies over a large fraction of
cosmic time. The SKA observations also will seek to address whether magnetic
fields are primordial and dating from the earliest times in the Universe or
generated much later by dynamo activity.
Strong Field Tests of Gravity Using Pulsars and Black Holes
With magnetic field strengths as large as $10^{14}$ G, rotation rates
approaching 1000 Hz, central densities exceeding $10^{14}$ g cm-3, and
normalized gravitational strengths of order 0.4, neutron stars represent
extreme laboratories. Their utility as fundamental laboratories has already
been demonstrated through results from observations of a number of objects.
The SKA will find many new millisecond pulsars and engage in high precision
timing of them in order to construct a Pulsar Timing Array for the detection
of nanohertz gravitational waves, probing the space-time environment around
black holes via both ultra- relativistic binaries (e.g., pulsar-black hole
binaries) and pulsars orbiting the central supermassive black hole in the
centre of the Milky Way, and probe the equation of state of nuclear matter.
The Cradle of Life
The existence of life elsewhere in the Universe has been a topic of
speculation for millennia. In the latter half of the $20^{\mathrm{th}}$
Century, these speculations began to be informed by observational data,
including organic molecules in interstellar space, and proto-planetary disks
and planets themselves orbiting nearby stars. With its sensitivity and
resolution, the SKA will be able to observe the centimeter-wavelength thermal
radiation from dust in the inner regions of nearby proto-planetary disks and
monitor changes as planets form, thereby probing a key regime in the planetary
formation process. On larger scales in molecular clouds, the SKA will search
for complex prebiotic molecules. Finally, detection of transmissions from
another civilization would provide immediate and direct evidence of life
elsewhere in the Universe, and the SKA will provide sufficient sensitivity to
enable, for the first time, searches for unintentional emissions or “leakage.”
In addition to the KSPs listed, and recognizing the long history of discovery
at radio wavelengths (pulsars, cosmic microwave background, quasars, masers,
the first extrasolar planets, etc.), the international science community also
recommended that the design and development of the SKA have “Exploration of
the Unknown” as a philosophy. Wherever possible, the design of the telescope
is being developed in a manner to allow maximum flexibility and evolution of
its capabilities to probe new parameter space (e.g., time-variable phenomena
that current telescopes are not well-equipped to detect). This philosophy is
essential as many of the outstanding questions of the 2020–2050 era—when the
SKA will be in its most productive years—are likely not even known today.
## 3 Opportunities for Panoramic SKA Science
Many of the papers in this volume illustrate far better than I could the
opportunities for Panoramic SKA Science, particularly in the areas of cosmic
magnetism and galaxy structure and evolution via H i observations.
Consequently, and similar to my approach in the conference itself, I shall
focus on opportunities for wide-field observations as they concern some of the
other SKA KSPs.
### 3.1 Emerging from the Dark Ages and the Epoch of Reionization
The primary focus of this KSP is tracking the transition from the Universe’s
largely neutral state to its currently nearly completed ionized state. Wide-
field observations will be both important and natural as the key observations
will be of the highly-redshifted H i line at frequencies below 200 MHz, which
will be carried out using dipole-based sparse aperture arrays. Dipoles have
fields of view that can exceed $\pi$ sr easily, but the relevant frequencies
are below the nominal focus of this conference.
A potential and important secondary observation that could be conducted near 1
GHz, however, would be of the synchrotron radiation from the first galaxies
[12]. Copious numbers of massive stars would have likely formed within these
first galaxies and then exploded soon thereafter as supernovae. If the
interstellar magnetic fields of these galaxies have developed sufficiently,
the galaxies will emit synchrotron radiation as a result of cosmic rays
accelerated by the supernova remnants from these first massive stars. While it
is not yet known if these galaxies will be detectable, the radio-far infrared
correlation for star-forming galaxies is now known to hold at least out to a
redshift $z\approx 3$ [14]. If it continues to hold to $z\approx 6$, then
radio observations would be a powerful means of probing dust-enshrouded first
galaxies and wide-field observations would naturally allow for large volumes
of the Universe to be sampled quickly.
### 3.2 Fundamental Physics Using Observations of Pulsars and Black Holes
Wide-field capabilities that enable the SKA to access a substantial solid
angle will be important for pulsar studies, even though pulsars are point
sources so that “panoramic imaging” per se of them is unlikely to be
profitable.
Fundamental physics constraints are derived from pulsar observations via long-
term timing programs that measure precisely the times of arrival of the
pulses. A significant constraint on the utility of radio pulsars is the
scarcity of “useful” pulsars. For instance, until recently, the most
significant constraints on the nuclear equation of state derived from radio
pulsars resulted from the first millisecond pulsar, PSR B1937$+$21, discovered
in the _early 1980s_. Similarly, many of the tests of theories of gravity and
for gravitational wave emission rely on one or a few objects. Recent surveys
have begun to demonstrate the potential for vastly increasing the number of
radio pulsars and thereby increasing the number of “useful” systems.
Perhaps the best example of the impact of increasing field of view for pulsar
surveys is the Parkes Multibeam Survey [10]. By installing a multiple feed
horn system on the Parkes antenna, the effective field of view was increased
by a factor of 13. The resulting survey essentially doubled the total number
of pulsars. (See also §3.4.) Future field-of-view expansion technologies
(e.g., phased array feeds or dense aperture arrays) coupled with the vastly
increased sensitivity of the SKA offer promise for an even larger yield.
The impact of a wide field of view for pulsar timing and monitoring programs
is less clear. In principle, a telescope with a sufficiently wide field of
view could time multiple pulsars simultaneously, yielding an improved
“throughput.” In practice, the current estimates of the density on the sky of
“useful” pulsars is sufficiently low that only dense aperture arrays are
likely to have a field of view that could be large enough to time multiple
pulsars simultaneously, except perhaps in special regions of the sky.
Moreover, in order to mitigate interstellar propagation effects, timing
observations have to be carried out over a relatively large frequency range
(e.g., 0.8–3 GHz), wider than what dense aperture arrays are currently thought
to be able to achieve.
### 3.3 Cradle of Life/Astrobiology
One of the key assumptions in the search for life elsewhere in the Universe,
particularly in searches for life within the solar system, is that other life
is likely to be based on carbon chemistry (i.e., “organic”), like life on
Earth is. Prime support for this approach is that the vast majority of multi-
atom molecules in interstellar space contain carbon, including a number of
complex organic species [15, 2, 1, and references within]. As the number of
atoms increases, the rotational and vibrational transitions tend to shift to
lower frequencies, and searches for and studies of complex organic molecules
have relied upon observations below 2 GHz. Consequently, a wide field-of-view
at frequencies around 1 GHz could be quite valuable for conducting surveys of
molecular clouds for complex organic molecules; such observations would find a
natural complement in ALMA observations.
Direct evidence for life elsewhere in the Universe would be the detection of
signals from another technological civilization. Two examples from our own
civilization are cell phone transmissions and aeronautical navigation, both of
which make use of frequencies around 1 GHz, though neither are strong enough
to be detectable over interstellar distances (even with the SKA!). More
generally, the “waterhole” between 1.4 and 1.7 GHz has been a focus of
numerous previous searches for extratestrial transmissions, as it has been
argued that any technological civilization capable of trying to communicate
over interstellar distances would certainly know about the H i line at 1.4 GHz
and the OH lines around 1.7 GHz.
One approach for searching for extraterrestial intelligence (SETI) is to
monitor a “habstar,” a star that might be orbited by a terrestrial planet(s)
within the star’s habitable zone [16]. Much like pulsar timing, being able to
monitor multiple habstars would increase the throughput of SETI observations;
the key contrast between pulsar and habstar observations is that the density
on the sky of suitable main sequence stars is sufficiently high that most, if
not all, fields of view will include more than one habstar.
### 3.4 The Dynamic Radio Sky
A series of discoveries over the past decade have both illustrated and
emphasized that the time domain has been explored only poorly at radio
wavelengths [3, 7, 9, 11]. Although time resolutions approaching 1 ns have
been achieved [8], typically these have been obtained only on relatively
narrow fields of view. The challenge and opportunity for the SKA, and
consistent with the “exploration of the unknown” design philosophy, is to
obtain both high time resolution and access to a significant solid angle.
Some of these observations might naturally happen in the course of pulsar
surveys (§3.2); indeed, the discovery of rotating radio transients, a new
class of radio-emitting neutron stars, resulted from the novel processing of a
pulsar survey [11]. Other types of transient surveys and exploration programs
might utilize wide fields of view in different manners, however.
We provide two examples to illustrate the potential range of applications of a
wide field of view:
1. 1.
Extreme scattering events (ESEs) are a class of dramatic flux density
variations ($\sim 50$%) of extragalactic sources caused by intervening plasma
lenses [5]. The initial surveys for ESEs observed only a relatively small
number of the strongest, most compact sources on the sky. Yet within even a
modest field of view, if the full field of view can be imaged, are potentially
tens to hundreds of sources. A potential ESE search program could be conducted
by surveying a significant solid angle with a regular cadence and constructing
light curves of all of the sources within the survey region. Clearly an
expanded field of view would determine the total number of sources that could
be monitored.
2. 2.
Many low-mass stars (spectral types K and M) show significant “radio
activity,” often with radio flares or bursts on short time scales [13]. This
radio emission is thought to be linked to coronal processes on the stars,
likely closely coupled to the magnetic field structure. Study of the coronal
processes in these extreme cases may provide understanding of solar processes,
which could impact not only astrophysics but aspects of the Earth-Sun
connection as well. Most, if not all, of the strongly “radio active” stars in
the solar neighborhood are known, but the typical separation on the sky is
fairly large. Similar to the case for pulsar timing, monitoring a large number
of low-mass stars for radio bursts would have a much higher throughput if
access to a wide field of view becomes possible.
We emphasize that these are only two possible examples, chosen to illustrate
the possible range of transient survey programs. The actual impact of the
field of view on any transient program will also depend upon the temporal
characteristics of the transients being targeted, their luminosity function,
and distribution on the sky, to the extent that these parameters are known.
## References
* [1] Belloche, A., Garrod, R. T., Müller, H. S. P., Menten, K. M., Comito, C., & Schilke, P. 2009, Astron. & Astrophys. 499, 215.
* [2] Belloche, A., Menten, K. M., Comito, C., Müller, H. S. P., Schilke, P., Ott, J., Thorwirth, S., & Hieret, C. 2009, Astron. & Astrophys. 482, 179.
* [3] Bower, G. C. et al. 2008, Astrophys. J. 666, 346.
* [4] C. L. Carilli and S. Rawlings, Science with the Square Kilometer Array, New Astron. Rev., 48, Elsevier, Amsterdam, 2004
* [5] Fiedler, R. L., Dennison, B., Johnston, K. J., & Hewish, A. 1987, Nature 326, 675.
* [6] Gaensler, B. M. 2004, “Key Science Projects for the SKA,” SKA Memorandum 44
* [7] Hallinan, G. et al. 2007, Astrophys. J. 663, L25.
* [8] Hankins, T. H., Kern, J. S., Weatherall, J. C., & Eilek, J. A. 2003, Nature 422, 141.
* [9] Hyman, S. D. et al. 2005, Nature 434, 50.
* [10] Manchester, R. N. et al. 2001, Mon. Not. R. Astron. Soc. 328, 17
* [11] McLaughlin, M. A. et al. 2006, Nature 439, 817.
* [12] Murphy, E. 2009, Astrophys. J., submitted
* [13] Osten, R. A., & Bastian, T. S. 2008, Astrophys. J. 674, 1078.
* [14] Seymour, N., Huynh, M., Dwelly, T., et al. 2009, Mon. Not. R. Astron. Soc., in press; arXiv:0906.1817
* [15] Snyder, L. E., Hollis, J. M., Jewell, P. R., Lovas, F. J., & Remijan, A. 2006, Astrophys. J. 647, 412.
* [16] Turnbull, M. C. ,& Tarter, J. C. 2003, Astrophys. J. Supp. 145, 181.
|
arxiv-papers
| 2009-10-04T19:18:36 |
2024-09-04T02:49:05.650980
|
{
"license": "Public Domain",
"authors": "Joseph Lazio",
"submitter": "Joseph Lazio",
"url": "https://arxiv.org/abs/0910.0632"
}
|
0910.0670
|
# Derivation of the Density Functional
via Effective Action
Yi-Kuo Yu National Center for Biotechnology Information, National Library of
Medicine
National Institutes of Health, Bethesda, MD 20894, USA
###### Abstract
A rigorous derivation of the density functional in the Hohenberg-Kohn theory
is presented. With no assumption regarding the magnitude of the electric
coupling constant $e^{2}$ (or correlation), this work provides a firm basis
for first-principles calculations. Using the auxiliary field method, in which
$e^{2}$ need not be small, we show that the bosonic loop expansion of the
exchange-correlation functional can be reorganized so as to be expressed
entirely in terms of the Kohn-Sham single-particle orbitals and energies. The
excitations of the many-particle system can be obtained within the same
formalism. We also explicitly demonstrate at zero-temperature the single-
particle limit, the weak-coupling limit of the energy functional, and its
application to homogeneous electron gas.
###### pacs:
71.15.Mb
## I Introduction
At low energy scale, interactions among electrons largely determine the
structure, phases, and stability of matter. Although this fact is well known,
pragmatic first-priniciples/quantum-mechanical calculations to determine
various properties of many-electron systems are often hindered by two factors.
First, in most condensed matter systems, the typical interaction energy
between electrons (the electric coupling constant $e^{2}$ divided by average
electron-electron separation) is often larger than the typical kinetic/Fermi
energy of electrons. The results is that a perturbative expansion using
$e^{2}$ as the expansion parameter may not be fruitful. This is particularly
true for strongly correlated systems. Second, there is an exponential increase
in the number of degrees of freedom as the number of electrons involved
increases. When the number of electrons becomes large, according to Kohn,Kohn
(1999) calculations based on constructing many-electron wave functions soon
lose accuracy and will be stopped by an “exponential wall”. It is thus
imperative to have a method that goes beyond the conventional perturbative
scheme using $e^{2}$ as the expansion parameter and whose computational
complexity does not grow exponentially with the number of electrons involved.
In 1964, Hohenberg and Kohn Hohenberg and Kohn (1964) proved a theorem stating
that there exists a unique description of the ground state of a many-body
system in terms of the expectation value of the particle-density operator.
This theorem started the development of the density functional theory (DFT),
which offers a possibility of finding the ground state energy $E_{g}$ by
minimizing the energy functional $E_{\upsilon}$ that depends on the charge
density $n$ only:
$E_{g}=\min\limits_{n}E_{\upsilon}\left[n\right],$ (1)
with $\upsilon$ representing the external one-particle potential of the
system. The electronic density $n_{g}$, which minimizes the energy functional
$E_{\upsilon}[n]$, is the ground state electronic density. Hohenberg and Kohn
showed that the energy functional $E_{\upsilon}[n]$ can be decomposed into
$E_{\upsilon}\left[n\right]=\int d{\bf r}\,\upsilon({\bf r})\,n({\bf
r})+{\mathcal{F}}\left[n\right]\;,$ (2)
with ${\mathcal{F}}\left[n\right]$ being a universal functional independent of
the external potential $\upsilon$. MerminMermin (1965) extended this theorem
to finite temperature with $E_{\upsilon}$ in (2) replaced by the grand
potential, and ${\mathcal{F}}[n]$ replaced by a different universal
functional. The electron density $n_{T}$, minimizing the grand potential
functional, corresponds to the electron density at thermal equilibrium.
To make practical use of the DFT, however, a recipe to compute the energy
functional is needed. Kohn and ShamKohn and Sham (1965) proposed a
decomposition scheme, aiming to express the energy functional
$E_{\upsilon}[n]$ via an auxiliary, noninteracting system that yields a
particle density identical to that of the physical ground state. For a typical
nonrelativistic many-fermion system, described by the Hamiltonian
$\displaystyle\hat{H}$ $\displaystyle=$ $\displaystyle\int d{\bf
x}{\hat{\psi}}^{{\dagger}}({{\bf
x}})\left(-\frac{1}{2m}\nabla^{2}+\upsilon_{\rm ion}({{\bf
x}})-\mu\right)\hat{\psi}({{\bf x}})$ (3) $\displaystyle\
+\frac{e^{2}}{2}\int\int\frac{{\hat{\psi}}^{{\dagger}}({{\bf
x}}){\hat{\psi}}^{{\dagger}}({{\bf y}})\hat{\psi}({{\bf y}})\hat{\psi}({{\bf
x}})}{|{{\bf x}}-{{\bf y}}|}d{{\bf x}}d{{\bf y}},$
the Kohn-Sham decomposition takes the form
$\displaystyle E_{\upsilon}\left[n\right]$ $\displaystyle=$ $\displaystyle
T_{0}\left[n\right]+\int\upsilon_{\rm ion}({\bf x})\,n\left({\bf
x}\right)d{{\bf x}}-\mu N_{e}$ (4)
$\displaystyle+\frac{e^{2}}{2}\int\int\frac{n({{\bf x}})n({{\bf y}})}{|{{\bf
x}}-{{\bf y}}|}d{\bf x}d{\bf y}+E_{xc}\left[n\right],$
where the chemical potential $\mu$ is introduced to ensure $\int\\!n({\bf
x})\,d{\bf x}=N_{e}$, with $N_{e}$ being the number of electrons. Here
$T_{0}\left[n\right]$ is the kinetic energy of an auxiliary system of
noninteracting fermions that yields the electron density $n\left({\bf
x}\right)$, and the density functional $E_{xc}\left[n\right]$ is the so-called
exchange-correlation energy functional. Given $E_{xc}\left[n\right]$ and
provided that it is differentiable, one may minimize the functional
$\left(\ref{KSfunctional}\right)$ to arrive at the familiar Kohn-Sham single-
particle equationsHohenberg et al. (1990)
$\displaystyle\left(-\frac{\nabla^{2}}{2m}+\upsilon_{\rm ion}({\bf
x})+\int\frac{n({\bf y})}{|{\bf x}-{\bf y}|}d{\bf y}+\frac{\delta
E_{xc}[n]}{\delta n({\bf x})}-\mu\right)\psi_{i}({\bf
x})=\epsilon_{i}\psi_{i}({\bf x})$ (5) $\displaystyle\hskip 36.135ptn({\bf
x})=\sum_{i=1}^{N_{e}}\psi_{i}^{*}({\bf x})\psi_{i}({\bf x})\;.$ (6)
All of the many-particle complexity is now completely hidden in the exchange-
correlation energy functional.
Although $T_{0}[n]+E_{xc}[n]$ is universal,Hohenberg and Kohn (1964) there
exists no simple means thus far to obtain it. As a consequence, various ad hoc
exchange-correlation density functionals have been suggested/needed to yield
acceptable results in different settings Andersson and Gru1ning (2004); Sousa
et al. (2007); Tekarli et al. (2009); Kosztyu and Lendvay (2009) when
employing the Kohn-Sham scheme. Limitations of these approximate functionals
have been discussed.Kümmel and Kronik (2008); Cohen et al. (2008) Some of the
failures while using ad hoc density functionals can be attributed to misuse of
the density-functional theory. For example, it is sometimes neglected that the
electron density $n({\bf x})$ achievable via introduction of a source
potential must obey $\int n({\bf x})d{\bf x}=N_{e}$ and does not cover the
functional space $\\{n({\bf x})\geq 0\\}$, a problem also known as the
$\upsilon$-representability. Kohn (1983); Lieb (1982); Levy (1982) As pointed
out in reference Valiev and Fernando, 1995, neglecting these constraints may
lead to conclusions Janak (1978) that are not always valid.
The main objective of the DFT is to describe a many-body system in terms of
the expectation value of the particle density operator. In fact, the use of
the expectation value of a suitable operator to describe a many-body system
via Legendre transformation, first introduced into quantum field theory by
Jona-Lasinio,Jona-Lasinio (1964) is known as the effective action formalism.
As the temperature approaches zero, the effective potential becomes the ground
state energy. This connection suggests that effective action formalism can be
used to achieve the general goal of the DFT: describing a many-particle system
in terms of the expectation value of the density operator. A number of
publications Fukuda et al. (1994, 1995); Valiev and Fernando (1997); Polonyi
and Sailer (2002) showed that at zero temperature the effective action plus
$\mu N_{e}$ is the ground state energy, linking effective action to the DFT.
Existing methods of expressing DFT via effective action formalism can be
classified roughly into two categories: either (a) by introducing an auxiliary
field or (b) by using a perturbative scheme assuming the electric coupling as
a small parameter. The former category includes a method developed by Fukuda
et al. Fukuda et al. (1994) and that developed by Polonyi and Sailer. Polonyi
and Sailer (2002) The latter scheme is used by Valiev and FernandoValiev and
Fernando (1997) and by Fukuda et al. Fukuda et al. (1995).
The strengths of the auxiliary field approach often come from the simplicity
of the effective action expression and from the fact that in principle each
term already includes infinitely many Feynman diagrams.Jackiw (1974) However,
as pointed out by Fukuda et al.,Fukuda et al. (1994) the auxiliary field
approach seems to lack a direct connection to the Kohn-Sham scheme. Valiev and
Fernando Valiev and Fernando (1996) introduced an auxiliary field to compute
the exchange-correlation energy. However, the source term they introduced is
coupled to the auxiliary field instead of the electron density operator.
Furthermore, as pointed out by the authors themselves, Valiev and Fernando
(1997) an artificial decomposition of the auxiliary field into a sum of the
Hartree potential, the exchange correlation potential, and the remaining
fluctuations is needed to write down the exchange-correlation energy.
There are two advantages when one uses electric coupling in the perturbative
(diagrammatic) expansion without introducing an auxiliary field. First, under
the effective action formalism of this type, a direct connection to the Kohn-
Sham scheme can be made. Valiev and Fernando (1997); Fukuda et al. (1995)
Second, there exist other $e^{2}$ expansion-based developments that can be
used to obtain $T_{0}[n]+E_{xc}[n]$, the universal density functional (UDF).
For example, with increasingly complex incorporation of KS orbitals and
energies at each order of $e^{2}$, Görling and Levy Görling and Levy (1994)
wrote $E_{g}$ as a perturbation series. Employing the Luttinger-Ward Luttinger
and Ward (1960) method that uses $e^{2}$ as the perturbative expansion
parameter to calculate electron self-energy, Sham and Schlüter expressed
$E_{xc}$ as a rather convoluted implicit functional of electron density.Sham
and Schlüter (1983); Sham (1985) As shown by Tokatly and Pankratov,Tokatly and
Pankratov (2001) these methods mentioned above can be expressed
diagrammatically. However, the problem associated with $e^{2}$ based expansion
remains. As described in reference Negele and Orland, 1988, the expansion
using $e^{2}$ is only good when $e^{2}$ is very small. Whether one can treat
$e^{2}$ as a small parameter or not depends on the kinetic/Fermi energy of the
electrons and the strength as well as the magnitude of variation of the
single-particle potential involved. As a matter of fact, the success Dahlen et
al. (2006) in employing the GW approximation Hedin (1965) indicates that not
treating $e^{2}$ as small may lead to results closer to experimental outcomes.
In principle, the problem associated with assuming $e^{2}$ small can be tamed
by summing an infinite subset of Feynman diagrams. However, as pointed out by
Hedin, Hedin (1965) it is nontrivial to devise a systematic resummation scheme
where each new term is free from divergence even if $e^{2}\gg 1$ and for which
the sum of the new terms, each containing an infinitely many Feynman diagrams,
accounts completely and non-redundantly for all conventional $e^{2}$ expansion
diagrams. In this paper, without assuming $e^{2}$ small we develop an
auxiliary field method that makes a direct connection to the Kohn-Sham scheme
and at the same time provides equivalently a systematic resummation scheme.
A commonly used approximation for the density functional is the so-called
local density approximation (LDA) in which the exchange-correlation energy is
approximated by a linear functional $E_{xc}[n]\approx\int d{\bf r}\,n({\bf
r})\;e_{xc}(n({\bf r}))$, where $e_{xc}(n)$ is a function of the local density
(not a functional of the density profile). See reference Kohn, 1999 for a
nontechnical review. This approximation ignores the nonlocal effect of the
density profile, i.e., it assumes that $\delta E_{xc}[n]/\delta n({\bf r})$
only depends on the value of $n$ at ${\bf r}$ but not on the density $n({\bf
r}^{\prime}\neq{\bf r})$ at locations other than ${\bf r}$. To complement the
LDA by incorporating nonlocal density dependence, Polonyi and Sailer Polonyi
and Sailer (2002) proposed the $l$-local approximation for the density
functional, based on an idea very similar to the cluster expansion in
statistical physics. Using this method to obtain explicit expressions for the
approximate functional with $l\geq 3$, however, becomes increasingly
challenging due to the necessity of going through the coupling constant
integration as required by the Hellmann-Feynman theorem.Hellmann (1937);
Feynman (1939)
Another route to developing density functionals is via the so-called optimal
effective potential (OEP) methods. Sharp and Horton (1953); Talman and
Shadwick (1976); Petersilka et al. (1996) These methods typically start by
introducing a priori an approximate, explicitly Kümmel and Perdew (2003)
orbital-dependent functional. (The approximate functional can be either
Hartree-Fock or a more elaborated form.) The procedure then continues with a
minimization of the functional via varying single-particle KS orbitals and
associated energies. Recently, the definition of OEP methods has been
generalized Casida (1995a); von Barth et al. (2005) to include functionals
dependent on either Green’s function, the self-energy, or the KS potential.
Since our effective action based functional is based on self-consistently
obtaining the KS potential, it falls exactly in the latter category. The
generalized definition of OEP methods is probably becoming the standard
definition now. A general characteristic of OEP methods is that the functional
arguments –be they KS orbitals/energies, Green’s functions, self energies, or
KS potentials– are obtained via self-consistent procedure. Therefore, even
with correction terms derived from pertrubative expansion, the self-
consistency condition for OEP methods distinguishes them from regular
pertrubative methods. The important matter here is whether an OEP functional
can be systematically improved and possibly be asymptotically exact or not.
Based on effective action formalism, the OEP functional proposed here is
asymptotically exact and can be shown to give rise to the desired UDF.
Containing all the $l$-local interaction vertices, our method can provide
equivalent approximate functionals for $l\geq 3$ without going through the
Hellmann-Feynman theorem. Since we focus on describing the proposed approach
in a manner as self-contained as possible, we have included a non-negligible
amount of standard materials available in existing literature/textbooks while
keeping only a small portion of the existing literature that we deem closely
related to the present manuscript. Readers interested in gaining a broad
background are referred to references Kümmel and Kronik, 2008 and Baroni et
al., 2001; Furnstahl, ; Jones and Gunnarsson, 1989; Bartlett and Musiał, 2007
for reviews on the extensive body of literature in the DFT and related many-
body approaches. Expert readers should note that new developments are mainly
provided in sections IIIB-D. Although section IV and end of section V also
contain some useful developments, they typically rederive/re-express known
results within our framework and/or provide contrast with existing methods.
Section VI contains some insight of problems shared in post Hartree
corrections.
This paper is otherwise organized as follows. We first establish the notation
in section II, followed by the development of the general formalism in section
III. The purpose of subsection III.7 is to provide a computational recipe and
to give some perspectives on computational complexity: no novelty is claimed
here. In section IV, we discuss a number of case studies: the emergence of the
universal functional ${\mathcal{F}}[n]$ in Eq. (2) at arbitrary temperature,
the behavior of the effective potential and the single electron limit at zero
temperature, the screening effect, as well as the case of homogeneous electron
gas. In section V, we then discuss the excitations of the system, and make
comparisons with existing studies along this direction. An alternative
formalism to obtain the effective action is then discussed in section VI. We
conclude with the discussion and future directions section, in which we also
provide some more relations/comparisons to other methods as well as some
technical remarks.
## II Notation
Let us first define useful notation to lighten the exposition of the
mathematical formulas.
We define a three dimensional integral contraction by a dot
$a{\cdot}b\equiv\int d{\bf x}\;a({\bf x})\,b({\bf x})$
where $a$ and $b$ may be single or composite fields. That is, with $m\geq 1$
and $n\geq 1$, $a({\bf x})$ and $b({\bf x})$ may represent
$\displaystyle a({\bf x})$ $\displaystyle=$ $\displaystyle a_{1}({\bf
x})\ldots a_{m}({\bf x})\;,$ $\displaystyle{\rm and\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ }b({\bf x})$ $\displaystyle=$
$\displaystyle b_{1}({\bf x})\ldots b_{n}({\bf x})\;.$
Similarly, with a kernel $M$, we may define
$a{\cdot}b{\cdot}M=a{\cdot}M{\cdot}b=M{\cdot}a{\cdot}b=\iint d{\bf x}d{\bf
y}\;M({\bf x},{\bf y})a({\bf x})\,b({\bf y})\;.$
Note that in all expressions $a$ is in front of $b$, and that is important
when there are Grassmann variables involved. Evidently, one may generalize
this notation to include higher order kernels. That is, one may have
$a{\cdot}b{\cdot}c{\cdot}M=a{\cdot}b{\cdot}M{\cdot}c=a{\cdot}M{\cdot}b{\cdot}c=M{\cdot}a{\cdot}b{\cdot}c=\iiint
d{\bf x}d{\bf y}d{\bf z}\;M({\bf x},{\bf y},{\bf z})\,a({\bf x})\,b({\bf
y})\,c({\bf z})\;.$
We define the four dimensional integral contraction by an open circle
$a{\scriptstyle\circ}b\equiv\int dx\;a(x)\,b(x)\;,$
with
$\displaystyle x$ $\displaystyle=$ $\displaystyle(\tau,{\bf x})\;,$
$\displaystyle 0$ $\displaystyle\leq$ $\displaystyle\tau\leq\beta\;,$
$\displaystyle{\rm and\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ }\int dx$ $\displaystyle=$
$\displaystyle\int_{0}^{\beta}d\tau\int d{\bf x}\;.$
Again, $a$ and $b$ may be single or composite fields. That is, with $m\geq 1$
and $n\geq 1$, $a(x)$ and $b(x)$ may represent
$\displaystyle a(x)$ $\displaystyle=$ $\displaystyle a_{1}(x)\ldots
a_{m}(x)\;,$ $\displaystyle{\rm and\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ }b(x)$ $\displaystyle=$ $\displaystyle b_{1}(x)\ldots
b_{n}(x)\;.$
Similarly, we may define
$a{\scriptstyle\circ}b{\scriptstyle\circ}M=a{\scriptstyle\circ}M{\scriptstyle\circ}b=M{\scriptstyle\circ}a{\scriptstyle\circ}b=\iint
dxdy\;M(x,y)a(x)\,b(y)\;,$
and
$M{\scriptstyle\circ}a{\scriptstyle\circ}b{\scriptstyle\circ}c=\iiint
dxdydz\;M(x,y,z)\,a(x)\,b(y)\,c(z)\;.$
## III Relevant formulation
Consider the following generic fermionic Hamiltonian with $s$ denoting the
spins
$\displaystyle\hat{H}[{\hat{\psi}}^{{\dagger}},\hat{\psi}]$ $\displaystyle=$
$\displaystyle\sum_{s}\int d{\bf x}\;{\hat{\psi}}^{{\dagger}}_{s}({\bf
x})\left(-\frac{\nabla^{2}}{2m}+\upsilon_{\rm ion}({{\bf
x},s})-\mu_{s}\right)\hat{\psi}_{s}({\bf x})$ (8) $\displaystyle\
+\frac{1}{2}\sum_{s,s^{\prime}}\iint{{\hat{\psi}}^{{\dagger}}_{s}({\bf
x}){\hat{\psi}}^{{\dagger}}_{s^{\prime}}({\bf y})U({\bf x}-{\bf
y})\hat{\psi}_{s^{\prime}}({\bf y})\hat{\psi}_{s}({\bf x})}d{\bf x}d{\bf y},$
$\displaystyle=$ $\displaystyle\sum_{s}\int d{\bf
x}\;{\hat{\psi}}^{{\dagger}}_{s}({\bf
x})\left(-\frac{\nabla^{2}}{2m}+\upsilon_{\rm ion}({{\bf
x},s})-\frac{U({\mathbf{0}})}{2}-\mu_{s}\right)\hat{\psi}_{s}({\bf x})$
$\displaystyle\
+\frac{1}{2}\iint{\left(\sum_{s}{\hat{\psi}}^{{\dagger}}_{s}({\bf
x})\hat{\psi}_{s}({\bf x})\right)U({\bf x}-{\bf
y})\left(\sum_{s}{\hat{\psi}}^{{\dagger}}_{s}({\bf y})\hat{\psi}_{s}({\bf
y})\right)}$ $\displaystyle=$ $\displaystyle\sum_{s}\int d{\bf
x}\;{\hat{\psi}}^{{\dagger}}_{s}({\bf
x})\left(-\frac{\nabla^{2}}{2m}+\upsilon_{\rm ion}({{\bf
x},s})-\frac{U({\mathbf{0}})}{2}-\mu_{s}\right)\hat{\psi}_{s}({\bf x})$
$\displaystyle\
+\frac{1}{2}(\sum_{s}{\hat{\psi}}^{{\dagger}}_{s}\hat{\psi}_{s}){\cdot}U{\cdot}(\sum_{s}{\hat{\psi}}^{{\dagger}}_{s}\hat{\psi}_{s})\;.$
From this point on, we absorb $-\frac{U({\mathbf{0}})}{2}$ into $\upsilon_{\rm
ion}({\bf x},s)$.
To lighten the notation, we will first ignore the spin degree of freedom but
will comment on its effect when clarifications are needed. Let $\beta$ be the
inverse temperature. The partition function
$Z\equiv\text{Tr}[e^{-\beta\hat{H}}]$ contains all the information one needs.
To probe the system in terms of the particle density, one often introduces a
classical source term $J({\bf x})$ coupled to ${\hat{\psi}}^{{\dagger}}({\bf
x})\hat{\psi}({\bf x})$. The partition function now becomes a functional of
the source $J$, and we write
$Z[J]\Rightarrow e^{-\beta
W[J]}=\text{Tr}\left[e^{-\beta\left[\hat{H}+J{\cdot}({\hat{\psi}}^{{\dagger}}\hat{\psi})\right]}\right]\;.$
It is easy to show that
$\frac{\delta W[J]}{\delta J({\bf
x})}=\frac{\text{Tr}\left[{\hat{\psi}}^{{\dagger}}({\bf x})\hat{\psi}({\bf
x})e^{-\beta\left[\hat{H}+J{\cdot}({\hat{\psi}}^{{\dagger}}\hat{\psi})\right]}\right]}{Z[J]}=\langle{\hat{n}}({\bf
x})\rangle_{J}\equiv n_{J}({\bf x})$ (9)
Eq. (9) expresses $n$ in terms of $J$, or more generally expresses $n_{s}$
(charge density of spin $s$) in terms of sources $J_{s^{\prime}}$ of all
spins. Given $\upsilon_{\rm ion}({\bf x})$ and $U({\bf x}-{\bf y})$ (for
Coulomb interaction $U({\bf x}-{\bf y})=\frac{e^{2}}{|{\bf x}-{\bf y}|}$),
each time-independent configuration of $\\{J({\bf x})\\}$ generates a time-
independent charge density distribution $\\{n({\bf x})\\}$. However, it is not
guaranteed that every configuration of $\\{n({\bf x})\\}$ is reachable by
varying $J$. The functional variation on $\\{n({\bf x})\\}$ is thus limited to
the subset of $\\{n({\bf x})\\}$ reachable by considering various $\\{J({\bf
x})\\}$. In this stationary case, the effective action $\Gamma[n]$ is defined
as the Legendre transformation of $W[J]$,
$\Gamma[n_{J}]=W[J]-J{\cdot}n_{J}\;,$
where the subscript $J$ indicates that the domain of $\Gamma[n]$ is the set of
density profiles reachable by varying $J$, or the so-called
$\upsilon$-representable Kohn (1983); Lieb (1982); Levy (1982) densities.
We now show the equivalence between the effective action and the energy
functional $E_{\upsilon}[n]$ in (1). Since
$e^{-\beta
W[J]}=\text{Tr}\left[e^{-\beta\left[\hat{H}+J{\cdot}({\hat{\psi}}^{{\dagger}}\hat{\psi})\right]}\right]\;,$
at zero temperature limit $W[J]$ is simply the ground state energy
corresponding to the Hamiltonian
$\hat{H}_{J}\equiv\hat{H}+J{\cdot}({\hat{\psi}}^{{\dagger}}\hat{\psi})$,
$\displaystyle\hat{H}_{J}$ $\displaystyle=$ $\displaystyle\int d{\bf
x}{\hat{\psi}}^{{\dagger}}({{\bf
x}})\left[-\frac{1}{2m}\nabla^{2}+\left(\upsilon_{\rm ion}({{\bf x}})+J({\bf
x})\right)-\mu\right]\hat{\psi}({{\bf x}})$ (10) $\displaystyle\
+\frac{e^{2}}{2}\int\int\frac{{\hat{\psi}}^{{\dagger}}({{\bf
x}}){\hat{\psi}}^{{\dagger}}({{\bf y}})\hat{\psi}({{\bf y}})\hat{\psi}({{\bf
x}})}{|{{\bf x}}-{{\bf y}}|}d{{\bf x}}d{{\bf y}},$
while the electron density $n_{J}({\bf x})$ is obtained by integrating all but
one spatial variable of the ground state wave function corresponding to
$\hat{H}_{J}$. Evidently, when $J=0$, $\Gamma[n]|_{n=n_{g}}=W[J]|_{J=0}=E_{g}$
where $E_{g}$ stands for the ground state energy corresponding to $\hat{H}$
and $n_{g}$ represents the electron density at the physical ($J=0$) ground
state. When $J\neq 0$ the corresponding electronic density $n[J\neq 0]$ is
different from $n_{g}$, and $\Gamma[n_{J}]$ represents the expectation value
of the original Hamiltonian $\hat{H}$, calculated using the ground state wave
function corresponding to a different Hamiltonian, namely,
$\hat{H}+J{\cdot}({\hat{\psi}}^{{\dagger}}\hat{\psi})$. Since $n_{J}\neq
n_{g}$, $\Gamma[n]|_{n=n[J]}>\Gamma[n]|_{n=n_{g}}$ by the definition of the
ground state. This means that $\Gamma[n]$ reaches its minimum at $n_{g}$, and
various other electron density profiles $n[J]$ producible by introducing
different $J$ form the domain of argument for $\Gamma[n]$. Thus $\Gamma[n]$
has all the properties attributed to the energy functional $E_{\upsilon}[n]$
in (1). Since the theorem of Hohenberg and Kohn states that this functional is
unique, it must be equal to $E_{\upsilon}[n]$. As we will show in section
IV.2, $\Gamma[n]$ can be decomposed exactly in the same manner as in (4).
Within allowable configurations of $\\{n({\bf x})\\}$, if one is able to
invert the relation (9) to obtain, say, $J[n]$, then the explicit construction
of the effective action becomes possible. In principle, this can be done via
an inversion method.Fukuda et al. (1995) Using this scheme, Valiev and
Fernando Valiev and Fernando (1997) proposed a perturbative expansion in terms
of $e^{2}$ to express the exchange-correlation functional as a sum of an
infinite number of Feynman diagrams and the diagrams’ derivatives with respect
to the Kohn-Sham potential.
We approach this problem from two different routes, both involving the
introduction of an auxiliary field.Fukuda et al. (1994); Polonyi and Sailer
(2002) As will be described in secton VI.1, the second route does not have an
exact correspondence to the Kohn-Sham decomposition, but has the advantage
that the correction terms may be obtained without further functional
derivatives. The first route, as will be described later in this section,
gives a recipe equivalent to the Kohn-Sham decomposition, together with a way
to calculate the exchange-correlation functional in a self-consistent manner.
This auxiliary field approach was pursued in an earlier publication, Fukuda et
al. (1994) but there it was concluded that it seems infeasible to make a
direct connection to the Kohn-Sham scheme. Using the inversion method, Fukuda
et al. (1995) we show explicitly how the connection to the Kohn-Sham scheme
can be made. One advantage of the auxiliary field method is that each Feynman
diagram here corresponds to the sum of infinitely many Feynman diagrams in
standard perturbative field theory calculations Jackiw (1974) such as used in
reference Valiev and Fernando, 1997. Subsections III.1 through III.6 detail
the proposed approach. Subsection III.7 lays out the computational procedure
to give some perspectives on computational complexity.
### III.1 Path Integral
To accommodate a time-dependent probe and to deal with excitations, we express
$Z$ as a path integral over Grassmann fields and we have
$e^{-\beta W[J]}\equiv Z[J]=\int
D\psi^{{\dagger}}D\psi\;\exp\left\\{-S\left[\psi^{{\dagger}},\psi\right]-J{\scriptstyle\circ}(\psi^{{\dagger}}\psi)\right\\}\;,$
(11)
with
$S\left[\psi^{\dagger},\psi\right]=\psi^{{\dagger}}{\scriptstyle\circ}G_{0}^{-1}{\scriptstyle\circ}\psi+\frac{1}{2}\left(\psi^{{\dagger}}\psi\right){\scriptstyle\circ}\,U{\scriptstyle\circ}\left(\psi^{{\dagger}}\psi\right)\;,$
(12)
where $\psi^{(\dagger)}$ denote Grassmann fields with
$\psi^{(\dagger)}(\beta,{\bf x})=-\psi^{(\dagger)}(0,{\bf x})$, and
$\displaystyle G_{0}^{-1}(x,x^{\prime})$ $\displaystyle\equiv$
$\displaystyle\langle x|G_{0}^{-1}|x^{\prime}\rangle$ $\displaystyle=$
$\displaystyle\left({\partial\tau}-\frac{\nabla^{2}}{2m}+\upsilon_{\rm
ion}({\bf x})-\mu\right)\langle
x|x^{\prime}\rangle=\left({\partial\tau}-\frac{\nabla^{2}}{2m}+\upsilon_{\rm
ion}({\bf x})-\mu\right)\delta(x-x^{\prime})\;,$
and
$\displaystyle U(x,x^{\prime})$ $\displaystyle=$ $\displaystyle
U(x-x^{\prime})=\delta\left(\tau-\tau^{\prime}\right)U({\bf x}-{\bf
x}^{\prime}),$ $\displaystyle\delta(x-x^{\prime})$ $\displaystyle=$
$\displaystyle\delta(\tau-\tau^{\prime})\delta({\bf x}-{\bf x}^{\prime})\;.$
For a time-independent source, $J(\tau,{\bf x})=J({\bf x})$ (i.e.,
$J{\scriptstyle\circ}(\psi^{{\dagger}}\psi)=\int dxJ({\bf
x})\psi^{{\dagger}}(x)\psi(x)$). For time-dependent probes, $J(x)$ becomes
$\tau$-dependent, and $J{\scriptstyle\circ}(\psi^{{\dagger}}\psi)=\int
dxJ(x)\psi^{{\dagger}}(x)\psi(x)$. It is straightforward to verify that
$\frac{\delta(\beta W\left[J\right])}{\delta
J(x)}=\langle{\hat{\psi}}^{{\dagger}}(x)\hat{\psi}(x)\rangle_{J}=\langle\hat{n}(x)\rangle_{J}\equiv
n_{J}(x)\;.$ (13)
This quantity is important for later development.
The quartic fermionic interaction in (12) can be disentangled via introducing
an auxiliary real field $\phi$ with
$D\phi\equiv\prod_{x}\frac{d\phi(x)}{\sqrt{2\pi}}$. Note that
$1=\sqrt{\det U}\int
D\phi\;e^{-\frac{1}{2}\phi{\scriptstyle\circ}\,U{\scriptstyle\circ}\phi}=\sqrt{\det
U}\int
D\phi\;e^{-\frac{1}{2}(\phi+Y){\scriptstyle\circ}\,U{\scriptstyle\circ}(\phi+Y)}\;,$
(14)
for an arbitrary field $Y$, provided that $Y(x)$ and $Y(x^{\prime})$ always
commute. Since $U(x,x^{\prime})$ is diagonal in $\tau$, it suffices that they
commute at equal Euclidean times. Let us set
$Y(x)=i\psi^{{\dagger}}(x)\psi(x)$, which satisfies the equal time commutation
requirement, and then multiply (11) by (14) to obtain
$Z\left[J\right]=\int D\phi
D\psi^{{\dagger}}D\psi\;\exp\left\\{-{S}\left[\phi,\psi^{{\dagger}},\psi\right]\right\\}\;,$
(15)
where
${S}\left[\phi,\psi^{{\dagger}},\psi\right]=-\frac{1}{2}\text{Tr}\ln(U)+\frac{1}{2}\phi{\scriptstyle\circ}\,U{\scriptstyle\circ}\phi+\psi^{{\dagger}}{\scriptstyle\circ}\,G^{-1}_{\\!(\phi-
iu^{-1}{\scriptstyle\circ}J)}\,{\scriptstyle\circ}\psi\;,$ (16)
with
$G^{-1}_{\\!(\phi-
iU^{-1}{\scriptstyle\circ}J)}(x,x^{\prime})=\left(\partial_{\tau}-\frac{\nabla^{2}}{2m}+\upsilon_{\rm
ion}({{\bf
x}})-\mu+i(U{\scriptstyle\circ}\phi)_{x}+J(x)\right)\delta(x-x^{\prime})\;.$
(17)
If we make a change of variable $\phi^{\prime}\equiv\phi-
iU^{-1}{\scriptstyle\circ}J$ and then rename $\phi^{\prime}$ by $\phi$, we may
rewrite (15-17) as
$\displaystyle Z\left[J\right]$ $\displaystyle=$ $\displaystyle
e^{\frac{1}{2}J{\scriptstyle\circ}U^{-1}{\scriptstyle\circ}J}\;\int D\phi
D\psi^{{\dagger}}D\psi\;\exp\left\\{-{S}_{J}\left[\phi,\psi^{{\dagger}},\psi\right]\right\\}\;,$
(18) $\displaystyle{S}_{J}\left[\phi,\psi^{{\dagger}},\psi\right]$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\text{Tr}\ln(U)+\frac{1}{2}\phi{\scriptstyle\circ}\,U{\scriptstyle\circ}\phi+i\phi\,{\scriptstyle\circ}J+\psi^{{\dagger}}{\scriptstyle\circ}{G}_{\phi}^{-1}{\scriptstyle\circ}\psi\;,$
(19) $\displaystyle\langle x|G_{\phi}^{-1}|x^{\prime}\rangle$
$\displaystyle\equiv$
$\displaystyle{G}_{\phi}^{-1}(x,x^{\prime})=\left(\partial_{\tau}-\frac{\nabla^{2}}{2m}+\upsilon_{\rm
ion}({{\bf
x}})-\mu+i(U{\scriptstyle\circ}\phi)_{x}\right)\delta(x-x^{\prime})\,.$ (20)
Integrating over the Grassmann fields in (18), we obtain an effective theory
in terms of $\phi$
$e^{-\beta
W[J]}=Z\left[J\right]=e^{\frac{1}{2}J{\scriptstyle\circ}\,U^{-1}{\scriptstyle\circ}J}\int
D\phi\;\exp\left\\{-I\left[\phi\right]-iJ{\scriptstyle\circ}\phi\right\\}$
(21)
where
$I[\phi]=-\frac{1}{2}\text{Tr}\ln(U)+\frac{1}{2}\phi{\scriptstyle\circ}\,U{\scriptstyle\circ}\phi-\text{Tr}\ln(G_{\phi}^{-1})\;.$
(22)
Let us introduce a new notation $W_{\phi}[J]$ via
$e^{-\beta W_{\phi}[J]}\equiv\int
D\phi\;\exp\left\\{-I\left[\phi\right]-iJ{\scriptstyle\circ}\phi\right\\}\;.$
(23)
We describe later how to evaluate (23) using well-developed functional
integral techniques. Evidently, we have
$\beta W\left[J\right]=\beta
W_{\phi}\left[J\right]-\frac{1}{2}J{\scriptstyle\circ}\,U^{-1}{\scriptstyle\circ}J\;.$
(24)
The expectation value, $n_{J}(x)$, of the density operator in the presence of
a source term $J$ is given by
$n_{J}(x)=\frac{\delta(\beta W[J])}{\delta J(x)}=\frac{\delta(\beta
W_{\phi}[J])}{\delta J(x)}-({U}^{-1}{\scriptstyle\circ}J)_{x}\equiv
i\varphi(x)-(U^{-1}{\scriptstyle\circ}J)_{x}\;,$ (25)
where
$(U^{-1}{\scriptstyle\circ}J)_{x}\equiv\int dy\,U^{-1}(x,y)J(y)\;,$
and the expectation value of the auxiliary field is defined by
$i\varphi(x)\equiv\langle i\phi(x)\rangle_{J}={\delta(\beta
W_{\phi})\over\delta J(x)}=\frac{\int
D\phi\left(i\phi(x)\right)e^{-I[\phi]-iJ{\scriptstyle\circ}\phi}}{\int D\phi
e^{-I[\phi]-iJ{\scriptstyle\circ}\phi}}\;,$ (26)
providing a relation between $J$ and $i\varphi$.
Eq. (25) tells us that at the physical limit $J\to 0$, $i\varphi$ is the same
as $n$. Since $n$ is a real number, this implies that the expectation value of
$\phi({\bf x})$ is an imaginary number, which also implies that when viewed in
the complex plane of $\phi({\bf x})$, the saddle point of the integrand is
located where the $\phi({\bf x})$s are imaginary numbers.
Now let us write down the effective action. At finite temperature, the
effective action is defined as the Legendre transform of $\beta W[J]$:
$\Gamma[n]\equiv\beta W\left[J\right]-{\delta(\beta W[J])\over\delta
J}{\scriptstyle\circ}J=\beta
W_{\phi}[J]-\frac{1}{2}J{\scriptstyle\circ}\,U^{-1}{\scriptstyle\circ}J-n{\scriptstyle\circ}J\;.$
(27)
Note that the functional derivative of $\Gamma[n]$ with respect to $n$ reads
$\frac{\delta\Gamma[n]}{\delta n}=\left[\frac{\delta(\beta W[J])}{\delta
J}-n\right]{\scriptstyle\circ}\frac{\delta J}{\delta n}-J=-J\;,$ (28)
because ${\delta(\beta W[J])}/{\delta J}=n$ by Eq. (13). The effective action
formalism requires one to express the probe $J$ in terms of an expectation
value of some sort, such as the electron density $n$, classical field
$i\varphi$, or some equivalent quantity. Below, we will first calculate $\beta
W_{\phi}[J]$, and then use the system’s electron density as the variable and
make an explicit connection to the Kohn-Sham decomposition.
### III.2 Evaluation of $e^{-\beta W_{\phi}[J]}$ via one-particle irreducible
diagrams
As shown by Jackiw,Jackiw (1974) it is possible to express $W_{\phi}[J]$ as a
diagrammatic expansion containing only one-particle irreducible diagrams. The
main idea is to shift the field $\phi$ by $\varphi$, $\phi\to\varphi+\phi$,
and note that $J$ is a functional of $\varphi$ via (26). We then rewrite
$e^{-\beta W_{\phi}[J]}\equiv
e^{-I[\varphi]-iJ{\scriptstyle\circ}\,\varphi}\int
D\phi\;e^{-I\left[\phi+\varphi\right]+I\left[\varphi\right]-iJ{\scriptstyle\circ}\phi}\equiv
e^{-I[\varphi]-iJ{\scriptstyle\circ}\,\varphi}Z_{*}[J]\;$
where
$-\ln Z_{*}[J]\equiv\beta W_{*}[J]=-\ln\left[\int
D\phi\;e^{-I\left[\phi+\varphi\right]+I\left[\varphi\right]-iJ{\scriptstyle\circ}\phi}\right]\;,$
(29)
leading to
$\beta W_{\phi}[J]=I[\varphi]+J{\scriptstyle\circ}(i\varphi)+\beta
W_{*}[\varphi]\;.$ (30)
Note that
$i\varphi(x)={\delta(\beta W_{\phi}[J])\over\delta J(x)}=i\varphi(x)+\int
dy\left[{\delta I\over\delta(i\varphi(y))}+{\delta(\beta
W_{*})\over\delta(i\varphi(y))}+J(y)\right]{\delta(i\varphi(y))\over\delta
J(x)}\;,$
leading to
${\delta I\over\delta(i\varphi(y))}+{\delta(\beta
W_{*})\over\delta(i\varphi(y))}=-J(y)\;.$ (31)
Using an implicit method and replacing $-J$ in (29) by the left-hand side
(LHS) of (31), Jackiw Jackiw (1974) showed that $\beta W_{*}[\varphi]$ is the
sum of all connected one-particle-irreducible (1PI) vacuum graphs governed by
the action
$-I[\phi+\varphi]+I[\varphi]+\phi\,{\scriptstyle\circ}{\delta
I[\varphi]\over\delta\varphi}\;.$
To evaluate the expression above, we first rewrite (20) as
$G_{\phi+\varphi}^{-1}(x,x^{\prime})=G_{\varphi}^{-1}(x,x^{\prime})+i\delta(x-x^{\prime})b(x)\equiv
G^{-1}_{\varphi}(x,x^{\prime})+V(x,x^{\prime})\;,$ (32)
with
$\displaystyle b$ $\displaystyle=$ $\displaystyle
U\,{\scriptstyle\circ}\,\phi\;,$ (33) $\displaystyle{\rm
and\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ }G_{\varphi}^{-1}(x,x^{\prime})$ $\displaystyle=$
$\displaystyle\left[\partial_{\tau}-\frac{\nabla^{2}}{2m}+\upsilon_{\rm
ion}({{\bf
x}})-\mu+U{\scriptstyle\circ}(i\varphi)\right]\delta(x-x^{\prime})\;.$ (34)
We may then write down
$G_{\phi+\varphi}^{-1}=G^{-1}_{\varphi}\left[\,{\mathbf{I}}+G_{\varphi}{\scriptstyle\circ}{\mathbf{V}}\right]\;,$
and
$\ln\left(G_{\phi+\varphi}^{-1}\right)=\ln\left(G^{-1}_{\varphi}\right)+\sum_{k=1}^{\infty}\frac{(-1)^{k-1}}{k}\left[G_{\varphi}{\scriptstyle\circ}{\mathbf{V}}\right]^{k}\;.$
(35)
Note also that
$\left[G_{\varphi}{\scriptstyle\circ}{\mathbf{V}}\right]_{x,z}=\int\\!\\!dy\,G_{\varphi}(x,y)V(y,z)=\int
dyG_{\varphi}(x,y)\delta(y-z)\left(ib(y)\right)=G_{\varphi}(x,z)\left(ib(z)\right)\;.$
Consequently,
$\displaystyle\text{Tr}\ln\left(G_{\phi+\varphi}^{-1}\right)$ $\displaystyle=$
$\displaystyle\text{Tr}\ln\left(G^{-1}_{\varphi}\right)+\int\\!dx_{1}\,G_{\varphi}(x_{1},x_{1})(ib(x_{1}))$
(36)
$\displaystyle-\frac{1}{2}\int\\!dx_{1}dx_{2}\,G_{\varphi}(x_{1},x_{2})G_{\varphi}(x_{2},x_{1})(ib(x_{1}))(ib(x_{2}))$
$\displaystyle+\sum_{k=3}^{\infty}\frac{(-1)^{k-1}}{k}\int\\!dx_{1}\ldots
dx_{k}G_{\varphi}(x_{k},x_{1})\ldots
G_{\varphi}(x_{k-1},x_{k})(ib(x_{1}))\ldots(ib(x_{k}))\;.$
Therefore,
$-I[\phi+\varphi]+I[\varphi]+\phi\,{\scriptstyle\circ}{\delta
I[\varphi]\over\delta\varphi}=-\frac{1}{2}b\,{\scriptstyle\circ}\tilde{\mathcal{D}}^{-1}{\scriptstyle\circ}\,b+\sum_{k=3}^{\infty}I^{(k)}[\varphi]{\scriptstyle\circ}\,b_{1}\ldots{\scriptstyle\circ}\,b_{k}$
(37)
with
$\tilde{\mathcal{D}}^{-1}=U^{-1}-D\;,$ (38)
$D(x,y)=G_{\varphi}(x,y)\,G_{\varphi}(y,x)\;,$ (39)
and
$I^{(k)}[\varphi]{\scriptstyle\circ}\,b_{1}\ldots{\scriptstyle\circ}\,b_{k}\equiv\frac{(-1)^{k-1}}{k}\int\\!dx_{1}\ldots
dx_{k}G_{\varphi}(x_{k},x_{1})\ldots
G_{\varphi}(x_{k-1},x_{k})\left[(ib(x_{1}))\ldots(ib(x_{k}))\right]\;.$ (40)
As a side note, the quantity $D(x,y)$ defined in (39) is also called the
polarization associated with the Green’s function $G_{\varphi}$, since it can
be shown, by using a derivation identical to that leading to (71), that
$D(x,y)=-\delta G_{\varphi}(x,x)/\delta J(y)$ represents the reaction rate of
density (given by $n(x)\equiv-G_{\varphi}(x,x)$) due to the influence of the
potential. According to Jackiw’s results,Jackiw (1974) Eq. (37) means that
$\beta W_{*}[\varphi]$ is given by
$\beta
W_{*}[\varphi]=\text{Tr}\ln(U)+\frac{1}{2}\text{Tr}\ln\left(\tilde{\mathcal{D}}^{-1}\right)-\sum_{n=1}^{\infty}\frac{1}{n!}\langle\left[\sum_{k=3}^{\infty}I^{(k)}[\varphi]{\scriptstyle\circ}\,b_{1}\ldots{\scriptstyle\circ}\,b_{k}\right]^{n}\rangle_{\rm
1PI,\leavevmode\nobreak\ conn.}$ (41)
where the $\text{Tr}\ln(U)$ term comes from the Jacobian of changing the
variable from $\phi$ to $b$ in (29), the angular bracket indicates the
following average
$\langle\hat{O}\rangle\equiv\frac{\int
D[b]\,\hat{O}\,\exp\left(-\frac{1}{2}b{\scriptstyle\circ}\tilde{\mathcal{D}}^{-1}{\scriptstyle\circ}b\right)}{\int
D[b]\,\exp\left(-\frac{1}{2}b{\scriptstyle\circ}\tilde{\mathcal{D}}^{-1}{\scriptstyle\circ}b\right)}\;,$
(42)
and the subscript “${\rm 1PI,\leavevmode\nobreak\ conn.}$” means to include
only connected, one-particle-irreducible diagrams. In our context, a one-
particle-irreducible diagrams refers to a diagram that cannot be separated
into two by cutting a propagator line representing $\tilde{\mathcal{D}}$.
Substituting (22) and (41) into (30), we obtain
$\displaystyle\beta W_{\phi}[J]$ $\displaystyle=$
$\displaystyle\frac{1}{2}\varphi{\scriptstyle\circ}\,U{\scriptstyle\circ}\varphi-\text{Tr}\ln\left(G_{\varphi}^{-1}\right)+iJ{\scriptstyle\circ}\varphi$
(43)
$\displaystyle+\frac{1}{2}\text{Tr}\ln\left(\tilde{\mathcal{D}}^{-1}{\scriptstyle\circ}\,U\right)-\sum_{n=1}^{\infty}\frac{1}{n!}\langle\left[\sum_{k=3}^{\infty}I^{(k)}[\varphi]{\scriptstyle\circ}\,b_{1}\ldots{\scriptstyle\circ}\,b_{k}\right]^{n}\rangle_{\rm
1PI,\leavevmode\nobreak\ conn.}\;.$
Note that Fukuda et al. Fukuda et al. (1994) obtained an expression similar to
(43) and used it to derive an effective action as a functional of $i\varphi$,
which coincides with $n_{J}$ only at $J=0$.
We wish to keep $n_{J}$ as the functional variable. Let us first note that
$\beta W[J]=\beta W_{\phi}[J]-{1\over
2}J{\scriptstyle\circ}\,U^{-1}{\scriptstyle\circ}J$. By employing the identity
(25)
$n_{J}=i\varphi-U^{-1}{\scriptstyle\circ}J\;,$
we can now write $\beta W[J]$ as
$\displaystyle\beta W[J]$ $\displaystyle=$
$\displaystyle-\text{Tr}\ln\left(G_{\varphi}^{-1}\right)-\frac{1}{2}n_{J}{\scriptstyle\circ}\,U{\scriptstyle\circ}n_{J}+\frac{1}{2}\text{Tr}\ln\left(\tilde{\mathcal{D}}^{-1}{\scriptstyle\circ}\,U\right)$
(44) $\displaystyle\hskip
15.0pt-\sum_{n=1}^{\infty}\frac{1}{n!}\langle\left[\sum_{k=3}^{\infty}I^{(k)}[\varphi]{\scriptstyle\circ}\,b_{1}\ldots{\scriptstyle\circ}\,b_{k}\right]^{n}\rangle_{\rm
1PI,\leavevmode\nobreak\ conn.}\;,$
and
$\displaystyle\Gamma[n]$ $\displaystyle=$
$\displaystyle-\text{Tr}\ln\left(G_{\varphi}^{-1}\right)-\left[J+\frac{1}{2}n_{J}{\scriptstyle\circ}\,U\right]{\scriptstyle\circ}n_{J}+\frac{1}{2}\text{Tr}\ln\left(\tilde{\mathcal{D}}^{-1}{\scriptstyle\circ}\,U\right)$
(45) $\displaystyle\hskip
15.0pt-\sum_{n=1}^{\infty}\frac{1}{n!}\langle\left[\sum_{k=3}^{\infty}I^{(k)}[\varphi]{\scriptstyle\circ}\,b_{1}\ldots{\scriptstyle\circ}\,b_{k}\right]^{n}\rangle_{\rm
1PI,\leavevmode\nobreak\ conn.}\;.$
For later convenience we introduce a parameter $\lambda$ to denote the order.
Specifically, we write
$\displaystyle\beta W[J]$ $\displaystyle=$
$\displaystyle-\text{Tr}\ln\left(G_{\varphi}^{-1}\right)-\frac{1}{2}n_{J}{\scriptstyle\circ}\,U{\scriptstyle\circ}n_{J}+\frac{\lambda}{2}\text{Tr}\ln\left(\tilde{\cal
D}^{-1}{\scriptstyle\circ}\,U\right)$ (46) $\displaystyle\hskip
15.0pt-\lambda\sum_{n=1}^{\infty}\frac{1}{n!}\langle\left[{1\over\lambda}\sum_{k=3}^{\infty}\lambda^{k\over
2}I^{(k)}[\varphi]{\scriptstyle\circ}\,b_{1}\ldots{\scriptstyle\circ}\,b_{k}\right]^{n}\rangle_{\rm
1PI,\leavevmode\nobreak\ conn.}\;.$ $\displaystyle\equiv$
$\displaystyle\beta\tilde{W}_{0}[J]+\beta\sum_{l=1}^{\infty}\lambda^{l}\,W_{l}[J]$
(47)
The bookkeeping parameter $\lambda$ will be set to $1$ in the end. The
exponent associated with the parameter $\lambda$ plays the role of the number
of loops introduced. For example, the term
$\frac{1}{2}\text{Tr}\ln\left(\tilde{\cal
D}^{-1}{\scriptstyle\circ}\,U\right)$ consists of one-loop contributions. The
last part of (46) contains diagrams of two loops or higher. The explicit
appearance of the $n_{J}{\scriptstyle\circ}\,U{\scriptstyle\circ}n_{J}$ term
in (44) and (46) suggests the possibility of a connection to the Kohn-Sham
decomposition of the DFT.
### III.3 Inversion Method by Loop Order
We now come to the point of departure from typical treatments of auxiliary
field approach and are ready to make a direct connection to the Kohn-Sham
scheme. Let us first define a new free fermion propagator
${\mathcal{G}}_{0}^{-1}(x,x^{\prime})=\left[\partial_{\tau}-\frac{\nabla^{2}}{2m}+\upsilon_{\rm
ion}({{\bf x}})-\mu+J_{0}(x)\right]\delta(x-x^{\prime})\;,$ (48)
where $J_{0}$ is chosen such that the free fermion system has the same
particle density as the physical system (where Coulomb interactions exist)
with source potential $J$
$-{\mathcal{G}}_{0}(x,x)=n_{J}(x)\;.$ (49)
The existence of $J_{0}$ is scrutinized below. From the perspective of the
Kohn-Sham decomposition, Eq. (49) corresponds to Eq. (6). In the presence of a
source term $J$, Eq. (10) shows that it is equivalent to making $\upsilon_{\rm
ion}\to\upsilon_{\rm ion}+J$. Comparing with Eq. (5), we see immediately if
one were to choose
$J_{0}=J+U{\cdot}n_{J}+\left.\frac{\delta E_{xc}[n]}{\delta
n}\right|_{n=n_{J}}\;,$ (50)
the requirement (49) can be fulfilled. We will therefore call the
corresponding free fermion system the Kohn-Sham system. The presence of
$J_{0}$, of course, depends crucially on the differentiability of $E_{xc}[n]$,
whose existence (not differentiability) was proven.Hohenberg and Kohn (1964)
To bring out the Kohn-Sham quantities (orbitals and energies) in our loop
expansion, let us first use a variant of (25)
$u{\scriptstyle\circ}(i\varphi)=J+U{\scriptstyle\circ}n_{J}$
and replace $U{\scriptstyle\circ}(i\varphi)$ by $J+U{\scriptstyle\circ}n_{J}$
in the expression of propagator $G_{\varphi}$. Specifically, we write (34) as
$G_{\varphi}^{-1}(x,x^{\prime})=\left[\partial_{\tau}-\frac{\nabla^{2}}{2m}+\upsilon_{\rm
ion}({{\bf
x}})-\mu+J(x)+U{\scriptstyle\circ}n_{J}\right]\delta(x-x^{\prime})\;.$ (51)
The critical step is to decompose the source $J$ in a particular way,
$J\equiv(J_{0}-U{\scriptstyle\circ}n_{J})+J^{\prime}\equiv\tilde{J}_{0}+J^{\prime}\;.$
(52)
Therefore, from Eqs. (25), (34) and (52) we have
$G_{\varphi}^{-1}(x,x^{\prime})={\mathcal{G}}_{0}^{-1}(x,x^{\prime})+J^{\prime}(x)\,\delta(x-x^{\prime})\;,$
(53)
and the Kohn-Sham propagator ${\mathcal{G}}_{0}$ appears as we expected. As
will be described in section III.5, ${\mathcal{G}}_{0}(x,x^{\prime})$ can be
expressed in terms of the Kohn-Sham quantities. Therefore, the idea is to
expand $G_{\varphi}$ around ${\mathcal{G}}_{0}$ provided that $J^{\prime}$ can
also be expressed via Kohn-Sham quantities. Comparing (52) with (50), we find
that formally speaking $J^{\prime}[n]=-\frac{\delta E_{xc}[n]}{\delta n}$.
This source decomposition is introduced here for the first time in the
auxiliary field approach. (A similar decomposition has been used Fukuda et al.
(1995); Valiev and Fernando (1997) in the perturbative expansion in powers of
$e^{2}$.)
The idea of inversion is to obtain $J[n]$, that is, to find the corresponding
$J$ for each configuration of electron density $n$ in the domain of
$\Gamma[n]$. One then substitutes $J[n]$ into $\Gamma[n]=\beta
W[J[n]]-J[n]{\scriptstyle\circ}n$ to express $\Gamma[n]$ using the density
profile $n$ as the natural variable. For each given density profile $n$ within
the domain of $\Gamma$, Eq. (49) determines the corresponding $\tilde{J}_{0}$.
The collection of such relations forms $\tilde{J}_{0}[n]$. Similarly, if for
every given $n$ one can find the corresponding $J^{\prime}$, one obtains
$J^{\prime}[n]$ and the goal of inversion is achieved. When evaluating
$\Gamma[n]$, one employs one density configuration at a time. That said, when
we expand $W[J[n]]=W[\tilde{J}_{0}[n]+J^{\prime}[n]]$ in powers of
$J^{\prime}$ within the expression $\Gamma[n]=\beta
W[J]-J{\scriptstyle\circ}n$, we will keep $n_{J}$ fixed, instead of treating
it as a functional of $\tilde{J}_{0}$.
We now examine the loop expansion of $W[J]$ carefully. Eq. (46) tells us that
$\beta
W[J]=\beta(\tilde{W}_{0}-W_{0})+\beta\sum_{l=0}^{\infty}\lambda^{l}\,W_{l}[J+U{\scriptstyle\circ}n_{J}]\;,$
(54)
where
$\beta(\tilde{W}_{0}-W_{0})=-\frac{1}{2}n_{J}{\scriptstyle\circ}\,U{\scriptstyle\circ}n_{J}$
and $\beta
W_{0}[J+U{\scriptstyle\circ}n_{J}]=-\text{Tr}\ln(G_{\varphi}^{-1})$. Note that
for any $W_{l}$ term, its $J$ dependence is through the propagator
$G_{\varphi}$, which always has $J+U{\scriptstyle\circ}n_{J}$ as the natural
variable. Our reasoning earlier indicates that when we expand
$W_{l}[J+U{\scriptstyle\circ}n_{J}]=W_{l}[\tilde{J}_{0}+U{\scriptstyle\circ}n_{J}+J^{\prime}]=W_{l}[J_{0}+J^{\prime}]$
in powers of $J^{\prime}$ within the expression $\Gamma[n]=\beta
W[J]-J{\scriptstyle\circ}n$, we may write the expansion in the following way
(and forget about needing to keep $n_{J}$ fixed)
$W_{l}[J_{0}+J^{\prime}]=W_{l}[J_{0}]+\frac{\delta W_{l}[J_{0}]}{\delta
J_{0}}{\scriptstyle\circ}J^{\prime}+\frac{1}{2!}J^{\prime}{\scriptstyle\circ}\frac{\delta^{2}W[J_{0}]}{\delta
J_{0}\,\delta J_{0}}{\scriptstyle\circ}J^{\prime}+\ldots\;\;.$ (55)
With (55), we may express $\beta W[J]$ as a double series
$\beta
W[J]=\beta(\tilde{W}_{00}-W_{00})+\beta\sum_{i,k}W_{ik}{J^{\prime}}^{k}\lambda^{i}\;,$
(56)
where each $W_{ik}$ involves the $k$’th derivative of $W_{i}$. In particular,
$\tilde{W}_{00}$ is given by (with $n_{J}\to n$ hereafter)
$\beta\tilde{W}_{00}=\beta
W_{00}-\frac{1}{2}n{\scriptstyle\circ}\,U{\scriptstyle\circ}n=-\text{Tr}\ln({\mathcal{G}}_{0}^{-1})-\frac{1}{2}n{\scriptstyle\circ}\,U{\scriptstyle\circ}n\;,$
(57)
and $W_{01}$ is given by
$\beta
W_{01}[J_{0}]=-\text{Tr}\left(\frac{\delta\ln({\mathcal{G}}_{0}^{-1})}{\delta
J_{0}(x)}\right)=-\\!\\!\int\\!\\!dzdy\,{\mathcal{G}}_{0}(z,y)\delta(y-x)\delta(y-z)=-{\mathcal{G}}_{0}(x,x)=n\;,$
(58)
in view of (49).
Instead of looking at the expansion of $W_{l}$ in powers of $J^{\prime}$
within the effective action expression, we now take a moment to look at the
$\tilde{W}_{00}$ term in the double series expansion of $\beta W[J]$. Consider
the functional derivative of $\beta\tilde{W}_{00}$ with respect to the source
term $\tilde{J}_{0}$. Here, one is asking the response of
$\beta\tilde{W}_{00}$ with respect to change in $\tilde{J}_{0}$. Evidently,
when $\tilde{J}_{0}$ changes, its corresponding density $n$ has to vary as
well. Using the chain rule of differentiation and (58), we obtain
$\frac{\delta(\beta\tilde{W}_{00})}{\delta\tilde{J}_{0}}=\frac{\delta(\beta
W_{00})}{\delta J_{0}}{\scriptstyle\circ}\frac{\delta
J_{0}}{\delta\tilde{J}_{0}}-n_{\scriptstyle\circ}\,U{\scriptstyle\circ}\frac{\delta
n}{\delta\tilde{J}_{0}}=n{\scriptstyle\circ}({\mathbf{I}}+U{\scriptstyle\circ}\frac{\delta
n}{\delta\tilde{J}_{0}})-n_{\scriptstyle\circ}\,U{\scriptstyle\circ}\frac{\delta
n}{\delta\tilde{J}_{0}}=n\;.$ (59)
This suggests that we define
$\tilde{\Gamma}_{0}[n]=\beta\tilde{W}_{00}[\tilde{J}_{0}]-\tilde{J}_{0}{\scriptstyle\circ}n=-\text{Tr}\ln({\mathcal{G}}_{0}^{-1})-\frac{1}{2}n{\scriptstyle\circ}\,U{\scriptstyle\circ}n-\tilde{J}_{0}{\scriptstyle\circ}n\;,$
(60)
the Legendre transformation of $\beta W_{00}[\tilde{J}_{0}]$, leading to
$\frac{\delta\tilde{\Gamma}_{0}[n]}{\delta n}=-\tilde{J}_{0}\;.$ (61)
Comparing (61) with (28), we find
$\frac{\delta(\Gamma[n]-\tilde{\Gamma}_{0}[n])}{\delta
n}=-J^{\prime}\;.\vspace*{-2pt}\phantom{12}$ (62)
The idea now is to develop a series for $\Gamma[n]$ led by
$\tilde{\Gamma}_{0}[n]$. Subtracting (60) from $\Gamma[n]=\beta
W[J]-J{\scriptstyle\circ}n$, we have
$\Gamma[n]-\tilde{\Gamma}_{0}[n]=\beta
W[J]-\beta\tilde{W}_{00}[\tilde{J}_{0}]-J^{\prime}{\scriptstyle\circ}n\;,\vspace*{-1pt}\phantom{12}$
(63)
in which the last two terms on the right-hand side (RHS) exactly cancel the
terms in $\tilde{W}_{00}$ and $W_{01}$ contributing to $\beta W[J]$. So the
series for $\Gamma-\tilde{\Gamma}_{0}$ is just (56) with those two terms
removed. Next we convert the double sum in (56) into a single sum by expanding
$J^{\prime}$ as a series in $\lambda$. We write
$J^{\prime}[n]=\sum_{l=1}^{\infty}J_{l}[n]\lambda^{l}\;,\vspace*{-2pt}\phantom{12}$
(64)
where the precise expressions for $J_{1},J_{2},\ldots$ are as yet undetermined
since (64) is not a loop expansion. We substitute (64) formally into (63) and
(56) to obtain a series 12
$\Gamma[n]-\tilde{\Gamma}_{0}[n]=\sum_{l=1}^{\infty}\Gamma_{l}[n]\;\lambda^{l}\;,\vspace*{-2pt}\phantom{12}$
(65)
in which each $\Gamma_{l}$ is defined explicitly in terms of the $J_{k}$,
$\beta W_{k\leq l}[J_{0}]$, and their derivatives. Because $W_{01}$ is missing
from (63), any occurrence of $J_{k}$ is accompanied by at least one other
factor $J_{k^{\prime}}$ or else by an occurrence of some $W_{i>0}$, and hence
by a power of $\lambda$ higher than the $k$’th. In other words, the expression
for $\Gamma_{l\geq 1}$ involves only $J_{k}$ with $k<l$. We finally remove the
indeterminacy in (64) by imposing (62) order by order in $\lambda$, leading to
(for $l\geq 1$)
$\frac{\delta\Gamma_{l}[n]}{\delta n}=-J_{l}\;.$ (66)
Since $\Gamma_{l\geq 1}$ involves only $J_{k<l}$, all the $J_{l}$ and
$\Gamma_{l}$ can be found explicitly by applying (65) and (66) alternately.
Evidently, it is the source decomposition (52) that allows us to obtain exact
correspondence to the Kohn-Sham scheme. Below we will provide an explicit
formula for $\Gamma_{l}[n]$ in terms of $W_{l}[J_{0}]$ and their functional
derivatives.
To obtain an explicit expression for $\Gamma_{l}[n]$, let us substitute the
LHS of (63) by the RHS of (65) and apply (56) as well as (64) to the RHS of
(63). Then, by equating the coefficients associated with $\lambda^{l}$ on both
sides of (63), we obtain (for $l\geq 1$)
$\displaystyle\Gamma_{l}\left[n\right]$ $\displaystyle=$ $\displaystyle\beta
W_{l}\left[J_{0}\right]+\sum_{k=1}^{l-1}\frac{\delta(\beta
W_{l-k}\left[J_{0}\right])}{\delta J_{0}}{\scriptstyle\circ}J_{k}$ (67)
$\displaystyle+\sum_{m=2}^{l}\frac{1}{m!}\sum_{k_{1},\ldots,k_{m}\geq
1}^{k_{1}+\ldots+k_{m}\leq l}\frac{\delta^{m}(\beta
W_{l-\left(k_{1}+\ldots+k_{m}\right)}\left[J_{0}\right])}{\delta
J_{0}\ldots\delta
J_{0}}{\scriptstyle\circ}J_{k_{1}}{\scriptstyle\circ}\cdots{\scriptstyle\circ}J_{k_{m}}\;.$
For $l=1$, we see that
$\Gamma_{1}[n]=\beta W_{1}[J_{0}]=\frac{1}{2}\text{Tr}\ln\left(\tilde{\cal
D}_{\\!\\!J\to\tilde{J}_{0}}^{-1}{\scriptstyle\circ}\,U\right)\equiv\text{Tr}\ln\left(\tilde{\cal
D}_{0}^{-1}{\scriptstyle\circ}\,U\right)\;.$ (68)
We observe that
$\tilde{\cal D}_{0}^{-1}\equiv\tilde{\cal
D}_{\\!\\!J\to\tilde{J}_{0}}^{-1}=U^{-1}-D_{\\!\\!J\to\tilde{J}_{0}}=U^{-1}-D_{0}$
We then have
$J_{1}=-\frac{\delta\Gamma_{1}[n]}{\delta n}=-\frac{\delta J_{0}}{\delta
n}{\scriptstyle\circ}\frac{\delta(\beta W_{1}[J_{0}])}{\delta J_{0}}\;.$
Note that $-\delta J_{0}/\delta n$ can be written as
$-\frac{\delta J_{0}(x)}{\delta n(y)}=-\left(\frac{\delta n(y)}{\delta
J_{0}(x)}\right)^{-1}=-\left(\frac{\delta^{2}(\beta W_{0}[J_{0}])}{\delta
J_{0}(x)\delta J_{0}(y)}\right)^{-1}=\frac{\delta^{2}\Gamma_{0}[n]}{\delta
n(x)\delta n(y)}$ (69)
where
$\Gamma_{0}[n]=\beta
W_{0}[J_{0}]-J_{0}{\scriptstyle\circ}n=\tilde{\Gamma}_{0}[n]-\frac{1}{2}n{\scriptstyle\circ}\,U{\scriptstyle\circ}n\;,$
(70)
and the inverse is in the functional matrix sense. Using (49), we can evaluate
$\delta n(x)/\delta J_{0}(y)$ by
$\displaystyle\frac{\delta n(x)}{\delta J_{0}(y)}$ $\displaystyle=$
$\displaystyle-\frac{\delta{\mathcal{G}}_{0}(x,x)}{\delta J_{0}(y)}=\int
dzdz^{\prime}{\mathcal{G}}_{0}(x,z)\frac{\delta{\mathcal{G}}_{0}^{-1}(z,z^{\prime})}{\delta
J_{0}(y)}{\mathcal{G}}_{0}(z^{\prime},x)$ (71) $\displaystyle=$
$\displaystyle{\mathcal{G}}_{0}(x,y){\mathcal{G}}_{0}(y,x)=D_{J\to\tilde{J}_{0}}(x,y)\equiv
D_{0}(x,y)\;.$
Therefore,
$J_{1}=-D_{0}^{-1}\;{\scriptstyle\circ}\frac{\delta(\beta
W_{1}[J_{0}])}{\delta J_{0}}\;.$
Since $-{\mathcal{G}}_{0}(x,x)=n(x)$ represents the electron density of the KS
system, we will call $D_{0}(x,y)$ the polarization associated with the KS
system. Note that if one were to approximate the effective action $\Gamma[n]$
by $\Gamma[n]=\tilde{\Gamma}_{0}[n]+\Gamma_{1}[n]$, the displayed equation
above is the OEP equation for GW-OEP Hellgren and von Barth (2007), while eq.
(68) is the corresponding exchange-correlation functional.
Once $J_{1}$ is known, one can find $\Gamma_{2}\left[n\right]$
$\displaystyle\Gamma_{2}\left[n\right]$ $\displaystyle=$ $\displaystyle\beta
W_{2}\left[J_{0}\right]+\frac{\delta\left(\beta
W_{1}\left[J_{0}\right]\right)}{\delta
J_{0}}{\scriptstyle\circ}J_{1}+\frac{1}{2}J_{1}{\scriptstyle\circ}\frac{\delta^{2}\left(\beta
W_{0}\left[J_{0}\right]\right)}{\delta J_{0}\delta
J_{0}}{\scriptstyle\circ}J_{1}$ (72) $\displaystyle=$ $\displaystyle\beta
W_{2}\left[J_{0}\right]-\frac{1}{2}\frac{\delta\left(\beta
W_{1}\left[J_{0}\right]\right)}{\delta
J_{0}}{\scriptstyle\circ}D_{0}^{-1}{\scriptstyle\circ}\frac{\delta\left(\beta
W_{1}\left[J_{0}\right]\right)}{\delta J_{0}}\;,$
and $J_{2}$ now can be computed via $-\delta\Gamma_{2}[n]/\delta n$
$J_{2}=D_{0}^{-1}{\scriptstyle\circ}\frac{\delta(\beta W_{2}[J_{0}])}{\delta
J_{0}}+D_{0}^{-1}{\scriptstyle\circ}\frac{\delta^{2}(\beta
W_{1}[J_{0}])}{\delta J_{0}\delta
J_{0}}{\scriptstyle\circ}J_{1}+\frac{1}{2}D_{0}^{-1}{\scriptstyle\circ}\frac{\delta^{3}\left(\beta
W_{0}[J_{0}]\right)}{\delta J_{0}\delta J_{0}\delta
J_{0}}{\scriptstyle\circ}J_{1}{\scriptstyle\circ}J_{1}\;.$ (73)
The explicit expression of $J_{2}$ leads to $\Gamma_{3}[n]$ and so on. It
should now be clear how the strategy goes. We are assuming that the
functionals $\\{W_{l}[J_{0}]\\}$ and their derivatives with respect to $J_{0}$
are known. This information can indeed be obtained by standard, albeit
tedious, many-body perturbation method. One then uses Eq. (67) to express
$\Gamma_{l}$ in terms of the known functionals and $J_{k}$ with $k\leq l-1$.
Once $\Gamma_{l}$ is obtained, one can then use Eq. (66) to obtain $J_{l}$,
which then facilitates the calculation of $\Gamma_{l+1}$ via (67) and so on.
We now take a moment to organize the terms of effective action $\Gamma[n]$.
From (65) and (70), we know that
$\Gamma[n]=\tilde{\Gamma}_{0}[n]+\sum_{l=1}^{\infty}\Gamma_{l}[n]=\Gamma_{0}[n]+\frac{1}{2}n{\scriptstyle\circ}\,U{\scriptstyle\circ}n+\sum_{l=1}^{\infty}\Gamma_{l}[n]\;.$
(74)
The first term on the RHS of (74) indeed corresponds to the effective action
of the free KS system, the second term is exactly the Hartree energy. It is
thus natural for us to define the last part, sum of $\Gamma_{l}[n]$, as
$\Gamma_{xc}[n]\equiv\sum_{l=1}^{\infty}\Gamma_{l}[n]\;.$ (75)
At the physical condition (i.e., when the source is absent), we should have
$\delta\Gamma/\delta n=-J=0$. Knowing that
$\frac{\delta\tilde{\Gamma}_{0}[n]}{\delta
n}=-\tilde{J}_{0}=-J_{0}+U{\scriptstyle\circ}n\;,$
we conclude that at the physical condition,
$\frac{\delta\Gamma_{xc}[n]}{\delta
n}=\tilde{J}_{0}=J_{0}-U{\scriptstyle\circ}n\;.$ (76)
Note that the potential $J_{0}$ together with the original non-interacting
part of the Hamiltonian leads to the exact particle density of the interacting
system. This implies that $J_{0}$ contains both the Hartree term and the
exchange-correlation potential. Since
$J_{0}=\tilde{J}_{0}+U{\scriptstyle\circ}n_{T}$, with
$U{\scriptstyle\circ}n_{T}$ being the Hartree term, the term $\tilde{J}_{0}$
plays the role of exchange-correlation potential, as evidenced by (76). The
quantum mechanical effects are completely contained in $\Gamma_{xc}[n]$,
defined in (75).
Note that a different density profile other than that corresponding to the
physical ground state can be induced by introducing a nonzero $J$. Let us
again call the corresponding density profile $n_{J}$. When $J\neq 0$, we learn
from (10) that it is equivalent to replacing $\upsilon$ by $\upsilon+J$. Eq.
(5) then tells us that $J_{0}$ must contain $J$, the Hartree term and the
exchange-correlational potential as shown in (50). In this case, one writes
$J_{0}=J+(\tilde{J}_{0}-J)+U{\scriptstyle\circ}n_{J}$. With
$U{\scriptstyle\circ}n_{J}$ being the Hartree term, $(\tilde{J}_{0}-J)$ must
represent the exchange-correlation potential corresponding to the
configuration $n_{J}$, and
$\frac{\delta\left(\Gamma_{xc}[n]\right)}{\delta n}=\tilde{J}_{0}-J\;.$ (77)
This then leads to
$\frac{\delta\Gamma[n]}{\delta n}=-J\;,$
what we expected when $J\neq 0$.
In terms of real computation, since the $n$ dependence is through $J_{0}$, we
may rewrite Eq. (76) as
$\frac{\delta J_{0}}{\delta
n}{\scriptstyle\circ}\frac{\delta\left(\Gamma_{xc}\left[J_{0}[n]\right]\right)}{\delta
J_{0}}=\tilde{J}_{0}=J_{0}-U{\scriptstyle\circ}n$
or
$0=D_{0}\,{\scriptstyle\circ}\,\tilde{J}_{0}-\frac{\delta\left(\Gamma_{xc}\left[J_{0}[n]\right]\right)}{\delta
J_{0}}=D_{0}\,{\scriptstyle\circ}\,\left(J_{0}-U{\scriptstyle\circ}n\right)-\frac{\delta\left(\Gamma_{xc}\left[J_{0}[n]\right]\right)}{\delta
J_{0}}\;,$ (78)
and the condition $\delta\Gamma[n]/\delta n=0$ is turned into
$\delta\Gamma[J_{0}[n]]/\delta J_{0}=0$ Since the effective action is a
strictly convex function of the electron density $n$, there exists no local
minima. One can therefore solve $\delta\Gamma[J_{0}[n]]/\delta n=0$ by
steepest descent. Effectively, we may define the direction of steepest descent
$\kappa(x)$ by
$\kappa(x)\equiv-\frac{\delta\Gamma\left[J_{0}[n]\right]}{\delta
J_{0}(x)}=D_{0}\,{\scriptstyle\circ}\,\left(J_{0}-U{\scriptstyle\circ}n\right)-\frac{\delta\left(\Gamma_{xc}\left[J_{0}[n]\right]\right)}{\delta
J_{0}}$ (79)
and then update $J_{0}(x)$ by $J_{0}(x)\to J_{0}(x)+\varsigma\kappa(x)$, with
$\varsigma>0$ being the step size, till convergence is reached, that is, when
$\kappa(x)\to 0$. Note that eq. (78) is the standard OEP-equation for
consistency. The iterative procedure described after (78) is largely identical
to the Kuemmel-Perdew Kümmel and Perdew (2003) procedure used for solving the
exchange-only OEP equation.
### III.4 Diagrammatic Expansion of the Density Functional
We now examine how one computes the effective action via diagrams. From Eqs.
(65), (68) and (70), we have
$\displaystyle\Gamma[n]$ $\displaystyle=$
$\displaystyle\tilde{\Gamma}_{0}[n]+\sum_{i=1}^{\infty}\Gamma_{i}[n]=\Gamma_{0}[n]+\frac{1}{2}n{\scriptstyle\circ}\,U{\scriptstyle\circ}n+\sum_{i=1}^{\infty}\Gamma_{i}[n]$
(80) $\displaystyle=$
$\displaystyle-\text{Tr}\ln\left({\mathcal{G}}_{0}^{-1}\right)-J_{0}{\scriptstyle\circ}n+\frac{1}{2}n{\scriptstyle\circ}\,U{\scriptstyle\circ}n+\frac{1}{2}\text{Tr}\ln\left(\tilde{\cal
D}_{0}^{-1}{\scriptstyle\circ}\,U\right)+\sum_{i=2}^{\infty}\Gamma_{i}[n]\;,$
where
$\displaystyle\tilde{\cal D}_{0}^{-1}$ $\displaystyle=$ $\displaystyle
U^{-1}-D_{0}$ (81) $\displaystyle D_{0}(x,y)$ $\displaystyle=$
$\displaystyle{\mathcal{G}}_{0}(x,y){\mathcal{G}}_{0}(y,x)\;.$ (82)
Evidently, we need diagrammatic symbols for $U$, ${\mathcal{G}}_{0}$ and
$\tilde{\mathcal{D}}_{0}$. To get to higher-order terms $\Gamma_{i\geq 2}$ of
the effective action, we see from Eqs. (71-73) and the text afterwards that it
is necessary to incorporate into the diagrams the inverse density correlator
$D_{0}^{-1}$ and to evaluate functional derivatives with respect to $n$ (or
$J_{0}$). We define the symbols for each line type below
The smaller dots associated with the $U$, $D_{0}^{-1}$, and
${\mathcal{G}}_{0}$ propagators are introduced to guide the eyes regarding the
starting and ending points of these propagators. The $\tilde{\mathcal{D}}_{0}$
propagator comes from contracting two $b$ fields, see (33), and thus the
bigger dots associated with $\tilde{\mathcal{D}}_{0}$ denote the
coordinates/locations of those $b$ fields. We will also use the bigger dots to
indicate the space-time coordinate of a point of interest.
To evaluate functional derivatives of $W_{l}[J_{0}[n]]$ with respect to $n$
(or $J_{0}$), we note from Eqs. (25), (34), (40) and (46) that the $J_{0}$
dependence comes from ${\mathcal{G}}_{0}(x,y)$ and the functional derivatives
associated with the formalism using Eqs. (67-79) necessarily require
evaluations of $\delta{\mathcal{G}}_{0}(x,x^{\prime})/\delta J_{0}(y)$. Using
a derivation similar to that in (71), we find that
$\frac{\delta{\mathcal{G}}_{0}(x,x^{\prime})}{\delta
J_{0}(y)}=-{\mathcal{G}}_{0}(x,y)\,{\mathcal{G}}_{0}(y,x^{\prime})\;.$ (83)
The propagators $D_{0}^{-1}$ and $\tilde{\mathcal{D}}_{0}$ also contain
${\mathcal{G}}_{0}$ and thus may be differentiated with respect to $J_{0}$.
Note that
$\tilde{\mathcal{D}}_{0}=({\mathbf{I}}-U{\scriptstyle\circ}D_{0})^{-1}{\scriptstyle\circ}\,U$
and $D_{0}(x,y)={\mathcal{G}}_{0}(x,y){\mathcal{G}}_{0}(y,x)$. Employing the
identity
$\frac{\delta M^{-1}}{\delta J_{0}}=-M^{-1}{\scriptstyle\circ}\frac{\delta
M}{\delta J_{0}}{\scriptstyle\circ}M^{-1}\;,$
we let $M=({\mathbf{I}}-U{\scriptstyle\circ}D_{0})$ for the case of
$\tilde{\mathcal{D}}_{0}$ and $M=D_{0}$ for the case of $D_{0}^{-1}$ to obtain
$\displaystyle\frac{\delta\tilde{\mathcal{D}}_{0}}{\delta J_{0}}$
$\displaystyle=$
$\displaystyle\tilde{\mathcal{D}}_{0}{\scriptstyle\circ}\frac{\delta
D_{0}}{\delta J_{0}}{\scriptstyle\circ}\tilde{\mathcal{D}}_{0}\;,$ (84)
$\displaystyle{\rm and\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ }\frac{\delta D_{0}^{-1}}{\delta
J_{0}}$ $\displaystyle=$ $\displaystyle-
D_{0}^{-1}{\scriptstyle\circ}\frac{\delta D_{0}}{\delta
J_{0}}{\scriptstyle\circ}D_{0}^{-1}\;.$ (85)
Eq. (83-85) may be expressed diagrammatically as follows
$\displaystyle\frac{\delta{\mathcal{G}}_{0}(x,x^{\prime})}{\delta
J_{0}(y)}=\frac{\delta}{\delta J_{0}(y)}\;\begin{picture}(10.0,40.0)(0.0,-3.0)
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \raise-34.0pt\hbox
to0.0pt{\kern 5.0pt\makebox(0.0,0.0)[tc]{$x^{\prime}$}\hss} \ignorespaces
\raise 34.0pt\hbox to0.0pt{\kern 5.0pt\makebox(0.0,0.0)[bc]{$x$}\hss}
\ignorespaces\end{picture}$ $\displaystyle=$
$\displaystyle-\;\begin{picture}(10.0,40.0)(0.0,-3.0) \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\raise-34.0pt\hbox to0.0pt{\kern 5.0pt\makebox(0.0,0.0)[tc]{$x^{\prime}$}\hss}
\ignorespaces \raise 34.0pt\hbox to0.0pt{\kern
5.0pt\makebox(0.0,0.0)[bc]{$x$}\hss} \ignorespaces \raise-4.0pt\hbox
to0.0pt{\kern 12.0pt\makebox(0.0,0.0)[bc]{$y$}\hss}
\ignorespaces\end{picture}\;,$
$\displaystyle\frac{\delta\tilde{\mathcal{D}}_{0}(x,x^{\prime})}{\delta
J_{0}(y)}=\frac{\delta}{\delta
J_{0}(y)}\;\begin{picture}(10.0,80.0)(0.0,-3.0)\put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \raise-34.0pt\hbox to0.0pt{\kern
5.0pt\makebox(0.0,0.0)[tc]{$x^{\prime}$}\hss} \ignorespaces \raise 34.0pt\hbox
to0.0pt{\kern 5.0pt\makebox(0.0,0.0)[bc]{$x$}\hss} \ignorespaces\end{picture}$
$\displaystyle=$
$\displaystyle-\;\begin{picture}(30.0,40.0)(0.0,-3.0)\put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \raise-34.0pt\hbox to0.0pt{\kern
15.0pt\makebox(0.0,0.0)[tc]{$x^{\prime}$}\hss} \ignorespaces \raise
34.0pt\hbox to0.0pt{\kern 15.0pt\makebox(0.0,0.0)[bc]{$x$}\hss} \ignorespaces
\raise-4.0pt\hbox to0.0pt{\kern 11.0pt\makebox(0.0,0.0)[bc]{$y$}\hss}
\ignorespaces\end{picture}-\;\begin{picture}(30.0,40.0)(0.0,-3.0)\put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \raise-34.0pt\hbox to0.0pt{\kern
15.0pt\makebox(0.0,0.0)[tc]{$x^{\prime}$}\hss} \ignorespaces \raise
34.0pt\hbox to0.0pt{\kern 15.0pt\makebox(0.0,0.0)[bc]{$x$}\hss} \ignorespaces
\raise-4.0pt\hbox to0.0pt{\kern 11.0pt\makebox(0.0,0.0)[bc]{$y$}\hss}
\ignorespaces\end{picture}\;,$ $\displaystyle\frac{\delta
D_{0}^{-1}(x,x^{\prime})}{\delta J_{0}(y)}=\frac{\delta}{\delta
J_{0}(y)}\;\begin{picture}(10.0,80.0)(0.0,-3.0) \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \raise-34.0pt\hbox to0.0pt{\kern
5.0pt\makebox(0.0,0.0)[tc]{$x^{\prime}$}\hss} \ignorespaces \raise 34.0pt\hbox
to0.0pt{\kern 5.0pt\makebox(0.0,0.0)[bc]{$x$}\hss} \ignorespaces\end{picture}$
$\displaystyle=$ $\displaystyle+\;\begin{picture}(30.0,40.0)(0.0,-3.0)
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \raise-34.0pt\hbox to0.0pt{\kern
15.0pt\makebox(0.0,0.0)[tc]{$x^{\prime}$}\hss} \ignorespaces \raise
34.0pt\hbox to0.0pt{\kern 15.0pt\makebox(0.0,0.0)[bc]{$x$}\hss} \ignorespaces
\raise-4.0pt\hbox to0.0pt{\kern 11.0pt\makebox(0.0,0.0)[bc]{$y$}\hss}
\ignorespaces\end{picture}+\;\begin{picture}(30.0,40.0)(0.0,-3.0)
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \raise-34.0pt\hbox to0.0pt{\kern
15.0pt\makebox(0.0,0.0)[tc]{$x^{\prime}$}\hss} \ignorespaces \raise
34.0pt\hbox to0.0pt{\kern 15.0pt\makebox(0.0,0.0)[bc]{$x$}\hss} \ignorespaces
\raise-4.0pt\hbox to0.0pt{\kern 11.0pt\makebox(0.0,0.0)[bc]{$y$}\hss}
\ignorespaces\end{picture}\;,$
12 where the $\mp$ signs come from
$\frac{\delta}{\delta J_{0}(y)}D_{0}(z,z^{\prime})=\frac{\delta}{\delta
J_{0}(y)}\begin{picture}(30.0,20.0)(0.0,-3.0) \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \raise-14.0pt\hbox to0.0pt{\kern
15.0pt\makebox(0.0,0.0)[tc]{$z^{\prime}$}\hss} \ignorespaces \raise
14.0pt\hbox to0.0pt{\kern 15.0pt\makebox(0.0,0.0)[bc]{$z$}\hss}
\ignorespaces\end{picture}=-\;\begin{picture}(30.0,20.0)(0.0,-3.0)
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \raise-14.0pt\hbox to0.0pt{\kern
15.0pt\makebox(0.0,0.0)[tc]{$z^{\prime}$}\hss} \ignorespaces \raise
14.0pt\hbox to0.0pt{\kern 15.0pt\makebox(0.0,0.0)[bc]{$z$}\hss} \ignorespaces
\raise-4.0pt\hbox to0.0pt{\kern 11.0pt\makebox(0.0,0.0)[bc]{$y$}\hss}
\ignorespaces\end{picture}-\;\begin{picture}(30.0,20.0)(0.0,-3.0)
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \raise-14.0pt\hbox to0.0pt{\kern
15.0pt\makebox(0.0,0.0)[tc]{$z^{\prime}$}\hss} \ignorespaces \raise
14.0pt\hbox to0.0pt{\kern 15.0pt\makebox(0.0,0.0)[bc]{$z$}\hss} \ignorespaces
\raise-4.0pt\hbox to0.0pt{\kern 11.0pt\makebox(0.0,0.0)[bc]{$y$}\hss}
\ignorespaces\end{picture}\;.$
12345
When combined with the inverse density correlator, the graphs above yield
$\displaystyle\frac{\delta{\mathcal{G}}_{0}(x,x^{\prime})}{\delta n(z)}$
$\displaystyle=$ $\displaystyle-\int
dy\,D_{0}^{-1}(z,y)\,{\mathcal{G}}_{0}(x,y)\,{\mathcal{G}}_{0}(y,x^{\prime})\;,$
$\displaystyle\frac{\delta\tilde{\mathcal{D}}_{0}(x,x^{\prime})}{\delta n(z)}$
$\displaystyle=$ $\displaystyle-\int
dydx_{1}dx_{2}D_{0}^{-1}(z,y)\,\tilde{\mathcal{D}}_{0}(x,x_{1})\left[{\mathcal{G}}_{0}(x_{2},x_{1}){\mathcal{G}}_{0}(x_{1},y){\mathcal{G}}_{0}(y,x_{2})+\right.$
$\displaystyle\hskip
20.0pt\left.+{\mathcal{G}}_{0}(x_{1},x_{2}){\mathcal{G}}_{0}(x_{2},y){\mathcal{G}}_{0}(y,x_{1})\right]\tilde{\mathcal{D}}_{0}(x_{2},x^{\prime})\;,$
$\displaystyle\frac{\delta D_{0}^{-1}(x,x^{\prime})}{\delta n(z)}$
$\displaystyle=$ $\displaystyle\int
dydx_{1}dx_{2}D_{0}^{-1}(z,y)\,D_{0}^{-1}(x,x_{1})\left[{\mathcal{G}}_{0}(x_{2},x_{1}){\mathcal{G}}_{0}(x_{1},y){\mathcal{G}}_{0}(y,x_{2})+\right.$
$\displaystyle\hskip
20.0pt\left.+{\mathcal{G}}_{0}(x_{1},x_{2}){\mathcal{G}}_{0}(x_{2},y){\mathcal{G}}_{0}(y,x_{1})\right]D_{0}^{-1}(x_{2},x^{\prime})\;,$
which are shown diagrammatically below
$\displaystyle\frac{\delta{\mathcal{G}}_{0}(x,x^{\prime})}{\delta
n(z)}=\frac{\delta}{\delta n(z)}\;\begin{picture}(10.0,40.0)(0.0,-3.0)
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \raise-34.0pt\hbox
to0.0pt{\kern 5.0pt\makebox(0.0,0.0)[tc]{$x^{\prime}$}\hss} \ignorespaces
\raise 34.0pt\hbox to0.0pt{\kern 5.0pt\makebox(0.0,0.0)[bc]{$x$}\hss}
\ignorespaces\end{picture}$ $\displaystyle=$
$\displaystyle-\;\begin{picture}(30.0,40.0)(-20.0,-3.0) \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \raise-34.0pt\hbox to0.0pt{\kern
5.0pt\makebox(0.0,0.0)[tc]{$x^{\prime}$}\hss} \ignorespaces \raise 34.0pt\hbox
to0.0pt{\kern 5.0pt\makebox(0.0,0.0)[bc]{$x$}\hss} \ignorespaces
\raise-9.0pt\hbox to0.0pt{\kern-14.0pt\makebox(0.0,0.0)[bc]{$z$}\hss}
\ignorespaces\end{picture}\;,$
$\displaystyle\frac{\delta\tilde{\mathcal{D}}_{0}(x,x^{\prime})}{\delta
n(z)}=\frac{\delta}{\delta
n(z)}\;\begin{picture}(10.0,80.0)(0.0,-3.0)\put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \raise-34.0pt\hbox to0.0pt{\kern
5.0pt\makebox(0.0,0.0)[tc]{$x^{\prime}$}\hss} \ignorespaces \raise 34.0pt\hbox
to0.0pt{\kern 5.0pt\makebox(0.0,0.0)[bc]{$x$}\hss} \ignorespaces\end{picture}$
$\displaystyle=$
$\displaystyle-\;\begin{picture}(50.0,40.0)(-20.0,-3.0)\put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \raise-34.0pt\hbox
to0.0pt{\kern 15.0pt\makebox(0.0,0.0)[tc]{$x^{\prime}$}\hss} \ignorespaces
\raise 34.0pt\hbox to0.0pt{\kern 15.0pt\makebox(0.0,0.0)[bc]{$x$}\hss}
\ignorespaces \raise-9.0pt\hbox
to0.0pt{\kern-14.0pt\makebox(0.0,0.0)[bc]{$z$}\hss}
\ignorespaces\end{picture}-\;\begin{picture}(50.0,40.0)(-20.0,-3.0)\put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \raise-34.0pt\hbox
to0.0pt{\kern 15.0pt\makebox(0.0,0.0)[tc]{$x^{\prime}$}\hss} \ignorespaces
\raise 34.0pt\hbox to0.0pt{\kern 15.0pt\makebox(0.0,0.0)[bc]{$x$}\hss}
\ignorespaces \raise-9.0pt\hbox
to0.0pt{\kern-14.0pt\makebox(0.0,0.0)[bc]{$z$}\hss}
\ignorespaces\end{picture}\;,$ $\displaystyle\frac{\delta
D_{0}^{-1}(x,x^{\prime})}{\delta n(z)}=\frac{\delta}{\delta
n(z)}\;\begin{picture}(10.0,80.0)(0.0,-3.0) \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \raise-34.0pt\hbox to0.0pt{\kern
5.0pt\makebox(0.0,0.0)[tc]{$x^{\prime}$}\hss} \ignorespaces \raise 34.0pt\hbox
to0.0pt{\kern 5.0pt\makebox(0.0,0.0)[bc]{$x$}\hss} \ignorespaces\end{picture}$
$\displaystyle=$ $\displaystyle+\;\begin{picture}(50.0,40.0)(-20.0,-3.0)
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\raise-34.0pt\hbox to0.0pt{\kern
15.0pt\makebox(0.0,0.0)[tc]{$x^{\prime}$}\hss} \ignorespaces \raise
34.0pt\hbox to0.0pt{\kern 15.0pt\makebox(0.0,0.0)[bc]{$x$}\hss} \ignorespaces
\raise-9.0pt\hbox to0.0pt{\kern-14.0pt\makebox(0.0,0.0)[bc]{$z$}\hss}
\ignorespaces\end{picture}+\;\begin{picture}(50.0,40.0)(-20.0,-3.0)
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\raise-34.0pt\hbox to0.0pt{\kern
15.0pt\makebox(0.0,0.0)[tc]{$x^{\prime}$}\hss} \ignorespaces \raise
34.0pt\hbox to0.0pt{\kern 15.0pt\makebox(0.0,0.0)[bc]{$x$}\hss} \ignorespaces
\raise-9.0pt\hbox to0.0pt{\kern-14.0pt\makebox(0.0,0.0)[bc]{$z$}\hss}
\ignorespaces\end{picture}\;.$
The diagrammatic differential rules of $\delta/\delta J_{0}$ and
$\delta/\delta n$ are needed not only for the calculations of higher-order
terms $\Gamma_{i\geq 2}$ of the effective action but also for the calculations
of excitations, which we will discuss in section V.
Before formally introducing the diagrammatic expansion, let us first set the
convention that we will use. In general, each Feynman graph will carry with it
a symmetry factor, which is inversely proportional to the number of ways to
label this graph without changing the topology of the graph. The convention in
quantum field theory usually leaves out the symmetry factor, as it may be
deduced from the graph. To avoid enumeration of the symmetry factors needed,
however, we will explicitly provide the symmetry factors for Feynman diagrams
to be investigated later.
Let us now look at the effective action (80) term by term. The Hartree term
$n{\scriptstyle\circ}\,U{\scriptstyle\circ}n/2$ is expressed diagrammatically
below
$\Gamma_{\rm Hartree}=\frac{1}{2}\;\;\begin{picture}(70.0,28.0)(0.0,11.0)
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \end{picture}\;\;.$
As remarked by Jackiw, Jackiw (1974) each term in the effective action
expansion represents an infinite number of Feynman diagrams in regular
perturbative field theoretic calculations. We use the
$\frac{1}{2}\text{Tr}\ln(\tilde{\mathcal{D}_{0}}^{-1}{\scriptstyle\circ}\,U)$
term as an explicit example
$\frac{1}{2}\text{Tr}\ln(\tilde{\mathcal{D}_{0}}^{-1}{\scriptstyle\circ}\,U)=\frac{1}{2}\text{Tr}\ln\left[{\mathbf{I}}-D_{0}{\scriptstyle\circ}\,U\right]=-\sum_{n=1}^{\infty}\frac{\text{Tr}\left(D_{0}{\scriptstyle\circ}\,U\right)^{n}}{2n}\;,$
where each term in the summation corresponds to a vacuum diagram, see Fig 1.
$-\frac{1}{2}$$-\frac{1}{4}$$-\frac{1}{6}$$-\frac{1}{8}$$+\cdots$ Figure 1:
Feynman diagrams corresponding to
$\frac{1}{2}\text{Tr}\ln(\tilde{\mathcal{D}}_{0}^{-1}{\scriptstyle\circ}\,U)$.
The first term corresponds to $-\frac{1}{2\cdot
1}\text{Tr}\,(D_{0}{\scriptstyle\circ}\,U)$, the second term corresponds to
$-\frac{1}{2\cdot 2}\text{Tr}\,(D_{0}{\scriptstyle\circ}\,U)^{2}$, the third
term corresponds to $-\frac{1}{2\cdot
3}\text{Tr}\,(D_{0}{\scriptstyle\circ}\,U)^{3}$, and the fourth term
corresponds to $-\frac{1}{2\cdot
4}\text{Tr}\,(D_{0}{\scriptstyle\circ}\,U)^{4}$. In general, the diagram
corresponding to $-\text{Tr}\,(D_{0}{\scriptstyle\circ}\,U)^{n}$ contains $n$
bubbles strung by $n$ $U$ propagators with the symmetry factor $\frac{1}{2n}$.
If we pull together the lowest order diagrams in $e^{2}$, we find the
combination
$\frac{1}{2}n{\scriptstyle\circ}\,U{\scriptstyle\circ}n-\frac{1}{2}\text{Tr}(D_{0}{\scriptstyle\circ}\,U)$
that corresponds to the Feynman diagrams in Fig. 2.
$\frac{1}{2}$$-\frac{1}{2}$($a$)($b$) Figure 2: The Feynman diagrams
corresponding to the Hartree term (a) and the lowest order exchange term (b),
the first graph from Fig. 1. The numerical factor associated with each diagram
is shown explicitly.
To compute $\Gamma_{2}$, we need to first calculate $J_{1}$. Since
$\Gamma_{1}[n]=\beta
W_{1}[J_{0}]=\frac{1}{2}\text{Tr}\ln({\mathbf{I}}-D_{0}{\scriptstyle\circ}\,U)$
$\displaystyle J_{1}(z)$ $\displaystyle=$
$\displaystyle-\frac{\delta\Gamma_{1}}{\delta
n(z)}=-\frac{1}{2}\text{Tr}\left[({\mathbf{I}}-D_{0}{\scriptstyle\circ}\,U)^{-1}{\scriptstyle\circ}\left(-\frac{\delta
D_{0}}{\delta n(z)}\right){\scriptstyle\circ}\,U\right]$ (86) $\displaystyle=$
$\displaystyle\frac{1}{2}\text{Tr}\left[U{\scriptstyle\circ}({\mathbf{I}}-D_{0}{\scriptstyle\circ}\,U)^{-1}{\scriptstyle\circ}\frac{\delta
D_{0}}{\delta
n(z)}\right]=\frac{1}{2}\text{Tr}\left[\tilde{\mathcal{D}}_{0}{\scriptstyle\circ}\frac{\delta
D_{0}}{\delta n(z)}\right]$ $\displaystyle=$ $\displaystyle-\int
dy\,dx\,dx^{\prime}\,D_{0}^{-1}(z,y)\tilde{\mathcal{D}}_{0}(x,x^{\prime}){\mathcal{G}}_{0}(x,x^{\prime}){\mathcal{G}}_{0}(x^{\prime},y){\mathcal{G}}_{0}(y,x)\;,$
which can also be expressed diagrammatically as
$J_{1}(z)=-\;\begin{picture}(48.0,20.0)(0.0,0.0) \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\raise-1.0pt\hbox to0.0pt{\kern 3.5pt\makebox(0.0,0.0)[tc]{$z$}\hss}
\ignorespaces\end{picture}\;\;\;.$
From Eq. (72), we know that $\Gamma_{2}[n]=\beta
W_{2}[J_{0}]-\frac{1}{2}J_{1}{\scriptstyle\circ}D_{0}^{-1}{\scriptstyle\circ}J_{1}$.
We note from Eq. (46) and (47) that $\beta W_{2}[J_{0}]$ corresponds to the
$\lambda^{2}$ diagrams in
$-\lambda\sum_{n=1}^{\infty}\frac{1}{n!}\langle\left[{1\over\lambda}\sum_{k=3}^{\infty}\lambda^{k\over
2}I^{(k)}[\varphi]{\scriptstyle\circ}\,b_{1}\ldots{\scriptstyle\circ}\,b_{k}\right]^{n}\rangle_{\rm
1PI,\leavevmode\nobreak\ conn.}\;,$
the last part of Eq. (46). There are two combinations of $n$ and $k$ that can
give rise to $\lambda^{2}$. The first one is to have $n=1$ and $k=4$, while
the second one is to have $n=2$ and $k=3$. The first possibility generates two
distinct graphs, while the second possibility generates three distinct
diagrams. For $n=1$ and $k=4$, we have the following diagrams with their
symmetry factors specified
$\begin{picture}(180.0,80.0)(0.0,0.0) \raise 72.0pt\hbox to0.0pt{\kern
30.0pt\makebox(0.0,0.0)[bc]{($a$)}\hss} \ignorespaces \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \raise 10.0pt\hbox
to0.0pt{\kern
30.0pt\makebox(0.0,0.0)[tc]{\large$-\frac{(i)^{4}}{1!}\frac{(-1)^{3}}{4}$}\hss}
\ignorespaces\put(0.0,0.0){} \put(0.0,0.0){} \raise 72.0pt\hbox to0.0pt{\kern
130.0pt\makebox(0.0,0.0)[bc]{($b$)}\hss} \ignorespaces \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \raise 10.0pt\hbox
to0.0pt{\kern
130.0pt\makebox(0.0,0.0)[tc]{{\large$-\frac{(i)^{4}}{1!}\frac{(-1)^{3}}{4}$}{\small$2$}}\hss}
\ignorespaces\put(0.0,0.0){} \put(0.0,0.0){} \end{picture}\;\;\;.$
For $n=2$ and $k=3$, we have the following diagrams with their symmetry
factors specified
$\displaystyle(a)$ \begin{picture}(100.0,22.5)(0.0,-4.0) \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \end{picture} $\displaystyle\hskip
10.0pt-\frac{3(i)^{6}}{2!}\frac{(-1)^{2}}{3}\frac{(-1)^{2}}{3}\;,$
$\displaystyle(b)$ \begin{picture}(100.0,42.5)(0.0,-4.0) \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \end{picture} $\displaystyle\hskip
10.0pt-\frac{3(i)^{6}}{2!}\frac{(-1)^{2}}{3}\frac{(-1)^{2}}{3}\;,$
$\displaystyle(c)$ \begin{picture}(100.0,42.5)(0.0,-4.0) \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \end{picture} $\displaystyle\hskip
10.0pt-\frac{3(i)^{6}}{2!}\frac{(-1)^{2}}{3}\frac{(-1)^{2}}{3}\;.$
12345
The first diagram(a) among the three will be discarded since one can cut a
$\tilde{\mathcal{D}}_{0}$ line and then separate it into two diagrams. Let us
display below the diagram corresponding to
$J_{1}{\scriptstyle\circ}D_{0}{\scriptstyle\circ}J_{1}$,
$J_{1}{\scriptstyle\circ}D_{0}{\scriptstyle\circ}J_{1}=(-J_{1}){\scriptstyle\circ}D_{0}{\scriptstyle\circ}(-J_{1})=\begin{picture}(48.0,20.0)(18.0,0.0)
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \end{picture}\hskip
5.0pt\left(\begin{picture}(17.0,8.0)(0.0,0.0) \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \end{picture}\right)^{-1}\begin{picture}(48.0,20.0)(0.0,0.0)
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \end{picture}\;=\;\;\begin{picture}(76.0,20.0)(0.0,0.0)
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \end{picture}\;\;.$
Therefore, the diagrammatic expression for $\Gamma_{2}[n]$ is given by
$\displaystyle\Gamma_{2}[n]$ $\displaystyle=$
$\displaystyle\frac{1}{4}\;\begin{picture}(40.0,20.0)(0.0,-4.0)
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){}
\end{picture}\;\;+\frac{1}{2}\;\begin{picture}(40.0,20.0)(0.0,-4.0)
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){}
\end{picture}\;\;-\frac{1}{2}\;\begin{picture}(100.0,20.0)(0.0,-4.0)
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \end{picture}$
$\displaystyle+\frac{1}{3!}\;\begin{picture}(100.0,42.5)(0.0,-4.0)
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\end{picture}\;+\frac{1}{3!}\;\begin{picture}(100.0,42.5)(0.0,-4.0)
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \end{picture}\;\;\;.$
\begin{picture}(20.0,20.0)(0.0,0.0)\end{picture} Figure 3: Diagrammatic
expression for $\Gamma_{2}[n]$.
Note that each $\tilde{\mathcal{D}}_{0}$ propagator line contains the sum of
infinitely many terms
$\displaystyle\tilde{\mathcal{D}}_{0}$ $\displaystyle=$
$\displaystyle\left(U^{-1}-D_{0}\right)^{-1}=\left({\mathbf{I}}-U{\scriptstyle\circ}D_{0}\right)^{-1}{\scriptstyle\circ}\,U=U+U{\scriptstyle\circ}D_{0}{\scriptstyle\circ}\,U+U{\scriptstyle\circ}D_{0}{\scriptstyle\circ}\,U{\scriptstyle\circ}D_{0}{\scriptstyle\circ}\,U+\ldots$
(87) \begin{picture}(20.0,30.0)(0.0,-3.0)\put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \end{picture} $\displaystyle=$
$\displaystyle\begin{picture}(20.0,30.0)(0.0,-3.0) \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){}
\end{picture}\;\;+\;\begin{picture}(60.0,30.0)(0.0,-3.0) \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\end{picture}\;\;+\;\begin{picture}(100.0,30.0)(0.0,-3.0) \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\end{picture}\;\;+\;\cdots\;.$
Since each $U$ carries a factor of $e^{2}$, the above expansion may be viewed
as the $e^{2}$ expansion of $\tilde{\mathcal{D}}_{0}$ with $U$ being the
leading order. Therefore, in the diagrams corresponding to $\Gamma_{2}$ (Fig.
3), if one were to expand $\tilde{\mathcal{D}}_{0}$ in $e^{2}$, the first
three diagrams are of order $e^{4}$ or higher, while the last two diagrams
contains terms of order $e^{6}$ or higher only.
It is now instructive to compare with the perturbative methods which use
$e^{2}$ as the expansion parameter. Among those methods, the approach of
Valiev and Fernando Valiev and Fernando (1997) is the closest to ours. Let us
pull out the diagrams contributing to order $e^{4}$ in $\Gamma_{i\leq 2}$ and
compare to the results of reference Valiev and Fernando, 1997. From
$\Gamma_{1}[n]$, we see that the second diagram in Fig. 1 corresponding to
$-\frac{1}{4}\text{Tr}\left[\left(D_{0}{\scriptstyle\circ}\,U\right)^{2}\right]$
will contribute to this order. The first three diagrams representing
$\Gamma_{2}[n]$ will contribute to the same order if we replace the
$\tilde{\mathcal{D}}$ propagator by $U$. Thus, one obtains the diagrams shown
in Fig. 4, which are identical to the results in reference Valiev and
Fernando, 1997.
$\frac{1}{4}$$-\frac{1}{4}$$+\frac{1}{2}$$-\frac{1}{2}$ Figure 4: The Feynman
diagrams of order $U^{2}$ (or $e^{4}$) of the effective action $\Gamma[n]$.
The correct symmetry factors are also provided.
We use this example to illustrate the difference between our diagrammatic
rules and those of reference Valiev and Fernando, 1997. In order to obtain
$e^{4}$ diagrams, the diagrammatic rules of reference Valiev and Fernando,
1997 require the generation of all $e^{4}$ diagrams of $\beta W[J]$, each of
which is shown in Fig. 5.
$\frac{1}{4}$$-\frac{1}{4}$$+\frac{1}{2}$$-$$+\frac{1}{2}$ Figure 5: The
Feynman diagrams of order $U^{2}$ (or $e^{4}$) of $\beta W[J]$. The correct
symmetry factors are also provided.
Then one deletes diagrams that can be cut into two parts by cutting a Coulomb
line $u$. This means that the last two diagrams above will be deleted from
consideration. One then needs to find in the remaining diagrams the two-
particle reducible (in the sense of fermion propagator) ones followed by an
iterative operation to construct $D_{0}^{-1}$ lines.Fukuda et al. (1995) In
this case, the only two-particle-reducible diagram is the third one and the
iterative procedure generates exactly the only diagram containing a
$D_{0}^{-1}$ line in $\Gamma_{2}$, with $\tilde{\mathcal{D}}$ replaced by $U$.
Therefore, the diagrammatic rules of reference Valiev and Fernando, 1997
require first one-particle-irreducibility of the Coulomb line followed by
searching for diagrams that are two-particle-reducible (in fermion propagators
sense).
For our case, when considering a diagram’s reducibility, we only consider the
$\tilde{\mathcal{D}}_{0}$ lines. Let us denote
$I^{(k)}[\varphi]{\scriptstyle\circ}\,b_{1}\ldots{\scriptstyle\circ}\,b_{k}$
by ${\mathcal{B}}^{(k)}$, and call it blob $k$. The one-particle-irreducible
criterion simply means that when one joins any two blobs, say
${\mathcal{B}}^{(k_{1})}$ and ${\mathcal{B}}^{(k_{2})}$, one must make sure
that at least two or more $\tilde{\mathcal{D}}_{0}$ lines are connecting
${\mathcal{B}}^{(k_{1})}$ and ${\mathcal{B}}^{(k_{2})}$ due to contraction of
$b$ fields. Therefore, it is very easy to find and exclude one-particle-
reducible diagrams in our method. For this particular example, only the
diagrams surviving in the end are present in our formalism. Since the leading
order of $\tilde{\mathcal{D}}_{0}$, in terms of expansion of $e^{2}$, is $U$
itself, our one-particle-irreducibility in $\tilde{\mathcal{D}}_{0}$ lines
covers the one-particle-irreducibility of the Coulomb lines in reference
Valiev and Fernando, 1997. Although one may show that the rule Fukuda et al.
(1995) of inserting $D_{0}^{-1}$ for two-particle-reducible (in terms of
fermion propagators) diagrams still applies , it is not essential to have.
However, one may choose to use this rule as a tool to ensure correct
generation of all distinct diagrams. Equipped with the diagrammatic expansion
rules, one may construct $\Gamma_{l\,\geq 2}$ following the inversion method
described in section III.3.
### III.5 Evaluation of ${\mathcal{G}}_{0}$ using single particle orbitals
To calculate ${\mathcal{G}}_{0}(x,x^{\prime})$, we define $v({\bf
x})\equiv\upsilon_{\rm ion}({\bf x})-\mu+J_{0}({\bf x})$. Since the evaluation
of the Green’s function is general, we use the symbol $G(x,x^{\prime})$ in
place of ${\mathcal{G}}_{0}(x,x^{\prime})$ in the following derivation.
Consider first a generic free fermion Hamiltonian,
$H[{\hat{\psi}}^{{\dagger}},\hat{\psi}]=\int d{\bf
x}\;{\hat{\psi}}^{{\dagger}}({\bf
x})\left(-\frac{\nabla^{2}}{2m}+\upsilon({{\bf x}})\right)\hat{\psi}({\bf
x})={\hat{\psi}}^{{\dagger}}{\cdot}\hat{h}{\cdot}\hat{\psi}$
where
$\hat{h}({\bf x},{\bf y})=\left(-\frac{\nabla_{{\bf
x}}^{2}}{2m}+\upsilon({{\bf x}})\right)\delta({\bf x}-{\bf y})$ (88)
and the corresponding Green’s function (with $Z\equiv\text{Tr}e^{-\beta H}$)
$\displaystyle G(x,y)$ $\displaystyle=$ $\displaystyle\langle
T\hat{\psi}(x){\hat{\psi}}^{{\dagger}}(y)\rangle=Z^{-1}\text{Tr}\left[e^{-\beta
H}T(\hat{\psi}(x){\hat{\psi}}^{{\dagger}}(y))\right]$ $\displaystyle=$
$\displaystyle\theta(\tau_{x}-\tau_{y}-\eta)Z^{-1}\text{Tr}\left[e^{-\beta
H}\hat{\psi}(x){\hat{\psi}}^{{\dagger}}(y)\right]-\theta(\tau_{y}-\tau_{x}+\eta)Z^{-1}\text{Tr}\left[e^{-\beta
H}{\hat{\psi}}^{{\dagger}}(y)\hat{\psi}(x)\right]\;.$
The positive infinitesimal quantity $\eta$ is introduced to ensure that in the
limit $\tau_{x}=\tau_{y}$, the time ordered product corresponds to normal
ordering (represented by the second term at equal time). Note that
$0<\tau_{x},\tau_{y}<\beta$, $-\beta<\tau_{x}-\tau_{y}<\beta$ and
$\hat{\psi}(x)=e^{\tau_{x}H}\hat{\psi}({\bf x})e^{-\tau_{x}H}$ and
${\hat{\psi}}^{{\dagger}}(y)=e^{\tau_{y}H}{\hat{\psi}}^{{\dagger}}({\bf
y})e^{-\tau_{y}H}$. When $\tau_{x}-\tau_{y}<0$, one must have
$\tau_{x}-\tau_{y}+\beta>0$. Observe that
$\displaystyle G(\tau_{x}-\tau_{y}<0)$ $\displaystyle=$
$\displaystyle-\text{Tr}\left[e^{-\beta
H}e^{\tau_{y}H}{\hat{\psi}}^{{\dagger}}({\bf
y})e^{-\tau_{y}H}e^{\tau_{x}H}\hat{\psi}({\bf x})e^{-\tau_{x}H}\right]/Z$ (89)
$\displaystyle=$ $\displaystyle-\text{Tr}\left[e^{\tau_{x}H}\hat{\psi}({\bf
x})e^{-\tau_{x}H}e^{-\beta H}e^{\tau_{y}H}{\hat{\psi}}^{{\dagger}}({\bf
y})e^{-\tau_{y}H}\right]/Z$ $\displaystyle=$
$\displaystyle-\text{Tr}\left[e^{-\beta H}e^{(\tau_{x}+\beta)H}\hat{\psi}({\bf
x})e^{-(\tau_{x}+\beta)H}e^{\tau_{y}H}{\hat{\psi}}^{{\dagger}}({\bf
y})e^{-\tau_{y}H}\right]/Z$ $\displaystyle=$
$\displaystyle-G(\tau_{x}-\tau_{y}+\beta>0),$
where the first equality results from $\text{Tr}(AB)=\text{Tr}(BA)$, while the
final equality results from the definition of the Green’s function with
positive time argument $\tau_{x}+\beta-\tau_{y}>0$. Similarly, one may easily
show that $G(\tau_{x}-\tau_{y}>0)=-G(\tau_{x}-\tau_{y}-\beta<0)$. Therefore,
the Green’s function is antiperiodic in the imaginary time $\tau$.
Letting $M(x,y)\equiv\left(\partial_{\tau_{x}}\delta(x-y)+\hat{h}({\bf x},{\bf
y})\delta(\tau_{x}-\tau_{y})\right)=\left(\partial_{\tau_{x}}-\frac{\nabla_{{\bf
x}}^{2}}{2m}+\upsilon({{\bf x}})\right)\delta(x-y)$, we find
$\displaystyle G(x,y)$ $\displaystyle=$ $\displaystyle\langle
T\hat{\psi}(x){\hat{\psi}}^{{\dagger}}(y)\rangle=-{\delta\over\delta\bar{\xi}(x)}{\delta\over\delta\xi(y)}\langle
Te^{\int\bar{\xi}\hat{\psi}+{\hat{\psi}}^{{\dagger}}\xi}\rangle_{\bar{\xi}\to
0,\xi\to 0}$ $\displaystyle=$
$\displaystyle-{\delta\over\delta\bar{\xi}(x)}{\delta\over\delta\xi(y)}\left.\frac{\int{\cal
D}[\psi^{\dagger},\psi]e^{-\psi^{\dagger}{\scriptstyle\circ}M{\scriptstyle\circ}\psi+\bar{\xi}{\scriptstyle\circ}\psi+\psi^{\dagger}{\scriptstyle\circ}\xi}}{\int{\cal
D}[\psi^{\dagger},\psi]e^{-\psi^{\dagger}{\scriptstyle\circ}M{\scriptstyle\circ}\psi}}\right|_{\bar{\xi}\to
0,\xi\to 0}$ $\displaystyle=$
$\displaystyle-{\delta\over\delta\bar{\xi}(x)}{\delta\over\delta\xi(y)}\left[e^{\bar{\xi}{\scriptstyle\circ}M^{-1}{\scriptstyle\circ}\xi}\right]_{\bar{\xi}\to
0,\xi\to 0}=M^{-1}(x,y)\;.$
This implies that $\int
dx^{\prime}M(x,x^{\prime})G(x^{\prime},y)=\left(\partial_{\tau_{x}}+\hat{h}_{{\bf
x}}\right)G(x,y)=\delta(x-y)$ with $\hat{h}_{{\bf x}}$ being a one particle
first quantized Hamiltonian $\hat{h}_{{\bf x}}=-\frac{\nabla_{{\bf
x}}^{2}}{2m}+\upsilon({{\bf x}})$. Note that $\delta(x-y)=\delta({\bf x}-{\bf
y})\delta(\tau_{x}-\tau_{y})$ and the latter delta function in time is defined
via
$\int_{0}^{\beta}g(\tau_{x})\delta(\tau_{x}-\tau_{y})d\tau_{x}=g(\tau_{y})$
for any antiperiodic function $g(\tau)$. With this understanding, one may
express $\delta(x-y)$ in the following way to obtain $G(x,y)$. Observing that
$\displaystyle(\partial_{\tau_{x}}+\hat{h}_{{\bf x}})G(x,y)$ $\displaystyle=$
$\displaystyle\langle x|y\rangle=\sum_{\omega_{n},\alpha}\langle
x|\omega_{n},\alpha\rangle\langle\omega_{n},\alpha|y\rangle$ $\displaystyle=$
$\displaystyle\sum_{\omega_{n},\alpha}\phi_{\alpha}({\bf
x})\phi_{\alpha}^{*}({\bf
y}){e^{-i\omega_{n}(\tau_{x}-\tau_{y})}\over\beta}\;,$
one sees that the action of $(\partial_{\tau_{x}}+\hat{h}_{{\bf x}})$ may be
compensated, leading to
$G(x,y)=\sum_{\alpha}\phi_{\alpha}({\bf x})\phi_{\alpha}^{*}({\bf
y})\left[{1\over\beta}\sum_{\omega_{n}}{e^{-i\omega_{n}(\tau_{x}-\tau_{y})}\over-i\omega_{n}+\varepsilon_{\alpha}-\mu}\right]\;,$
(90)
where
$\displaystyle\langle x|\omega_{n},\alpha\rangle$ $\displaystyle=$
$\displaystyle\phi_{\alpha}({\bf
x}){e^{-i\omega_{n}\tau_{x}}\over\sqrt{\beta}}\;,$ (91)
$\displaystyle\omega_{n}$ $\displaystyle=$
$\displaystyle{\pi(2n+1)\over\beta}\;,$ (92) $\displaystyle{\rm
and\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ }\hat{h}_{{\bf x}}\,\phi_{\alpha}({\bf
x})=\left[-\frac{\nabla^{2}}{2m}+\upsilon({\bf x})\right]\phi_{\alpha}({\bf
x})$ $\displaystyle=$
$\displaystyle(\varepsilon_{\alpha}-\mu)\phi_{\alpha}({\bf x})\;.$ (93)
Eq. (93) implies that
$\left[-\frac{\nabla^{2}}{2m}+\upsilon_{\rm ion}({\bf x})+J_{0}({\bf
x})\right]\phi_{\alpha}({\bf x})=\varepsilon_{\alpha}\phi_{\alpha}({\bf
x})\;.$ (94)
Only frequencies of the type ${\pi(2n+1)\over\beta}$ is included in the
expansion to ensure the antiperiodicity of the fermionic Green’s function. To
proceed further in (90), one may sum the frequency by introducing a function
$-\beta/(e^{\beta\omega}+1)$ which has poles at
$\omega={i\pi(2n+1)\over\beta}$ with residue $1$. Evidently, poles for the
function $-\beta/(e^{\beta\omega}+1)$ occur whenever $e^{\beta\omega}+1=0$. To
investigate the strength of each pole, let’s rewrite $e^{\beta\omega}+1$ in
the limit when $\omega\to i\pi(2n+1)/\beta$ as
$\displaystyle
1+\exp\left(i\pi(2n+1)+\beta(\omega-{i\pi(2n+1)\over\beta})\right)=1-\exp\left(\beta(\omega-{i\pi(2n+1)\over\beta})\right)$
$\displaystyle=-\beta\left(\omega-{i\pi(2n+1)\over\beta}\right)+{\cal
O}\left[\left(\omega-{i\pi(2n+1)\over\beta}\right)^{2}\right]\;.$
Therefore, $-\beta/(e^{\beta\omega}+1)$ indeed has residue strength $1$ at
each of the allowable frequencies.
To evaluate
${1\over\beta}\sum_{\omega_{n}}{e^{-i\omega_{n}(\tau_{x}-\tau_{y})}\over-i\omega_{n}+\varepsilon_{\alpha}-\mu}$
when $\tau_{x}<\tau_{y}$, one integrates over a circular contour (with radius
$|\omega|\to\infty$) on the complex $\omega$-plane
${1\over\beta}\oint_{|\omega|\to\infty}{e^{-\omega(\tau_{x}-\tau_{y})}\over-\omega+\varepsilon_{\alpha}-\mu}{-\beta\over
e^{\beta\omega}+1}d\omega\;.$
Because the line integral along the infinite circle vanishes, the sum of
residues must vanish, meaning that
${1\over\beta}\sum_{\omega_{n}}{e^{-i\omega_{n}(\tau_{x}-\tau_{y})}\over-i\omega_{n}+\varepsilon_{\alpha}-\mu}+e^{-(\varepsilon_{\alpha}-\mu)(\tau_{x}-\tau_{y})}{1\over
e^{\beta(\varepsilon_{\alpha}-\mu)}+1}=0\;.$
Thus, when $\tau_{x}\leq\tau_{y}$ (because when $\tau_{x}=\tau_{y}$, it must
agree with $\tau_{x}-\tau_{y}\to 0^{-}$) one finds
$G(x,y)=\sum_{\alpha}\phi_{\alpha}({\bf x})\phi_{\alpha}^{*}({\bf
y})e^{-(\varepsilon_{\alpha}-\mu)(\tau_{x}-\tau_{y})}(-n_{\alpha})\;.$
On the other hand, when $\tau_{x}>\tau_{y}$, one considers
${1\over\beta}\oint_{|\omega|\to\infty}{e^{\omega(\tau_{x}-\tau_{y})}\over\omega+\varepsilon_{\alpha}-\mu}{-\beta\over
e^{\beta\omega}+1}d\omega\;.$
The residue sum then turns into
${1\over\beta}\sum_{\omega_{n}}{e^{-i\omega_{n}(\tau_{x}-\tau_{y})}\over-i\omega_{n}+\varepsilon_{\alpha}-\mu}-e^{-(\varepsilon_{\alpha}-\mu)(\tau_{x}-\tau_{y})}{1\over
e^{-\beta(\varepsilon_{\alpha}-\mu)}+1}=0\;.$
Thus, when $\tau_{x}>\tau_{y}$ one obtains
$G(x,y)=\sum_{\alpha}\phi_{\alpha}({\bf x})\phi_{\alpha}^{*}({\bf
y})e^{-(\varepsilon_{\alpha}-\mu)(\tau_{x}-\tau_{y})}(1-n_{\alpha})\;.$
Therefore, we have
$G(x,y)=\sum_{\alpha}\phi_{\alpha}({\bf x})\phi_{\alpha}^{*}({\bf
y})e^{-(\varepsilon_{\alpha}-\mu)(\tau_{x}-\tau_{y})}\left\\{\begin{array}[]{l
r}(-n_{\alpha})&{\rm if\leavevmode\nobreak\ }\tau_{x}\leq\tau_{y}\\\
(1-n_{\alpha})&{\rm if\leavevmode\nobreak\
}\tau_{x}>\tau_{y}\end{array}\right.\;\;,$ (95)
where $n_{\alpha}=1/(e^{\beta(\varepsilon_{\alpha}-\mu)}+1)$. Note that in the
expression involving $\varepsilon_{\alpha}$, it is always
$\varepsilon_{\alpha}$ minus the chemical potential $\mu$.
To evaluate ${\mathcal{G}}_{0}(x,y)$, we need to solve the eingensystem (93).
It requires evaluations of the RHS of Eq. (79) and self-consistency is reached
when $\kappa(x)\to 0$. Of course, one cannot evaluate all the terms and must
truncate the series on the RHS of Eq. (79) at some stage. This implies that
the density profile obtained through the self-consistency condition in this
manner depends on the number of terms one includes on the RHS of Eq. (79).
Nevertheless, the self-consistent solution obtained when keeping $k$ terms on
the RHS of Eq. (79) can be used as the starting point when one wishes to
include $k+1$ terms on the RHS of Eq. (79).
### III.6 Functional derivative of ${\mathcal{G}}_{0}(x,x^{\prime})$ and
$\int d\tau_{y}\;{\mathcal{G}}_{0}(x,y){\mathcal{G}}_{0}(y,x^{\prime})$
Especially when $J_{0}$ is time-independent, we need to evaluate
$\displaystyle\frac{\delta{\mathcal{G}}_{0}(x,x^{\prime})}{\delta J_{0}({\bf
y})}$ $\displaystyle=$
$\displaystyle-\int{\mathcal{G}}_{0}(x,z)\,\left[\frac{\delta{\mathcal{G}}_{0}^{-1}(z,z^{\prime})}{\delta
J_{0}({\bf
y})}\right]\,{\mathcal{G}}_{0}(z^{\prime},x^{\prime})\;dzdz^{\prime}$ (96)
$\displaystyle=$
$\displaystyle-\int{\mathcal{G}}_{0}(x,z)\,\delta(z-z^{\prime})\delta({\bf
y}-{\bf z})\,{\mathcal{G}}_{0}(z^{\prime},x^{\prime})\;dzdz^{\prime}$
$\displaystyle=$
$\displaystyle-\int_{0}^{\beta}{\mathcal{G}}_{0}(x,y)\,{\mathcal{G}}_{0}(y,x^{\prime})\,d\tau_{y},$
where the central expression
$\int_{0}^{\beta}{\mathcal{G}}_{0}(x,y)\,{\mathcal{G}}_{0}(y,x^{\prime})\,d\tau_{y}$
may be re-expressed by single-particle orbitals as will be shown below.
From Eq. (90), we see that
$\displaystyle G(x,y)$ $\displaystyle=$
$\displaystyle\frac{1}{\beta}\sum_{n,\alpha}\frac{e^{-i\omega_{n}(\tau_{x}-\tau_{y}-\eta)}}{-i\omega_{n}+\varepsilon_{\alpha}-\mu}\phi_{\alpha}({\bf
x})\phi^{*}_{\alpha}({\bf y})\;,$ $\displaystyle{\rm and\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
}G(y,x^{\prime})$ $\displaystyle=$
$\displaystyle\frac{1}{\beta}\sum_{n^{\prime},\rho}\frac{e^{-i\omega_{n^{\prime}}(\tau_{y}-\tau_{x^{\prime}}-\eta)}}{-i\omega_{n^{\prime}}+\varepsilon_{\rho}-\mu}\phi_{\rho}({\bf
y})\phi^{*}_{\rho}({\bf x}^{\prime})\;,$
and therefore
$\displaystyle\int_{0}^{\beta}d\tau_{y}\,G(x,y)\,G(y,x^{\prime})$
$\displaystyle=$
$\displaystyle\frac{\beta}{\beta^{2}}\sum_{n,n^{\prime},\alpha,\rho}\frac{\phi_{\alpha}({\bf
x})\phi_{\alpha}^{*}({\bf y})\phi_{\rho}({\bf y})\phi_{\rho}^{*}({\bf
x}^{\prime})}{(-i\omega_{n}+\varepsilon_{\alpha}-\mu)(-i\omega_{n^{\prime}}+\varepsilon_{\rho}-\mu)}\delta_{n,n^{\prime}}\,e^{-i\omega_{n}(\tau_{x}-\tau_{x^{\prime}}-2\eta)}$
$\displaystyle=\sum_{\alpha,\rho}\left(\frac{1}{\beta}\sum_{n}\frac{e^{-i\omega_{n}(\tau_{x}-\tau_{x^{\prime}}-2\eta)}}{(-i\omega_{n}+\varepsilon_{\alpha}-\mu)(-i\omega_{n}+\varepsilon_{\rho}-\mu)}\right)\phi_{\alpha}({\bf
x})\phi_{\alpha}^{*}({\bf y})\phi_{\rho}({\bf y})\phi_{\rho}^{*}({\bf
x}^{\prime})\;.$
Let us now concentrate on the portion inside the parentheses. Assuming that
there is no energy degeneracy, we rewrite the denominator of this fctor when
$\alpha\neq\rho$
$\frac{1}{(-i\omega_{n}+\varepsilon_{\alpha}-\mu)(-i\omega_{n}+\varepsilon_{\rho}-\mu)}=\frac{1}{\varepsilon_{\rho}-\varepsilon_{\alpha}}\left(\frac{1}{-i\omega_{n}+\varepsilon_{\alpha}-\mu}-\frac{1}{-i\omega_{n}+\varepsilon_{\rho}-\mu}\right)$
and when $\alpha=\rho$
$\frac{1}{(-i\omega_{n}+\varepsilon_{\alpha}-\mu)(-i\omega_{n}+\varepsilon_{\rho}-\mu)}\to\frac{1}{(-i\omega_{n}+\varepsilon_{\alpha}-\mu)^{2}}=-\frac{\partial}{\partial\varepsilon_{\alpha}}\left(\frac{1}{-i\omega_{n}+\varepsilon_{\alpha}-\mu}\right)\;.$
Therefore, we need to evaluate
$\sum_{n}\frac{e^{i\omega_{n}(\tau_{x^{\prime}}-\tau_{x})}}{-i\omega_{n}+\varepsilon_{\alpha}-\mu}$.
We distinguish the two cases: $\tau_{x}\leq\tau_{x^{\prime}}$ and
$\tau_{x}>\tau_{x^{\prime}}$. When $\tau_{x}\leq\tau_{x^{\prime}}$, we
consider the following integral over the infinite circle
$\oint\frac{e^{\omega(\tau_{x^{\prime}}-\tau_{x})}}{-\omega+\varepsilon_{\alpha}-\mu}\frac{-\beta}{e^{\beta\omega}+1}\frac{d\omega}{2\pi
i}=\sum_{n}\frac{e^{i\omega_{n}(\tau_{x^{\prime}}-\tau_{x})}}{-i\omega_{n}+\varepsilon_{\alpha}-\mu}+\frac{\beta
e^{(\varepsilon_{\alpha}-\mu)(\tau_{x^{\prime}}-\tau_{x})}}{e^{\beta(\varepsilon_{\alpha}-\mu)}+1}\;.$
Since the integral over the infinite circle vanishes, we have
$\frac{1}{\beta}\sum_{n}\frac{e^{i\omega_{n}(\tau_{x^{\prime}}-\tau_{x}})}{-i\omega_{n}+\varepsilon_{\alpha}-\mu}=-\frac{e^{-(\varepsilon_{\alpha}-\mu)(\tau_{x}-\tau_{x^{\prime}})}}{e^{\beta(\varepsilon_{\alpha}-\mu)}+1}=-e^{-(\varepsilon_{\alpha}-\mu)(\tau_{x}-\tau_{x^{\prime}})}n_{\alpha}\;.$
(97)
To evaluate the expression
$\frac{1}{\beta}\sum_{n}\frac{e^{i\omega_{n}(\tau_{x^{\prime}}-\tau_{x})}}{(-i\omega_{n}+\varepsilon_{\alpha}-\mu)^{2}}$,
we consider
$-\frac{\partial}{\partial\varepsilon_{\alpha}}\left(\frac{1}{\beta}\sum_{n}\frac{e^{i\omega_{n}(\tau_{x^{\prime}}-\tau_{x}})}{-i\omega_{n}+\varepsilon_{\alpha}-\mu}\right)=e^{-(\varepsilon_{\alpha}-\mu)(\tau_{x}-\tau_{x^{\prime}})}\left[-\beta
n_{\alpha}(1-n_{\alpha})-(\tau_{x}-\tau_{x^{\prime}})n_{\alpha}\right]\;.$
Therefore, when $\tau_{x}\leq\tau_{x^{\prime}}$,
$\displaystyle\int_{0}^{\beta}\\!\\!d\tau_{y}\,G(x,y)\,G(y,x^{\prime})$
$\displaystyle=$ $\displaystyle\sum_{\alpha}\phi_{\alpha}({\bf
x})n_{\alpha}({\bf y})\phi_{\alpha}^{*}({\bf
x}^{\prime})e^{-(\varepsilon_{\alpha}-\mu)(\tau_{x}-\tau_{x^{\prime}})}\left[-\beta
n_{\alpha}(1-n_{\alpha})-(\tau_{x}-\tau_{x^{\prime}})n_{\alpha}\right]$ (98)
$\displaystyle+\sum_{\alpha\neq\rho}\phi_{\alpha}({\bf
x})\phi_{\alpha}^{*}({\bf y})\phi_{\rho}({\bf y})\phi_{\rho}^{*}({\bf
x}^{\prime})\left[\frac{e^{-\varepsilon_{\rho}(\tau_{x}-\tau_{x^{\prime}})}n_{\rho}-e^{-\varepsilon_{\alpha}(\tau_{x}-\tau_{x^{\prime}})}n_{\alpha}}{\varepsilon_{\rho}-\varepsilon_{\alpha}}\right]\;,$
where $n_{\alpha}({\bf y})=\phi_{\alpha}^{*}({\bf y})\phi_{\alpha}({\bf y})$.
On the other hand, when $\tau_{x}>\tau_{x^{\prime}}$, we consider the integral
$\oint\frac{e^{\omega(\tau_{x}-\tau_{x^{\prime}})}}{\omega+\varepsilon_{\alpha}-\mu}\frac{-\beta}{e^{\beta\omega}+1}\frac{d\omega}{2\pi
i}=\sum_{n}\frac{e^{-i\omega_{n}(\tau_{x}-\tau_{x^{\prime}})}}{-i\omega_{n}+\varepsilon_{\alpha}-\mu}-\frac{\beta
e^{-(\varepsilon_{\alpha}-\mu)(\tau_{x}-\tau_{x^{\prime}})}}{e^{-\beta(\varepsilon_{\alpha}-\mu)}+1}\;,$
and obtain (since the integral over the infinite circle vanishes)
$\frac{1}{\beta}\sum_{n}\frac{e^{-i\omega_{n}(\tau_{x}-\tau_{x^{\prime}})}}{-i\omega_{n}+\varepsilon_{\alpha}-\mu}=\frac{e^{-(\varepsilon_{\alpha}-\mu)(\tau_{x}-\tau_{x^{\prime}})}}{e^{-\beta(\varepsilon_{\alpha}-\mu)}+1}=e^{-(\varepsilon_{\alpha}-\mu)(\tau_{x}-\tau_{x^{\prime}})}(1-n_{\alpha})\;.$
(99)
Similarly,
$-\frac{\partial}{\partial\varepsilon_{\alpha}}\left(\frac{1}{\beta}\sum_{n}\frac{e^{-i\omega_{n}(\tau_{x}-\tau_{x^{\prime}}})}{-i\omega_{n}+\varepsilon_{\alpha}-\mu}\right)=e^{-(\varepsilon_{\alpha}-\mu)(\tau_{x}-\tau_{x^{\prime}})}\left[-\beta
n_{\alpha}(1-n_{\alpha})+(\tau_{x}-\tau_{x^{\prime}})(1-n_{\alpha})\right]\;.$
Therefore, when $\tau_{x}>\tau_{x^{\prime}}$,
$\displaystyle\int_{0}^{\beta}\\!\\!d\tau_{y}\,G(x,y)\,G(y,x^{\prime})$
$\displaystyle=$ $\displaystyle\sum_{\alpha}\phi_{\alpha}({\bf
x})n_{\alpha}({\bf y})\phi_{\alpha}^{*}({\bf
x}^{\prime})e^{-(\varepsilon_{\alpha}-\mu)(\tau_{x}-\tau_{x^{\prime}})}\left[-\beta
n_{\alpha}(1-n_{\alpha})+(\tau_{x}-\tau_{x^{\prime}})(1-n_{\alpha})\right]$
(100) $\displaystyle+\sum_{\alpha\neq\rho}\phi_{\alpha}({\bf
x})\phi_{\alpha}^{*}({\bf y})\phi_{\rho}({\bf y})\phi_{\rho}^{*}({\bf
x}^{\prime})\left[\frac{e^{-(\varepsilon_{\rho}-\mu)(\tau_{x}-\tau_{x^{\prime}})}n_{\rho}-e^{-(\varepsilon_{\alpha}-\mu)(\tau_{x}-\tau_{x^{\prime}})}n_{\alpha}}{\varepsilon_{\rho}-\varepsilon_{\alpha}}\right]\;.$
We may now write down the full expression for
$\delta{\mathcal{G}}_{0}(x,x^{\prime})/\delta J_{0}({\bf y})$ as
$\displaystyle\frac{\delta{\mathcal{G}}_{0}(x,x^{\prime})}{\delta J_{0}({\bf
y})}$ $\displaystyle=$
$\displaystyle-\int_{0}^{\beta}{\mathcal{G}}_{0}(x,y)\,{\mathcal{G}}_{0}(y,x^{\prime})\,d\tau_{y}$
(101) $\displaystyle=$ $\displaystyle\sum_{\alpha}\phi_{\alpha}({\bf
x})\,n_{\alpha}({\bf y})\,\phi_{\alpha}^{*}({\bf
x}^{\prime})\,e^{-(\varepsilon_{\alpha}-\mu)(\tau_{x}-\tau_{x^{\prime}})}\left[\beta
n_{\alpha}(1-n_{\alpha})+(\tau_{x}-\tau_{x^{\prime}})n_{\alpha}\right]$
$\displaystyle-(\tau_{x}-\tau_{x^{\prime}})\,\theta(\tau_{x}-\tau_{x^{\prime}}-\eta)\sum_{\alpha}\phi_{\alpha}({\bf
x})\,n_{\alpha}({\bf y})\,\phi_{\alpha}^{*}({\bf
x}^{\prime})\,e^{-(\varepsilon_{\alpha}-\mu)(\tau_{x}-\tau_{x^{\prime}})}$
$\displaystyle-\sum_{\alpha\neq\rho}\phi_{\alpha}({\bf
x})\phi_{\alpha}^{*}({\bf y})\phi_{\rho}({\bf y})\phi_{\rho}^{*}({\bf
x}^{\prime})\left[\frac{e^{-(\varepsilon_{\rho}-\mu)(\tau_{x}-\tau_{x^{\prime}})}n_{\rho}-e^{-(\varepsilon_{\alpha}-\mu)(\tau_{x}-\tau_{x^{\prime}})}n_{\alpha}}{\varepsilon_{\rho}-\varepsilon_{\alpha}}\right]\;.$
In the absence of time-dependence, we have
$n({\bf x})=\frac{1}{\beta}\frac{\delta(\beta W[J_{0}])}{\delta J_{0}({\bf
x})}=-\frac{1}{\beta}\\!\int\\!\\!dzdy\,{\mathcal{G}}_{0}(z,y)\delta({\bf
y}-{\bf x})\delta(y-z)=-{\mathcal{G}}_{0}(x,x)\;.$
Therefore, using Eq. (101) we have
$\displaystyle D_{0}({\bf x},{\bf y})=\frac{\delta n({\bf x})}{\delta
J_{0}({\bf y})}$ $\displaystyle=$
$\displaystyle\sum_{\alpha\neq\rho}\phi_{\alpha}({\bf
x})\phi_{\alpha}^{*}({\bf y})\phi_{\rho}({\bf y})\phi_{\rho}^{*}({\bf
x})\left[\frac{n_{\rho}-n_{\alpha}}{\varepsilon_{\rho}-\varepsilon_{\alpha}}\right]$
$\displaystyle+\sum_{\alpha}n_{\alpha}({\bf x})\,n_{\alpha}({\bf
y})\,\left[\beta n_{\alpha}(1-n_{\alpha})\right]\;.$
Note that the expression $\beta n_{\alpha}(1-n_{\alpha})$ vanishes as
$\beta\to\infty$, because $n_{\alpha}(1-n_{\alpha})$ decays exponentially with
$\beta$ when $\mu\neq\varepsilon_{\alpha}$. That is, at the zero temperature
limit, one has
$D_{0}({\bf x},{\bf y})\rightarrow\sum_{\alpha\neq\rho}\phi_{\alpha}({\bf
x})\phi_{\alpha}^{*}({\bf y})\phi_{\rho}({\bf y})\phi_{\rho}^{*}({\bf
x})\left[\frac{n_{\rho}-n_{\alpha}}{\varepsilon_{\rho}-\varepsilon_{\alpha}}\right]$
as long as $\mu$ is not equal to any eigenenergy of the orbital.
### III.7 The Computational Procedure
The recipe for computation goes as follows. Starting with a reasonably guessed
$J_{0}({\bf x})$, one first obtains single particle wave functions
$\phi_{\alpha}$ and energies $\varepsilon_{\alpha}$ through (94). Note that
the occupation number of state $\alpha$ is given by
$n_{\alpha}=\frac{1}{e^{\beta(\varepsilon_{\alpha}-\mu)}+1}$
and the chemical potential is chosen such that
$g_{s}\sum_{\alpha}n_{\alpha}=N_{e}\;,$
where $g_{s}$ denotes the spin degeneracy ($g_{s}=2$ for spin $1/2$
electrons).
Through eqs. (95), (82), and (81), one constructs respectively the
${\mathcal{G}}_{0}$, $D_{0}^{-1}$, and $\tilde{\mathcal{D}}_{0}$ propagators.
Combined with their differentiation rules (83-85) with respect to $J_{0}$,
these propagators are used to compute $\delta\Gamma/\delta J_{0}$.
As the third step, one obtains a new estimate of $J_{0}$ given by
$J_{0}-\varsigma\,\delta\Gamma/\delta J_{0}$ with $\varsigma>0$ being the step
size. See eq. (79) and the text nearby for details.
Finally, one starts again with the new $J_{0}$ and goes through the other
steps, iteratively until the convergence condition $\delta\Gamma/\delta
J_{0}=0$ is reached. This simultaneously determines $J_{0}$ (sum of the
Hartree potential and the KS potential), the ground state charge density and
the ground state energy, as well as the KS orbitals and energies.
## IV Case Studies
### IV.1 The Universal Functional ${\mathcal{F}}[n]$ at Arbitrary Temperature
There are vast discussions Fukuda et al. (1994); Valiev and Fernando (1997);
Polonyi and Sailer (2002) on the equivalence between $E_{\upsilon}[n]$ in
(1-2) and $\lim_{\beta\to\infty}\frac{1}{\beta}\Gamma[n]$. However, not much
attention was given to the emergence of the universal functional
${\mathcal{F}}[n]$ (at any given temperature) resulting from the effective
action formalism. We address here for the first time how ${\mathcal{F}}[n]$
arises naturally from our effective action formulation.
Fukuda et al. Fukuda et al. (1994) showed that it is possible to eliminate the
$\upsilon$ dependence in the functional to obtain (when translated into our
terms)
$\Gamma_{\upsilon}[i\varphi]=\Gamma_{\upsilon=0}[i\varphi]+\upsilon{\scriptstyle\circ}(i\varphi)+\frac{1}{2}\upsilon{\scriptstyle\circ}\,U^{-1}{\scriptstyle\circ}\upsilon\;.$
They interpret $i\varphi$ as some sort of electron density since it coincides
with the real electron density when $J=0$. The appearance of the quadratic
term in $\upsilon$, however, disagrees with (2) where it is clearly stated
that in addition to a term that is linear in $\upsilon$, the remainder should
be $\upsilon$-independent. We settle this discrepancy below by showing that
the appearance of the quadratic $\upsilon$ dependence is due to the fact that
reference Fukuda et al., 1994 did not use the system’s electron density as the
natural variable. Once the system’s electron density is used as the natural
variable, the universal functional ${\mathcal{F}}[n]$ emerges naturally.
Note that (10) indicates that one may view $\upsilon_{\rm ion}({\bf x})$ as a
nonvanishing source term. In the change of variable
$\phi\to\phi+iU^{-1}{\scriptstyle\circ}J$ right after (17), if we make
$\phi\to\phi+iU^{-1}{\scriptstyle\circ}(J+\upsilon)$ instead, we see that
$\upsilon$ is then bonded with $J$ from that point on. That said, we may view
$W[J]\equiv W^{\prime}[J+\upsilon]$. That is, the generating functional in the
presence of the one-body potential $\upsilon$ with source $J$ is equivalent to
the generating functional of a system without a one-body potential but with
source $J+\upsilon$. Following the algebra in Eqs. (21-46), one sees that
$\beta W[J]=\beta
W^{\prime}_{\phi}[J+\upsilon]-\frac{1}{2}(J+\upsilon){\scriptstyle\circ}\,U^{-1}(J+\upsilon)\;,$
(102)
where
$\displaystyle\beta W^{\prime}_{\phi}[J+\upsilon]$ $\displaystyle=$
$\displaystyle\frac{1}{2}\varphi{\scriptstyle\circ}\,U{\scriptstyle\circ}\varphi-\text{Tr}\ln\left(\bar{G}_{\varphi}^{-1}\right)+i(J+\upsilon){\scriptstyle\circ}\varphi$
$\displaystyle+\frac{1}{2}\text{Tr}\ln\left(\tilde{\mathcal{D}}^{-1}{\scriptstyle\circ}\,U\right)-\sum_{n=1}^{\infty}\frac{1}{n!}\langle\left[\sum_{k=3}^{\infty}I^{(k)}[\varphi]{\scriptstyle\circ}\,b_{1}\ldots{\scriptstyle\circ}\,b_{k}\right]^{n}\rangle_{\rm
1PI,\leavevmode\nobreak\ conn.}\;$
$\displaystyle\bar{G}_{\varphi}^{-1}(x,x^{\prime})$ $\displaystyle=$
$\displaystyle\left(\partial_{\tau}-\frac{\nabla^{2}}{2m}-\mu+i(U{\scriptstyle\circ}\varphi)_{x}\right)\delta(x-x^{\prime})\;.$
Upon using the following variant of (25)
$n=i\varphi-U^{-1}{\scriptstyle\circ}(J+\upsilon)\;,$
the quadratic term in $J+\upsilon$ in (102) gets absorbed into the density of
the electron and one arrives at the following variant of (44)
$\displaystyle\beta W[J]=\beta W^{\prime}[J+\upsilon]$ $\displaystyle=$
$\displaystyle-\text{Tr}\ln\left(\bar{G}_{\varphi}^{-1}\right)-\frac{1}{2}n{\scriptstyle\circ}\,U{\scriptstyle\circ}n+\frac{1}{2}\text{Tr}\ln\left(\tilde{\mathcal{D}}^{-1}{\scriptstyle\circ}\,U\right)$
$\displaystyle\hskip
15.0pt-\sum_{n=1}^{\infty}\frac{1}{n!}\langle\left[\sum_{k=3}^{\infty}I^{(k)}[\varphi]{\scriptstyle\circ}\,b_{1}\ldots{\scriptstyle\circ}\,b_{k}\right]^{n}\rangle_{\rm
1PI,\leavevmode\nobreak\ conn.}\;.$
Note that here $n=n_{J}=n^{\prime}_{J+\upsilon}$ represents the electron
density of the system.
The effective action $\Gamma[n]=\beta W[J]-J{\scriptstyle\circ}n$ can thus be
rewritten as
$\Gamma[n]=\beta
W^{\prime}[J+\upsilon]-J{\scriptstyle\circ}n=\upsilon{\scriptstyle\circ}n+\beta
W^{\prime}[J+\upsilon]-(J+\upsilon){\scriptstyle\circ}n\;,$
where the first term on the RHS represents $\beta\int n({\bf x})\upsilon({\bf
x})d{\bf x}$, and the last two terms represent the effective action,
$\Gamma^{\prime}[n]=\beta
W^{\prime}[J+\upsilon]-(J+\upsilon){\scriptstyle\circ}n$, of a system in the
absence of one-body potential. We thus identify ${\mathcal{F}}[n]$ as
${\mathcal{F}}[n]=\frac{1}{\beta}\left\\{\beta
W^{\prime}[J+\upsilon]-(J+\upsilon){\scriptstyle\circ}n\right\\}\;.$
This also serves as an alternative exposition of what Mermin proved.Mermin
(1965)
### IV.2 Effective Potential near Zero Temperature
The effective potential divided by $\beta$ is the ground state energy plus
$\mu N_{e}$ in the $T\to 0$ limit. Below, we will show how the Hartree-Fock
terms appear in this formulation.
Equations (47-73) provide a systematic expansion for calculating the effective
potential. In the expression (60), the term
$-\text{Tr}\ln({\mathcal{G}}_{0}^{-1})$ is equivalent to
$-\ln[\det({\mathcal{G}}_{0}^{-1})]$. There are many ways to obtain
$\det({\mathcal{G}}_{0}^{-1})$: one may obtain the results directly through
the definition of $e^{-\beta W_{0}[J_{0}]}$ or one may express $e^{-\beta
W_{0}[J_{0}]}$ as a path integral to obtain the determinant in discrete time.
We shall denote $-\text{Tr}\ln({\mathcal{G}}_{0}^{-1})$ by $\beta
W_{0}[J_{0}]$ with
$\displaystyle e^{-\beta W_{0}[J_{0}]}$ $\displaystyle=$
$\displaystyle\text{Tr}\left\\{e^{-\beta\left[{\hat{\psi}}^{{\dagger}}{\cdot}(\hat{h}+J_{0}){\cdot}\hat{\psi}\right]}\right\\}=\int{\cal
D}[\psi^{\dagger},\psi]e^{-\int
dx\psi^{\dagger}(x)\left[\partial_{\tau}+\hat{h}+J_{0}({\bf
x})\right]\psi(x)}$ $\displaystyle=$ $\displaystyle\int{\cal
D}[\psi^{\dagger},\psi]e^{-\psi^{\dagger}{\scriptstyle\circ}{\mathcal{G}}_{0}^{-1}{\scriptstyle\circ}\psi}=\det({\mathcal{G}}_{0}^{-1})\;.$
Direct evaluation using the trace definition leads to
$\det({\mathcal{G}}_{0}^{-1})=\prod_{\alpha}\left(1+e^{-\beta(\varepsilon_{\alpha}-\mu)}\right)\;,$
and consequently
$W_{0}[J_{0}]=-{1\over\beta}\text{Tr}\ln\left({\mathcal{G}}_{0}^{-1}\right)=-{1\over\beta}\sum_{\alpha}\ln\left(1+e^{-\beta(\varepsilon_{\alpha}-\mu)}\right)={1\over\beta}\sum_{\alpha}\ln\left(1-n_{\alpha}\right)\;.$
Note that the energy is measured with respect to the chemical potential $\mu$.
Therefore, in the limit of $\beta\to\infty$, we have
$\lim_{\beta\to\infty}W_{0}[J_{0}]=-\lim_{\beta\to\infty}{1\over\beta}\text{Tr}\ln\left({\mathcal{G}}_{0}^{-1}\right)=\sum_{\alpha=1}^{N_{e}}(\varepsilon_{\alpha}-\mu)\;,$
(103)
with $\varepsilon_{\alpha}\leq\mu$ for $1\leq\alpha\leq N_{e}$.
Using Eqs. (80) and (103), we obtain the low temperature limit of the
effective action
$\displaystyle\lim_{\beta\to\infty}\left({1\over\beta}\Gamma[n]\right)$
$\displaystyle=$
$\displaystyle\sum_{\alpha=1}^{N_{e}}(\varepsilon_{\alpha}-\mu)-J_{0}{\cdot}n+\frac{1}{2}n{\cdot}\,U{\cdot}n$
(104) $\displaystyle\hskip
12.0pt+\lim_{\beta\to\infty}\left[\frac{1}{2\beta}\text{Tr}\ln\left(\tilde{\cal
D}_{0}^{-1}{\scriptstyle\circ}\,U\right)+\frac{1}{\beta}\sum_{i=2}^{\infty}\Gamma_{i}[n]\right]\;.$
The first term in (104) can be expressed in a different way if we multiply
both sides of Eq. (93) by $\phi_{\alpha}^{*}({\bf x})$, sum $\alpha$ over the
lowest $N_{e}$ states, and integrate over ${\bf x}$. Upon doing this, we
obtain
$\displaystyle\sum_{\alpha=1}^{N_{e}}(\varepsilon_{\alpha}-\mu)$
$\displaystyle=$ $\displaystyle\sum_{\alpha=1}^{N_{e}}\int d{\bf
x}\phi_{\alpha}^{*}({\bf x})\left[-\frac{\nabla^{2}}{2m}+\upsilon_{\rm
ion}({\bf x})-\mu+J_{0}({\bf x})\right]\phi_{\alpha}({\bf x})$
$\displaystyle\equiv$ $\displaystyle T_{0}[n]-\mu N_{e}+\int d{\bf
x}\left[\upsilon_{\rm ion}({\bf x})+J_{0}({\bf x})\right]\,n({\bf x})\;,$
where
$n({\bf x})=\sum_{\alpha=1}^{N_{e}}\phi_{\alpha}^{*}({\bf
x})\phi_{\alpha}({\bf x})\;,$
and
$T_{0}[n]\equiv\sum_{\alpha=1}^{N_{e}}\int d{\bf x}\phi_{\alpha}^{*}({\bf
x})\left[-\frac{\nabla^{2}}{2m}\right]\phi_{\alpha}({\bf x})\;.$
Therefore, the first two terms in (104) give us
$\sum_{\alpha=1}^{N_{e}}(\varepsilon_{\alpha}-\mu)-J_{0}{\cdot}n=T_{0}[n]-\mu
N_{e}+\int d{\bf x}\upsilon_{\rm ion}({\bf x})\,n({\bf x})\;.$
Evidently, the third term in (104) is nothing but the Hartree energy
$\frac{1}{2}n{\cdot}\,U{\cdot}n=\frac{1}{2}\iint d{\bf x}d{\bf y}\,n({\bf
x})\frac{e^{2}}{|{\bf x}-{\bf y}|}n({\bf y})\;.$
Therefore, we have
$\displaystyle\lim_{\beta\to\infty}\left({1\over\beta}\Gamma[n]\right)$
$\displaystyle=$ $\displaystyle T_{0}[n]+\int\upsilon_{\rm ion}({\bf
x})\,n({\bf x})\,d{\bf x}-\mu N_{e}+\frac{1}{2}\iint d{\bf x}d{\bf y}\,n({\bf
x})\frac{e^{2}}{|{\bf x}-{\bf y}|}n({\bf y})$ (105)
$\displaystyle+\lim_{\beta\to\infty}\frac{1}{2\beta}\text{Tr}\ln\left({\mathbf{I}}-D_{0}{\scriptstyle\circ}\,U\right)+\lim_{\beta\to\infty}\left[\frac{1}{\beta}\sum_{i=2}^{\infty}\Gamma_{i}[n]\right]\;,$
with the last two terms combined to form the exchange-correlation energy
functional when compared with the Kohn-Sham decomposition (4).
If $D_{0}{\scriptstyle\circ}\,U$ (or $e^{2}$) may be treated as a small
quantity, one may expand ${1\over
2\beta}\text{Tr}\ln({\mathbf{I}}-D_{0}{\scriptstyle\circ}\,U)$ as
${1\over
2\beta}\text{Tr}\ln\left({\mathbf{I}}-D_{0}{\scriptstyle\circ}\,U\right)=-{1\over
2\beta}\sum_{n=1}^{\infty}\frac{\text{Tr}\;\left[(D_{0}{\scriptstyle\circ}\,U)^{n}\right]}{n}\;,$
(106)
with the leading term (in the static limit)
${-1\over 2\beta}\int dxdyD_{0}(x,y)u(y,x)={-e^{2}\over 2}\int d{\bf x}d{\bf
y}{n({\bf x},{\bf y})n({\bf y},{\bf x})\over|{\bf x}-{\bf y}|}\;,$
where $n({\bf x},{\bf y})=\sum_{m=1}^{N_{e}}\phi_{\alpha}({\bf
x})\phi_{\alpha}^{*}({\bf y})=-{\mathcal{G}}_{0}({\bf x},\tau;{\bf y},\tau)$.
Consequently, when $D_{0}{\scriptstyle\circ}\,U$ (or $e^{2}$) can be treated
as a small quantity, we may write the leading terms of the effective potential
as
$\displaystyle\lim_{\beta\to\infty}{1\over\beta}\Gamma[n]$ $\displaystyle=$
$\displaystyle T_{0}[n]+\int\upsilon_{\rm ion}({\bf x})n({\bf x})d{\bf x}-\mu
N_{e}+{e^{2}\over 2}\int d{\bf x}d{\bf y}{n({\bf x})n({\bf y})-n({\bf x},{\bf
y})n({\bf y},{\bf x})\over|{\bf x}-{\bf y}|}$ (107)
$\displaystyle+\lim_{\beta\to\infty}\left[{-1\over
2\beta}\sum_{n=2}^{\infty}\frac{1}{n}\text{Tr}\;\left[(D_{0}{\scriptstyle\circ}\,U)^{n}\right]+\frac{1}{\beta}\sum_{i=2}^{\infty}\Gamma_{i}[n]\right]\;,$
where the fourth term is nothing but the Hartree-Fock term. The higher-order
terms inside the square brackets encode the remaining contributions of
exchange and correlation. In the case when the Coulomb interaction is strong,
one may choose not to use the expansion in Eq. (106), but to use (105) for the
zero temperature limit and (80) for finite temperature.
### IV.3 Single Electron at Zero Temperature
When the system contains only one electron and when the energy difference
between the first excited state and the ground state is much larger than
$k_{B}T$, there will be no particle-hole pairs. Consequently, there should be
no exchange-correlation energy as well as the Hartree energy. When one employs
empirical density functionals, this feature is unlikely to be preserved – an
issue known as the self-interaction problem. It is customary to define the
exchange-correlation energy $E_{xc}$ as the sum of the Fock exchange energy
$E_{x}$ and the correlation energy $E_{c}$. Since it is easy to show that the
exchange energy $E_{x}$ exactly cancels the Hartree energy in the case of one
electron, one easily concludes that $E_{c}=0$ for the one electron case. This
argument has been used, for example, by Perdew and Zunger.Perdew and Zunger
(1981) However, from the diagrammatic expansion point of view, the Hartree
term and the Fock exchange term correspond only to the first order (in terms
of $e^{2}$) diagrams. The cancellation of the first order terms does not imply
that the higher order diagrams, making up $E_{c}$, will give no contribution.
That is, although the fact that $E_{c}=0$ for one electron system can be
argued, a formal diagrammatic exposition incorporating all higher orders is
needed to justify the asymptotic exactness of the proposed functional.
The purpose of this section is to illustrate how $E_{c}=0$ can be derived
formally within our formalism. It should be noted, however, that the self-
interaction will remain if one truncate the series in eq. (80). While the
exchange-only functional will have no self-interaction problem, as we will
show below it is not because the exchange-only functional is an approach with
more correct physics, but because the simplification it makes is equivalent to
completely ignoring the correlation energy.
When $N_{e}=1$ and when $T\to 0$, we find from Eq. (95) that the Green’s
function ${\mathcal{G}}_{0}(x,y)$ takes the following form
${\mathcal{G}}_{0}(x,y)=\left\\{\begin{array}[]{l r}\phi_{1}({\bf
x})\phi_{1}^{*}({\bf y})e^{-(\varepsilon_{1}-\mu)(\tau_{x}-\tau_{y})}(-1)&{\rm
if\leavevmode\nobreak\ }\tau_{x}\leq\tau_{y}\\\ \sum\limits_{\alpha\geq
2}\phi_{\alpha}({\bf x})\phi_{\alpha}^{*}({\bf
y})e^{-(\varepsilon_{\alpha}-\mu)(\tau_{x}-\tau_{y})}&{\rm
if\leavevmode\nobreak\ }\tau_{x}>\tau_{y}\end{array}\right.\;,$ (108)
where $\varepsilon_{1}<\mu<\varepsilon_{\alpha\geq 2}$ and
$\phi_{1(\rho)}({\bf x})$ is the ground ($\rho^{\rm th}$) state wave function
of the single particle Hamiltonian
$\hat{h}({\bf x})=-\frac{\nabla^{2}}{2m}+\upsilon_{\rm ion}({\bf
x})+J_{0}({\bf x})-\mu\;$
with eigenvalue $\varepsilon_{1(\rho)}-\mu$.
The disappearance of the Hartree energy and the exchange-correlation energy is
best seen by grouping terms with the same number of Coulomb lines $U$. We will
explicitly show the first few calculations followed by a sketch of the general
proof.
Let us first show that the first term ($n=1$) on the RHS of (106) cancels the
Hartree term exactly. When there is only one electron, the Hartree term
becomes
$\frac{e^{2}}{2}\int d{\bf x}d{\bf y}\frac{n({\bf x})n({\bf y})}{|{\bf x}-{\bf
y}|}=\frac{e^{2}}{2}\int d{\bf x}d{\bf y}\frac{\phi_{1}({\bf
x})\phi_{1}^{*}({\bf x})\phi_{1}({\bf y})\phi_{1}^{*}({\bf y})}{|{\bf x}-{\bf
y}|}\;.$
The Fock term ($n=1$ term of order $U$ in (106)), when there is only one
electron, reads,
${-1\over 2\beta}\text{Tr}\left[D_{0}{\scriptstyle\circ}\,U\right]={-1\over
2\beta}\int dxdyD_{0}(x,y)U(y,x)={-e^{2}\over 2}\int d{\bf x}d{\bf
y}{\phi_{1}({\bf x})\phi_{1}^{*}({\bf x})\phi_{1}({\bf y})\phi_{1}^{*}({\bf
y})\over|{\bf x}-{\bf y}|}\;.$
The cancellation between the Fock term and the Hartree term is thus apparent.
If one were to approximate the exchange-correlation functional by the Fock
exchange functional, correlation energy can never be accounted for. That is,
this approximation coincidently leads to the expected result $E_{c}=0$ at the
single electron limit simply because it never takes $E_{c}$ into account.
In terms of diagrammatic expression, the Hartree term is given by diagram (a)
of Fig. 2, while the lowest order exchange term ($n=1$ term of (106) )
corresponds to diagram (b) of Fig. 2. That is, the order $U$ diagrams in
$\frac{1}{\beta}\Gamma[n]$ are identical to the order $U$ diagrams in
$W[J_{0}]$ at zero temperature. The perfect cancellation between the Hartree
term and the lowest order exchange term implies that the sum of these two
terms remains zero when one makes a derivative with respect to either $J_{0}$
or $n$. Diagrammatically speaking, this means that when the system contains
only one electron, we have
$\left\\{\begin{array}[]{l}0=\frac{\delta 0}{\delta J_{0}}\\\ 0=\frac{\delta
0}{\delta n}\end{array}\right.=\left\\{\begin{array}[]{l}\frac{\delta}{\delta
J_{0}}\\\ \frac{\delta}{\delta
n}\end{array}\right.\left[\begin{picture}(136.0,20.0)(0.0,17.0) \raise
20.0pt\hbox to0.0pt{\kern 0.0pt\makebox(0.0,0.0)[lc]{\large$\frac{1}{2}$}\hss}
\ignorespaces \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \raise 20.0pt\hbox to0.0pt{\kern
84.0pt\makebox(0.0,0.0)[lc]{\large$-\frac{1}{2}$}\hss} \ignorespaces
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \end{picture}\right]\;.$ (109)
Terms of the next order in $U$ are contained in $\Gamma_{i\leq 2}$. See Fig. 4
for a diagrammatic expression of $\Gamma_{i\leq 2}$ of order $U^{2}$ (or of
order $e^{4}$). We show below that in Fig. 4 the first two graphs cancel each
other and the last two diagrams also cancel each other. For illustration, we
will work out the explicit cancellation of the first two graphs by labelling
the space-time points. The cancellation of the the last two graphs require
additional operations which we will turn to later. The first two graphs in
Fig. 4 with the space-time points labelled appear to be
$\displaystyle\hskip
2.0pt=\int\\!\\!d\tau_{x}\\!d\tau_{y}\\!\prod_{i=1}^{2}d{\bf x}_{i}d{\bf
y}_{i}\frac{{\mathcal{G}}_{0}(x_{1},y_{1})({\mathcal{G}}_{0}(x_{2},y_{2})}{4|{\bf
x}_{1}-{\bf x}_{2}||{\bf y}_{1}-{\bf y}_{2}|}\times$ $\displaystyle\hskip
10.0pt\times\large[{\mathcal{G}}_{0}(y_{1},x_{2}){\mathcal{G}}_{0}(y_{2},x_{1})-{\mathcal{G}}_{0}(y_{1},x_{1}){\mathcal{G}}_{0}(y_{2},x_{2})\large]\;.$
(110)
We have used $\tau_{x}$ to denote the time associated with both ${\bf x}_{1}$
and ${\bf x}_{2}$, while denoting by $\tau_{y}$ the time associated with both
${\bf y}_{1}$ and ${\bf y}_{2}$. When $\tau_{y}\leq\tau_{x}$, the quantity
inside the square brackets of Eq. (110) vanishes because
${\mathcal{G}}_{0}(y_{i},x_{j})\propto\phi_{1}({\bf y}_{i})\phi^{*}_{1}({\bf
x}_{j})\;,$
and thus
${\mathcal{G}}_{0}(y_{1},x_{2}){\mathcal{G}}_{0}(y_{2},x_{1})={\mathcal{G}}_{0}(y_{1},x_{1}){\mathcal{G}}_{0}(y_{2},x_{2})$.
When $\tau_{y}>\tau_{x}$, we simply swap the dummy variable $y_{1}$ and
$y_{2}$ in the second graph above to arrive at
$\int\\!\\!d\tau_{x}\\!d\tau_{y}\\!\prod_{i=1}^{2}d{\bf x}_{i}d{\bf
y}_{i}\frac{{\mathcal{G}}_{0}(y_{1},x_{2})({\mathcal{G}}_{0}(y_{2},x_{1})}{4|{\bf
x}_{1}-{\bf x}_{2}||{\bf y}_{1}-{\bf
y}_{2}|}\large[{\mathcal{G}}_{0}(x_{1},y_{1}){\mathcal{G}}_{0}(x_{2},y_{1})-{\mathcal{G}}_{0}(x_{1},y_{2}){\mathcal{G}}_{0}(x_{2},y_{1})\large]\;.$
And again the quantity inside the square brackets vanishes due to the same
reason as before. As for the cancellation of the last two graphs in Fig. 4, we
first use the bottom portion of (109) to obtain
$0=\frac{\delta}{\delta n}\left[\begin{picture}(136.0,20.0)(0.0,17.0) \raise
20.0pt\hbox to0.0pt{\kern 0.0pt\makebox(0.0,0.0)[lc]{\large$\frac{1}{2}$}\hss}
\ignorespaces \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \raise 20.0pt\hbox to0.0pt{\kern
84.0pt\makebox(0.0,0.0)[lc]{\large$-\frac{1}{2}$}\hss} \ignorespaces
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \end{picture}\right]=\begin{picture}(116.0,20.0)(0.0,17.0)
\raise 20.0pt\hbox to0.0pt{\kern 0.0pt\makebox(0.0,0.0)[lc]{\large$-$}\hss}
\ignorespaces \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\raise 20.0pt\hbox to0.0pt{\kern 60.0pt\makebox(0.0,0.0)[lc]{\large$+$}\hss}
\ignorespaces \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \end{picture}\;,$
and then
$0=\frac{1}{2}\left[\begin{picture}(116.0,20.0)(0.0,17.0) \raise 20.0pt\hbox
to0.0pt{\kern 0.0pt\makebox(0.0,0.0)[lc]{\large$-$}\hss} \ignorespaces
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \raise
20.0pt\hbox to0.0pt{\kern 60.0pt\makebox(0.0,0.0)[lc]{\large$+$}\hss}
\ignorespaces \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \end{picture}\right]\left(\begin{picture}(17.0,8.0)(0.0,0.0)
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\end{picture}\right)^{-1}\left[\begin{picture}(116.0,20.0)(0.0,17.0) \raise
20.0pt\hbox to0.0pt{\kern 0.0pt\makebox(0.0,0.0)[lc]{\large$-$}\hss}
\ignorespaces \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\raise 20.0pt\hbox to0.0pt{\kern 60.0pt\makebox(0.0,0.0)[lc]{\large$+$}\hss}
\ignorespaces \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \end{picture}\right]\;,$
or
$-\frac{1}{2}\;\;\begin{picture}(60.0,20.0)(78.0,17.0) \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){}
\end{picture}\;\;=\;\frac{1}{2}\;\;\begin{picture}(96.0,20.0)(0.0,-3.0)
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){}
\end{picture}\;\;-\;\begin{picture}(60.0,20.0)(0.0,-3.0) \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \end{picture}\;,$ (111)
when there is only one electron in the system.
Equation (111) means that at the single electron limit, the last two graphs in
Fig. 4 are equivalent to the last three graphs in Fig. 5. In other words, at
the single electron limit and at zero temperature, the set of order $U^{2}$
diagrams in $\Gamma[n]$ is equivalent to the set of order $U^{2}$ diagrams in
$\beta W[J_{0}]$. We shall pause at this point and elucidate the general
situation by looking at these cancellations via Hugenholtz diagrams. Negele
and Orland (1988)
With a two-body interaction term such as the Coulomb interaction, typical
Feynman diagrams treat the direct (Coulomb) and exchange matrix element
separately. It is not surprising that one may simplify the calculation by
combining the direct and exchange matrix elements into a single
antisymmetrized matrix element. The basic idea is to combine the following two
scenarios into one
$\displaystyle\begin{picture}(50.0,40.0)(-5.0,15.0) \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\raise-5.0pt\hbox to0.0pt{\kern-5.0pt\makebox(0.0,0.0)[lc]{$\alpha$}\hss}
\ignorespaces \raise-5.0pt\hbox to0.0pt{\kern
45.0pt\makebox(0.0,0.0)[rc]{$\rho$}\hss} \ignorespaces \raise 45.0pt\hbox
to0.0pt{\kern-5.0pt\makebox(0.0,0.0)[lc]{$\gamma$}\hss} \ignorespaces \raise
45.0pt\hbox to0.0pt{\kern 45.0pt\makebox(0.0,0.0)[rc]{$\delta$}\hss}
\ignorespaces\end{picture}-\begin{picture}(50.0,40.0)(-5.0,15.0)
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \raise-5.0pt\hbox
to0.0pt{\kern-5.0pt\makebox(0.0,0.0)[lc]{$\alpha$}\hss} \ignorespaces
\raise-5.0pt\hbox to0.0pt{\kern 45.0pt\makebox(0.0,0.0)[rc]{$\rho$}\hss}
\ignorespaces \raise 45.0pt\hbox
to0.0pt{\kern-5.0pt\makebox(0.0,0.0)[lc]{$\gamma$}\hss} \ignorespaces \raise
45.0pt\hbox to0.0pt{\kern 45.0pt\makebox(0.0,0.0)[rc]{$\delta$}\hss}
\ignorespaces\end{picture}$ $\displaystyle\equiv$
$\alpha$$\rho$$\gamma$$\delta$
$\displaystyle\left(\gamma\delta|\upsilon|\alpha\rho\right)-\left(\gamma\delta|\upsilon|\rho\alpha\right)$
$\displaystyle\equiv$ $\displaystyle\\{\gamma\delta|\upsilon|\alpha\rho\\}\;,$
(112)
where
$\left(\gamma\delta|\upsilon|\alpha\rho\right)\equiv\int
d\tau_{x}d\tau_{y}d{\bf x}d{\bf y}\phi_{\gamma}^{*}({\bf
x})\phi^{*}_{\delta}({\bf y})\upsilon(x,y)\phi_{\alpha}({\bf
x})\phi_{\rho}({\bf y})\;,$
and
$\left\\{\gamma\delta|\upsilon|\alpha\rho\right\\}\equiv\int
d\tau_{x}d\tau_{y}d{\bf x}d{\bf y}\phi_{\gamma}^{*}({\bf
x})\phi^{*}_{\delta}({\bf y})\upsilon(x,y)\left[\phi_{\alpha}({\bf
x})\phi_{\rho}({\bf y})-\phi_{\rho}({\bf x})\phi_{\alpha}({\bf y})\right]\;,$
with $\phi_{\alpha}({\bf x})$ being the single-particle wave function
described earlier in Eq. (93). The resulting diagrams with bullet dots as the
new vertices are called Hugenholtz diagrams. The appearance of those vertex
matrix elements comes from the following. When one evaluates a Feynman
diagram, a propagator ${\mathcal{G}}_{0}(x,x^{\prime})$ going from
$x^{\prime}$ to $x$ connects two vertices located at $x^{\prime}$ and $x$
respectively. From Eq. (95), we know that
${\mathcal{G}}_{0}(x,x^{\prime})=\sum_{\alpha}\phi_{\alpha}({\bf
x})\phi_{\alpha}^{*}({\bf
x}^{\prime})e^{-(\varepsilon_{\alpha}-\mu)(\tau_{x}-\tau_{x^{\prime}})}\left\\{\begin{array}[]{l
r}(-n_{\alpha})&{\rm if\leavevmode\nobreak\ }\tau_{x}\leq\tau_{x^{\prime}}\\\
(1-n_{\alpha})&{\rm if\leavevmode\nobreak\
}\tau_{x}>\tau_{x^{\prime}}\end{array}\right.\;.$
Upon integration of the space-time coordinates, we see that
$\phi_{\alpha}({\bf x})$ will be integrated with vertex $x$, whither the
propagator leads, while $\phi^{*}_{\alpha}({\bf x}^{\prime})$ will be
integrated with vertex $x^{\prime}$, whence the propagator originates.
Therefore, when evaluating a Feynman diagram, one may associate each
propagator going from time $\tau^{\prime}$ to time $\tau$ with a state index
$\alpha$ with
${\mathcal{G}}_{0}(\alpha,\tau-\tau^{\prime})=e^{-(\varepsilon_{\alpha}-\mu)(\tau-\tau^{\prime})}\left[(1-n_{\alpha})\theta(\tau-\tau^{\prime}-\eta)-n_{\alpha}\theta(\tau^{\prime}-\tau+\eta)\right]\;.$
Each vertex will contribute a numerical factor equivalent to its vertex matrix
element. Note that each vertex carries a time index. At zero temperature, for
a vertex with time index $\tau$, if the two incoming lines originate from
vertices with times larger than or equal to $\tau$, the vertex matrix element
associated with $\tau$ becomes zero in the single electron limit. This is
because both incoming lines each carry only the $\alpha=1$ index. Upon
antisymmetrization, the Hugenholtz vertex matrix element becomes zero.
For an arbitrary Hugenholtz diagram with $n$ vertices, one may always name
their time index such that
$\tau_{1}\leq\tau_{2}\leq\tau_{3}\leq\cdots\leq\tau_{n}$. In this case, the
vertex associated with $\tau_{1}$ gives zero matrix element since its two
incoming lines must come from two other time indices that are larger than or
equal to $\tau_{1}$. Consequently, each Hugenholtz diagram yields value zero
at zero temperature when there is only one electron present. As a matter of
fact, the order $U$ diagrams of $W[J_{0}]$ are represented by the Hugenholtz
diagram \begin{picture}(24.0,6.0)(0.0,-3.0) \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \end{picture} . On the other hand, the first two graphs of
Fig. 4 (order $U^{2}$ terms of $\Gamma[n]$) correspond to the Hugenholtz
diagram \begin{picture}(20.0,5.0)(0.0,-3.0) \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \end{picture}
, while the last two graphs of Fig. 4 (order $U^{2}$ terms of $\Gamma[n]$) are
equivalent to the last three graphs of Fig. 5 (order $U^{2}$ terms of $\beta
W[J_{0}]$) and correspond to the Hugenholtz diagram
\begin{picture}(38.0,10.0)(0.0,-3.0) \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \end{picture}
(see reference Negele and Orland, 1988). Therefore, we see that all the order
$U^{2}$ terms cancel each other out in the effective action $\Gamma[n]$.
If one were to expand the effective action (80) in powers of $U$, our approach
reduces to performing the inversion method using $e^{2}$ as the expansion
parameter.Valiev and Fernando (1997); Fukuda et al. (1995) In this case,
$W_{l\geq 1}[J_{0}]$ in Eqs. (47-73) contains exactly all the order $U^{l}$
diagrams in $W[J_{0}]$ and can be expressed as Hugenholtz diagrams of order
$U^{l}$ as well. Since all the Hugenholtz diagrams give value zero, all the
derivatives of $W_{l}$ (all the order $u^{l}$ diagrams) with respect to the
density vanish as well. This implies that all the $J_{l\geq 1}$ vanish and
consequently in the effective action the sum of Hartree energy and the
exchange-correlation energy equals zero at zero temperature when there is only
one electron present.
The perfect cancellation of the Hartree energy and exchange-correlation energy
means that one is the negative of the other. This means that
$\sum_{i=1}^{\infty}\Gamma_{i}=-\frac{1}{2}n{\scriptstyle\circ}\,U{\scriptstyle\circ}n$.
Employing Eq. (77) for the most general case, we obtain
$\tilde{J}_{0}-J=\frac{\delta\left(\sum_{i=1}\Gamma_{i}[n]\right)}{\delta
n}=-\frac{1}{2}\frac{\delta\left(n{\scriptstyle\circ}\,U{\scriptstyle\circ}n\right)}{\delta
n}=-U{\scriptstyle\circ}n\;.$
Since $J_{0}=\tilde{J}_{0}+U{\scriptstyle\circ}n$, we find that $J_{0}=J$ at
the single electron limit at zero temperature. The fact that $J_{0}=J$ means
that the final effective potential $\varepsilon_{1}-\mu-J{\cdot}n$ is nothing
but the lowest eigenenergy of the following single-particle Hamiltonian
$\hat{h}({\bf x})=-\frac{\nabla^{2}}{2m}+\upsilon_{\rm ion}({\bf x})-\mu+J$
less the expectation value of $J$, which is exactly what one expected.
### IV.4 Screening of Coulomb Interaction
In the physical limit, $n=i\varphi$. In our formulation, lumps of charge
fluctuation around the configuration $i\varphi$ interact with each other via
$U{\scriptstyle\circ}\tilde{\cal
D}_{0}^{-1}{\scriptstyle\circ}\,U=\left(U-U{\scriptstyle\circ}D_{0}{\scriptstyle\circ}\,U\right)\;.$
As will be described in the discussion section,
$U{\scriptstyle\circ}(i\phi)=ib$ plays the role of the photon field here.
Therefore,
$-\phi{\scriptstyle\circ}\left(U-U{\scriptstyle\circ}D_{0}{\scriptstyle\circ}\,U\right){\scriptstyle\circ}\phi=(i\phi{\scriptstyle\circ}\,U){\scriptstyle\circ}\tilde{\mathcal{D}}_{0}^{-1}{\scriptstyle\circ}(U{\scriptstyle\circ}i\phi)=(ib){\scriptstyle\circ}\left(U^{-1}-D_{0}\right){\scriptstyle\circ}(ib)\;.$
Since $U(x,y)=\delta(\tau_{x}-\tau_{y})U({\bf x},{\bf y})$ and
${\mathcal{G}}_{0}(x,y)$ only depends on the relative time difference
$\tau_{x}-\tau_{y}$, we expect $D_{0}(x,y)$ to depend only on
$\tau\equiv\tau_{x}-\tau_{y}$. Furthermore, since ${\mathcal{G}}_{0}(x,y)$ is
antiperiodic in $\tau_{x}$, $\tau_{y}$, and $\tau_{x}-\tau_{y}$, one expects
that ${\mathcal{G}}_{0}(x,y){\mathcal{G}}_{0}(y,x)$ to be periodic in
$\tau_{x}-\tau_{y}$.
Introducing the spatial momenta ${\mathbf{p}}$ and ${\mathbf{q}}$ conjugate to
the spatial variables ${\bf x}$ and ${\bf y}$, we may write
$\left[U^{-1}-D_{0}\right](\nu_{n},{\mathbf{p}},{\mathbf{q}})=\int
e^{i{\mathbf{p}}{\bf x}+i{\mathbf{q}}{\bf y}}d{\bf x}d{\bf
y}\int_{0}^{\beta}\,d\tau\;e^{i\nu_{n}\tau}\left[U^{-1}({\bf x},{\bf
y})\delta(\tau)-{\mathcal{G}}_{0}(x,y){\mathcal{G}}_{0}(y,x)\right]\;.$
When $U({\bf x},{\bf y})={e^{2}\over|{\bf x}-{\bf y}|}$, we find that
$U^{-1}(x,y)=-{1\over 4\pi e^{2}}\delta(\tau_{x}-\tau_{y})\nabla^{2}_{\bf
x}\delta({\bf x}-{\bf y})$ and
$U^{-1}(\nu_{n},{\mathbf{p}},{\mathbf{q}})={(2\pi)^{3}\over 4\pi
e^{2}}{\mathbf{q}}^{2}\delta({\mathbf{p}}+{\mathbf{q}})={L^{3}\over 4\pi
e^{2}}{\mathbf{q}}^{2}\delta_{{\mathbf{p}}+{\mathbf{q}}}\;,$
where $L^{3}$ is the spatial volume of the system. Let us now write
$D_{0}(\nu_{n},{\mathbf{p}},{\mathbf{q}})$ as
$D_{0}(\nu_{n},{\mathbf{p}},{\mathbf{q}})=\int e^{i{\mathbf{p}}{\bf
x}+i{\mathbf{q}}{\bf y}}d{\bf x}d{\bf
y}\int_{0}^{\beta}\,d\tau\;e^{i\nu_{n}\tau}\,{\mathcal{G}}_{0}(x,y){\mathcal{G}}_{0}(y,x)\;.$
In momentum space,
$U^{-1}\propto{{\mathbf{q}}}^{2}\delta({\mathbf{p}}+{\mathbf{q}})$ and it is
well known that this type of Coulomb interaction leads to infrared divergence
(that occurs near ${{\mathbf{q}}}\to 0$). In the presence of $-D_{0}$, one may
wish to calculate the zero momentum contribution of $-D_{0}$. Let us define
$D_{0}(\nu_{n})\equiv D_{0}(\nu_{n},{\mathbf{p}}=0,{\mathbf{q}}=0)$. Using Eq.
(95), we find
$D_{0}(\nu_{n})=-\sum_{\alpha,\alpha^{\prime}}\delta_{\alpha,\alpha^{\prime}}\int_{0}^{\beta}d\tau
e^{i\nu_{n}\tau}\left[n_{\alpha}(1-n_{\alpha^{\prime}})\right]=-\beta\delta_{\nu_{n},0}\sum_{\alpha}n_{\alpha}(1-n_{\alpha})\;.$
(113)
From Eq. (101), we see that when $x=x^{\prime}$,
$\beta\sum_{\alpha}n_{\alpha}(1-n_{\alpha})=-\int d{\bf
x}\frac{\partial{\mathcal{G}}_{0}(x,x)}{\partial\mu}=\frac{\partial
N_{e}}{\partial\mu}\;.$
We therefore have
$D_{0}(\nu_{n})=-\delta_{\nu_{n},0}{L^{3}}\frac{\partial\bar{n}}{\partial\mu}\;,$
where $\bar{n}=N_{e}/L^{3}$ is the average electron density.
Therefore at the stationary limit, where $\nu_{n}=0$, the inverse propagator
with nearly zero momentum behaves as
$L^{3}\left[\frac{q^{2}}{4\pi
e^{2}}+\frac{\partial\bar{n}}{\partial\mu}\right]\Rightarrow\frac{L^{3}}{4\pi
e^{2}}\left[q^{2}+4\pi e^{2}\frac{\partial\bar{n}}{\partial\mu}\right]\;,$
analogous to the Thomas-Fermi results for electric charge screening.
Except during the intermediate steps of computing the excitation spectrum, one
deals with the time independent system. Note that $4\pi
e^{2}\partial\bar{n}/\partial\mu$ plays the role of $m^{2}$ in the screened
potential $e^{-mr}/r$. This shows that the static interaction between charge
fluctuations in the long wave length limit is a screened one instead of the
bare Coulomb interaction. It should be noted that the use of this screened
propagator dates back half a century. DuBois DuBois (1959) replaced the bare
Coulomb interaction by the screened interaction in his study of electron
interactions. Hedin Hedin (1965) also used it to replace the bare Coulomb
interaction in the expansion of the Luttinger-Ward functional, Luttinger and
Ward (1960) resulting in the so-called GW approximation. The differences among
the three mentioned approaches should be described. In both DuBois’s and
Hedin’s formalism, they use the full polarization of the interacting system.
The difference between their formalism is in the electron propagators
employed. In DuBois’s approach, the free electron propagator is used, while in
Hedin’s method, similar to the Luttinger-Ward functional, Luttinger and Ward
(1960) the full electron propagator is employed. In our approach, it is the
polarization of the KS system that enters the calculation and the electron
propagator entering the diagrammatic calculation is the noninteracting KS
Green’s function.
### IV.5 Homogeneous Electron Gas
Being the foundation for the LDA of the DFT, the homogeneous electron gas
(HEG) is also a simplified model for a metal or a plasma. The HEG system is an
interacting electron gas placed in a uniformly distributed positive background
chosen to ensure that the total system is neutral. Due to the translational
symmetry of the HEG system, the one-particle propagator functions only depend
on the coordinate difference between two variables (space-time points) instead
of on both variables. Consequently, each propagator (Green’s function) carries
a definite momentum even if the Coulomb interaction among electrons is fully
taken into consideration.
Investigation of the HEG system dates back to 1930s by Wigner,Wigner (1934)
who coined the term “correlation energy” to represent the ground state energy
per electron after subtracting away the average kinetic and exchange energy.
(The Hartree energy is cancelled exactly by the interaction between electrons
and the positive background, and by the Coulomb energy among the uniform
positive charges.) Apparently, after choosing the Rydberg (the negative of the
Hydrogen atom ground state energy) as the energy unit, one may express either
the correlation energy or the ground state energy of the HEG system in terms
of the dimensionless quantity $r_{s}\equiv r_{0}/r_{B}$, where ${4\pi\over
3}r_{0}^{3}=\frac{V}{N_{e}}$, $V$ being the volume of the system considered,
$N_{e}$ being the total number of electrons, and
$r_{B}=\frac{\hbar^{2}}{me^{2}}$ representing the Bohr radius. In fact,
efforts have been invested to express the correlation energy as a power series
in $r_{s}$ (and $r_{s}\ln r_{s}$ as well) at both the high density and low
density limits. Lacking real experimental results to compare with, however, it
is hard to assess how much improvement each increasing order of $r_{s}$ (or
$1/r_{s}$) can bring.
Since $r_{s}\propto 1/r_{B}\propto e^{2}$, a small $r_{s}$ expansion is most
naturally performed by treating the Coulomb interaction as a perturbation.
Equivalent to the method of computing the vacuum amplitude,Fetter and Walecka
(1971); Jones and March (1973); Negele and Orland (1988) the time-ordered
perturbative expansion developed by Goldstone Goldstone formalized the
Brueckner theory Brueckner and Levinson (1955) and made a direct connection to
diagrammatic expansion in calculation of the ground state energy. The electron
propagator in diagrams under this formalism is that of the noninteracting
electrons. An alternative formalism to compute the ground state energy (or the
grand potential) is to utilize the full propagator (i.e., the one with self-
energy included) in diagrammatic calculation. Under this alternative
formalism, only two-particle irreducible diagrams contribute. Indeed, the work
of Luttinger and Ward Luttinger and Ward (1960) (as well as that of Klein
Klein (1961)) was exactly along this line. Since the self-energy is not known
a priori, this type of energetic expression is a functional of the self-energy
(or the full Green’s function), which must be determined via a stationary
condition.Luttinger and Ward (1960)
An equivalent of the Brueckner-Goldstone formalism Goldstone ; Brueckner and
Gammel (1958) was used by Gell-Mann and Brueckner Gell-Mann and Brueckner
(1957) to compute the correlation energy of the HEG system. Given the long-
range nature of the Coulomb interaction, it is not surprising that Gell-Mann
and Brueckner identified the occurrence of divergence as early as in the
second order of $e^{2}$-based perturbative calculation. To circumvent this
unphysical divergence, they summed an infinite subset of diagrams to arrive at
a contribution proportional to $\ln r_{s}$. To eliminate unphysical
divergence, DuBois DuBois (1959) replaced the bare Coulomb interaction by the
screened one. Starting with calculating the vacuum amplitude, he expressed the
ground state energy in terms of integration of the full electronic
polarization over the Coulomb coupling strength, using a version of the
Feynman-Hellmann theorem. While the full polarization is used, the electron
propagator entering DuBois’s formalism is the free electron propagator instead
of the full propagator. Unfortunately, DuBois made a mistake in extracting the
higher order terms of the ground state energy at the high density (small
$r_{s}$) limit. Although this error was later corrected by Carr and
Maradudin,Carr and Maradudin (1964) according to Hedin,Hedin (1965) this high
density expansion violated at moderate $r_{s}$ values Ferrell’s
condition,Ferrell (1958) which is based on the simple fact that the second
order perturbation contribution to the ground state energy is always negative.
Except for using the screened Coulomb interaction to replace the bare one and
working directly at zero temperature, Hedin’s approach is largely similar to
the finite-temperature formalism of Luttinger and Ward Luttinger and Ward
(1960) in the sense that the full electron propagator is used as the
fundamental variable. As a consequence, the higher order diagrams contributing
to the full polarization appear different in DuBois’s work and in Hedin’s
formalism. Instead of solving his own self-consistent equations, however,
Hedin approximated the full Green’s function within his formalism by the non-
interacting Green’s function to compute the energetics of the HEG system.
Nevertheless, it is still possible to maintain the exactness within Hedin’s
approach if two-particle reducible diagrams are properly included. We will
provide an example of such a two-particle reducible diagram that would not be
included in Hedin’s approximate calculation but should be incorporated for
theoretical soundness.
Another important issue in obtaining the ground state energy of a many-
electron system has to do with whether the finite temperature formalism
(setting ${V\to\infty}$ first followed by ${T\to 0}$) or the zero temperature
formalism (setting $T\to 0$ and then followed by $V\to\infty$) is used. In
terms of diagrammatic expansion, there exist diagrams (termed anomalous
diagrams by Luttinger et al.Kohn and Luttinger (1960); Luttinger and Ward
(1960)) that are present (giving finite contribution) within the finite
temperature formalism but are absent (giving zero contribution) within the
zero temperature formalism. To be specific, a diagram is anomalous if within
it there exist two electron propagators linking two different times (say
$\tau_{1}$, $\tau_{2}$ and of course $\tau_{1}\neq\tau_{2}$) in the opposite
orders: ${\mathcal{G}}({\bf x},\tau_{1},{\bf
y},\tau_{2}){\mathcal{G}}({\mathbf{w}},\tau_{2},{\mathbf{z}},\tau_{1})$.
Zero temperature methods typically rely on the Gell-Mann and Low theorem,
Gell-Mann and Low (1951) which assures that adiabatic transformation of the
noninteracting ground state by gradually switching on the interaction leads to
an eigenstate of the interacting system.Fetter and Walecka (1971); Negele and
Orland (1988) The energetic difference of this eigenstate and the
noninteracting ground state is what the zero temperature formalism obtains.
This approach therefore makes sense only if the adiabatically transformed
state is the ground state of the interacting system, which occurs only if the
noninteracting Fermi surface is identical to that of the interacting system.
Jones and March (1973) For the HEG system, the perturbed Fermi surface remains
spherical (identical to that of the unperturbed one) since the Coulomb
interaction respects spherical symmetry and the background positive charge
distribution also has spherical symmetry. Consequently, for the HEG system,
the anomalous diagrams should end up giving no contribution. Indeed, Luttinger
et al. Kohn and Luttinger (1960); Luttinger and Ward (1960) illustrated how
the contributions from the anomalous diagrams in this case are cancelled by
the chemical potential shift. Luttinger and Ward Luttinger and Ward (1960)
also showed how one may avoid anomalous contributions from appearing by
expressing the grand potential as a sum of all possible linked diagrams
(including two-particle reducible ones). The key step there is to subtract
from each self-energy part a number, which is given by that self-energy part
evaluated at the Fermi surface. As we will illustrate later, under our finite
temperature formalism it is not necessary to implement such an elaborate
subtraction scheme because each two-particle reducible diagram is
automatically accompanied by another appropriate diagrammatic subtraction.
What sets our self-consistent equation (78) apart from that of Luttinger and
Ward and of Hedin is the variable to be solved for. It is the KS potential,
instead of the physical Green’s function, that enters our exact, self-
consistent equation. In general, one needs to first solve $\tilde{J}_{0}$
self-consistently using eq. (78) prior to the evaluation of the grand
potential (or the ground state energy). When limiting to a homogeneous system
with constant electron density, however, $\tilde{J}_{0}={\rm const.}$ becomes
the only possibility. Whether or not such a choice satisfies the self-
consistent equation (78) depends on whether a uniform electron density truly
represent the lowest energy configuration. For the present purpose of
considering a high density HEG, we assume a constant $\tilde{J}_{0}$ to
proceed. Because a constant $\tilde{J}_{0}$ can be easily absorbed into the
chemical potential, the resulting KS Green’s function carries a corresponding
energy that consists of kinetic energy only. That is, rather than being an
approximation as in Hedin’s case, the single-particle Green’s function
carrying only kinetic energy represents exactly the self-consistent KS Green’s
function under our formalism. Consequently, for the HEG system, the grand
potential (or the ground state energy) may be calculated using eq. (80). In
the remaining part of this section, we will first show how our formalism
naturally avoids divergence and how one may use it to obtain the ground state
energy of the HEG as a series in $r_{s}$ (and $\ln r_{s}$). We will then
illustrate with an explicit example how the anomalous contributions are
cancelled within our formalism, followed by a brief description of diagrams
that would be missed within Hedin’s approximation Hedin (1965) in computing
the ground state energy of the HEG system. To provide an easier comparison
with existing results, we restore in this section the electron spins that have
been suppressed thus far to simplify the exposition.
In our definition of the energy functional $E_{\upsilon}$, see eq. (4), the
$-\mu N_{e}$ term is included but the interaction among background charges is
not included. Since most zero temperature formalism does not include the $-\mu
N_{e}$ term and for the HEG system the interaction between background charges
is also included, the ground state energy in the literature will correspond to
$\lim_{\beta\to\infty}\left[\frac{1}{\beta}\Gamma[n]+\mu
N_{e}+\frac{1}{2}n_{\rm bg}{\cdot}\,U{\cdot}n_{\rm bg}\right]$
in our formalism. Eq. (104) can then be used to arrive at
$\displaystyle E_{g}$ $\displaystyle=$
$\displaystyle\lim_{\beta\to\infty}\left[\frac{1}{\beta}\Gamma[n]+\mu
N_{e}+\frac{1}{2}n_{\rm bg}{\cdot}\,U{\cdot}n_{\rm bg}\right]$ (114)
$\displaystyle=$
$\displaystyle\sum_{\alpha=1}^{N_{e}}\varepsilon_{\alpha}-J_{0}{\cdot}n+\frac{1}{2}n{\cdot}\,U{\cdot}n+\frac{1}{2}n_{\rm
bg}{\cdot}\,U{\cdot}n_{\rm
bg}+\lim_{\beta\to\infty}\left[\frac{1}{2\beta}\text{Tr}\ln\left(\tilde{\cal
D}_{0}^{-1}{\scriptstyle\circ}\,U\right)+\frac{1}{\beta}\sum_{i=2}^{\infty}\Gamma_{i}[n]\right]$
$\displaystyle=$
$\displaystyle\sum_{\alpha=1}^{N_{e}}(\varepsilon_{\alpha}-\tilde{J}_{0})+\lim_{\beta\to\infty}\left[\frac{1}{2\beta}\text{Tr}\ln\left(\tilde{\cal
D}_{0}^{-1}{\scriptstyle\circ}\,U\right)+\frac{1}{\beta}\sum_{i=2}^{\infty}\Gamma_{i}[n]\right]\;,$
where $J_{0}=\tilde{J}_{0}+U{\cdot}n$ is used and $n_{\rm bg}=n$ for the HEG
system is also employed. Note that the state label
$\alpha=({\mathbf{p}},\sigma)$ includes both momentum and spin. For the HEG
system, $\upsilon_{\rm ion}+U{\cdot}n=0$ and thus eq. (94) leads to
$\varepsilon_{\alpha}={\mathbf{p}}^{2}/2m+\tilde{J}_{0}$. Therefore, the
ground state energy of the HEG system may be written as
$E_{g}=2\sum_{{\mathbf{p}}}^{|{\mathbf{p}}|\leq
p_{F}}\frac{{\mathbf{p}}^{2}}{2m}+\lim_{\beta\to\infty}\frac{1}{\beta}\left[\frac{1}{2}\text{Tr}\ln\left(\tilde{\cal
D}_{0}^{-1}{\scriptstyle\circ}\,U\right)+\sum_{i=2}^{\infty}\Gamma_{i}[n]\right]\;,$
(115)
where $p_{F}$ indicates the Fermi momentum and the factor of $2$ in the first
term of the right hand side comes from noting that there are two spin states
associated with each momentum. Furthermore, the KS Green’s function in this
case may be written as
${\mathcal{G}}_{0}(x,\sigma;y,\sigma^{\prime})=\frac{1}{\beta
V}\sum_{{\mathbf{p}},n}\frac{e^{-i\omega_{n}(\tau_{x}-\tau_{y})}e^{-i{\mathbf{p}}\cdot({\bf
x}-{\bf
y})}}{-i\omega_{n}+\varepsilon_{\mathbf{p}}+\tilde{J}_{0}-\mu}\,\delta_{\sigma,\sigma^{\prime}}\,,$
(116)
where $\varepsilon_{\mathbf{p}}={\mathbf{p}}^{2}/2m$ is the kinetic energy of
an electron carrying momentum ${\mathbf{p}}$. By absorbing the constant
$\tilde{J}_{0}$ into the chemical potential in (116), one sees that the KS
Green’s function in the HEG system is indeed the free electron propagator and
at the zero temperature, the new chemical potential (the original one
subtracted by $\tilde{J}_{0}$) simply becomes $p_{F}^{2}/2m$.
The diagrammatic expression of our first correction term
$\Gamma_{1}[n]=\frac{1}{2}\text{Tr}\ln(\tilde{\mathcal{D}_{0}}^{-1}{\scriptstyle\circ}\,U)=\frac{1}{2}\text{Tr}\ln\left[{\mathbf{I}}-D_{0}{\scriptstyle\circ}\,U\right]$
is shown in Fig. 1. It can be seen that, upon being divided by the inverse
temperature $\beta$, our $\Gamma_{1}[n]$ contains the exchange energy
$\epsilon_{x}$ and all the ring-like correlation energy $\epsilon^{\prime}$
discussed by Gell-Mann and Brueckner Gell-Mann and Brueckner (1957) via the
relation $\Gamma_{1}[n]/\beta=N_{e}(\epsilon_{x}+\epsilon^{\prime})$, with
$N_{e}$ being the total number of electrons. Since both $D_{0}$ and $U$ are
diagonal in momentum space for the HEG system, we may write
$\frac{1}{\beta}\Gamma_{1}[n]=\frac{1}{2\beta}\sum_{{\mathbf{q}},\nu_{n}}\ln\left[1-D_{0}(q,\nu_{n})\,U(q)\right]=\frac{V}{2\beta}\int\frac{d{\mathbf{q}}}{(2\pi)^{3}}\sum_{\nu_{n}}\ln\left[1-D_{0}(q,\nu_{n})\,U(q)\right]\;.$
(117)
By comparing this equation to a derived result (eq. (30.16) of reference
Fetter and Walecka, 1971), it is also evident that our $\Gamma_{1}[n]/(\beta
N_{e})$ indeed gives $\epsilon_{x}+\epsilon^{\prime}$ of reference Gell-Mann
and Brueckner, 1957 as $\beta\to\infty$. In Fig. 1, except for the first
diagram, all other diagrams when evaluated individually exhibit infrared
divergence due to the piling up of $1/{\mathbf{q}}^{2}$ propagators.Gell-Mann
and Brueckner (1957) To obtain the leading contribution, one must pay
particular attention to the small $|{\mathbf{q}}|$ region.
The polarization $D_{0}\equiv\delta n(x)/\delta J_{0}(y)$ is defined in eq.
(71). When the electron spins are included,
$n(x)=-\sum_{\sigma}{\mathcal{G}}_{0}(x,\sigma;x,\sigma)$. Because the Coulomb
interaction does not flip spins, the KS propagator is diagonal in the electron
spins and thus
$D_{0}(x,y)=\sum_{\sigma}{\mathcal{G}}_{0}(x,\sigma;y,\sigma){\mathcal{G}}_{0}(y,\sigma;x,\sigma)=\,2\;{\mathcal{G}}_{0}(x,y){\mathcal{G}}_{0}(y,x)\;.$
(118)
We therefore have
$\displaystyle D_{0}({\mathbf{q}},\nu_{n})$ $\displaystyle\equiv$
$\displaystyle 2\int d({\bf x}-{\bf
y})d(\tau_{x}-\tau_{y})e^{i{\mathbf{q}}.({\bf x}-{\bf
y})}d^{i\nu_{n}(\tau_{x}-\tau_{y})}\left[{\mathcal{G}}_{0}(x,y){\mathcal{G}}_{0}(y,x)\right]$
$\displaystyle=$
$\displaystyle\frac{2}{\beta}\sum_{n^{\prime}}\int\frac{d{\mathbf{p}}}{(2\pi)^{3}}\frac{1}{(-i\omega_{n^{\prime}}-i\nu_{n}+\varepsilon_{{\mathbf{p}}+{\mathbf{q}}}-\mu)(-i\omega_{n^{\prime}}+\varepsilon_{\mathbf{p}}-\mu)}$
$\displaystyle=$ $\displaystyle
2\int\frac{d{\mathbf{p}}}{(2\pi)^{3}}\frac{n_{{\mathbf{p}}+{\mathbf{q}}}-n_{{\mathbf{p}}}}{-i\nu_{n}+\varepsilon_{{\mathbf{p}}+{\mathbf{q}}}-\varepsilon_{{\mathbf{p}}}}$
$\displaystyle=$
$\displaystyle-2\int\frac{d{\mathbf{p}}}{(2\pi)^{3}}\frac{n_{\mathbf{p}}(1-n_{{\mathbf{p}}+{\mathbf{q}}})-n_{{\mathbf{p}}+{\mathbf{q}}}(1-n_{{\mathbf{p}}})}{-i\nu_{n}+\varepsilon_{{\mathbf{p}}+{\mathbf{q}}}-\varepsilon_{{\mathbf{p}}}}$
$\displaystyle=$
$\displaystyle-2\int\frac{d{\mathbf{p}}}{(2\pi)^{3}}n_{\mathbf{p}}(1-n_{{\mathbf{p}}+{\mathbf{q}}})\left[\frac{1}{-i\nu_{n}+\varepsilon_{{\mathbf{p}}+{\mathbf{q}}}-\varepsilon_{{\mathbf{p}}}}+\frac{1}{i\nu_{n}+\varepsilon_{{\mathbf{p}}+{\mathbf{q}}}-\varepsilon_{{\mathbf{p}}}}\right]$
$\displaystyle=$
$\displaystyle-2\int\frac{d{\mathbf{p}}}{(2\pi)^{3}}\;n_{\mathbf{p}}\left[\frac{1}{-i\nu_{n}+\varepsilon_{{\mathbf{p}}+{\mathbf{q}}}-\varepsilon_{{\mathbf{p}}}}+\frac{1}{i\nu_{n}+\varepsilon_{{\mathbf{p}}+{\mathbf{q}}}-\varepsilon_{{\mathbf{p}}}}\right]$
$\displaystyle=$
$\displaystyle-4\int\frac{d{\mathbf{p}}}{(2\pi)^{3}}\frac{n_{\mathbf{p}}(1-n_{{\mathbf{p}}+{\mathbf{q}}})(\varepsilon_{{\mathbf{p}}+{\mathbf{q}}}-\varepsilon_{\mathbf{p}})}{(\varepsilon_{{\mathbf{p}}+{\mathbf{q}}}-\varepsilon_{\mathbf{p}})^{2}+\nu_{n}^{2}}=-4\int\frac{d{\mathbf{p}}}{(2\pi)^{3}}\frac{n_{\mathbf{p}}(\varepsilon_{{\mathbf{p}}+{\mathbf{q}}}-\varepsilon_{\mathbf{p}})}{(\varepsilon_{{\mathbf{p}}+{\mathbf{q}}}-\varepsilon_{\mathbf{p}})^{2}+\nu_{n}^{2}}$
where the third line of the above equation is obtained using the technique
described in section III.5, the fourth line is obtained by rewriting the
numerator of the third line as
$-n_{\mathbf{p}}(1-n_{{\mathbf{p}}+{\mathbf{q}}})+n_{{\mathbf{p}}+{\mathbf{q}}}(1-n_{\mathbf{p}})$,
the fifth line is obtained by a change of variable
(${\mathbf{p}}+{\mathbf{q}}\to-{\mathbf{p}}$) in the second half and assuming
the spherical property of $n_{\mathbf{p}}$, and the sixth line comes from the
fact that $n_{\mathbf{p}}n_{{\mathbf{p}}+{\mathbf{q}}}$ is symmetric with
respect to $({\mathbf{p}}+{\mathbf{q}})\leftrightarrow{\mathbf{p}}$ while the
quantity inside the square bracket is antisymmetric with respect to
$({\mathbf{p}}+{\mathbf{q}})\leftrightarrow{\mathbf{p}}$. Since
$\varepsilon_{{\mathbf{p}}}\propto|{\mathbf{p}}|^{2}$ is a monotonic
increasing function of $|{\mathbf{p}}|$, while both
$n_{{\mathbf{p}}}=1/(e^{\beta(\varepsilon_{\mathbf{p}}-\mu)}+1)$ and
$1-n_{\mathbf{p}}$ are nonnegative, we see that $D_{0}({\mathbf{q}},\nu_{n})$
is a negative definite quantity.
It turns out that for the HEG system one can further simplify the expression
of $D_{0}(q,\nu)$ by introducing a new variable $u$ via $\nu\equiv
u|{\mathbf{q}}|/m$. We then have (by choosing $\hat{q}$ as the $\hat{z}$
direction in ${\mathbf{p}}$ and using the fact that
$n_{\mathbf{p}}=n(|{\mathbf{p}}|)=n(p)$)
$\displaystyle D_{0}(q,\frac{uq}{m})$ $\displaystyle=$
$\displaystyle-2\int\frac{d{\mathbf{p}}}{(2\pi)^{3}}\;n(p)\left[\frac{1}{-iuq/m+\varepsilon_{{\mathbf{p}}+{\mathbf{q}}}-\varepsilon_{{\mathbf{p}}}}+\frac{1}{iuq/m+\varepsilon_{{\mathbf{p}}+{\mathbf{q}}}-\varepsilon_{{\mathbf{p}}}}\right]$
(120) $\displaystyle=$
$\displaystyle-2\frac{m}{q}\int_{0}^{\infty}\frac{n(p)\,p^{2}\,dp}{(2\pi)^{2}}\;\int_{-1}^{1}d\cos\theta\left[\frac{1}{-iu+q/2+p\cos\theta}+\frac{1}{iu+q/2+p\cos\theta}\right]$
$\displaystyle=$
$\displaystyle-2\frac{m}{q}\int_{0}^{\infty}\frac{n(p)\,dp}{(2\pi)^{2}}\;p\left[\ln\left(\frac{q}{2}+p+iu\right)+\ln\left(\frac{q}{2}+p-iu\right)\right.$
$\displaystyle\hskip
90.0pt\left.-\ln\left(\frac{q}{2}-p+iu\right)-\ln\left(\frac{q}{2}-p-iu\right)\right]$
$\displaystyle=$
$\displaystyle\frac{2m}{(2\pi)^{2}}\int_{0}^{\infty}\frac{\partial
n(p)}{\partial
p}\,dp\,\left[p+\frac{1}{2q}(p^{2}+u^{2}-\frac{q^{2}}{4})\ln\frac{(p+\frac{q}{2})^{2}+u^{2}}{(p-\frac{q}{2})^{2}+u^{2}}\right.$
$\displaystyle\hskip
115.0pt\left.-u\tan^{-1}\frac{\frac{q}{2}+p}{u}+u\tan^{-1}\frac{\frac{q}{2}-p}{u}\right]\;.$
As $\beta\to\infty$, $n(p)=\theta(p_{F}-p)$ and therefore $\partial
n(p)/\partial p=-\delta(p-p_{F})$, leading to
$\displaystyle D_{0}(q,\frac{uq}{m})$
$\displaystyle\xrightarrow[\beta\to\infty]{}$
$\displaystyle-\frac{2m}{(2\pi)^{2}}\left[p_{F}+\frac{1}{2q}(p_{F}^{2}+u^{2}-\frac{q^{2}}{4})\ln\frac{(p_{F}+\frac{q}{2})^{2}+u^{2}}{(p_{F}-\frac{q}{2})^{2}+u^{2}}\right.$
(121) $\displaystyle\hskip
75.0pt\left.-u\tan^{-1}\frac{\frac{q}{2}+p_{F}}{u}+u\tan^{-1}\frac{\frac{q}{2}-p_{F}}{u}\right]\;.$
The exact expression (121) allows one to extract the limits of $q\to\infty$
and $q\to 0$, both of which are important for determining the convergence
properties of the energy expansion. We have
$\displaystyle D_{0}(q\gg 1,\frac{uq}{m})$
$\displaystyle\xrightarrow[\beta\to\infty]{}$
$\displaystyle-\frac{4m\,p_{F}^{3}}{3\pi^{2}}\frac{1}{q^{2}+4u^{2}}+{\cal
O}\left((q^{2}+4u^{2})^{-2}\right)\;,$ (122) $\displaystyle D_{0}(q\ll
1,\frac{uq}{m})$ $\displaystyle\xrightarrow[\beta\to\infty]{}$
$\displaystyle-\frac{m}{\pi^{2}}\left[R_{0}(u)+R_{1}(u)\,q^{2}+R_{2}(u)\,q^{4}\right]+{\cal
O}\left(q^{6}\right)\;,$ (123)
where
$\displaystyle R_{0}(u)$ $\displaystyle\equiv$ $\displaystyle
p_{F}-u\tan^{-1}\frac{p_{F}}{u}\;,$ (124) $\displaystyle R_{1}(u)$
$\displaystyle=$ $\displaystyle-\frac{p_{F}^{3}}{12(p_{F}^{2}+u^{2})^{2}}\;,$
(125) $\displaystyle R_{2}(u)$ $\displaystyle=$
$\displaystyle-\frac{p_{F}^{3}(p_{F}^{2}-5u^{2})}{240(p_{F}^{2}+u^{2})^{4}}\;.$
(126)
With these asymptotic behaviors, we see from eq. (117) that $\Gamma_{1}[n]$ is
finite. Methods for extracting coefficients associated with the $e^{2}$-based
expansion for $\epsilon^{\prime}$ can be found in references Gell-Mann and
Brueckner, 1957 and Carr and Maradudin, 1964. The
$\epsilon^{\prime}=\lim_{\beta\to\infty}\Gamma_{1}[n]/(N_{e}\,\beta)-\epsilon_{x}$
part contains in general $r_{s}^{n\geq 0}\ln r_{s}$ and $r_{s}^{(n\geq 0)}$
terms when the energy is expressed in Rydbergs and expanded in power of
$r_{s}$ (or $e^{2}$).
We now turn our attention to $\Gamma_{2}[n]$, whose diagrammatic expression is
given in Figure 3. As mentioned earlier, within our formalism, $\Gamma_{i\geq
2}[n]$ correspond to diagrams containing only $\tilde{\cal D}_{0}$ propagator
and the KS electron propagator. Since the $\tilde{\cal D}_{0}$ propagator
retains a finite value even when its associated momentum approaches zero, each
of the diagrams corresponding to $\Gamma_{i\geq 2}[n]$ takes a finite value.
The absence of divergence no longer holds when one perform high density (
small $r_{s}$) expansion, as we will illustrate explicitly later. To make easy
comparison with existing HEG studies, one employs Dyson’s equation,
$\displaystyle\tilde{\cal D}_{0}$ $\displaystyle=$ $\displaystyle
U+U{\scriptstyle\circ}D_{0}{\scriptstyle\circ}\tilde{\cal D}_{0}$ (127)
\begin{picture}(20.0,30.0)(0.0,-3.0)\put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \end{picture} $\displaystyle=$
$\displaystyle\;\;\begin{picture}(20.0,30.0)(0.0,-3.0) \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){}
\end{picture}\;\;+\;\begin{picture}(60.0,30.0)(0.0,-3.0) \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \end{picture}\;\;$
to decompose the propagator $\tilde{\cal D}_{0}$ (of order $e^{2}$ and higher)
into the sum of a bare Coulomb propagator $U$ (of order $e^{2}$) and
$U{\scriptstyle\circ}D_{0}{\scriptstyle\circ}\tilde{\cal D}_{0}$ (of order
$e^{4}$ and higher). The reason that one should not expand further and write
$\tilde{\cal
D}_{0}=U+U{\scriptstyle\circ}D_{0}{\scriptstyle\circ}\,U+U{\scriptstyle\circ}D_{0}{\scriptstyle\circ}\,U{\scriptstyle\circ}D_{0}{\scriptstyle\circ}\tilde{\cal
D}_{0}$ is because the $U{\scriptstyle\circ}D_{0}{\scriptstyle\circ}\,U$ term
causes infrared divergence due to the momentum integral $\int
d{\mathbf{q}}/q^{4}$. In fact, this is exactly what causes (in the second
diagram of Figure 1) the infrared divergence, motivating the summation of the
ring diagrams.
To illustrate the main points, let us begin by examining the first three
diagrams of $\Gamma_{2}[n]$ and employ the decomposition rule mentioned above:
$\displaystyle\frac{1}{4}\;\;\;\;\;\;\;\;\;\;\;\;\;\begin{picture}(80.0,20.0)(0.0,-4.0)\put(-16.0,15.0){\makebox(0.0,0.0)[]{${\mathbf{p}}_{1}+{\mathbf{q}}$}}
\put(45.0,15.0){\makebox(0.0,0.0)[]{${\mathbf{p}}_{1}$}}
\put(-8.0,-15.0){\makebox(0.0,0.0)[]{$-{\mathbf{p}}_{2}$}}
\put(58.0,-15.0){\makebox(0.0,0.0)[]{$-{\mathbf{p}}_{2}-{\mathbf{q}}$}}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \end{picture}\;\;$ $\displaystyle=$
$\displaystyle\;\;\frac{1}{4}\;\begin{picture}(40.0,20.0)(0.0,-4.0)
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
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\put(0.0,0.0){} \put(0.0,0.0){} \raise-34.0pt\hbox to0.0pt{\kern
20.0pt\makebox(0.0,0.0)[cc]{$(a)$}\hss}
\ignorespaces\end{picture}\;\;+\frac{1}{2}\;\begin{picture}(50.0,25.0)(0.0,-4.0)
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\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\raise-34.0pt\hbox to0.0pt{\kern 25.0pt\makebox(0.0,0.0)[cc]{$(b)$}\hss}
\ignorespaces\end{picture}\;\;+\frac{1}{4}\;\begin{picture}(50.0,25.0)(0.0,-4.0)
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \raise-34.0pt\hbox to0.0pt{\kern
25.0pt\makebox(0.0,0.0)[cc]{$(c)$}\hss} \ignorespaces\end{picture}$ (128)
$\displaystyle\frac{1}{2}\;\;\;\;\;\;\;\;\;\;\;\;\;\begin{picture}(80.0,60.0)(0.0,-4.0)\put(-17.5,19.0){\makebox(0.0,0.0)[]{$-{\mathbf{p}}_{1}-{\mathbf{q}}$}}
\put(50.0,15.0){\makebox(0.0,0.0)[]{$-{\mathbf{p}}_{1}$}}
\put(-16.0,-19.0){\makebox(0.0,0.0)[]{$-{\mathbf{p}}_{2}-{\mathbf{q}}$}}
\put(60.0,-15.0){\makebox(0.0,0.0)[]{$-{\mathbf{p}}_{1}-{\mathbf{q}}$}}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \end{picture}\;\;$ $\displaystyle=$
$\displaystyle\;\;\frac{1}{2}\;\begin{picture}(40.0,20.0)(0.0,-4.0)
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \raise-34.0pt\hbox to0.0pt{\kern
20.0pt\makebox(0.0,0.0)[cc]{$(a)$}\hss}
\ignorespaces\end{picture}\;\;+\;\;\;\;\begin{picture}(50.0,25.0)(0.0,-4.0)
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\raise-34.0pt\hbox to0.0pt{\kern 25.0pt\makebox(0.0,0.0)[cc]{$(b)$}\hss}
\ignorespaces\end{picture}\;\;+\frac{1}{2}\;\begin{picture}(50.0,25.0)(0.0,-4.0)
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \raise-34.0pt\hbox to0.0pt{\kern
25.0pt\makebox(0.0,0.0)[cc]{$(c)$}\hss} \ignorespaces\end{picture}$ (129)
$\displaystyle-\frac{1}{2}\;\;\;\;\;\;\;\begin{picture}(108.0,60.0)(0.0,-4.0)\put(-5.5,15.0){\makebox(0.0,0.0)[]{${\mathbf{p}}_{2}$}}
\put(-6.5,8.0){\makebox(0.0,0.0)[]{$+$}}
\put(-6.5,2.0){\makebox(0.0,0.0)[]{${\mathbf{q}}^{\prime}$}}
\put(43.0,18.0){\makebox(0.0,0.0)[]{${\mathbf{p}}_{2}$}}
\put(43.0,-18.0){\makebox(0.0,0.0)[]{${\mathbf{p}}_{2}$}}
\put(59.0,18.0){\makebox(0.0,0.0)[]{${\mathbf{p}}_{1}$}}
\put(60.0,-18.0){\makebox(0.0,0.0)[]{${\mathbf{p}}_{1}$}}
\put(106.0,15.0){\makebox(0.0,0.0)[]{${\mathbf{p}}_{1}$}}
\put(107.0,8.0){\makebox(0.0,0.0)[]{$+$}}
\put(107.0,1.0){\makebox(0.0,0.0)[]{${\mathbf{q}}$}} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \end{picture}\;\;$ $\displaystyle=$
$\displaystyle\;\;-\frac{1}{2}\;\begin{picture}(100.0,60.0)(0.0,-4.0)
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \raise-20.0pt\hbox
to0.0pt{\kern 50.0pt\makebox(0.0,0.0)[cc]{$(a)$}\hss}
\ignorespaces\end{picture}\;\;-\;\;\begin{picture}(100.0,60.0)(0.0,-4.0)
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \raise-20.0pt\hbox to0.0pt{\kern
50.0pt\makebox(0.0,0.0)[cc]{$(b)$}\hss} \ignorespaces\end{picture}\;$ (130)
$\displaystyle\;\;-\frac{1}{2}\;\;\begin{picture}(100.0,50.0)(0.0,-4.0)
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \raise-20.0pt\hbox
to0.0pt{\kern 50.0pt\makebox(0.0,0.0)[cc]{$(c)$}\hss}
\ignorespaces\end{picture}$
Let us start with the diagrams to the left of the equal signs in (128-130). We
will for now keep the bosonic propagators general, that is, allowing them to
carry frequencies. After that, we will discuss the (a) diagrams in (128-130),
followed by the (b) diagrams and then the (c) diagrams.
In (128), let the vertical boson propagator, denoted by $B_{1}$, carry
momentum ${\mathbf{q}}$ (upward) and frequency $\nu$. Let the horizontal boson
propagator, denoted by $B_{2}$, carry momentum
${\mathbf{q}}^{\prime}\equiv-({\mathbf{p}}_{1}+{\mathbf{p}}_{2}+{\mathbf{q}})$
(leftward) and frequency $\nu^{\prime}$. Using techniques described in section
III.5 for frequency summation, one obtains the following generic result
$\displaystyle\frac{\beta
V}{4}\frac{2}{\beta^{2}}\sum_{\nu,\nu^{\prime}}\int\frac{d{\mathbf{p}}_{1}}{(2\pi)^{3}}\frac{d{\mathbf{p}}_{2}}{(2\pi)^{3}}\frac{d{\mathbf{q}}}{(2\pi)^{3}}\frac{B_{1}({\mathbf{q}},\nu)}{-i\nu+\varepsilon_{{\mathbf{p}}_{1}+{\mathbf{q}}}-\varepsilon_{{\mathbf{p}}_{1}}}\frac{B_{2}({\mathbf{q}}^{\prime},\nu^{\prime})}{i\nu+\varepsilon_{{\mathbf{p}}_{2}+{\mathbf{q}}}-\varepsilon_{{\mathbf{p}}_{2}}}\left\\{\frac{n_{{\mathbf{p}}_{2}}-n_{{\mathbf{p}}_{1}}}{\varepsilon_{{\mathbf{p}}_{2}}-\varepsilon_{{\mathbf{p}}_{1}}-i(\nu+\nu^{\prime})}\right.$
$\displaystyle\left.\hskip
15.0pt+\frac{n_{{\mathbf{p}}_{1}}-n_{{\mathbf{p}}_{1}+{\mathbf{q}}^{\prime}}}{\varepsilon_{{\mathbf{p}}_{1}+{\mathbf{q}}^{\prime}}-\varepsilon_{{\mathbf{p}}_{1}}-i\nu^{\prime}}+\frac{n_{{\mathbf{p}}_{2}}-n_{{\mathbf{p}}_{2}+{\mathbf{q}}^{\prime}}}{\varepsilon_{{\mathbf{p}}_{2}+{\mathbf{q}}^{\prime}}-\varepsilon_{{\mathbf{p}}_{2}}+i\nu^{\prime}}+\frac{n_{{\mathbf{p}}_{2}+{\mathbf{q}}}-n_{{\mathbf{p}}_{1}+{\mathbf{q}}}}{\varepsilon_{{\mathbf{p}}_{2}+{\mathbf{q}}}-\varepsilon_{{\mathbf{p}}_{1}+{\mathbf{q}}}+i(\nu-\nu^{\prime})}\right\\}\;,$
(131)
or equivalently,
$\displaystyle\frac{\beta
V}{4}\frac{2}{\beta^{2}}\sum_{\nu,\nu^{\prime}}\int\frac{d{\mathbf{p}}_{1}}{(2\pi)^{3}}\frac{d{\mathbf{p}}_{2}}{(2\pi)^{3}}\frac{d{\mathbf{q}}}{(2\pi)^{3}}\frac{B_{1}({\mathbf{q}},\nu)}{-i\nu^{\prime}+\varepsilon_{{\mathbf{p}}_{1}+{\mathbf{q}}^{\prime}}-\varepsilon_{{\mathbf{p}}_{1}}}\frac{B_{2}({\mathbf{q}}^{\prime},\nu^{\prime})}{i\nu^{\prime}+\varepsilon_{{\mathbf{p}}_{2}+{\mathbf{q}}^{\prime}}-\varepsilon_{{\mathbf{p}}_{2}}}\left\\{\frac{n_{{\mathbf{p}}_{2}}-n_{{\mathbf{p}}_{1}}}{\varepsilon_{{\mathbf{p}}_{2}}-\varepsilon_{{\mathbf{p}}_{1}}-i(\nu+\nu^{\prime})}\right.$
$\displaystyle\left.\hskip
15.0pt+\frac{n_{{\mathbf{p}}_{1}}-n_{{\mathbf{p}}_{1}+{\mathbf{q}}}}{\varepsilon_{{\mathbf{p}}_{1}+{\mathbf{q}}}-\varepsilon_{{\mathbf{p}}_{1}}-i\nu}+\frac{n_{{\mathbf{p}}_{2}}-n_{{\mathbf{p}}_{2}+{\mathbf{q}}}}{\varepsilon_{{\mathbf{p}}_{2}+{\mathbf{q}}}-\varepsilon_{{\mathbf{p}}_{2}}+i\nu}+\frac{n_{{\mathbf{p}}_{2}+{\mathbf{q}}}-n_{{\mathbf{p}}_{1}+{\mathbf{q}}}}{\varepsilon_{{\mathbf{p}}_{2}+{\mathbf{q}}}-\varepsilon_{{\mathbf{p}}_{1}+{\mathbf{q}}}+i(\nu-\nu^{\prime})}\right\\}\;.$
(132)
Note that the factor $2$ associated with $\frac{2}{\beta^{2}}$ comes from the
two possible spin states for an electron.
In (129), let the boson propagator on top, denoted by $B_{1}$, carry momentum
${\mathbf{q}}$ (towards lower-right) and frequency $\nu$. Let the boson
propagator at the bottom, denoted by $B_{2}$, carry momentum
${\mathbf{q}}^{\prime}\equiv({\mathbf{p}}_{1}-{\mathbf{p}}_{2})$ (towards
upper-left) and frequency $\nu^{\prime}$. Using techniques described in
section III.5 for frequency summation, one obtains the following generic
result
$\displaystyle\frac{\beta
V}{2}\frac{2}{\beta^{2}}\sum_{\nu,\nu^{\prime}}\int\frac{d{\mathbf{p}}_{1}}{(2\pi)^{3}}\frac{d{\mathbf{p}}_{2}}{(2\pi)^{3}}\frac{d{\mathbf{q}}}{(2\pi)^{3}}\frac{B_{1}({\mathbf{q}},\nu)}{-i\nu+\varepsilon_{{\mathbf{p}}_{1}+{\mathbf{q}}}-\varepsilon_{{\mathbf{p}}_{1}}}\frac{B_{2}({\mathbf{q}}^{\prime},\nu^{\prime})}{-i\nu^{\prime}+\varepsilon_{{\mathbf{p}}_{1}+{\mathbf{q}}}-\varepsilon_{{\mathbf{p}}_{2}+{\mathbf{q}}}}\left\\{-\beta
n_{{\mathbf{p}}_{1}+{\mathbf{q}}}(1-n_{{\mathbf{p}}_{1}+{\mathbf{q}}})\right.$
$\displaystyle\left.\hskip
15.0pt-\frac{n_{{\mathbf{p}}_{1}+{\mathbf{q}}}-n_{{\mathbf{p}}_{2}+{\mathbf{q}}}}{-i\nu^{\prime}+\varepsilon_{{\mathbf{p}}_{1}+{\mathbf{q}}}-\varepsilon_{{\mathbf{p}}_{2}+{\mathbf{q}}}}+\frac{n_{{\mathbf{p}}_{1}}-n_{{\mathbf{p}}_{1}+{\mathbf{q}}}}{-i\nu+\varepsilon_{{\mathbf{p}}_{1}+{\mathbf{q}}}-\varepsilon_{{\mathbf{p}}_{1}}}+\frac{n_{{\mathbf{p}}_{1}}-n_{{\mathbf{p}}_{2}+{\mathbf{q}}}}{i(\nu-\nu^{\prime})+\varepsilon_{{\mathbf{p}}_{1}}-\varepsilon_{{\mathbf{p}}_{2}+{\mathbf{q}}}}\right\\}\;.$
(133)
In (130), let the boson propagator at the right hand side, denoted by $B_{1}$,
carry momentum ${\mathbf{q}}$ (downwards) and frequency $\nu$. Let the boson
propagator at the left hand side, denoted by $B_{2}$, carry momentum
${\mathbf{q}}^{\prime}$ (also downwards) and frequency $\nu^{\prime}$. Using
techniques described in section III.5 for frequency summation, one obtains the
following generic result
$\displaystyle-\frac{\beta
V}{2}\frac{2^{2}}{\beta^{2}}\frac{-1}{2\beta\int\frac{d{\mathbf{p}}}{(2\pi)^{3}}n_{{\mathbf{p}}}(1-n_{{\mathbf{p}}})}\sum_{\nu,\nu^{\prime}}\int\frac{d{\mathbf{p}}_{1}}{(2\pi)^{3}}\frac{d{\mathbf{p}}_{2}}{(2\pi)^{3}}\frac{d{\mathbf{q}}}{(2\pi)^{3}}\frac{d{\mathbf{q}}^{\prime}}{(2\pi)^{3}}\frac{B_{1}({\mathbf{q}},\nu)}{-i\nu+\varepsilon_{{\mathbf{p}}_{1}+{\mathbf{q}}}-\varepsilon_{{\mathbf{p}}_{1}}}\frac{B_{2}({\mathbf{q}}^{\prime},\nu^{\prime})}{-i\nu^{\prime}+\varepsilon_{{\mathbf{p}}_{2}+{\mathbf{q}}^{\prime}}-\varepsilon_{{\mathbf{p}}_{2}}}$
$\displaystyle\hskip
15.0pt\left\\{\beta^{2}n_{{\mathbf{p}}_{1}}(1-n_{{\mathbf{p}}_{1}})n_{{\mathbf{p}}_{2}}(1-n_{{\mathbf{p}}_{2}})+\beta
n_{{\mathbf{p}}_{1}}(1-n_{{\mathbf{p}}_{1}})\frac{n_{{\mathbf{p}}_{2}+{\mathbf{q}}^{\prime}}-n_{{\mathbf{p}}_{2}}}{-i\nu^{\prime}+\varepsilon_{{\mathbf{p}}_{2}+{\mathbf{q}}^{\prime}}-\varepsilon_{{\mathbf{p}}_{2}}}\right.$
$\displaystyle\hskip 20.0pt\left.+\beta
n_{{\mathbf{p}}_{2}}(1-n_{{\mathbf{p}}_{2}})\frac{n_{{\mathbf{p}}_{1}+{\mathbf{q}}}-n_{{\mathbf{p}}_{1}}}{-i\nu+\varepsilon_{{\mathbf{p}}_{1}+{\mathbf{q}}}-\varepsilon_{{\mathbf{p}}_{1}}}+\frac{n_{{\mathbf{p}}_{2}+{\mathbf{q}}^{\prime}}-n_{{\mathbf{p}}_{2}}}{-i\nu^{\prime}+\varepsilon_{{\mathbf{p}}_{2}+{\mathbf{q}}^{\prime}}-\varepsilon_{{\mathbf{p}}_{2}}}\frac{n_{{\mathbf{p}}_{1}+{\mathbf{q}}}-n_{{\mathbf{p}}_{1}}}{-i\nu+\varepsilon_{{\mathbf{p}}_{1}+{\mathbf{q}}}-\varepsilon_{{\mathbf{p}}_{1}}}\right\\}\;,$
(134)
where the factor
$-1/\left\\{2\beta\int\frac{d{\mathbf{p}}}{(2\pi)^{3}}n_{\mathbf{p}}(1-n_{\mathbf{p}})\right\\}=D_{0}^{-1}({\mathbf{q}}=0,\nu=0)$
arises from the fact that in a translationally invariant system, momentum
conservation at each vertex demands that the inverse density correlator must
carry zero momentum and frequency.
For the diagram (a) on the right hand side of eq. (128),
$B_{1}({\mathbf{q}},\nu)=4\pi e^{2}/{\mathbf{q}}^{2}$ and
$B_{2}({\mathbf{q}}^{\prime},\nu^{\prime})=4\pi
e^{2}/({\mathbf{p}}_{1}+{\mathbf{p}}_{2}+{\mathbf{q}})^{2}$ are both frequency
independent. Thus, one may sum over both $\nu$ and $\nu^{\prime}$. And upon
doing so, we obtain
$\frac{1}{4\beta}\;\begin{picture}(40.0,20.0)(0.0,-4.0) \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){}
\end{picture}\;=\;{V}\int\frac{d{\mathbf{p}}_{1}}{(2\pi)^{3}}\frac{d{\mathbf{p}}_{2}}{(2\pi)^{3}}\frac{d{\mathbf{q}}}{(2\pi)^{3}}\frac{(4\pi
e^{2})^{2}}{{\mathbf{q}}^{2}({\mathbf{p}}_{1}+{\mathbf{p}}_{2}+{\mathbf{q}})^{2}}\frac{n_{{\mathbf{p}}_{1}}n_{{\mathbf{p}}_{2}}(1-n_{{\mathbf{p}}_{1}+{\mathbf{q}}})(1-n_{{\mathbf{p}}_{2}+{\mathbf{q}}})}{\varepsilon_{{\mathbf{p}}_{1}+{\mathbf{q}}}+\varepsilon_{{\mathbf{p}}_{2}+{\mathbf{q}}}-\varepsilon_{{\mathbf{p}}_{1}}-\varepsilon_{{\mathbf{p}}_{2}}}\;.\vspace*{8pt}$
If we divide the quantity above by $N_{e}$ and then take the zero temperature
limit, it gives rise to (using $p_{F}$ as the unit for momentum)
$\displaystyle\frac{mp_{F}^{3}V}{N_{e}(2\pi)^{9}}(4\pi
e^{2})^{2}\int_{|{\mathbf{p}}_{i}|<1,{\mathbf{p}}_{i}+{\mathbf{q}}|>1}\;d{\mathbf{q}}\;d{\mathbf{p}}_{1}\;d{\mathbf{p}}_{2}\frac{1}{{\mathbf{q}}^{2}({\mathbf{p}}_{1}+{\mathbf{p}}_{2}+{\mathbf{q}})^{2}}\frac{1}{{\mathbf{q}}^{2}+{\mathbf{q}}\cdot({\mathbf{p}}_{1}+{\mathbf{p}}_{2})}$
$\displaystyle=\left[\frac{3}{16\pi^{5}}\int_{|{\mathbf{p}}_{i}|<1,{\mathbf{p}}_{i}+{\mathbf{q}}|>1}\;d{\mathbf{q}}\;d{\mathbf{p}}_{1}\;d{\mathbf{p}}_{2}\frac{1}{{\mathbf{q}}^{2}({\mathbf{p}}_{1}+{\mathbf{p}}_{2}+{\mathbf{q}})^{2}}\frac{1}{{\mathbf{q}}^{2}+{\mathbf{q}}\cdot({\mathbf{p}}_{1}+{\mathbf{p}}_{2})}\right]\;\;{\rm
Rydberg},$
which is exactly the $\epsilon_{b}^{(2)}$ term of Gell-Mann and
Brueckner.Gell-Mann and Brueckner (1957)
Diagram (a) on the right hand side of eq. (129) is one of the anomalous
diagramsKohn and Luttinger (1960); Luttinger and Ward (1960) that give rise to
finite contribution as the $T\to 0$ limit is taken within finite temperature
formalism but are absent within zero temperature formalism. For this diagram,
$B_{1}({\mathbf{q}},\nu)=4\pi e^{2}/{\mathbf{q}}^{2}$ and
$B_{2}({\mathbf{q}}^{\prime},\nu^{\prime})=4\pi
e^{2}/({\mathbf{p}}_{1}-{\mathbf{p}}_{2})^{2}$ are both frequency independent.
Thus, one may evaluate its zero temperature contribution by summing over both
$\nu$ and $\nu^{\prime}$. Upon doing so, one obtains from diagram (a) of (129)
the following anomalous contribution
$\displaystyle\frac{1}{2\beta}\;\begin{picture}(40.0,20.0)(0.0,-4.0)
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \end{picture}\;\;$ $\displaystyle=$
$\displaystyle-(V)\int\frac{d{\mathbf{p}}_{1}\,d{\mathbf{p}}_{2}\,d{\mathbf{q}}}{(2\pi)^{9}}\frac{4\pi
e^{2}}{{\mathbf{q}}^{2}}\frac{4\pi
e^{2}}{({\mathbf{p}}_{1}-{\mathbf{p}}_{2})^{2}}\,\beta
n_{{\mathbf{p}}_{1}+{\mathbf{q}}}(1-n_{{\mathbf{p}}_{1}+{\mathbf{q}}})n_{{\mathbf{p}}_{2}+{\mathbf{q}}}n_{{\mathbf{p}}_{1}}$
(135) $\displaystyle=$
$\displaystyle-(V)\int\frac{d{\mathbf{p}}_{1}\,d{\mathbf{q}}\,d{\mathbf{q}}^{\prime}}{(2\pi)^{9}}\frac{4\pi
e^{2}}{{\mathbf{q}}^{2}}\frac{4\pi e^{2}}{({\mathbf{q}}^{\prime})^{2}}\,\beta
n_{{\mathbf{p}}_{1}}(1-n_{{\mathbf{p}}_{1}})n_{{\mathbf{p}}_{1}+{\mathbf{q}}^{\prime}}n_{{\mathbf{p}}_{1}+{\mathbf{q}}}\;,$
where the last expression is obtained via the following change of variables:
${\mathbf{p}}_{1(2)}\to-{\mathbf{p}}_{1(2)}-{\mathbf{q}}$, eliminating
${\mathbf{p}}_{2}$ by
${\mathbf{q}}^{\prime}\equiv{\mathbf{p}}_{1}-{\mathbf{p}}_{2}$, and
${\mathbf{q}}^{\prime}\to-{\mathbf{q}}^{\prime}$. This expression can be
further simplified via the following definition
$f({\mathbf{p}})\equiv\int\frac{d{\mathbf{q}}}{(2\pi)^{3}}\frac{1}{{\mathbf{q}}^{2}}n_{{\mathbf{p}}+{\mathbf{q}}}\;,$
leading to
$\displaystyle\frac{1}{2\beta}\;\begin{picture}(40.0,20.0)(0.0,-4.0)
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \end{picture}\;\;$ $\displaystyle=$
$\displaystyle-V(4\pi e^{2})^{2}\int\frac{d{\mathbf{p}}_{1}}{(2\pi)^{3}}\beta
n_{{\mathbf{p}}_{1}}(1-n_{{\mathbf{p}}_{1}})f^{2}({\mathbf{p}}_{1})$ (136)
$\displaystyle=$ $\displaystyle-V(4\pi
e^{2})^{2}\int\frac{d{\mathbf{p}}_{1}}{(2\pi)^{3}}\frac{\partial
n_{{\mathbf{p}}_{1}}}{\partial\mu}f^{2}({\mathbf{p}}_{1})\;.\vspace*{18pt}$
As mentioned earlier, each two-particle reducible diagram within our formalism
is accompanied by a corresponding diagram that will eliminate anomalous
contribution when applicable. Diagrams in eq. (130), appearing only under the
effective action formalism, are such diagrams. They are neither present in the
zero temperature formalism of Goldstone Goldstone and Brueckner Brueckner and
Levinson (1955) nor in the finite temperature formalism of Luttinger and
Ward.Luttinger and Ward (1960) Diagram (a) on the right hand side of (130)
corresponds to the case $B_{1}({\mathbf{q}},\nu)=4\pi e^{2}/{\mathbf{q}}^{2}$
and $B_{2}({\mathbf{q}}^{\prime},\nu^{\prime})=4\pi
e^{2}/({\mathbf{q}}^{\prime})^{2}$, both being frequency independent. Thus,
upon summing over both $\nu$ and $\nu^{\prime}$, one obtains
$\displaystyle-\frac{1}{2\beta}\;\begin{picture}(100.0,40.0)(0.0,-4.0)
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \end{picture}$
$\displaystyle=$ $\displaystyle\frac{2(-1)^{2}V(4\pi
e^{2})^{2}}{2\beta\int\frac{d{\mathbf{p}}}{(2\pi)^{3}}n_{\mathbf{p}}(1-n_{\mathbf{p}})}\int\frac{d{\mathbf{p}}_{1}\,d{\mathbf{p}}_{2}\,d{\mathbf{q}}\,d{\mathbf{q}}^{\prime}}{(2\pi)^{12}}\frac{4\pi
e^{2}}{q^{2}}\frac{4\pi e^{2}}{{q^{\prime}}^{2}}\times$ (137)
$\displaystyle\times\beta^{2}n_{{\mathbf{p}}_{1}}(1-n_{{\mathbf{p}}_{1}})n_{{\mathbf{p}}_{2}}(1-n_{{\mathbf{p}}_{2}})n_{{\mathbf{p}}_{1}+{\mathbf{q}}}n_{{\mathbf{p}}_{2}+{\mathbf{q}}^{\prime}}$
$\displaystyle=$ $\displaystyle\frac{V(4\pi
e^{2})^{2}}{\int\frac{d{\mathbf{p}}}{(2\pi)^{3}}\frac{\partial
n_{\mathbf{p}}}{\partial\mu}}\left[\int\frac{d{\mathbf{p}}}{(2\pi)^{3}}\frac{\partial
n_{\mathbf{p}}}{\partial\mu}f({\mathbf{p}})\right]^{2}\;.$ (138)
When combined with eq. (136), one obtains
$\frac{1}{2\beta}\;\begin{picture}(40.0,20.0)(0.0,-4.0) \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){}
\end{picture}\;\;-\frac{1}{2\beta}\;\begin{picture}(100.0,30.0)(0.0,-4.0)
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \end{picture}=-V(4\pi
e^{2})^{2}\left(\int\frac{d{\mathbf{p}}}{(2\pi)^{3}}\frac{\partial
n_{\mathbf{p}}}{\partial\mu}\right)\overline{\left(f-\overline{f}\right)^{2}}\;,\vspace*{10pt}$
(139)
where the overline symbol is defined as
$\overline{f}\equiv\int\frac{d{\mathbf{p}}}{(2\pi)^{3}}\frac{\partial
n_{\mathbf{p}}}{\partial\mu}f({\mathbf{p}})$. Apparently, the expression (139)
is in general negative unless $f=\overline{f}$. We will show that this happens
only at zero temperature and only if the system has spherical symmetry.
In order to have $f({\mathbf{p}})=\overline{f}$, ${\mathbf{p}}$ can only have
support at a constant $f({\mathbf{p}})$ surface. This is achieved when $T\to
0$, where $n_{\mathbf{p}}\to\theta(\mu-\varepsilon_{\mathbf{p}})$ and
$\partial n_{\mathbf{p}}/\partial\mu=\delta(\mu-\varepsilon_{\mathbf{p}})$
forcing ${\mathbf{p}}$ to lie on a constant $\varepsilon_{\mathbf{p}}$
surface. Furthermore, $f({\mathbf{p}})=\overline{f}$ demands that
$f({\mathbf{p}})$ depends only on the magnitude of ${\mathbf{p}}$, i.e.,
$f({\mathbf{p}})=f(p)$. This can only be achieved if $n_{\mathbf{p}}$ depends
only on $|{\mathbf{p}}|=p$. For the HEG system,
$\varepsilon_{\mathbf{p}}={\mathbf{p}}^{2}/2m$, thus $n_{\mathbf{p}}=n(p)$ and
$\mu=p_{F}^{2}/2m$. We therefore have $\partial
n_{\mathbf{p}}/\partial\mu=m\delta(p-p_{F})/p_{F}$, fixing the length of
${\mathbf{p}}$. Now
$n_{{\mathbf{p}}+{\mathbf{q}}}=\theta(\mu-({\mathbf{p}}+{\mathbf{q}})^{2}/2m)|_{|{\mathbf{p}}|=p_{F}}=\theta(-\cos\vartheta-\frac{q}{2p_{F}})$,
with $\vartheta$ being the angle between ${\mathbf{q}}$ and ${\mathbf{p}}$.
Thus, in the integral defining $f({\mathbf{p}})$, although $\hat{\mathbf{p}}$
defines the $\hat{z}$ direction of vector ${\mathbf{q}}$ the integral is
independent of the choice of $\hat{\mathbf{p}}$. Therefore, the anomalous
contribution (136) may be written as (when $T\to 0$)
$-(V)\left[\int\frac{d{\mathbf{p}}_{1}}{(2\pi)^{3}}\frac{m}{p_{F}}\delta(p_{1}-p_{F})\right]\left[\int_{-1}^{1}dx\int_{0}^{\infty}\frac{q^{2}\,dq}{(2\pi)^{2}}\frac{4\pi
e^{2}}{q^{2}}\;\theta\big{(}-x-\frac{q}{2p_{F}}\big{)}\right]^{2}\;,$ (140)
while the corresponding subtraction term (138) may be written as
$\displaystyle\frac{V}{\int\frac{d{\mathbf{p}}_{1}}{(2\pi)^{3}}\frac{m}{p_{F}}\delta(p_{1}-p_{F})}\left[\int\frac{d{\mathbf{p}}_{1}}{(2\pi)^{3}}\frac{m}{p_{F}}\delta(p_{1}-p_{F})\right]^{2}\left[\int_{-1}^{1}dx\int_{0}^{\infty}\frac{q^{2}\,dq}{(2\pi)^{2}}\frac{4\pi
e^{2}}{q^{2}}\;\theta\big{(}-x-\frac{q}{2p_{F}}\big{)}\right]^{2}\;$
$\displaystyle=V\left[\int\frac{d{\mathbf{p}}_{1}}{(2\pi)^{3}}\frac{m}{p_{F}}\delta(p_{1}-p_{F})\right]\left[\int_{-1}^{1}dx\int_{0}^{\infty}\frac{q^{2}\,dq}{(2\pi)^{2}}\frac{4\pi
e^{2}}{q^{2}}\;\theta\big{(}-x-\frac{q}{2p_{F}}\big{)}\right]^{2}\;,$ (141)
cancelling exactly the anomalous contribution (140) in the HEG case.
Within the framework of Luttinger et al.,Kohn and Luttinger (1960); Luttinger
and Ward (1960) the anomalous contribution is cancelled by the chemical
potential shift, the difference between $\mu(T\to 0)$, under finite
temperature formalism, and $\mu(T=0)=p_{F}^{2}/2m$, under zero temperature
formalism. Within the current finite temperature framework, however, the
$\mu(T\to 0)$ is identical to $\mu(T=0)$ and the cancellation of anomalous
contribution is explicit. As we will show later, however, diagram (b) of (129)
can’t be cancelled by diagram (b) of (130). This is because diagram (b) of
(129) is not an anomalous diagram.
Before moving onto (b) diagrams in (128-130), let us remark that diagrams
within our formalism, i.e., diagrams on the left hand side of (128-130) with
$B_{1(2)}\to\tilde{\cal D}_{0}$ yield only finite contribution. The
decomposition made in (128-130), however, may introduce divergence as we will
illustrate using the (b) diagrams that correspond to the main results of
DuBois.DuBois (1959)
For the (b) diagram of (128), $B_{2}({\mathbf{q}}^{\prime},\nu^{\prime})=4\pi
e^{2}/({\mathbf{p}}_{1}+{\mathbf{p}}_{2}+{\mathbf{q}})^{2}$ while
$B_{1}({\mathbf{q}},\nu)=(4\pi
e^{2})^{2}D_{0}({\mathbf{q}},\nu)/\left\\{{\mathbf{q}}^{2}[{\mathbf{q}}^{2}-4\pi
e^{2}D_{0}({\mathbf{q}},\nu)]\right\\}$. Therefore, one may sum over
$\nu^{\prime}$ to simplify the expression. Using again the methods described
in section III.5 for frequency summation, one obtains
$\displaystyle\frac{1}{2\beta}\;\;\begin{picture}(50.0,25.0)(0.0,-4.0)
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\end{picture}\;$ $\displaystyle=$ $\displaystyle\;2\frac{V}{4}\frac{(4\pi
e^{2})^{3}}{(2\pi)^{9}}\frac{2}{\beta}\sum_{\nu}\int
d{\mathbf{p}}_{1}d{\mathbf{p}}_{2}d{\mathbf{q}}\frac{D_{0}({\mathbf{q}},\nu)}{{\mathbf{q}}^{2}({\mathbf{q}}^{2}-4\pi
e^{2}D_{0}({\mathbf{q}},\nu))}\frac{1}{({\mathbf{p}}_{1}+{\mathbf{p}}_{2}+{\mathbf{q}})^{2}}\times$
(142) $\displaystyle\hskip
60.0pt\times\frac{n_{{\mathbf{p}}_{1}+{\mathbf{q}}}-n_{{\mathbf{p}}_{1}}}{-i\nu+\varepsilon_{{\mathbf{p}}_{1}+{\mathbf{q}}}-\varepsilon_{{\mathbf{p}}_{1}}}\,\frac{n_{{\mathbf{p}}_{2}+{\mathbf{q}}}-n_{{\mathbf{p}}_{2}}}{i\nu+\varepsilon_{{\mathbf{p}}_{2}+{\mathbf{q}}}-\varepsilon_{{\mathbf{p}}_{2}}}$
$\displaystyle=$ $\displaystyle\;\frac{V}{\beta}\frac{(4\pi
e^{2})^{3}}{(2\pi)^{9}}\sum_{\nu}\int
d{\mathbf{p}}_{1}d{\mathbf{p}}_{2}d{\mathbf{q}}\frac{D_{0}({\mathbf{q}},\nu)}{{\mathbf{q}}^{2}({\mathbf{q}}^{2}-4\pi
e^{2}D_{0}({\mathbf{q}},\nu))}\frac{1}{({\mathbf{p}}_{1}-{\mathbf{p}}_{2})^{2}}\times$
$\displaystyle\hskip
60.0pt\times\frac{n_{{\mathbf{p}}_{1}+{\mathbf{q}}}-n_{{\mathbf{p}}_{1}}}{-i\nu+\varepsilon_{{\mathbf{p}}_{1}+{\mathbf{q}}}-\varepsilon_{{\mathbf{p}}_{1}}}\,\frac{n_{{\mathbf{p}}_{2}+{\mathbf{q}}}-n_{{\mathbf{p}}_{2}}}{-i\nu+\varepsilon_{{\mathbf{p}}_{2}+{\mathbf{q}}}-\varepsilon_{{\mathbf{p}}_{2}}}\;,$
where the last expression is obtained by changing variables:
${\mathbf{p}}_{2}+{\mathbf{q}}\to-{\mathbf{p}}_{2}^{\prime}$ followed by
${\mathbf{p}}_{2}^{\prime}\to{\mathbf{p}}_{2}$. For the high density
expansion, where $e^{2}$ is treated as a small parameter, the major
contribution in (142) comes from the region ${\mathbf{q}}\to 0$. One thus
writes
$n_{{\mathbf{p}}+{\mathbf{q}}}-n_{\mathbf{p}}\xrightarrow[|{\mathbf{q}}|\to
0]{}(\varepsilon_{{\mathbf{p}}+q}-\varepsilon_{{\mathbf{p}}})\frac{\partial
n_{{\mathbf{p}}}}{\partial\varepsilon_{{\mathbf{p}}}}=-\beta
n_{{\mathbf{p}}}(1-n_{{\mathbf{p}}})(\varepsilon_{{\mathbf{p}}+{\mathbf{q}}}-\varepsilon_{{\mathbf{p}}})\xrightarrow[T\to
0]{}-q\cos\vartheta\,\delta(p-p_{F})\;,$
where $\vartheta$ is the angle between ${\mathbf{p}}$ and ${\mathbf{q}}$. As
$T\to 0$,
$\frac{1}{\beta}\sum_{n}F(\nu_{n})\to\int_{-\infty}^{\infty}\frac{d\nu}{2\pi}F(\nu)$
if $F(\nu)$ does not have pole strength greater than one. Making a change of
variable $\nu\equiv u|{\mathbf{q}}|/m$ and treating $q$ as a small quantity,
the major contribution of the (b) diagram of (128) is given by
$\displaystyle\frac{1}{2\beta}\;\;\begin{picture}(50.0,25.0)(0.0,-4.0)
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \end{picture}$
$\displaystyle\approx$ $\displaystyle\frac{V}{m}\frac{(4\pi
e^{2})^{3}}{(2\pi)^{9}}\int_{-\infty}^{\infty}\frac{du}{2\pi}\int
d{\mathbf{p}}_{1}\,d{\mathbf{p}}_{2}\frac{m\delta(p_{1}-p_{F})\cos\vartheta_{1}}{-iu+p_{F}\cos\vartheta_{1}}\frac{m\delta(p_{2}-p_{F})\cos\vartheta_{2}}{iu+p_{F}\cos\vartheta_{2}}\times$
(143) $\displaystyle\hskip
10.0pt\frac{1}{({\mathbf{p}}_{1}-{\mathbf{p}}_{2})^{2}}\int_{0}^{q_{c}}4\pi
qdq\frac{D_{0}(q,uq/m)}{q^{2}-4\pi e^{2}D_{0}(q,uq/m)}$ $\displaystyle\approx$
$\displaystyle Vm\,\frac{(4\pi
e^{2})^{3}}{(2\pi)^{9}}\int_{-\infty}^{\infty}\frac{du}{2\pi}\int_{-1}^{1}\frac{\cos\vartheta_{1}\,d\cos\vartheta_{1}}{-iu+p_{F}\cos\vartheta_{1}}\,\frac{\cos\vartheta_{2}\,d\cos\vartheta_{2}}{-iu+p_{F}\cos\vartheta_{2}}\times$
$\displaystyle\hskip
10.0pt\frac{(2\pi)^{2}p_{F}^{2}/2}{|\cos\vartheta_{1}-\cos\vartheta_{2}|}\int
4\pi qdq\frac{D_{0}(q,uq/m)}{q^{2}-4\pi e^{2}D_{0}(q,uq/m)}\;.$
Apparently, the $1/|\cos\vartheta_{1}-\cos\vartheta_{2}|$ factor in the
integrand causes an undesirable divergence in the $\vartheta$ integral that is
absent if one does not decompose the diagrams using (127). A finite result
emerges only when one combines (143) with the (b) diagram of (129), which we
soon turn to.
Before elaborating on the evaluation of the (b) diagram of (129), we wish to
point out that this is the type of diagram (two-particle reducible ones) that
was missed in Hedin’s approximation Hedin (1965) but should have been included
for the exactness of the theory. For this diagram,
$B_{2}({\mathbf{q}}^{\prime},\nu^{\prime})=4\pi
e^{2}/({\mathbf{p}}_{1}-{\mathbf{p}}_{2})^{2}$ while
$B_{1}({\mathbf{q}},\nu)=(4\pi
e^{2})^{2}D_{0}({\mathbf{q}},\nu)/\left\\{{\mathbf{q}}^{2}[{\mathbf{q}}^{2}-4\pi
e^{2}D_{0}({\mathbf{q}},\nu)]\right\\}$. Summing over $\nu^{\prime}$ via
methods described in section III.5, one obtains
$\displaystyle\frac{1}{\beta}\;\begin{picture}(50.0,25.0)(0.0,-4.0)
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \end{picture}$
$\displaystyle=$ $\displaystyle V\frac{(4\pi
e^{2})^{3}}{(2\pi)^{9}}\frac{2}{\beta}\sum_{\nu}\int
d{\mathbf{p}}_{1}d{\mathbf{p}}_{2}\,d{\mathbf{q}}\frac{1}{({\mathbf{p}}_{1}-{\mathbf{p}}_{2})^{2}}\frac{D_{0}(q,\nu)}{q^{2}(q^{2}-4\pi
e^{2}D_{0}(q,\nu))}\times$ $\displaystyle\hskip
10.0pt\left[-\frac{n_{{\mathbf{p}}_{2}+{\mathbf{q}}}(n_{{\mathbf{p}}_{1}+{\mathbf{q}}}-n_{{\mathbf{p}}_{1}})}{(-i\nu+\varepsilon_{{\mathbf{p}}_{1}+{\mathbf{q}}}-\varepsilon_{{\mathbf{p}}_{1}})^{2}}-\frac{\beta
n_{{\mathbf{p}}_{1}}(1-n_{{\mathbf{p}}_{1}})n_{{\mathbf{p}}_{2}}}{-i\nu+\varepsilon_{{\mathbf{p}}_{1}}-\varepsilon_{{\mathbf{p}}_{1}+{\mathbf{q}}}}\right]$
$\displaystyle=$ $\displaystyle\frac{V}{\beta}\frac{(4\pi
e^{2})^{3}}{(2\pi)^{9}}\sum_{\nu}\int
d{\mathbf{p}}_{1}d{\mathbf{p}}_{2}\,d{\mathbf{q}}\frac{1}{({\mathbf{p}}_{1}-{\mathbf{p}}_{2})^{2}}\frac{D_{0}(q,\nu)}{q^{2}(q^{2}-4\pi
e^{2}D_{0}(q,\nu))}\times$ $\displaystyle\hskip
10.0pt\left[-\frac{(n_{{\mathbf{p}}_{2}+{\mathbf{q}}}-n_{{\mathbf{p}}_{2}})(n_{{\mathbf{p}}_{1}+{\mathbf{q}}}-n_{{\mathbf{p}}_{1}})}{(-i\nu+\varepsilon_{{\mathbf{p}}_{1}+{\mathbf{q}}}-\varepsilon_{{\mathbf{p}}_{1}})^{2}}+\frac{2\beta
n_{{\mathbf{p}}_{1}}(1-n_{{\mathbf{p}}_{1}})n_{{\mathbf{p}}_{2}}}{-i\nu+\varepsilon_{{\mathbf{p}}_{1}+{\mathbf{q}}}-\varepsilon_{{\mathbf{p}}_{1}}}\right]$
$\displaystyle\xrightarrow[T\to 0]{|{\mathbf{q}}|\to 0}$
$\displaystyle(-Vm)\,\frac{(4\pi
e^{2})^{3}}{(2\pi)^{9}}\int_{-\infty}^{\infty}\frac{du}{2\pi}\int_{-1}^{1}\frac{\cos\vartheta_{1}\,\cos\vartheta_{2}\,d\cos\vartheta_{1}\,d\cos\vartheta_{2}}{(-iu+p_{F}\cos\vartheta_{1})^{2}}\times$
$\displaystyle\hskip
30.0pt\frac{(2\pi)^{2}p_{F}^{2}/2}{|\cos\vartheta_{1}-\cos\vartheta_{2}|}\int
4\pi qdq\frac{D_{0}(q,uq/m)}{q^{2}-4\pi e^{2}D_{0}(q,uq/m)}$
$\displaystyle+(2V)\frac{(4\pi
e^{2})^{3}}{(2\pi)^{3}}\int_{-\infty}^{\infty}\frac{du}{2\pi}\int_{-1}^{1}\frac{4\pi
mp_{F}\,d\\!\cos\vartheta}{-iu+p_{F}\cos\vartheta}\int\frac{dq}{(2\pi)^{2}}\frac{D_{0}(q,uq/m)}{q^{2}-4\pi
e^{2}D_{0}(q,uq/m)}\times$ $\displaystyle\hskip
15.0pt\left[\int\frac{q^{\prime
2}dq^{\prime}d\\!\cos\vartheta^{\prime}}{(2\pi)^{2}}\frac{1}{({\mathbf{q}}^{\prime})^{2}}\theta(-\cos\vartheta^{\prime}-\frac{q^{\prime}}{2p_{F}})\right]\;$
$\displaystyle\equiv$ $\displaystyle(-Vm)L_{1}+(2V)L_{2}\;,$
where the first term inside the square brackets after the second equal sign is
obtained by adding to it an equivalent expression with
${\mathbf{p}}_{1}+{\mathbf{q}}\to-{\mathbf{p}}_{1}$,
${\mathbf{p}}_{2}+{\mathbf{q}}\to-{\mathbf{p}}_{2}$, $\nu\to-\nu$, and then
taking the average, while the second term inside the same square brackets
results from changing $\nu\to-\nu$. The $L_{2}$ part, where the dummy variable
of the integral is switched from ${\mathbf{p}}_{2}$ to
${\mathbf{q}}^{\prime}\equiv{\mathbf{p}}_{2}-{\mathbf{p}}_{1}$, can be
cancelled by one of the terms contributing to the (b) diagram of (130). The
$L_{1}$ part, however, upon taking the $T\to 0$ and $q\to 0$ limits, may be
combined with (143) to yield a finite expression
$\displaystyle\frac{Vmp_{F}^{3}}{2}\,\frac{(4\pi
e^{2})^{3}}{(2\pi)^{7}}\int_{-\infty}^{\infty}\frac{du}{2\pi}\int_{-1}^{1}\frac{\cos\vartheta_{1}\,d\cos\vartheta_{1}}{(-iu+p_{F}\cos\vartheta_{1})^{2}}\,\frac{\cos\vartheta_{2}\,d\cos\vartheta_{2}}{-iu+p_{F}\cos\vartheta_{2}}\times$
$\displaystyle\hskip
10.0pt\frac{\cos\vartheta_{1}-\cos\vartheta_{2}}{|\cos\vartheta_{1}-\cos\vartheta_{2}|}\int
2\pi\;d(q^{2})\frac{D_{0}(q,uq/m)}{q^{2}-4\pi e^{2}D_{0}(q,uq/m)}\;.$ (145)
Note that this expression, in agreement with Carr and Maradudin,Carr and
Maradudin (1964) carries a different sign when compared to the original result
of DuBois.DuBois (1959)
It was conjectured before Valiev and Fernando (1997) that cancellation of
diagrams of the following form always hold true for HEG
$\begin{picture}(30.0,15.0)(0.0,-4.0) \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){}
\end{picture}\;\;\;+\;\;\;\;\;\begin{picture}(80.0,15.0)(0.0,-4.0)
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \end{picture}\;=\;0\;.\vspace*{4pt}$
Here, black circles denote parts of the diagram that are connected to each
other via two propagators. If this is the case, then the contributions of the
(b) diagrams of (129) and (130) will cancel each other. We don’t expect this
to happen since the (b) diagram of (129) is not anomalous. To illustrate that
the (b) diagram of (130) does not eliminate the (b) diagram of (129), we now
proceed to evaluate the (b) diagram of (130).
Here, $B_{2}({\mathbf{q}}^{\prime},\nu^{\prime})=4\pi
e^{2}/({\mathbf{q}}^{\prime})^{2}$ while $B_{1}({\mathbf{q}},\nu)=(4\pi
e^{2})^{2}D_{0}({\mathbf{q}},\nu)/\left\\{{\mathbf{q}}^{2}[{\mathbf{q}}^{2}-4\pi
e^{2}D_{0}({\mathbf{q}},\nu)]\right\\}$. Summing over $\nu^{\prime}$ via
methods described in section III.5, one obtains
$\displaystyle-\frac{1}{\beta}\;\begin{picture}(90.0,40.0)(0.0,-4.0)
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){} \put(0.0,0.0){}
\put(0.0,0.0){} \end{picture}$ $\displaystyle=$ $\displaystyle-(2V)\frac{(4\pi
e^{2})^{3}/(2\pi)^{12}}{2\beta\int\frac{d{\mathbf{p}}}{(2\pi)^{3}}n_{\mathbf{p}}(1-n_{\mathbf{p}})}\frac{2}{\beta}\sum_{\nu}\int
d{\mathbf{p}}_{1}d{\mathbf{p}}_{2}\,d{\mathbf{q}}\,d{\mathbf{q}}^{\prime}\frac{n_{{\mathbf{p}}_{2}+{\mathbf{q}}^{\prime}}}{({\mathbf{q}}^{\prime})^{2}}\times$
(146) $\displaystyle\frac{D_{0}(q,\nu)\beta
n_{{\mathbf{p}}_{2}}(1-n_{{\mathbf{p}}_{2}})}{q^{2}(q^{2}-4\pi
e^{2}D_{0}(q,\nu))}\left[\frac{\beta
n_{{\mathbf{p}}_{1}}(1-n_{{\mathbf{p}}_{1}})}{-i\nu+\varepsilon_{{\mathbf{p}}_{1}+{\mathbf{q}}}-\varepsilon_{{\mathbf{p}}_{1}}}+\frac{n_{{\mathbf{p}}_{1}+{\mathbf{q}}}-n_{{\mathbf{p}}_{1}}}{(-i\nu+\varepsilon_{{\mathbf{p}}_{1}+{\mathbf{q}}}-\varepsilon_{{\mathbf{p}}_{1}})^{2}}\right]$
$\displaystyle\xrightarrow[T\to 0]{|{\mathbf{q}}|\to 0}$
$\displaystyle\frac{-(2V)(4\pi
e^{2})^{3}/(2\pi)^{3}}{2\int\frac{d{\mathbf{p}}}{(2\pi)^{3}}\delta(\mu-\varepsilon_{\mathbf{p}})}\int\frac{du}{2\pi}\int
d{\mathbf{p}}_{1}\frac{d{\mathbf{q}}}{(2\pi)^{3}}\frac{q}{m}\frac{D_{0}(q,uq/m)}{q^{2}(q^{2}-4\pi
e^{2}D_{0}(q,uq/m))}\times$ $\displaystyle\hskip
5.0pt\left[2\int\frac{d{\mathbf{p}}_{2}\,d{\mathbf{q}}^{\prime}}{(2\pi)^{6}}\delta(\mu-\varepsilon_{{\mathbf{p}}_{2}})\frac{n_{{\mathbf{p}}_{2}+{\mathbf{q}}^{\prime}}}{({\mathbf{q}}^{\prime})^{2}}\right]\times$
$\displaystyle\hskip
5.0pt\left[\frac{\delta(\mu-\varepsilon_{{\mathbf{p}}_{1}})m/q}{-iu+p_{F}\cos\vartheta_{1}}+\frac{\delta(\mu-\varepsilon_{{\mathbf{p}}_{1}})m^{2}/q^{2}}{(-iu+p_{F}\cos\vartheta_{1})^{2}}\frac{qp_{F}\cos\vartheta_{1}}{m}\right]$
$\displaystyle=$ $\displaystyle\frac{-(2V)(4\pi
e^{2})^{3}/(2\pi)^{3}}{2\int\frac{d{\mathbf{p}}}{(2\pi)^{3}}\delta(\mu-\varepsilon_{\mathbf{p}})}\int\frac{du}{2\pi}\int
d{\mathbf{p}}_{1}\frac{d{\mathbf{q}}}{(2\pi)^{3}}\frac{D_{0}(q,uq/m)}{q^{2}(q^{2}-4\pi
e^{2}D_{0}(q,uq/m))}\times$ $\displaystyle\hskip
5.0pt\left[2\int\frac{d{\mathbf{p}}_{2}}{(2\pi)^{3}}\delta(\mu-\varepsilon_{{\mathbf{p}}_{2}})\right]\left[\int\frac{d{\mathbf{q}}^{\prime}}{(2\pi)^{3}}\frac{\theta(-\cos\vartheta^{\prime}-\frac{q^{\prime}}{2p_{F}})}{({\mathbf{q}}^{\prime})^{2}}\right]\times$
$\displaystyle\hskip
5.0pt\left[\frac{\delta(p_{1}-p_{F})m/p_{F}}{-iu+p_{F}\cos\vartheta_{1}}+\frac{\delta(p_{1}-p_{F})m\cos\vartheta_{1}}{(-iu+p_{F}\cos\vartheta_{1})^{2}}\right]$
$\displaystyle=$ $\displaystyle-(2V)\frac{(4\pi
e^{2})^{3}}{(2\pi)^{3}}\int_{-\infty}^{\infty}\frac{du}{2\pi}\int_{-1}^{1}\frac{4\pi
mp_{F}d\cos\vartheta_{1}}{-iu+p_{F}\cos\vartheta_{1}}\left(1+\frac{p_{F}\cos\vartheta_{1}}{-iu+p_{F}\cos\vartheta_{1}}\right)\times$
$\displaystyle\hskip
5.0pt\left[\int\frac{dq}{(2\pi)^{2}}\frac{D_{0}(q,uq/m)}{q^{2}-4\pi
e^{2}D_{0}(q,uq/m)}\right]\left[\int\frac{d{\mathbf{q}}^{\prime}}{(2\pi)^{3}}\frac{\theta(-\cos\vartheta^{\prime}-\frac{q^{\prime}}{2p_{F}})}{({\mathbf{q}}^{\prime})^{2}}\right]\;.$
The contribution of (146) can be easily divided in two by explicitly expanding
the two parts inside the round parentheses. It is obvious that the
contribution associated with $1$ cancels exactly the $(2V)L_{2}$ part of
(IV.5) while the contribution associated with
$\frac{p_{F}\cos\vartheta_{1}}{-iu+p_{F}\cos\vartheta_{1}}$ cannot cancel the
$(-Vm)L_{1}$ part of (IV.5).
For the (c) diagrams of (128-130), both $B_{1}$ and $B_{2}$ are of order
$(4\pi e^{2})^{2}$, leading to contributions of order $(e^{2})^{4}$ and
higher. The (c) diagrams thus already lead us beyond what was studied by
DuBois DuBois (1959) and by Carr and Maradudin.Carr and Maradudin (1964) The
last two diagrams of $\Gamma_{2}[n]$ (shown in Figure 3) gives rise to the
$E^{\prime}_{3}$ term of Carr and Maradudin Carr and Maradudin (1964) when the
$\tilde{\mathcal{D}}_{0}$ lines are each replaced by $U$, the first term in
the decomposition of $\tilde{\mathcal{D}}_{0}$. In principle, one may go on to
study terms of order $(e^{2})^{4}$; we will not, however, delve into this
endeavor since this is not the primary aim here. We would like to emphasize
the following points. First, unlike conventional $e^{2}$ based perturbation
theory, the formalism presented here naturally avoids divergence. This is
shown by the fact that each diagram in our formalism contains no singularity
while attempts to perform $e^{2}$ based purturbation necessarily require
further re-grouping, such as combining the (b) diagrams of (128) and (129), to
tame the divergence. Second, even if one were to pursue $e^{2}$ expansion,
using Dyson’s equation (127) within our formalism still makes the task
straightforward. In fact, as shown in this section, the
$\tilde{\Gamma}_{0}[n]+\Gamma_{1}[n]+\Gamma_{2}[n]$ part when applied to the
HEG already contain the celebrated results of Carr and Maradudin.Carr and
Maradudin (1964) Third, the removal of anomalous contributions that required
special attention within the formalism of Luttinger et al. Kohn and Luttinger
(1960); Luttinger and Ward (1960) becomes automatic under this formalism.
## V Excitations
To obtain information regarding the excitations, one needs a time-dependent
probe, although one with infinitesimal amplitude is sufficient. Runge and
Gross Runge and Gross (1984) extended the correspondence between the external
probe potential and the ground state charge density of the DFT to the time-
dependent case. This relationship provides the foundation for studying
excitation energies under DFT. Fukuda et al. Fukuda et al. (1994) expressed
the excitation energy condition using effective action formalism, without
explicit connection to the Kohn-Sham formalism. By introducing time-dependent
Kohn-Sham orbitals while assuming time idependence of the orbital occupation
numbers, Casida Casida (1995b) derived via linear response theory a self-
consistent condition on the density matrix response that leads to
determination of excitation energies. Along a similar line, Petersilka et al.
Petersilka et al. (1996) proposed the so-called optimized effective potential
(expanded in time-dependent Kohn-Sham orbitals) to tackle the problem of
excitation energies. Since there exist formalisms to extract the excitation
energies of the system provided that the UDF is known, our result for
excitation energies should not be considered novel. The reasons for this
section are twofold. First, we would like to explicitly show that the
excitation energies can be obtained using the formalism of section III without
introducing time-dependent orbitals. Second, although it is possible to find
derivations of formulas Valiev and Fernando (1997) similar to those that will
be shown here, they do appear slightly different and thus a self-contained
exposition may be helpful.
Intuitively speaking, by varying the frequency of the probe, one seeks the
frequency/energy where the amplitude of the response function diverges.
Indeed, it is known that the spectral representation of the correlation
function has poles at the excitation energies of the system Kobe (1962).
Having obtained the effective action $\Gamma[n]$, we also note that the second
(functional) derivative of $\Gamma[n]$ with respect to the local electron
density is the inverse of the density-density correlation function. Therefore,
any pole associated with the correlation function becomes a root of the
effective action. It can be shown that upon analytic continuation the
correlation function, obtained using the imaginary time (finite temperature)
formalism, can be turned into the response function of the real time. Below we
briefly illustrate this point. Readers interested in more details can find an
extensive exposition in reference Fetter and Walecka, 1971.
From Eqs. (13) and (28), we know that
$\displaystyle n(x)$ $\displaystyle=$ $\displaystyle{\delta(\beta
W[J])\over\delta J(x)}\;,$ $\displaystyle{\rm and\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ }J(x)$
$\displaystyle=$ $\displaystyle-\frac{\delta\Gamma[n]}{\delta n(x)}\;.$
One then considers
$\delta(x-y)=\frac{\delta J(x)}{\delta J(y)}=-\int
dz\frac{\delta^{2}\Gamma[n]}{\delta n(x)\delta n(z)}\frac{\delta n(z)}{\delta
J(y)}=-\int dz\frac{\delta^{2}\Gamma[n]}{\delta n(x)\delta
n(z)}\frac{\delta^{2}(\beta W[J])}{\delta J(z)\delta J(y)}\;.$ (147)
Although a time-dependence of $J$ is introduced to probe the excitations, in
the end we will return to a time-independent source ($J(x)\to J({\bf x})$)
while computing the excitation energies. As we will show below, it is most
convenient to go to the zero temperature limit to compute the excitation
energies.
Note that
$\displaystyle\frac{\delta^{2}(\beta W[J])}{\delta J(x)\delta J(y)}$
$\displaystyle=$
$\displaystyle-\left[\hat{n}(x)\hat{n}(y)\right]_{T}+\left[\hat{n}(x)\right]_{T}\left[\hat{n}(y)\right]_{T}=-\left[\hat{n}(x)\hat{n}(y)\right]_{T}+n(x)n(y)$
(148) $\displaystyle=$
$\displaystyle-\left[(\hat{n}(x)-n(x))(\hat{n}(y)-n(y))\right]_{T}\equiv-\left[\tilde{n}(x)\,\tilde{n}(y)\right]_{T}$
where $\hat{n}(x)=\psi^{{\dagger}}(x)\psi(x)$ is the electron density operator
and the square bracket $[]_{T}$ indicates an (imaginary) time-ordered thermal
average such that
$\left[\hat{\mathcal{O}}_{1}(t_{1})\hat{\mathcal{O}}_{2}(t_{2})\right]_{T}=\frac{\text{Tr}\left[e^{-\beta
H[J]}T\left(\hat{\mathcal{O}}_{1}(t_{1})\hat{\mathcal{O}}_{2}(t_{2})\right)\right]}{\text{Tr}\left[e^{-\beta
H[J]}\right]}=\frac{\text{Tr}\left[e^{-\beta
H[J]}T\left(\hat{\mathcal{O}}_{1}(t_{1})\hat{\mathcal{O}}_{2}(t_{2})\right)\right]}{Z[J]}\;.$
Let us denote by $\\{|\ell\rangle_{ex}\\}_{\ell=0}^{\infty}$ the eigenstates
of the Hamiltonian $H[J]$ with the corresponding eigenenergies
$\\{{\mathcal{E}}_{\ell}\\}_{\ell=0}^{\infty}$. The spectral representation is
obtained by first writing the time-ordered product (assuming that operators
$\hat{\mathcal{O}}_{1}(t_{1})$ and $\hat{\mathcal{O}}_{2}(t_{2})$ are bosonic)
as
$\phantom{}{}_{ex}\\!\langle\ell|T\left(\hat{\mathcal{O}}_{1}(t_{1})\hat{\mathcal{O}}_{2}(t_{2})\right)|\ell\rangle_{ex}=\theta(t_{1}-t_{2})\,\phantom{}_{ex}\\!\langle\ell|\hat{\mathcal{O}}_{1}(t_{1})\hat{\mathcal{O}}_{2}(t_{2})|\ell\rangle_{ex}+\theta(t_{2}-t_{1})\,\phantom{}_{ex}\\!\langle\ell|\hat{\mathcal{O}}_{2}(t_{2})\hat{\mathcal{O}}_{1}(t_{1})|\ell\rangle_{ex}\;,$
then by inserting the identity operator
$\sum_{\ell\,^{\prime}}|\ell\,^{\prime}\rangle_{ex}\;\phantom{}{}_{ex}\\!\langle\ell\,^{\prime}|$
between the two operators, and finally by multiplying by
$e^{i\omega(t_{1}-t_{2})}$ and then integrating over the time variable
$t_{1}-t_{2}$. Proceeding in this way, one will then obtain information on
${\mathcal{E}}_{\ell\,^{\prime}}-{\mathcal{E}}_{\ell}$. Since knowing
$\Omega_{\ell\,^{\prime}}\equiv{\mathcal{E}}_{\ell\,^{\prime}}-{\mathcal{E}}_{0}$
also provides complete information on
${\mathcal{E}}_{\ell\,^{\prime}}-{\mathcal{E}}_{\ell}$, one may also focus on
$\ell=0$ by taking the limit $\beta\to\infty$. Let
$-W^{(2)}(x,y)\equiv-\frac{\delta^{2}(\beta W[J])}{\delta J(x)\delta J(y)}\;,$
(149)
and
$-W^{(2)}({\bf x},{\bf
y},i\nu_{n})\equiv\int_{0}^{\beta}d(\tau_{x}-\tau_{y})e^{i\nu_{n}(\tau_{x}-\tau_{y})}\;W^{(2)}(x,y)\equiv\int_{-\infty}^{\infty}\frac{d\omega^{\prime}}{2\pi}\frac{A(\omega^{\prime})}{i\nu_{n}-\omega^{\prime}}\;,$
(150)
with $\nu_{n}=2\pi n/\beta$ and
$A(\omega)=e^{\beta
W[J]}\sum_{\ell,m}e^{-\beta{\mathcal{E}}_{\ell}}\left(e^{-\beta\omega}-1\right)2\pi\delta(\omega+{\mathcal{E}}_{\ell}-{\mathcal{E}}_{m})\,\phantom{}_{ex}\\!\langle\ell|\tilde{n}({\bf
x})|m\rangle_{ex}\,\phantom{}{}_{ex}\\!\langle m|\tilde{n}({\bf
y})|\ell\rangle_{ex}\;.$ (151)
Note that $\tilde{n}({\bf x})$ measures the deviation from the thermally
averaged electronic density $n_{T}({\bf x})$. Since the expression (149) is
evaluated at static $J({\bf x})$, the Hamiltonian contains no time dependence.
We may thus write $\tilde{n}(x)=e^{H\tau_{x}}\tilde{n}({\bf
x})e^{-H\tau_{x}}$.
Let’s now express using real time the retarded correlation function
(${\mathcal{R}}$), also called the response function, and the advanced
correlation function (${\mathcal{A}}$) as follows (with
$n(x)=e^{it_{x}H}n({\bf x})e^{-it_{x}H}$)
$\displaystyle\left(\begin{array}[]{c}{\cal R}(x,y)\\\
{\mathcal{A}}(x,y)\end{array}\right)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{r}-i\theta(t_{x}-t_{y})\\\
i\theta(t_{y}-t_{x})\end{array}\right)e^{\beta W[J]}\text{Tr}\left(e^{-\beta
H}\left[\tilde{n}(x),\tilde{n}(y)\right]\right)$ (156) $\displaystyle=$
$\displaystyle\left(\begin{array}[]{r}-i\theta(t_{x}-t_{y})\\\
i\theta(t_{y}-t_{x})\end{array}\right)\sum_{\ell,m}\,e^{\beta(W[J]-{\mathcal{E}}_{\ell})}\left[e^{i(t_{x}-t_{y})({\mathcal{E}}_{\ell}-{\mathcal{E}}_{m})}\,\phantom{}_{ex}\\!\langle\ell|\tilde{n}({\bf
x})|m\rangle_{ex}\,\phantom{}{}_{ex}\\!\langle m|\tilde{n}({\bf
y})|\ell\rangle_{ex}\right.$ (160) $\displaystyle\left.\hskip
90.0pt-e^{-i(t_{x}-t_{y})({\mathcal{E}}_{\ell}-{\mathcal{E}}_{m})}\phantom{}_{ex}\\!\langle\ell|\tilde{n}({\bf
y})|m\rangle_{ex}\,\phantom{}{}_{ex}\\!\langle m|\tilde{n}({\bf
x})|\ell\rangle_{ex}\right]\;.$
Taking the Fourier transform of the response function, we consider
$\displaystyle\left(\\!\\!\begin{array}[]{c}{\cal R}({\bf x},{\bf
y},\omega)\\\ {\mathcal{A}}({\bf x},{\bf y},\omega)\end{array}\\!\\!\right)$
$\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}d(t_{x}-t_{y})\;e^{i\omega(t_{x}-t_{y})}\
\left(\begin{array}[]{c}{\cal R}(x,y)\\\ {\mathcal{A}}(x,y)\end{array}\right)$
$\displaystyle=$ $\displaystyle e^{\beta
W[J]}\sum_{\ell,m}e^{-\beta{\mathcal{E}}_{\ell}}\frac{e^{\beta({\mathcal{E}}_{\ell}-{\mathcal{E}}_{m})}-1}{\omega+{\mathcal{E}}_{\ell}-{\mathcal{E}}_{m}\pm
i\eta}\,\phantom{}_{ex}\\!\langle\ell|\tilde{n}({\bf
x})|m\rangle_{ex}\,\phantom{}{}_{ex}\\!\langle m|\tilde{n}({\bf
y})|\ell\rangle_{ex}$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}\frac{d\omega^{\prime}}{2\pi}\frac{A(\omega^{\prime})}{\omega-\omega^{\prime}\pm
i\eta}\;.$ (166)
Comparing Eqs. (150) and (166), one finds that substituting
$i\nu_{n}\to\omega+i\eta$ ($\omega-i\eta$) in the imaginary time-ordered
correlation function leads to the retarded (advanced) correlation function.
The validity of this analytic continuation was discussed by Baym and Mermin
Baym and Mermin (1961).
As $T\to 0$, $e^{-\beta
W[J]}=e^{-\beta{\mathcal{E}}_{0}}\left[1+{\mathcal{O}}(e^{-\beta({\mathcal{E}}_{1}-{\mathcal{E}}_{0})})\right]$,
and $e^{\beta
W[J]}=e^{\beta{\mathcal{E}}_{0}}\left[1-{\mathcal{O}}(e^{-\beta({\mathcal{E}}_{1}-{\mathcal{E}}_{0})})\right]$.
Under this limit, we may rewrite the spectral weight $A(\omega)$ as
$\displaystyle\lim_{\beta\to\infty}\frac{A(\omega)}{2\pi}$ $\displaystyle=$
$\displaystyle\sum_{\ell,m}\left(e^{-\beta({\mathcal{E}}_{m}-{\mathcal{E}}_{0})}-e^{-\beta({\mathcal{E}}_{\ell}-{\mathcal{E}}_{0})}\right)\delta(\omega+{\mathcal{E}}_{\ell}-{\mathcal{E}}_{m})\phantom{}_{ex}\\!\langle\ell|\tilde{n}({\bf
x})|m\rangle_{ex}\,\phantom{}{}_{ex}\\!\langle m|\tilde{n}({\bf
y})|\ell\rangle_{ex}$ (167)
$\displaystyle+{\mathcal{O}}(e^{-\beta({\mathcal{E}}_{1}-{\mathcal{E}}_{0})})$
$\displaystyle=$
$\displaystyle\sum_{\ell}\,\left[\,\delta(\omega+{\mathcal{E}}_{\ell}-{\mathcal{E}}_{0})\phantom{}_{ex}\\!\langle
0|\tilde{n}({\bf
y})|\ell\rangle_{ex}\,\phantom{}{}_{ex}\\!\langle\ell|\tilde{n}({\bf
x})|0\rangle_{ex}\right.$ $\displaystyle\hskip
10.0pt\left.-\delta(\omega-({\mathcal{E}}_{\ell}-{\mathcal{E}}_{0}))\phantom{}_{ex}\\!\langle
0|\tilde{n}({\bf
x})|\ell\rangle_{ex}\,\phantom{}{}_{ex}\\!\langle\ell|\tilde{n}({\bf
y})|0\rangle_{ex}\right]$ $\displaystyle\equiv$
$\displaystyle\sum_{\ell}\left[\delta(\omega+{\mathcal{E}}_{\ell}-{\mathcal{E}}_{0})n_{\ell,e}^{*}({\bf
y})n_{\ell,e}({\bf
x})-\delta(\omega-({\mathcal{E}}_{\ell}-{\mathcal{E}}_{0}))n_{\ell,e}^{*}({\bf
x})n_{\ell,e}({\bf y})\right]$
where $n_{\ell,e}({\bf y})\equiv\phantom{}_{ex}\\!\langle\ell|\tilde{n}({\bf
y})|0\rangle_{ex}$ .
When continued to the retarded correlation function (response function), the
density correlation function $W^{(2)}$ reads
$\lim_{\beta\to\infty}W^{(2)}({\bf x},{\bf
y},\omega)=\sum_{\ell}\left[\frac{n^{*}_{\ell,e}({\bf x})\,n_{\ell,e}({\bf
y})}{\omega-\Omega_{\ell}+i\eta}-\frac{n^{*}_{\ell,e}({\bf
y})\,n_{\ell,e}({\bf x})}{\omega+\Omega_{\ell}+i\eta}\right]\;,$ (168)
with
$\Omega_{\ell}\equiv{\mathcal{E}}_{\ell}-{\mathcal{E}}_{0}\;.$ (169)
Provided that the amplitudes $\phantom{}{}_{ex}\\!\langle
0|\tilde{n}|\ell\rangle_{ex}$ are nonzero, we see from Eq. (168) that
$\omega+i\eta=\pm\left({\mathcal{E}}_{l}-{\mathcal{E}}_{0}\right)$ are simple
poles of $W^{(2)}({\bf x},{\bf y},\omega)$. Furthermore, we also see that
$\left(W^{(2)}({\bf x},{\bf y},\omega)\right)^{*}=W^{(2)}({\bf x},{\bf
y},-\omega^{*})\;,$ (170)
and when $\omega$ is real
$\left(W^{(2)}({\bf x},{\bf y},\omega)\right)^{*}=W^{(2)}({\bf x},{\bf
y},-\omega)\;.$
Let us also define
$\Gamma^{(2)}(x,y)\equiv\frac{\delta^{2}\Gamma[n]}{\delta n(x)\delta n(y)}\;.$
Eq. (147) may thus be rewritten as
$-\delta(x-y)=\int dz\;\Gamma^{(2)}(x,z)\,W^{(2)}(z,y)\;.$ (171)
Since eventually, $\tau_{x}$ must agree with $\tau_{y}$ in the equation above,
if $\tau_{z}>\tau_{y}$, we must have $\tau_{z}>\tau_{x}$ as well. Similarly to
Eq. (150), the Fourier transform for $\Gamma^{(2)}$can be written as
$\Gamma^{(2)}({\bf x},{\bf
z},i\nu_{n})\equiv\int_{0}^{\beta}\\!\\!d(\tau_{z}-\tau_{x})\,e^{i\nu_{n}(\tau_{z}-\tau_{x})}\,\Gamma^{(2)}(x,z)\;.$
(172)
The inverse transform of (150) and (172) can be written as
$\displaystyle W^{(2)}(z,y)$ $\displaystyle=$
$\displaystyle\frac{1}{\beta}\sum_{n}e^{-i\nu_{n}(\tau_{z}-\tau_{y})}W^{(2)}({\bf
z},{\bf y},i\nu_{n})$ $\displaystyle\Gamma^{(2)}(x,z)$ $\displaystyle=$
$\displaystyle\frac{1}{\beta}\sum_{n}e^{-i\nu_{n}(\tau_{z}-\tau_{x})}\Gamma^{(2)}({\bf
x},{\bf z},i\nu_{n})$
and
$\int_{0}^{\beta}\\!\\!\\!\\!d\tau_{z}\\!\\!\int\\!\\!d{\bf
z}\;\Gamma^{(2)}(x,z)W^{(2)}(z,y)=\int d{\bf
z}\,\frac{1}{\beta}\sum_{n}\,e^{-i\nu_{n}(\tau_{x}-\tau_{y})}\Gamma^{(2)}({\bf
x},{\bf z},-i\nu_{n})\,W^{(2)}({\bf z},{\bf y},i\nu_{n})\;.$
Since
$\delta(\tau_{x}-\tau_{y})=\frac{1}{\beta}\sum_{n}e^{-i\nu_{n}(\tau_{x}-\tau_{y})}$,
one obtains
$-\delta({\bf x}-{\bf y})=\int\\!\\!d{\bf z}\;\Gamma^{(2)}({\bf x},{\bf
z},-i\nu_{n})W^{(2)}({\bf z},{\bf y},i\nu_{n})=\int\\!\\!d{\bf
z}\;W^{(2)}({\bf x},{\bf z},-i\nu_{n})\Gamma^{(2)}({\bf z},{\bf
y},i\nu_{n})\;,$ (173)
which under analytic continuation becomes
$-\delta({\bf x}-{\bf y})=\int\\!\\!d{\bf z}\;\Gamma^{(2)}({\bf x},{\bf
z},-\omega)W^{(2)}({\bf z},{\bf y},\omega)=\int\\!\\!d{\bf z}\;W^{(2)}({\bf
x},{\bf z},-\omega)\Gamma^{(2)}({\bf z},{\bf y},\omega)\;.$ (174)
From Eqs. (170) and (174), one has
$\left(\Gamma^{(2)}({\bf x},{\bf z},\omega)\right)^{*}=\Gamma^{(2)}({\bf
x},{\bf z},-\omega^{*})\;.$ (175)
Multiplying the LHS and the middle expression of Eq. (174) by
$(\omega\mp\Omega_{\ell}+i\eta)$ and then setting
$\omega\to\pm\,\Omega_{\ell}-i\eta$, we see that
$\displaystyle\int\\!\\!d{\bf z}\,\Gamma^{(2)}({\bf x},{\bf
z},-\omega\to-\Omega_{\ell}-i\eta)\,n^{*}_{\ell,e}({\bf z})$ $\displaystyle=$
$\displaystyle 0\;,$ (176) $\displaystyle{\rm and\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
}\int\\!\\!d{\bf z}\,\Gamma^{(2)}({\bf x},{\bf
z},-\omega\to+\Omega_{\ell}-i\eta)\,n_{\ell,e}({\bf z})$ $\displaystyle=$
$\displaystyle 0\;.\;,$ (177)
That is, $n_{\ell,e}^{(*)}({\bf y})$ become eigenvectors of $\Gamma^{(2)}({\bf
x},{\bf y},-\omega)$ with zero eigenvalues. The matter of finding excitation
energies and the corresponding electronic densities thus reduces to finding
for $\Gamma^{(2)}$ the eigenvectors with zero eigenvalue.Fukuda et al. (1994)
Since we are interested in obtaining the excitation energy under the physical
condition, $J({\bf x})\to 0$, this also means that the derivatives of $\Gamma$
above are evaluated at the ground state electronic density in the zero
temperature limit.
From the expressions (65) and (70), one sees that the effective action is
split into the free particle part $\Gamma_{0}$, the Hartree functional and the
exchange-correlational functional
$\Gamma[n]=\Gamma_{0}[n]+\frac{1}{2}n{\scriptstyle\circ}\,U{\scriptstyle\circ}n+\Gamma_{xc}[n]\equiv\Gamma_{0}[n]+\Gamma_{\rm
int}[n],$ (178)
where $\Gamma_{xc}[n]=\sum_{l=1}^{\infty}\Gamma_{l}[n]$. Letting $\Delta(x)$
be the eigenvector, the excitation condition becomes
$\int d{\bf z}\Gamma^{(2)}({\bf x},{\bf z})\Delta({\bf z})=0\;.$ (179)
After splitting the effective action into $\Gamma_{0}$ and $\Gamma_{\rm int}$
the eigenvalue equations (176-177) may be expressed as
$-\int\\!\\!d{\bf z}\,\Gamma_{0}^{(2)}({\bf y},{\bf z},-\omega)\Delta({\bf
z})=\int\\!\\!d{\bf z}\,\Gamma_{\rm int}^{(2)}({\bf y},{\bf
z},-\omega)\Delta({\bf z})\;.$ (180)
Multiplying both sides of (180) by $W_{0}^{(2)}({\bf x},{\bf y},\omega)$ and
then integrating over $d{\bf y}$, one obtains
$\Delta({\bf x})=\int d{\bf y}d{\bf z}\;W_{0}^{(2)}({\bf x},{\bf
y},\omega)\Gamma_{\rm int}^{(2)}({\bf y},{\bf z},-\omega)\Delta({\bf z})\;.$
(181)
Note that $W_{0}[J_{0}]$ describes our Kohn-Sham system, a constructed non-
interacting system that produces the same ground state electron density as
that of the physical system considered. From Eq. (168), one can write down
$W^{(2)}_{0}({\bf x},{\bf y},\omega)$ in terms of the excitation energies
associated with the Kohn-Sham non-interacting system:
$W^{(2)}_{0}({\bf x},{\bf y},\omega)=\sum_{\ell}\left[\frac{n^{*}_{\ell}({\bf
x})\,n_{\ell}({\bf y})}{\omega-\omega_{\ell}+i\eta}-\frac{n^{*}_{\ell}({\bf
y})\,n_{\ell}({\bf x})}{\omega+\omega_{\ell}+i\eta}\right]\;,$ (182)
where
$\omega_{\ell}\equiv E_{\ell}-E_{0}\;,$ (183)
and $E_{\ell}$ is the energy of $|\ell\rangle_{ks}$, the $\ell$th state of the
many-particle Kohn-Sham system with $J_{0}({\bf x})$ chosen to generate the
correct ground state electron density. Therefore, for any $\ell$, $E_{\ell}$
is simply the sum of single-particle energies $\varepsilon_{m}$. Note that in
Eq. (182), $n_{\ell}({\bf x})$ is defined by
$\phantom{}{}_{ks}\\!\langle\ell|\tilde{n}({\bf x})|0\rangle_{ks}$ except that
$|\ell\rangle_{ks}$ now describes the $\ell$th state of the Kohn-Sham system,
not the physical system considered.
We seek a general solution for $\Delta({\bf x})$ of the form
$\Delta({\bf x})=\sum_{\ell}\,\left[a_{\ell}\,n_{\ell}({\bf
x})+b_{\ell}\,n^{*}_{\ell}({\bf x})\right]\;.$ (184)
Evidently, if a frequency $\hat{\omega}$ leads to a solution $\\{b_{\ell}\\}$,
then $-\hat{\omega}^{*}$ should lead to a solution $\\{a_{\ell}\\}$, which
plays the role of $\\{b_{\ell}^{*}\\}$. Substituting Eqs. (182) and (184) into
Eq. (181), we find that
$\displaystyle\sum_{\ell}\left[a_{\ell}n_{\ell}({\bf
x})+b_{\ell}n^{*}_{\ell}({\bf x})\right]$
$\displaystyle=\sum_{\ell,\ell\,^{\prime}}\int\\!\\!d{\bf y}d{\bf
z}\left[\frac{n^{*}_{\ell}({\bf x})n_{\ell}({\bf
y})}{\omega-\omega_{\ell}+i\eta}-\frac{n_{\ell}({\bf x})n^{*}_{\ell}({\bf
y})}{\omega+\omega_{\ell}+i\eta}\right]\Gamma^{(2)}_{\rm int}({\bf y},{\bf
z},-\omega)\left[a_{\ell\,^{\prime}}n_{\ell\,^{\prime}}({\bf
z})+b_{\ell\,^{\prime}}n^{*}_{\ell\,^{\prime}}({\bf z})\right].$ (185)
Equating the coefficients associated with $n_{\ell}({\bf x})$ and
$n_{\ell}^{*}({\bf x})$, we find that
$\displaystyle a_{\ell}$ $\displaystyle=$
$\displaystyle-\sum_{\ell\,^{\prime}}\int d{\bf y}d{\bf
z}\frac{n_{\ell}^{*}({\bf y})}{\omega+\omega_{\ell}+i\eta}\Gamma^{(2)}_{\rm
int}({\bf y},{\bf z},-\omega)\left[a_{\ell\,^{\prime}}n_{\ell\,^{\prime}}({\bf
z})+b_{\ell\,^{\prime}}n^{*}_{\ell\,^{\prime}}({\bf z})\right]\;,$
$\displaystyle b_{\ell}$ $\displaystyle=$
$\displaystyle\sum_{\ell\,^{\prime}}\int d{\bf y}d{\bf z}\frac{n_{\ell}({\bf
y})}{\omega-\omega_{\ell}+i\eta}\Gamma^{(2)}_{\rm int}({\bf y},{\bf
z},-\omega)\left[a_{\ell\,^{\prime}}n_{\ell\,^{\prime}}({\bf
z})+b_{\ell\,^{\prime}}n^{*}_{\ell\,^{\prime}}({\bf z})\right]\;.$
Let us define
$\displaystyle Y_{\ell,\ell\,^{\prime}}(\omega)$ $\displaystyle=$
$\displaystyle\int d{\bf y}d{\bf z}\;n_{\ell}^{*}({\bf y})\Gamma^{(2)}_{\rm
int}({\bf y},{\bf z},-\omega)n_{\ell\,^{\prime}}({\bf
z})\;+\omega_{\ell}\,\delta_{\ell,\ell\,^{\prime}}\;,$ (186) $\displaystyle
K_{\ell,\ell\,^{\prime}}(\omega)$ $\displaystyle=$ $\displaystyle\int d{\bf
y}d{\bf z}\;n_{\ell}^{*}({\bf y})\Gamma^{(2)}_{\rm int}({\bf y},{\bf
z},-\omega)n^{*}_{\ell\,^{\prime}}({\bf z})\;,\;$ (187)
and we obtain the following matrix equation
$\left(\begin{array}[]{cc}{\mathbf{Y}}(\omega)&{\mathbf{K}}(\omega)\\\
{\mathbf{K}}^{*}(-\omega^{*})&{\mathbf{Y}}^{*}(-\omega^{*})\end{array}\right)\left(\begin{array}[]{c}A\\\
B\end{array}\right)=\left(\omega+i\eta\right)\left(\begin{array}[]{rr}-1&0\\\
0&1\end{array}\right)\left(\begin{array}[]{c}A\\\ B\end{array}\;,\right)$
(188)
where $(A)_{\ell}=a_{\ell}$ and $(B)_{\ell}=b_{\ell}$. Evidently, one seeks
$\hat{\omega}$ such that
$\det\left[\left(\begin{array}[]{cc}{\mathbf{Y}}(\hat{\omega})&{\mathbf{K}}(\hat{\omega})\\\
{\mathbf{K}}^{*}(-\hat{\omega}^{*})&{\mathbf{Y}}^{*}(-\hat{\omega}^{*})\end{array}\right)-\left(\hat{\omega}+i\eta\right)\left(\begin{array}[]{rr}-1&0\\\
0&1\end{array}\right)\right]=0\;.$ (189)
As mentioned earlier, one anticipates
$\left(A(\hat{\omega})\right)=\left(B^{*}(-\hat{\omega}^{*})\right)$. To see
this, we perform the change $\omega\Leftrightarrow-\omega^{*}$ in (188) and
find that one can then rearrange the resulting equation into
$\left(\begin{array}[]{cc}{\mathbf{Y}}(\omega)&{\mathbf{K}}(\omega)\\\
{\mathbf{K}}^{*}(-\omega^{*})&{\mathbf{Y}}^{*}(-\omega^{*})\end{array}\right)\left(\begin{array}[]{c}B^{*}(-\omega^{*})\\\
A^{*}(-\omega^{*})\end{array}\right)=\left(\omega+i\eta\right)\left(\begin{array}[]{rr}-1&0\\\
0&1\end{array}\right)\left(\begin{array}[]{c}B^{*}(-\omega^{*})\\\
A^{*}(-\omega^{*})\end{array}\right)\;,$ (190)
which is identical to Eq. (188) except with
$\left(B^{*}(-\hat{\omega}^{*})\right)$ playing the role of
$\left(A(\hat{\omega})\right)$ and $\left(A^{*}(-\hat{\omega}^{*})\right)$
playing the role of $\left(B(\hat{\omega})\right)$ .
We now compare Eq. (188) with similar existing results. In references
Bauernschmitt and Ahlrichs, 1996 and Valiev and Fernando, 1997, equations
similar to (188) were obtained, and those will be identical to Eq. (188)
provided that ${\mathbf{K}}^{*}(-\omega^{*})={\mathbf{K}}^{*}(\omega)$. This
will happen if $\Gamma({\bf x},{\bf y},-\omega)=\Gamma({\bf x},{\bf
y},\omega)$ for real $\omega$.
## VI Saddle-Point as an Alternative Formalism
Below, we will obtain the effective action using a classical variable
$i\varphi_{c}$ that corresponds to the saddle-point of the auxiliary field
path integral. At the physical condition $J=0$, $i\varphi_{c}$ is interpreted
as the electron density of a self-consistent Hartree solution. Our Hartee
problem is not of the conventional type, but rather similar to what Kohn
described in his Nobel lecture. Kohn (1999) In the conventional Hartree
calculation, the wave functions obtained may not be orthogonal to each other
due to the fact that each particle’s wave function is solved with a different
potential. Slater (1930) In the method below and mentioned by Kohn Kohn
(1999), the electric potential experienced by every electron is the same.
Another difference between the method below and the aforementioned Hartree
methods Slater (1930); Kohn (1999) is that the integral of the Hartree density
in our method is not necessarily an integer due to the possibility that the
density correction term may have a nonzero integral.
The saddle-point method below is quite different from what was described in
the previous sections. First, although the diagrams in the saddle-point method
are all connected diagrams, they are not one-particle irreducible (1PI).
Second, unlike the method presented in previous sections, the computation of
the effective action now requires no further functional derivatives of $\beta
W[J]$ with respect to $J$ evaluated at $J_{c}$, while in the formalism
mentioned in previous sections, one needs to compute higher order derivatives
of $\beta W[J_{0}]$ with respect to $J_{0}$ (see Eqs. (67-73) ).
### VI.1 Evaluation of $e^{-\beta W_{\phi}[J]}$ via expansion around the
saddle-point
The path integral (23)
$e^{-\beta W_{\phi}[J]}\equiv\int
D\phi\;\exp\left\\{-I\left[\phi\right]-iJ{\scriptstyle\circ}\phi\right\\}\;.$
may be evaluated by the saddle point method. The extrema condition gives
$\left.\frac{\delta
I\left[\phi\right]}{\delta\phi(x)}\right|_{\varphi_{c}}=-iJ(x)\;.$
Since
$\displaystyle\left.\frac{\delta
I\left[\phi\right]}{\delta\phi(x)}\right|_{\varphi_{c}}$ $\displaystyle=$
$\displaystyle\left(U{\scriptstyle\circ}\varphi_{c}\right)_{x}-\text{Tr}\left(G_{\phi}\frac{\delta
G_{\phi}^{-1}}{\delta\phi(x)}\right)_{\phi\to\varphi_{c}}$ (191)
$\displaystyle=$
$\displaystyle\left(U{\scriptstyle\circ}\varphi_{c}\right)_{x}-\int
dy\;{\mathcal{G}}_{c}(y,y)\left(iU(y,x)\right)\;,$
where ${\mathcal{G}}_{c}(y,y)\equiv G_{\phi\to\varphi_{c}}(y,y)$, we obtain
with $U(x,y)=U(y,x)$
$\displaystyle J(x)=\left(U{\scriptstyle\circ}(i\varphi_{c})\right)_{x}+\int
dy\;U(x,y){\mathcal{G}}_{c}(y,y)=\int
dy\;U(x,y)\left[i\varphi_{c}(y)+{\mathcal{G}}_{c}(y,y)\right]\;.$ (192)
As $J\to 0$ (the physical condition), when $U$ is invertible such as Coulomb
interaction, one must have $i\varphi_{c}(x)=-{\mathcal{G}}_{c}(x,x)$. Note
that the negative of the diagonal element of the Green’s function
$-{\mathcal{G}}_{c}(x,x)$ is the particle density corresponding to the
following Hamiltonian
$\int
dx\;{\hat{\psi}}^{{\dagger}}(x)\left[-\frac{\nabla^{2}}{2m}+\upsilon_{\rm
ion}({{\bf
x}})-\mu+i(U{\scriptstyle\circ}\varphi_{c})_{x}\right]\hat{\psi}(x)\;.$
The saddle-point equation therefore produces a Hartree-like equation: the
Green’s function depends on the input particle density $i\varphi_{c}(x)$ and
is required to produce the same particle density $i\varphi_{c}(x)$ in the end.
When $J\neq 0$, one can still view (192) as a generalized Hartree equation in
the following sense. Remember that the inverse Green’s function
${\mathcal{G}}^{-1}_{c}(x,y)$ is given by (with
$\delta(x-x^{\prime})=\delta(\tau-\tau^{\prime})\delta({\bf x}-{\bf
x}^{\prime})$)
${\mathcal{G}}^{-1}_{c}(x,x^{\prime})=\left[\partial_{\tau}-\frac{\nabla^{2}}{2m}+\upsilon_{\rm
ion}({{\bf
x}})-\mu+U{\scriptstyle\circ}(i\varphi_{c})\right]\delta(x-x^{\prime})\;,$
(193)
and may be rewritten as
${\mathcal{G}}^{-1}_{c}(x,x^{\prime})=\left[\partial_{\tau}-\frac{\nabla^{2}}{2m}+\upsilon_{\rm
ion}({{\bf
x}})+J(x)-\mu+U{\scriptstyle\circ}n_{H}\right]\delta(x-x^{\prime})\;.$ (194)
That is, we now view the potential as given by $\upsilon_{\rm ion}({\bf
x})+J(x)$ and the Hartree particle density
$n_{H}(x)=i\varphi_{c}(x)-(U^{-1}{\scriptstyle\circ}J)_{x}$. This
interpretation indeed agrees with our equation (192) which can be written as
$0=\int dy\;U(x,y)\left[\left(i\varphi_{c}(y)-\int
dz\;U^{-1}(y,z)J(z)\right)+{\mathcal{G}}_{c}(y,y)\right]\;.$ (195)
That is, for a given $J(x)\neq 0$, one will solve as before the Hartree
equation but with $\upsilon_{\rm ion}({{\bf x}})\to\upsilon_{\rm ion}({{\bf
x}})+J(x)$. Once the Hartree particle density $n_{H}$ is obtained, one obtains
$i\varphi_{c}=n_{H}+U^{-1}{\scriptstyle\circ}J$. Since
$U{\scriptstyle\circ}(i\varphi_{c})$ always appears as a unit inside the
Green’s function ${\mathcal{G}}_{c}(x,x^{\prime})$, for convenience, we define
$J_{c}\equiv\,U{\scriptstyle\circ}(i\varphi_{c})=J+U{\scriptstyle\circ}n_{H}\;.$
Once $\varphi_{c}(x)$ is obtained, we shift $\phi$ by $\varphi_{c}$ and re-
express the exponent in the integrand as
$-I[\phi]-iJ{\scriptstyle\circ}\phi\Rightarrow-I[\varphi_{c}+\phi]-iJ{\scriptstyle\circ}(\varphi_{c}+\phi)\;,$
and then expand around $\varphi_{c}$. To do the expansion, we first rewrite
(20) as
$G_{\phi}^{-1}(x,x^{\prime})={\mathcal{G}}_{c}^{-1}(x,x^{\prime})+i\,b(x)\,\delta(x-x^{\prime})\equiv{\mathcal{G}}^{-1}_{c}(x,x^{\prime})+V(x,x^{\prime})\;,$
(196)
where $b=U{\scriptstyle\circ}\phi$. We may then write down
$G_{\phi}^{-1}={\mathcal{G}}^{-1}_{c}\left[{\mathbf{I}}+{\mathcal{G}}_{c}{\scriptstyle\circ}{\mathbf{V}}\right]\;,$
and
$\ln\left(G_{\phi}^{-1}\right)=\ln\left({\mathcal{G}}^{-1}_{c}\right)+\sum_{k=1}^{\infty}\frac{(-1)^{k-1}}{k}\left[{\mathcal{G}}_{c}{\scriptstyle\circ}{\mathbf{V}}\right]^{k}\;.$
(197)
Note that
$\left[{\mathcal{G}}_{c}{\scriptstyle\circ}{\mathbf{V}}\right]_{x,z}=\int\\!\\!dy\,{\mathcal{G}}_{c}(x,y)V(y,z)=\int
dy{\mathcal{G}}_{c}(x,y)\delta(y-z)\left(i\,b(y)\,\right)={\mathcal{G}}_{c}(x,z)\left(ib(z)\right)\;.$
Consequently,
$\displaystyle\text{Tr}\ln\left(G_{\phi}^{-1}\right)$ $\displaystyle=$
$\displaystyle\text{Tr}\ln\left({\mathcal{G}}^{-1}_{c}\right)+\int\\!dx_{1}\,{\mathcal{G}}_{c}(x_{1},x_{1})(ib(x_{1}))$
(198)
$\displaystyle-\frac{1}{2}\int\\!dx_{1}dx_{2}\,{\mathcal{G}}_{c}(x_{1},x_{2}){\mathcal{G}}_{c}(x_{2},x_{1})(ib(x_{1}))(ib(x_{2}))$
$\displaystyle+\sum_{k=3}^{\infty}\frac{(-1)^{k-1}}{k}\int\\!dx_{1}\ldots
dx_{k}\;{\mathcal{G}}_{c}(x_{k},x_{1})\ldots{\mathcal{G}}_{c}(x_{k-1},x_{k})(ib(x_{1}))\ldots(ib(x_{k}))\;.$
Note that in the final expression of the exponent of the integrand in (23) the
terms linear in $\phi$ (or $b$) cancel out as one may verify and we arrive at
$\displaystyle-I[\varphi_{c}+\phi]-iJ{\scriptstyle\circ}(\varphi_{c}+\phi)$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\varphi_{c}{\scriptstyle\circ}\,U{\scriptstyle\circ}\varphi_{c}-\varphi_{c}{\scriptstyle\circ}b-\frac{1}{2}b{\scriptstyle\circ}\,U^{-1}{\scriptstyle\circ}b+\text{Tr}\ln(G_{\phi}^{-1})$
(199)
$\displaystyle+\frac{1}{2}\text{Tr}\ln(U)-iJ{\scriptstyle\circ}\varphi_{c}-iJ{\scriptstyle\circ}\phi$
$\displaystyle=$
$\displaystyle\frac{1}{2}\text{Tr}\ln(U)-\frac{1}{2}\varphi_{c}{\scriptstyle\circ}\,U{\scriptstyle\circ}\varphi_{c}+\text{Tr}\ln({\mathcal{G}}_{c}^{-1})-iJ{\scriptstyle\circ}\varphi_{c}$
$\displaystyle-\frac{1}{2}b{\scriptstyle\circ}\tilde{\cal
D}_{c}^{-1}{\scriptstyle\circ}b+\sum_{k=3}^{\infty}I^{(k)}[\varphi_{c}]{\scriptstyle\circ}\,b_{1}\ldots{\scriptstyle\circ}\,b_{k}\;,$
where
$\tilde{\cal D}_{c}^{-1}=U^{-1}-D_{c}\;,$
$D_{c}(x,y)={\mathcal{G}}_{c}(x,y)\,{\mathcal{G}}_{c}(y,x)\;,$
and
$I^{(k)}[\varphi_{c}]{\scriptstyle\circ}\,b_{1}\ldots{\scriptstyle\circ}\,b_{k}\equiv\frac{(-1)^{k-1}}{k}\int\\!dx_{1}\ldots
dx_{k}{\mathcal{G}}_{c}(x_{k},x_{1})\ldots{\mathcal{G}}_{c}(x_{k-1},x_{k})\left[(ib(x_{1}))\ldots(ib(x_{k}))\right]\;.$
(200)
We therefore have (based on the linked-cluster theorem)
$\displaystyle\beta W_{\phi}[J]$ $\displaystyle=$
$\displaystyle\frac{1}{2}\varphi_{c}{\scriptstyle\circ}\,U{\scriptstyle\circ}\varphi_{c}-\text{Tr}\ln\left({\mathcal{G}}_{c}^{-1}\right)+iJ{\scriptstyle\circ}\varphi_{c}$
(201) $\displaystyle+\frac{1}{2}\text{Tr}\ln\left(\tilde{\cal
D}_{c}^{-1}{\scriptstyle\circ}\,U\right)-\sum_{n=1}^{\infty}\frac{1}{n!}\langle\left[\sum_{k=3}^{\infty}I^{(k)}[\varphi_{c}]{\scriptstyle\circ}\,b_{1}\ldots{\scriptstyle\circ}\,b_{k}\right]^{n}\rangle_{\rm
conn.}\;,$
where the subscript “${\rm conn.}$” is used to indicate connected Feynman
diagrams.
Consequently, the effective action, defined by
$\Gamma[n]=\beta
W_{\phi}[J]-\frac{1}{2}J{\scriptstyle\circ}\,U^{-1}{\scriptstyle\circ}J-J{\scriptstyle\circ}n=\beta
W_{\phi}[J]+\frac{1}{2}J{\scriptstyle\circ}\,U^{-1}{\scriptstyle\circ}J-iJ{\scriptstyle\circ}\varphi\;$
and with
$\varphi=\varphi_{c}+\langle\phi\rangle\equiv\varphi_{c}+\tilde{\varphi}$ as
well as with $J=J_{c}-U{\scriptstyle\circ}n_{H}$, can now be written as
$\displaystyle\Gamma[n]$ $\displaystyle=$
$\displaystyle-\text{Tr}\ln\left({\mathcal{G}}_{c}^{-1}\right)-J_{c}{\scriptstyle\circ}n_{H}+\frac{1}{2}n_{H}{\scriptstyle\circ}\,U{\scriptstyle\circ}n_{H}+\frac{1}{2}\text{Tr}\ln\left(\tilde{\cal
D}_{c}^{-1}{\scriptstyle\circ}\,U\right)$ (202) $\displaystyle-
iJ{\scriptstyle\circ}{\tilde{\varphi}}-\sum_{n=1}^{\infty}\frac{1}{n!}\langle\left[\sum_{k=3}^{\infty}I^{(k)}[\varphi_{c}]{\scriptstyle\circ}\,b_{1}\ldots{\scriptstyle\circ}\,b_{k}\right]^{n}\rangle_{\rm
conn.}\;.$
This expression should be contrasted with Eq. (45) where the true particle
density is introduced as the natural variable.
From the definition of $\varphi$ (see Eq. (26) we have
$i\varphi(x)=\frac{\int
D\phi\left(i\phi(x)\right)e^{-I[\phi]-iJ{\scriptstyle\circ}\phi}}{\int
D\phi\,e^{-I[\phi]-iJ{\scriptstyle\circ}\phi}}=i\varphi_{c}(x)+\int
dy\,U^{-1}(x,y)\frac{\int
Db\left(ib(y)\right)e^{-I[\varphi_{c}+\phi]-iJ{\scriptstyle\circ}(\varphi_{c}+\phi)}}{\int
Db\,e^{-I[\varphi_{c}+\phi]-iJ{\scriptstyle\circ}(\varphi_{c}+\phi)}}\;,$
(203)
which means that
${i\tilde{\varphi}}=\frac{\int
D\,b\left(iU^{-1}{\scriptstyle\circ}b\right)e^{-\frac{1}{2}b{\scriptstyle\circ}\tilde{\cal
D}_{c}^{-1}{\scriptstyle\circ}b+\sum_{k=3}^{\infty}\frac{1}{k!}I^{(k)}[\varphi_{c}]{\scriptstyle\circ}\,b_{1}\ldots{\scriptstyle\circ}\,b_{k}}}{\int
Db\,e^{-\frac{1}{2}b{\scriptstyle\circ}\tilde{\cal
D}_{c}^{-1}{\scriptstyle\circ}b+\sum_{k=3}^{\infty}\frac{1}{k!}I^{(k)}[\varphi_{c}]{\scriptstyle\circ}\,b_{1}\ldots{\scriptstyle\circ}\,b_{k}}}\;.$
(204)
On the basis of a simple replica argument Negele and Orland (1988), one sees
that the above expression may be written as
$\displaystyle{i\tilde{\varphi}}(x)$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\frac{1}{n!}\int
dy\,U^{-1}(x,y)\langle\left[\sum_{k=3}^{\infty}I^{(k)}[\varphi_{c}]{\scriptstyle\circ}\,b_{1}\ldots{\scriptstyle\circ}\,b_{k}\right]^{n}ib(y)\rangle_{\rm
conn.}$ (205) $\displaystyle=$
$\displaystyle\sum_{n=1}^{\infty}\frac{1}{n!}\int
dy\,U^{-1}(x,y)\langle\left[\sum_{k=3}^{\infty}I^{(k)}[\varphi_{c}]{\scriptstyle\circ}\,b_{1}\ldots{\scriptstyle\circ}\,b_{k}\right]^{n}ib(y)\rangle_{\rm
conn.}\;,$
where the $n=0$ term vanishes because $\int
Db\,e^{-\frac{1}{2}b{\scriptstyle\circ}\tilde{\cal
D}_{c}^{-1}{\scriptstyle\circ}b}ib(x)=0$. Eq. (205) implies that
$i\tilde{\varphi}(x)$ at each point $x$ is a functional of $\varphi_{c}$.
Returning to (202), this implies that one may express the effective action
$\Gamma$ in terms of $i\varphi_{c}$ although its canonical argument is
supposed to be $n$. The relation between $n$ and $i\varphi_{c}$ is obtained
through
$n=i\varphi_{c}+i\tilde{\varphi}[i\varphi_{c}]-U^{-1}{\scriptstyle\circ}J[i\varphi_{c}]=n_{H}+i\tilde{\varphi}[i\varphi_{c}]$
(206)
with $i\tilde{\varphi}(x)$ expressed as a functional of $i\varphi_{c}$. This
implies that given an $i\varphi_{c}$, we may obtain $n$ through (i) the
diagrammatic expansion of (205) to produce the corresponding $i\varphi$, and
(ii) equation (192) to find $-U^{-1}{\scriptstyle\circ}J$. Since in general,
we have
$\int d{\bf x}\,\left(i\tilde{\varphi}({\bf x})\right)\neq 0\;,$
one cannot expect the integral of the Hartree electron density to be $N_{e}$,
the total number of electrons. Instead, we have
$N_{e}=\int d{\bf x}\,n({\bf x})=\int d{\bf x}\,\left[n_{H}({\bf
x})+i\tilde{\varphi}({\bf x})\right]$ (207)
and $\int d{\bf x}\,n_{H}({\bf x})$ is not necessarily an integer. In
particular, when $\int d{\bf x}\,n_{H}({\bf x})$ deviates significantly from
$N_{e}$ or $\int d{\bf x}\,|i\tilde{\varphi}({\bf x})|\gg 1$, forcing $\int
d{\bf x}\,n_{H}({\bf x})=N_{e}$ may lead to the occurrence of a self-
consistent solution that differs significantly from the true solution.
Note that $n_{H}({\bf x})=-{\mathcal{G}}_{c}(x,x)$ and in the absence of the
source term $n_{H}({\bf x})=i\varphi_{c}({\bf x})$. Furthermore, since
$(U^{-1}{\scriptstyle\circ}J)_{x}\propto\nabla_{{\bf x}}^{2}J(x)$, $\int
n_{H}({\bf x})\,d{\bf x}=\int(i\varphi_{c}({\bf x}))\,d{\bf x}$. The
constraint (207) for the total number of electrons can also be written as
$N_{e}=\int d{\bf x}\,n({\bf x})=\int d{\bf x}\,\left[n_{H}({\bf
x})+i\tilde{\varphi}({\bf x})\right]=\int d{\bf x}\,\left[i\varphi_{c}({\bf
x})+i\tilde{\varphi}({\bf x})\right]\;.$ (208)
The Hartree-like Green’s function ${\mathcal{G}}_{c}(x,x^{\prime})$ shown in
(193) may be viewed as a functional of $i\varphi_{c}(x)$. When expressing
${\mathcal{G}}_{c}(x,x^{\prime})$ using only single particle orbitals, we
define in Eq. (88) $v({\bf x})\equiv\upsilon_{\rm ion}({\bf
x})-\mu+U{\cdot}(i\varphi_{c})_{{\bf x}}$. For the evaluation of
${\mathcal{G}}_{c}(x,y)$, we solve first the eigensystem (93). The single-
particle wave functions (91) associated with $\hat{h}$ are to be obtained
self-consistently. Basically, one starts with a guess for the electronic
density $i\varphi_{c}({\bf x})$ satisfying $\int d{\bf x}(i\varphi_{c}({\bf
x}))\approx N_{e}$, where $N_{e}$ is the number of electrons. One then obtains
the single-particle wave functions, and then computes the corresponding
Green’s function ${\mathcal{G}}_{c}$, obtains $\int d{\bf
x}(i\tilde{\varphi}({\bf x}))$ and tunes the chemical potential to ensure that
$-\int d{\bf x}\;{\mathcal{G}}_{c}({\bf x},{\bf x})=N_{e}-\int d{\bf
x}(i\tilde{\varphi}({\bf x}))$. One then takes $-{\mathcal{G}}_{c}({\bf
x},{\bf x})$ in place of $i\varphi_{c}({\bf x})$ in the next round of
iteration until convergence is reached.
The procedure of the saddle-point method is now obvious. One starts with an
external potential, and then determines the Hartree electron density via
$n_{H}({\bf x})=-{\mathcal{G}}_{c}(x,x)$,
$n_{H}=i\varphi_{c}-U^{-1}{\scriptstyle\circ}J$ and Eq. (208). This self-
consistent procedure will also provide the physical electron density $n({\bf
x})$. The ground state energy is obtained by using (202) to calculate the
effective action, which is the ground state energy times $\beta$ in the limit
$T\to 0$. When one wishes to obtain the density functional $\Gamma[n]$ at an
electron density other than $n_{T}$, one adds the source term into the
potential $\upsilon_{\rm ion}({\bf x})$ and then solves for the Hartree
density, shown in (194), as outlined above.
### VI.2 Remark on the single-particle limit
The single electron limit for the Hartree-method is more complicated than for
the method presented in section IV.3. Because $\int n_{H}({\bf x})\,d{\bf x}$
is not necessarily $1$ in the single electron limit, due to Eq. (208), in
general one needs to obtain $n_{1}$ self-consistently. The other issue is that
the diagrammatic expansion contained in (202) does not cover all the diagrams
for a given order of $U$. This means that one can’t use the Hugenholtz diagram
argument to eliminate vertex matrix element even when $n_{1}\leq 1$. To see
this point explicitly, we rewrite (202) as
$\displaystyle\Gamma[n]$ $\displaystyle=$
$\displaystyle-\text{Tr}\ln\left({\mathcal{G}}_{c}^{-1}\right)+(J-J_{c}){\scriptstyle\circ}n_{H}+\frac{1}{2}n_{H}{\scriptstyle\circ}\,U{\scriptstyle\circ}n_{H}+\frac{1}{2}\text{Tr}\ln\left(\tilde{\cal
D}_{c}^{-1}{\scriptstyle\circ}\,U\right)$ (209)
$\displaystyle-\sum_{n=1}^{\infty}\frac{1}{n!}\langle\left[\sum_{k=3}^{\infty}I^{(k)}[\varphi_{c}]{\scriptstyle\circ}\,b_{1}\ldots{\scriptstyle\circ}\,b_{k}\right]^{n}\rangle_{\rm
conn.}-J{\scriptstyle\circ}n\;.$
Evidently, except for the last term above, the rest of the terms must
constitute $\beta W[J]$. Since $J-J_{c}=-U{\scriptstyle\circ}n_{H}$, one can
also write $\beta W[J]$ as
$\displaystyle\beta W[J]$ $\displaystyle=$
$\displaystyle-\text{Tr}\ln\left({\mathcal{G}}_{c}^{-1}\right)-n_{H}{\scriptstyle\circ}\,U{\scriptstyle\circ}n_{H}+\frac{1}{2}n_{H}{\scriptstyle\circ}u{\scriptstyle\circ}n_{H}+\frac{1}{2}\text{Tr}\ln\left(\tilde{\cal
D}_{c}^{-1}{\scriptstyle\circ}\,U\right)$
$\displaystyle-\sum_{n=1}^{\infty}\frac{1}{n!}\langle\left[\sum_{k=3}^{\infty}I^{(k)}[\varphi_{c}]{\scriptstyle\circ}\,b_{1}\ldots{\scriptstyle\circ}\,b_{k}\right]^{n}\rangle_{\rm
conn.}\;\;.$
Diagrammatic expansion of the last two terms shows that the following diagrams
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of order $U^{2}$ are absent, when compared to the regular field theoretic
perturbation calculation. This is not a disadvantage of the method. Instead,
what our derivation shows is that the missing diagrams eventually will be
compensated by the $-n_{H}{\scriptstyle\circ}\,U{\scriptstyle\circ}n_{H}$
term. However, it is obvious that the saddle-point formalism makes the single-
electron limit hard to analyze.
When $n_{1}>0$ but $n_{1}\ll 1$ at the low temperature limit, we know that
$D_{c}(x,y)={\mathcal{G}}_{c}(x,y){\mathcal{G}}_{c}(y,x)$ will be of order
$n_{1}$. This is because
${\mathcal{G}}_{c}(x,y)=\sum_{\alpha}\phi_{\alpha}({\bf
x})\phi_{\alpha}^{*}({\bf
y})e^{-(\varepsilon_{\alpha}-\mu)(\tau_{x}-\tau_{y})}\times\left\\{\begin{array}[]{l
r}(-n_{\alpha})&{\rm if\leavevmode\nobreak\ }\tau_{x}\leq\tau_{y}\\\
(1-n_{\alpha})&{\rm if\leavevmode\nobreak\
}\tau_{x}>\tau_{y}\end{array}\right.\;,$
and whenever $\tau_{x}\leq\tau_{y}$, the propagator is of order $n_{1}$. Since
$D_{c}(x,y)={\mathcal{G}}_{c}(x,y){\mathcal{G}}_{c}(y,x)$, one of the
propagators in the product must be of order $n_{1}$. In principle, one needs
to solve for the occupation number $n_{1}$ of the lowest energy state using
$i\varphi_{c}({\bf
x})-\left(U^{-1}{\scriptstyle\circ}J\right)_{x}=-{\mathcal{G}}_{c}(x,x)$ and
Eq. (208). Nevertheless, the correct one particle limit can be seen if one
starts with a chemical potential $\mu$ such that $n_{1}\approx 0$. In this
case, we have at the $J=0$ limit $i\varphi_{c}(x)=n_{H}({\bf x})\approx 0$ as
well as $D_{c}(x,y)\propto n_{1}\approx 0$. This way, the higher order
exchange-correlation terms may be viewed as the having smaller contributions
and one may control the accuracy by controlling the number of higher order
terms included. Of course, one then has $n({\bf x})\approx
i\tilde{\varphi}({\bf x})$ and the condition $\int d{\bf
x}\left(i\tilde{\varphi}({\bf x})\right)=1-n_{1}$ must be satisfied.
### VI.3 Obtaining excitations using the Hartree method
In general the excitations are determined by Eqs. (176) and (177). Under the
Hartree formalism described in this section, the natural variable used is the
Hartree density $n_{H}$ rather than the true particle density $n_{T}$. One
thus must transform the variable used in Eqs. (176) and (177) from $n_{T}$ to
$n_{H}$. We describe below how this can be achieved.
At the physical condition ($J=0$), one has
$0=\left.\frac{\delta\Gamma}{\delta n(x)}\right|_{n=n_{T}}=\left.\int
dx_{1}dx_{2}\frac{\delta\Gamma}{\delta
n_{H}(x_{2})}\right|_{n=n_{T}}\left.\frac{\delta n_{H}(x_{2})}{\delta
J_{c}(x_{1})}\right|_{n=n_{T}}\left.\frac{\delta J_{c}(x_{1})}{\delta
n(x)}\right|_{n=n_{T}}\;.$
Note that $\delta n_{H}/\delta J_{c}$ contains no zero mode because
$n_{H}=\delta W_{H}/\delta J_{c}$, where
$W_{H}[J_{c}]\equiv-\text{Tr}\ln\left({{\mathcal{G}}_{c}}^{-1}\right)$, and
$\delta^{2}W_{H}/\delta J_{c}\delta J_{c}$ is known to be strictly negative
from a theorem proved by Valiev and Fernando. Valiev and Fernando (1997) The
strict negative-definiteness of $\delta^{2}W/\delta J\delta J$ can also be
interpreted as the stability condition
$\int dx\,\delta n(x)\,\delta J(x)<0\;,$
which means raising the local one-particle potential leads to an average
decrease of the local particle concentration and vice versa. The strict
negative-definiteness means that $\delta n_{H}/\delta J_{c}$ contains no zero
modes and is invertible. Also, because $n=n_{H}+i\tilde{\varphi}_{c}$, $\delta
n/\delta J_{c}=\delta n_{H}/\delta J_{c}+\delta(i\tilde{\varphi})/\delta
J_{c}$ exists via diagrammatic expansion of $i\tilde{\varphi}$ in terms of
$J_{c}$. The existence of $\delta n/\delta J_{c}$ implies that $\delta
J_{c}/\delta n$ is invertible (i.e., has no zero eigenvalues). Therefore,
after multiplying the inverse of $\delta J_{c}/\delta n$ and $\delta
n_{H}/\delta J_{c}$ on both sides of the above equation, one has
$\left.\frac{\delta\Gamma}{\delta n_{H}(x)}\right|_{n=n_{T}}=0\;,$
which then leads to
$\displaystyle\Gamma^{(2)}(x,y)$ $\displaystyle=$
$\displaystyle\left.\frac{\delta^{2}\Gamma}{\delta n(x)\delta
n(y)}\right|_{n=n_{T}}$ $\displaystyle=$ $\displaystyle\left[\int
dx_{1}dx_{2}dy_{1}dy_{2}\frac{\delta J_{c}(x_{1})}{\delta n(x)}\frac{\delta
n_{H}(x_{2})}{\delta J_{c}(x_{1})}\frac{\delta^{2}\Gamma}{\delta
n_{H}(x_{2})\delta n_{H}(y_{2})}\frac{\delta n_{H}(y_{2})}{\delta
J_{c}(y_{1})}\frac{\delta J_{c}(y_{1})}{\delta n(y)}\right]_{n=n_{T}}\;.$
Using Eq. (172), one may write $\Gamma^{(2)}({\bf x},{\bf y},\omega)$ as the
analytic continuation of the integral of $\Gamma^{(2)}(x,y)$ over time. To
achieve this goal, one first writes down
$\displaystyle\Gamma^{(2)}({\bf x},{\bf y},i\nu_{n})$ $\displaystyle=$
$\displaystyle\int_{0}^{\beta}d(\tau_{y}-\tau_{x})\,e^{i\nu_{n}(\tau_{y}-\tau_{x})}\,\Gamma^{(2)}(x,y)$
(210) $\displaystyle=$
$\displaystyle\int_{0}^{\beta}d(\tau_{y}-\tau_{x})\,e^{i\nu_{n}(\tau_{y}-\tau_{y_{1}}+\tau_{y_{1}}-\tau_{y_{2}}+\tau_{y_{2}}-\tau_{x_{2}}+\tau_{x_{2}}-\tau_{x_{1}}+\tau_{x_{1}}-\tau_{x})}\,\Gamma^{(2)}(x,y)$
$\displaystyle\equiv$ $\displaystyle\int d{\bf x}_{1}d{\bf x}_{2}d{\bf
y}_{1}d{\bf y}_{2}f^{{\bf x}_{1}}_{{\bf x}}\\!(-i\nu_{n})g^{{\bf x}_{2}}_{{\bf
x}_{1}}\\!(-i\nu_{n})\tilde{\Gamma}^{(2)}({\bf x}_{2},{\bf
y}_{2},i\nu_{n})g^{{\bf y}_{2}}_{{\bf y}_{1}}\\!(i\nu_{n})f^{{\bf
y}_{1}}_{{\bf y}}\\!(i\nu_{n})$ $\displaystyle=$ $\displaystyle\int d{\bf
x}_{2}d{\bf y}_{2}h^{{\bf x}_{2}}_{{\bf
x}}(-i\nu_{n})\tilde{\Gamma}^{(2)}({\bf x}_{2},{\bf y}_{2},i\nu_{n})h^{{\bf
y}_{2}}_{{\bf y}}(i\nu_{n})\;,$
where
$\displaystyle\tilde{\Gamma}^{(2)}({\bf x},{\bf y},i\nu_{n})$ $\displaystyle=$
$\displaystyle\int_{0}^{\beta}d(\tau_{y}-\tau_{x})\frac{\delta^{2}\Gamma}{\delta
n_{H}(x)\delta n_{H}(y)}$ $\displaystyle g^{{\bf y}_{2}}_{{\bf
y}_{1}}\\!(i\nu_{n})$ $\displaystyle\equiv$
$\displaystyle\left.\int_{0}^{\beta}d(\tau_{y_{1}}-\tau_{y_{2}})\,e^{i\nu_{n}(\tau_{y_{1}}-\tau_{y_{2}})}\,\frac{\delta
n_{H}(y_{2})}{\delta J_{c}(y_{1})}\right|_{n=n_{T}}$ $\displaystyle f^{{\bf
y}_{1}}_{{\bf y}}\\!(i\nu_{n})$ $\displaystyle\equiv$
$\displaystyle\left.\int_{0}^{\beta}d(\tau_{y}-\tau_{y_{1}})\,e^{i\nu_{n}(\tau_{y}-\tau_{y_{1}})}\,\frac{\delta
J_{c}(y_{1})}{\delta n(y)}\right|_{n=n_{T}}$ $\displaystyle h^{{\bf
y}_{2}}_{{\bf y}}(i\nu_{n})$ $\displaystyle\equiv$ $\displaystyle\int d{\bf
y}_{1}\;g^{{\bf y}_{2}}_{{\bf y}_{1}}\\!(i\nu_{n})f^{{\bf y}_{1}}_{{\bf
y}}\\!(i\nu_{n})\;.$
Therefore, Eq. (179) for finding the excitations becomes
$\int\\!\\!d{\bf z}\,\tilde{\Gamma}^{(2)}({\bf x},{\bf
z},-\omega)\,\tilde{\Delta}({\bf z})=0\;,$ (211)
with $\tilde{\Delta}({\bf z})$ given by
$\tilde{\Delta}({\bf z})=\int d{\bf y}\,h^{{\bf z}}_{{\bf
y}}(-\omega)\Delta({\bf y})\;.$
One therefore obtains the excitation energy via solving Eq. (211). Since
$h^{{\bf z}}_{{\bf y}}(-\omega)$ is invertible, one may also obtain
$\Delta({\bf z})$ via $\tilde{\Delta}({\bf z})$ if desired. Within this
framework, the first two terms of (202) correspond to the non-interacting part
$\Gamma_{0}$ of the effective action in section V. The protocol for obtaining
the excitation energy is then the same as described in section V with the one
exception that the ground state of the Hartree system is not the same as the
ground state of the real system.
## VII Discussion and Future Directions
In this paper we focus on the auxiliary field method applied to the
development of the density functional. It is natural to inquire into the
physical meaning of the auxiliary, bosonic field $\phi$ introduced in eqs
(15-20). In the path integral treatment of relativistic quantum
electrodynamics, if one were to integrate out the photon field, one generates
the current-current interaction which is quartic in fermionic field. When
viewing this process backwards, one sees that the quartic fermionic
interaction is disentangled by introducing the photon field. The $\phi$ field
here thus plays a similar role to the photon field as it disentangles the
quartic fermionic interaction term. As we will argue below, the $\phi$
variable is closely related to the “time” component of the photon field, i.e.,
the electric potential.
To make the connection between the photon field and $\phi$, let us first seek
the nonrelativistic limit of the Lagrangian density,
${\mathcal{L}}_{EM}=-F^{\mu\nu}F_{\mu\nu}/(16\pi)$ , associated with the
relativistic photon field. Note that in the limit when the speed of light
approaches infinity (removal of terms involving time derivative of the three-
vector potential), the Lagrangian density turns into $(\nabla
A_{0})^{2}/8\pi$, which contains no quadratic derivative with respect to time.
Therefore, at finite temperature (with $it\to\tau$) this implies that the
exponent associated with the photon field path integral will behave as
$i\int dt\,d{\bf x}\;{\mathcal{L}}_{EM}\to\int\\!d\tau\\!\int\\!d{\bf
x}\frac{(\nabla A_{0})^{2}}{8\pi}=-\int\\!d\tau\\!\int\\!d{\bf
x}\frac{1}{8\pi}\,A_{0}\nabla^{2}A_{0}\;.$ (212)
Setting $U({\bf x}-{\bf y})=e^{2}/|{\bf x}-{\bf y}|$ [thus
$U^{-1}(x,y)=-\frac{1}{4\pi e^{2}}\nabla^{2}_{{\bf x}}\delta({\bf x}-{\bf
y})$] and comparing (212) with Eqs. (15-16), we make the identification
$(iU{\scriptstyle\circ}\phi)/e=A_{0}$, because now
$-\frac{1}{2}\phi{\scriptstyle\circ}\,U{\scriptstyle\circ}\phi\to\frac{1}{2}A_{0}\,{\scriptstyle\circ}(e^{2}u^{-1})\,{\scriptstyle\circ}A_{0}=-\int
d\tau\int d{\bf x}\frac{1}{8\pi}\,A_{0}\nabla^{2}A_{0}\;.$
In the non-relativistic limit, the electric part and the magnetic part of
electromagnetism are decoupled. Starting with a non-relativistic many–electron
system, one may ask what is the quantum mechanical analog of the Poisson
equation that forms the basis of electrostatics. It turns out that this is
easily obtained via computing $\left.\delta Z[J]/\delta J\right|_{J=0}$ in two
different ways. First, upon taking the derivative of (15) and using Eqs. (16)
and (17), one obtains
$\left.\frac{\delta Z[J]}{\delta J({\bf x})}\right|_{J=0}=-\beta
Z_{J=0}\langle\psi^{{\dagger}}({\bf x})\psi({\bf x})\rangle=-\beta
Z_{J=0}\frac{\langle e\psi^{{\dagger}}({\bf x})\psi({\bf x})\rangle}{e}\;.$
Second, while taking the derivative of (15), if using Eqs. (19) and (20) one
arrives at
$\left.\frac{\delta Z[J]}{\delta J({\bf x})}\right|_{J=0}=-\beta
Z_{J=0}\langle i\phi\rangle=-\beta
Z_{J=0}\,e\;U^{-1}{\scriptstyle\circ}\langle A_{0}\rangle=\beta
Z_{J=0}\frac{1}{4\pi e}\nabla^{2}_{{\bf x}}\langle A_{0}({\bf x})\rangle\;.$
We therefore obtain the thermal quantum-mechanical analog of the Poisson
equation
$\nabla^{2}_{{\bf x}}\langle A_{0}({\bf x})\rangle=-4\pi\langle
e\psi^{{\dagger}}({\bf x})\psi({\bf x})\rangle\;,$
a result also obtained in reference Valiev and Fernando, 1996. This connection
to classical electrostatics is essential since it provides the quantum-
mechanical correspondence of an important ingredient in (bio)molecular
interactions that have been extensively studied in the presence of
dielectrics. Yu (2003); Doerr and Yu (2004, 2006); Obolensky et al. (2009)
The UDF described in this paper is systematically constructed, uniquely
determined, and in principle exact. However, in terms of real computations,
one can only keep $\Gamma_{i}$ terms up to some order in $\lambda$. A natural
question thus arises. How well will the truncated version work? In general,
this question can only be answered with numerical results. However, by
providing theoretical arguments and comparisons to other approaches, we wish
to convey that this method is likely to produce good results and thus to
attract computational efforts towards using the proposed approach.
It is worth pointing out the relation between the expression (46) and the
scheme Polonyi and Sailer (2002) motivated by the renormalization-group. The
vertex functions $I^{(j\leq l)}$ in (46) will contribute to the so-called
$l$-local approximation of reference Polonyi and Sailer, 2002. The absence of
$I^{(2)}$ manifests the absence of the correction term due to the bi-local
contribution, as shown in reference Polonyi and Sailer, 2002. Furthermore,
with the bilocal approximation Polonyi and Sailer (2002) included, Polonyi and
Sailer obtained an approximate energy functional which corresponds exactly to
our $\Gamma_{0}+\Gamma_{1}$. To reach an equivalent form of the proposed
$l$-local approximation of reference Polonyi and Sailer, 2002, we simply keep
terms up to $\Gamma_{l/2}$. Therefore, our formulation provides an explicit
means for achieving an $l$-local approximation without resorting to the
Hellmann-Feynman theorem.
As shown in (87), the propagator $\tilde{\mathcal{D}}_{0}$ can be expanded as
$\tilde{\mathcal{D}}_{0}=U+U{\scriptstyle\circ}D_{0}{\scriptstyle\circ}\,U+U{\scriptstyle\circ}D_{0}{\scriptstyle\circ}\,U{\scriptstyle\circ}D_{0}{\scriptstyle\circ}\,U+\ldots\;,$
when $U$ can be viewed as a small quantity and treated perturbatively. When
one is not allowed to treat $U$ as a small parameter (say in the strong
coupling regime), or when one needs to treat $U^{-1}$ as a small parameter
instead, the conventional perturbative expansion in $e^{2}$ breaks down
completely while our approach is still applicable. In the case when $U^{-1}$
must be treated as small, we expand $\tilde{\mathcal{D}}_{0}$ as
$\tilde{\mathcal{D}}_{0}=-D_{0}^{-1}-D_{0}^{-1}{\scriptstyle\circ}\,U^{-1}{\scriptstyle\circ}D_{0}^{-1}-D_{0}^{-1}{\scriptstyle\circ}\,U^{-1}{\scriptstyle\circ}D_{0}^{-1}{\scriptstyle\circ}\,U^{-1}{\scriptstyle\circ}D_{0}^{-1}-\ldots\;.$
(213)
And in this case, $U\gg 1$, our effective action expansion does have the
Hartree term $\frac{1}{2}n{\scriptstyle\circ}\,U{\scriptstyle\circ}n$ as the
leading order, followed by terms of order $U^{0}$ and then the expansion of
$\tilde{\mathcal{D}}_{0}$ provides series in powers of $U^{-1}$. Note that in
this case, the exchange-correlation functional is not led by order $U$ at all,
but is led by order $U^{0}$. This feature is not present in the conventional
perturbative approach using $U$ (or $e^{2}$) as the expansion parameter.
As mentioned earlier, there also exist different functional methods for many
electron systems. For example, the exchange correlation functional outlined by
Sham Sham (1985) is founded on the perturbative functional approach developed
by Luttinger and Ward Luttinger and Ward (1960) or equivalently by Klein.Klein
(1961) A succinct review of the Luttinger-Ward/Klein functional and its
applications can be found in reference Kotliar et al., 2006. The Luttinger-
Ward/Klein functional yields the grand-potential/ground-state-energy only when
the functional argument is equal to the fully-interacting, physical, one-
particle Green’s function. Instead of allowing the physical, full, one-
particle Green’s function, Dahlen et al. Dahlen et al. (2006) proposed to find
that stationary point of the Luttinger-Ward/Klein functional while restricting
the argument to the Hartree-Fock Green’s functions or the Green’s functions of
non-interacting systems. However, even if the Luttinger-Ward/Klein functional
is computed to all orders, the error in the value of the grand potential (or
the ground state energy) due to restriction on the Green’s functions remains
unknown.
Furthermore, it should be noted that when the functional argument is equal to
the physical one-particle Green’s function, the Luttinger-Ward/Klein
functional reaches a stationary point, not the minimum.Klein (1961) This means
that it is possible for the Klein/Luttinger-Ward functional to assume an even
lower value than the ground state energy (or the grand potential) when the
functional argument deviates from the physical Green’s function. In other
words, the Klein/Luttinger-Ward functional only retains its meaning as the
ground state energy (or grand potential) when the Green’s function takes the
value of the true (physical) Green’s function. Our effective action expression
of the energy functional, on the other hand, truly represents energy of the
system. Our effective action energy functional, when no truncation on the
series is made, reaches its minimum when the electron density assumes the true
(physical) density, and for any other $\upsilon$-representable density profile
prescribed, it represents the lowest energy possible associated with that
prescribed density profile.
The method of Hedin Hedin (1965) is largely identical to that of Luttinger and
Ward.Luttinger and Ward (1960) This includes the fact that the energy
functional reaches a stationary point, rather than the minimum, when the
functional argument is the fully-interacting one-particle Green’s function.
However, Hedin aims to replace the $e^{2}$-based (bare Coulomb) perturbative
expansion of the electron self energy by another expansion using a screened
interaction ${\mathcal{W}}$. Hedin expresses the electron self energy and the
screened interaction as a functional of the electron Green’s function of the
interacting system. Interestingly, the first order result, also termed the GW
approximation, of Hedin Hedin (1965) has been shown Dahlen et al. (2006) to
produce good results when compared to other density functionals. This suggests
that not treating $e^{2}$ as small might have some advantage. It is worth
pointing out that the first order term,
$\Gamma_{1}=-\frac{1}{2}\text{Tr}\ln(\tilde{\mathcal{D}}_{0}^{-1}{\scriptstyle\circ}\,U)$,
of the UDF described here is equivalent to the celebrated GW approximation.
Like the ${\mathcal{W}}$ propagator of Hedin,Hedin (1965) the
$\tilde{\mathcal{D}}_{0}$ propagator introduced here also corresponds to that
of a screened interaction (see section IV.4), thereby avoiding any possible
infrared divergence associated with perturbative expansion based on bare
Coulomb interactions. However, the screening associated with
$\tilde{\mathcal{D}}_{0}$ is from the KS particles and thus keeps the same
form no matter how many orders one wishes to include. This is different from
that of Hedin’s where the expression of ${\mathcal{W}}$ in terms of the
electron Green’s function changes with the order included. The other
difference between the proposed approach and reference Hedin, 1965 is that the
UDF proposed here depends on $J_{0}$, a function of three spatial variables
(and possibly with one additional time variable), while the method of
reference Hedin, 1965 expresses via ${\mathcal{W}}$ the electron self energy
as a functional of the Green’s function, a function of six spatial variables
(and possibly with two additional time variables).
It is well known that a loopwise expansion may also be viewed as an $\hbar$
expansion Itzykson and Zuber (1980), that is, an expansion of quantum-
mechanical effects. By first integrating out the fermionic degrees of freedom
completely, the proposed method is an expansion of bosonic loops formed by
$\tilde{\mathcal{D}}_{0}$ propagators associated with the auxiliary field $b$.
The $b$ field describes the potential produced by electron density
fluctuations around $n_{g}$. Since the ground state charge density $n_{g}$
captures the full quantum information of the ground state thanks to the HK
theorem, one anticipates a weaker quantum effect associated with the auxiliary
$b$ field than with the fermionic field. This makes the auxiliary $b$ field a
suitable candidate for loop (or quantum effect) expansion, the approach
pursued in this paper.
Finally, let us remark on the issue of convexity. The full $\Gamma[n]$ is
supposed to be convex, Valiev and Fernando (1997) thus guaranteeing a unique
solution without any local minima when searching for the minimum of
$\Gamma[n]$. However, in real computations only a finite number of terms of
the effective action can be kept. This approximate/truncated expression may
not warrant convexity and thus it is not guaranteed to be free of local minima
while numerically searching for the ground state density $n_{g}$ (or thermal
averaged density $n_{T}$ at finite temperature). In the near future, we plan
to implement numerically the methods presented in this paper, and will
describe in a separate publication the results obtained as well as the
investigation on the issue of convexity.
## Acknowledgement
This research was supported by the Intramural Research Program of the National
Library of Medicine of the National Institutes of Health. The author thanks
Dr. Oleg Obolensky and Professor John Neumeier for useful comments. He is
particularly indebted to Professor Richard Friedberg, who has provided
numerous useful suggestions and correspondence during the writing of the
paper. The Feynman diagrams are made using the style file contained in the
package developed in reference Binosi and Theußl, 2004
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|
arxiv-papers
| 2009-10-05T03:55:24 |
2024-09-04T02:49:05.661994
|
{
"license": "Public Domain",
"authors": "Yi-Kuo Yu",
"submitter": "Yi-Kuo Yu",
"url": "https://arxiv.org/abs/0910.0670"
}
|
0910.0808
|
# Comment on “Nonlinear photoluminescence spectra from a quantum dot-cavity
system revisited” by Yao _et al._ and “Photoluminescence from Microcavities
Strongly Coupled to Single Quantum Dots” by Ridolfo _et al._
F. P. Laussy School of Physics and Astronomy. University of Southampton,
Southampton, SO171BJ, United Kingdom. E. del Valle School of Physics and
Astronomy. University of Southampton, Southampton, SO171BJ, United Kingdom.
C. Tejedor Física Teórica de la Materia Condensada, Universidad Autónoma de
Madrid, 28049, Spain.
###### Abstract
We refute the criticisms of our work on strong-coupling in the presence of an
incoherent pumping.
###### pacs:
42.50.Ct, 78.67.Hc, 42.55.Sa, 32.70.Jz
Our description of strong-coupling in semiconductors Laussy et al. (2008,
2009); del Valle et al. (2009) has been recently criticized by two groups:
Ridolfo _et al._ Ridolfo et al. (2009) and Yao _et al._ Yao et al. (2009).
Both groups of authors make the same statements: they claim that our master
equation is flawed, on the ground that its domain of convergence is bounded,
they both propose to use exclusively a thermal bath for the reservoir of
excitations of the cavity instead of our most general case (Yao _et al._
require a thermal bath also for the excitonic reservoir whereas Ridolfo _et
al._ allow independent pumping and decay coefficients for the Quantum Dot
(QD)) and they all claim a better agreement than our model with experimental
data.
We show in this comment that their criticisms are erroneous and that the
alternatives they propose to fulfill them are well known particular cases
that, in their approximations, are also erroneous.
Both groups focus on our boson model only, and in fact only on our Letter on
the topic Laussy et al. (2008). Many of their statements are already addressed
in our full boson text Laussy et al. (2009) and more crucially in its fermion
counterpart del Valle et al. (2009). Throughout, we made clear that the boson
model is adequate either in the limit of small pumping, or in the limit of
bosonic 0D system. The interest of the boson case is in its analytical
solutions, that allow to explain transparently fundamental features of
pumping, such as the effect of the effective quantum state on the spectral
shapes. We will show that the analytical solutions for the fermion models from
Ridolfo et al. (2009) and Yao et al. (2009) are in fact valid only in trivial
cases (namely, the boson limit or the uncoupled limit).
In the following, we shall clarify, on the one hand, that our approach does
not suffer from any inconsistency or pathology, and on the other hand, that in
the particular cases where the reservoirs of excitations are thermal baths,
our model also applies (by enforcing the ad-hoc constrain in the equation) and
show the errors made by the approximations of Refs. Ridolfo et al. (2009); Yao
et al. (2009).
## I Validity of our master equation
Our model describes the linear regime (of vanishing excitations) analytically
and the nonlinear regime semi-analytically. Its master equation reads:
$\displaystyle\frac{d\rho}{dt}$ $\displaystyle=$ $\displaystyle
i[\rho,H_{a}+H_{\sigma}+H_{a\sigma}]$ (1a) $\displaystyle+$
$\displaystyle\frac{\gamma_{a}}{2}(2a\rho{a^{\dagger}}-{a^{\dagger}}a\rho-\rho{a^{\dagger}}a)$
(1b) $\displaystyle+$ $\displaystyle\frac{P_{a}}{2}(2{a^{\dagger}}\rho
a-a{a^{\dagger}}\rho-\rho a{a^{\dagger}})$ (1c) $\displaystyle+$
$\displaystyle\frac{\gamma_{\sigma}}{2}(2\sigma\rho{\sigma^{\dagger}}-{\sigma^{\dagger}}\sigma\rho-\rho{\sigma^{\dagger}}\sigma)$
(1d) $\displaystyle+$
$\displaystyle\frac{P_{\sigma}}{2}(2{\sigma^{\dagger}}\rho\sigma-\sigma{\sigma^{\dagger}}\rho-\rho\sigma{\sigma^{\dagger}})\,.$
(1e)
where $a$ is the cavity mode and $\sigma$ the exciton in the QD. The cavity is
always a bosonic mode. In Ref. Laussy et al. (2008) and Laussy et al. (2009),
$\sigma$ is also a Bose operator, that we note $b$ for clarity, while in Ref.
del Valle et al. (2009), $\sigma$ is a fermion operator, describing a two-
level system 111We have been aware of and considering the Fermi dynamics from
the start, as one can see from the initial version arXiv:0711.1894v1 of our
manuscript Laussy et al. (2008) although it eventually retained only the boson
results, postponing the general but non-analytical case to Ref. del Valle et
al. (2009). All our statements in the linear and boson models, even in
November 2007, were confronted with the fermion case for validity, in contrast
with our critics, as is shown in this comment.. This choice of $\sigma$ as a
Bose operator addresses two important cases: $i$) the limit of vanishing
pumpings (even if the QD is indeed a fermion emitter) and $ii$) the case where
the excitons follow bose statistics. The latter case could be realized in
large QDs, that recover the physics of quantum wells where excitons are known
to behave as good bosons Kavokin et al. (2007).
The main reproach of Ridolfo et al. (2009); Yao et al. (2009) is that in Eq.
(1), the effective pumping rates $P_{a,\sigma}$ can vary independently of the
effective decay rates $\gamma_{a,\sigma}$ 222They both also make the reproach
that $\sigma$ is limited to Bose statistics but this is because they are
unaware—or do not want to consider—our main line of work, for which we can
only point them to Ref. del Valle et al. (2009).. They observe that this can
lead to some divergences beyond some critical values of pumping, as we have
ourselves discussed before, and they conclude that the model is flawed. They
propose instead to use thermal reservoirs, that do not exhibit such
divergences, at all values of pumping.
A thermal reservoir for a bosonic mode $a$ (with frequency $\omega_{a}$ and
$H_{a}=\omega_{a}{a^{\dagger}}a$) at temperature $T$ leads to the effective
rate of excitation:
$P_{a}=\kappa_{a}\bar{n}_{T}\,.$ (2)
with $\bar{n}_{T}$ given by the reservoir Bose-Einstein distribution. It
vanishes at $T=0$. In thermal equilibrium, the system is loosing excitations
at a larger rate of:
$\gamma_{a}=\kappa_{a}(1+\bar{n}_{T})=\kappa_{a}+P_{a}\,.$ (3)
The parameter $\kappa_{a}$ is the _spontaneous emission_ (SE) rate at $T=0$.
The steady state thermal equilibrium reads
$n_{a}=\langle{a^{\dagger}}a\rangle=\frac{P_{a}}{\gamma_{a}-P_{a}}=\frac{P_{a}}{\kappa_{a}}=\bar{n}_{T}\,.$
(4)
At very high temperatures, as the effective income of particles approaches the
outcome, $P_{a}\approx\gamma_{a}$, the number of particles remains finite,
since $P_{a}<\gamma_{a}$. As long as $\gamma_{a}\neq 0$, any combination of
parameters $\gamma_{a},P_{a}$ corresponds to a given thermal bath (with
$\kappa_{a}=\gamma_{a}-P_{a}$ and $T>0$).
The linewidth of the optical spectrum of emission is:
$\Gamma_{a}\equiv\gamma_{a}-P_{a}=\kappa_{a}\,.$ (5)
and is independent of temperature (i.e., of the population of the mode), since
it is always equal to the spontaneous emission decay rate $\kappa_{a}$.
It is clear from the above results, that a bosonic thermal bath cannot provide
gain and does not exhibit any line-narrowing in its luminescence spectrum. A
thermal bath is a medium of _loss_ as $P_{a}<\gamma_{a}$ by definition.
A thermal reservoir is, however, a particular case. In out-of-equilibrium
conditions, especially under externally applied pumping, one can expect
deviations from the thermal paradigm.
In contrast to the thermal case, a _gain_ medium can be derived with bosonic
baths out-of-equilibrium. This is discussed in textbooks, e.g., in Chapter 7
of Gardiner and Zoller’s text Gardiner and Zoller (2000). A linear gain can be
obtained with an “inverted” harmonic oscillator maintained at a negative
temperature $-T^{\prime}$. The effect in the master equation is that the
effective parameters are now given by a relation opposite to Eqs. (2-3):
$\gamma_{a}=G_{a}\bar{m}_{-T^{\prime}}\,,\quad
P_{a}=G_{a}(1+\bar{m}_{-T^{\prime}})=G_{a}+\gamma_{a}\,.$ (6)
In this way, $G_{a}$ is the gain or input of particles into the mode at zero
temperature. Obviously, given that now $P_{a}>\gamma_{a}$, there is no
stationary solution to the master or rate equations. Since the absence of this
stationary solution is the source of confusion in Refs Ridolfo et al. (2009);
Yao et al. (2009), we quote here Gardiner and Zoller’s comment, p°216:
> _If $\gamma<\kappa$_ [that is $\gamma_{a}<P_{a}$ with our notation]_, there
> is no stationary situation, and the amplifier gives a signal that increases
> without limit. Essentially, the power being fed into the cavity cannot
> escape fast enough. (Of course the idea of an inverted medium which
> maintains its inversion independent of power output is not exactly valid,
> and depletion effects will then need to be considered. The system is then
> essentially a laser)._
Obviously, an ever-growing population will always be stopped by some external
physical effect (the sample will have burnt, the reservoir will be depleted,
etc…). There is however nothing pathologic in this behaviour. In particular,
this does not invalidate the results for values below the critical pumping
rates. In the case where $\gamma_{c}>P_{c}$ ($c=a,b$), there is a physical
solution to the dynamical master equation, starting at $t=0$: at all
(_finite_) times, there is a valid master equation, with positive trace,
normed to unity, etc… However, indeed, the system diverges with time. There is
no unphysical behaviour or flaw of some sort. Not all dynamical systems have a
steady state, some because they are oscillatory, others because they increase
without bounds.
In our work Laussy et al. (2008, 2009); del Valle et al. (2009), we have
naturally considered the configurations which admit a steady state. We have
even given analytical solutions for their domain of convergence, supported by
a clear physical picture 333The limit of validity of the boson master equation
is given by a total outcome of excitation larger than the income
$\gamma_{a}+\gamma_{b}>P_{a}+P_{b}$, that is, $\Gamma_{a}+\Gamma_{b}>0$ (in
the notations of Laussy et al. (2009)) If one of the oscillators has a net
gain ($\Gamma_{b}<0$), there is an extra condition for convergence that has a
clear physical motive: not only mode $a$ is constrained to be neatly lossy,
$\Gamma_{a}>0$, but also the income through the “inverted” mode $b$, given by
$P_{b}$, must be smaller than its total outcome, which is given by
$\gamma_{b}$ plus the effective (Purcell) decay through cavity emission. That
is, $P_{b}<\gamma_{b}+\frac{4(g^{\mathrm{eff}})^{2}}{\Gamma_{a}}$ where
$g^{\mathrm{eff}}=g/\sqrt{1+[2\Delta/(\Gamma_{a}+\Gamma_{b})]^{2}}$, and $=g$
at resonance..
In a microcavity QED system (a QD in a microcavity), which is a complicated
solid state system, open to many sources of excitations Gies et al. (2007);
Winger et al. (2009), one should also include gain effects in the most general
case. Granted all together, the microscopic coefficients are likely terms of
the form:
$\displaystyle\gamma_{a}=\kappa_{a}(1+\bar{n}_{T})+G_{a}\bar{m}_{-T^{\prime}}\,,$
(7a) $\displaystyle
P_{a}=\kappa_{a}\bar{n}_{T}+G_{a}(1+\bar{m}_{-T^{\prime}})\,,$ (7b)
that is, including loss media and gain media. Net losses in the case of a
cavity mode comes, among other reasons, from the fact that the photons can
escape the cavity through the imperfect mirrors. Net gain could come from
surrounding off-resonance or weakly coupled QDs, high energy QD levels or the
wetting layer. A gain medium can be obtained by the very configuration
realized with self-assembled QDs in a microcavity. Gardiner and Zoller, p°140
(ibid) study a bath of uncorrelated two-level emitters that are kept in
average in the excited state. Quoting them again:
> _Note that there is no restriction on $N^{+}_{a}$ and $N^{-}_{a}$_ [that is,
> in their notations, the number of emitters in the excited and ground states,
> respectively, that provide the pump and decay terms for the bosonic
> mode]_—this means that $N^{+}_{a}>N^{-}_{a}$ is permissible. Physically such
> a population inversion could only be maintained by some kind of pumping, as
> indeed happens in a laser._
and as indeed could happen in an incoherently pumped microcavity. The previous
discussion on gain-media that applies for the cavity mode can also be extended
for bosonic emitters (QDs, in our case). In this case, for example, the
electron-hole pairs that form excitons inside the QD decay from the wetting
layer (at higher energy) by, e.g., emitting phonons with the corresponding
energy difference. Such phonons will not be reabsorbed to bring back the
electron-hole pair to the wetting layer, leading to a net source of particles.
Net losses take place through the spontaneous decay into leaky modes.
In general, there is thus no reason to restrict the master equation (1)
designed to account effectively for the largest possible amount of physical
effects, to a given type of reservoirs (namely, thermal baths). Our study aims
at the greatest level of applicability and generality and provides the tools
for the understanding of any case. Therefore, better than trying to apply at
the theoretical level any criteria (other than convergence) to choose
$\gamma_{a}$, $P_{a}$, we prefer to consider them independent. So far, the
dynamics of lines (1b-1c) found its most important domain of applicability
with atom lasers and polariton lasers Holland et al. (1996); Ĭmamoḡlu and Ram
(1996); Porras and Tejedor (2003); Rubo et al. (2003); Laussy et al. (2004a);
Schwendimann and Quattropani (2006, 2008); Doan et al. (2008), that is,
systems where a condensate (or coherent state) is formed by scattering of
bosons into the final state from another state rather than by emission. In
both cases, scattering or emission, the process is stimulated. In this case
the income and outcome of particles is a complicated function of the
distribution of excitons (or polaritons) in the higher $k$-states. See, e.g.,
Ref Ĭmamoḡlu and Ram (1996). In a pulsed experiment, it is typically time
dependent (see, e.g., Fig. 2 of Ref. Laussy et al. (2004b)). Line narrowing is
a natural feature of this dynamics in such systems.
It remains, of course, possible that a particular experiment corresponds to a
thermal bath of excitation. In this case, a fitting analysis with our model
should indicate this constrain in correlations between the $\gamma$ and $P$
coefficients. As a result of this analysis, one will then understand that the
given system refers to the particular cases of Eqs. (2) and (3).
The above discussion concerns the bosonic mode. We now turn to the fermion
mode.
The thermal equilibrium case, at temperature $T$, gives the counterpart of the
boson case (given above):
$\displaystyle\gamma_{\sigma}$
$\displaystyle=\kappa_{\sigma}(1+\bar{n}_{T})=\kappa_{\sigma}+P_{\sigma}\,,$
(8a) $\displaystyle P_{\sigma}$ $\displaystyle=\kappa_{\sigma}\bar{n}_{T}\,,$
(8b)
where $\kappa_{\sigma}$ is the Einstein $A$-coefficient and $P_{\sigma}$ is
the Einstein $B$-coefficient. The steady state is the _Fermi-Dirac
distribution_ :
$n_{\sigma}=\frac{P_{\sigma}}{\gamma_{\sigma}+P_{\sigma}}=\frac{P_{\sigma}}{\kappa_{\sigma}+2P_{\sigma}}=\frac{\bar{n}_{T}}{2\bar{n}_{T}+1}=\Big{(}e^{\frac{\omega_{\sigma}}{k_{B}T}}+1\Big{)}^{-1}\,.$
(9)
The maximum occupation for the emitter is $1/2$, at infinite temperature. It
is, therefore, not possible to invert the two-level system with a thermal
bath, (where, again, $\gamma_{\sigma}>P_{\sigma}$), which is a well known
result. In the fermion case, even a gain-medium does not lead to a divergence
thanks to the intrinsic saturation of a two-level system. The emission
spectrum is also a Lorentzian, with effective linewidth:
$\Gamma_{\sigma}\equiv\gamma_{\sigma}+P_{\sigma}=\kappa_{\sigma}+2P_{\sigma}$
(10)
which broadens from the decay rate at zero temperature, $\kappa_{\sigma}$ when
the temperature increases.
This elementary discussion illustrates the notorious fact that thermal
reservoirs are unable to achieve population inversion of a two-level system
Briegel and Englert (1993). Therefore, the choices of excitation reservoirs of
Yao _et al._ Yao et al. (2009), forbids lasing in such systems, which is
already contradicted by experiments Nomura et al. (2009).
The issue of achieving gain with a master equation for a fermion emitter has
been extensively discussed in the lasing literature, in particular in the one-
atom laser case Agarwal and Dutta Gupta (1990); Cirac et al. (1991); Mu and
Savage (1992); Horak et al. (1995); Briegel et al. (1996); Löffler et al.
(1997); Meyer and Briegel (1998); Benson and Yamamoto (1999); Koganov and
Shuker (2000); Florescu et al. (2004); Karlovich and Kilin (2008); Li et al.
(2009), and it is typically described theoretically also by considering an
effective negative-temperature thermal bath. A popular notation to consider
gain and dissipation is Briegel et al. (1996):
$P_{\sigma}=\Gamma_{\sigma}s\,,\quad\gamma_{\sigma}=\Gamma_{\sigma}(1-s)\,.$
(11)
With $\Gamma_{\sigma}>0$ (the linewidth broadening) and $0\leq s\leq 1$,
including both the situations with net losses ($s<1/2$) and gain ($s>1/2$). In
this case:
$n_{\sigma}=\frac{P_{\sigma}}{\gamma_{\sigma}+P_{\sigma}}=s\,.$ (12)
This parameterization is not linked to any particular experimental realization
but is designed to separate the physical effects that lead to line broadening,
$\Gamma_{\sigma}$, from those that change the population, $s$. Apart from
that, it is equivalent to consider directly the effect of varying the
effective decay and pumping parameters, $\gamma_{\sigma}$ and $P_{\sigma}$, as
we have done in Refs del Valle et al. (2009) and Gonzalez-Tudela et al.
(2009).
The Jaynes-Cummings model couples the fermionic and bosonic modes. It has been
extensively studied in the case of thermal cavity bath and some gain-medium
for the emitter (which is also the case of Ref. Ridolfo et al. (2009)). We
studied it in its most general form, using the master equation (1). Again, if
a particular constrain arises from a given realization of the reservoirs of
excitations, such as (2-3), (6), (7), (8) or some other case, this would
appear in our unconstrained case from correlations following these trends. We
expect, as we previously discussed, that a successful statistical analysis
would indeed inform about underlying microscopic details of the excitation
scheme.
## II Validity of the proposed substitutes to our work
In the case of Ridolfo _et al._ Ridolfo et al. (2009), only the photonic mode
was excited thermally while the excitonic pump was still allowing an
unconstrained pumping and decay. Yao _et al._ Yao et al. (2009), on the other
hand, require both modes to be excited by thermal baths.
Thermal baths in the linear (boson) model reduce to results identical to the
spontaneous emission of an initial state that is a mixture of excitons and
photons in the ratio of population $n_{a}(t=0)/n_{b}(t=0)=P_{a}/P_{b}$. The
thermal character of the bath merely prevents renormalization of the
linewidths and of the Rabi frequencies. The ratio $P_{a}/P_{b}$ still
determines the possibility to resolve the line-splitting. This fundamental
consequence of the effective quantum state is independent of any choice of the
reservoirs. It is a general result that we have amply discussed before Laussy
et al. (2008, 2009); del Valle et al. (2009) and that is “rediscovered” by
Ridolfo _et al._
Ridolfo _et al._ Ridolfo et al. (2009) otherwise have mistakes in their
formulas, that certainly bias their analysis. For instance, their parameters
$n_{a}$ and $C$ should read, in their notations: (cf. their Eq. (7) & (8))
$\displaystyle n_{a}$ $\displaystyle=$
$\displaystyle\frac{P_{a}}{\gamma_{a}}+\frac{g^{2}}{\gamma_{a}}\frac{(\gamma_{a}+\gamma_{x})(\gamma_{a}P_{x}-\gamma_{x}P_{a})}{g^{2}(\gamma_{a}+\gamma_{x})^{2}+\gamma_{a}\gamma_{x}|\tilde{\omega}_{a}^{*}-\tilde{\omega}_{x}|^{2}}\,,$
(13a) $\displaystyle C$ $\displaystyle=$
$\displaystyle\frac{g}{\tilde{\omega}_{a}^{*}-\tilde{\omega}_{x}}(n_{a}-n_{x})\,.$
(13b)
Also, their spectra are normalized to $\sqrt{2\pi}n_{a}$, again apparently as
an error since they compare them directly to ours which are normalized to
unity. These mistakes do not seem to be a misprint, given that the authors
state in their paper:
> _Although at low pump intensities, our approach and that of Ref. [Laussy et
> al.] essentially represent models of a linear Bose-like dynamics of two
> coupled harmonic oscillators, nontrivial differences can be appreciated._
There should be no difference in the limit of vanishing pump. In fact, once
corrected as above, their formulas and lineshapes do converge to our results
at low pumps.
In the _non-vanishing_ case, of course, the thermal reservoir gives a
different result than unconstrained parameters of our Eq. (1), even in the
linear regime (that is, when $n_{\sigma}\ll 1$, although $n_{a}$ is not
compulsorily also vanishing). The illustration of this fact was attempted in
Fig. (1a) of Ref. Ridolfo et al. (2009), although here also the plot is wrong.
With non-vanishing cavity pumping, the linear regime can be maintained only if
the modes are almost uncoupled. This is shown in our corrected version (Fig.
1).
Figure 1: Cavity emission for the parameters of Fig. 1(a) of Ridolfo _et al._
Ridolfo et al. (2009): $\Delta=-3.6g$, $\gamma_{\sigma}=1.48g$, $P_{a}=0.49g$,
$P_{\sigma}=0.0078g$. Like in their figure—but with the correct formulas
(13)—we compare: in solid-blue, the case where $\gamma_{a}=1.96g$ and in
dashed-purple $\kappa_{a}=1.96g$ ($\gamma_{a}=\kappa_{a}+P_{a}$, thermal
bath). Both cases are indeed in the linear regime ($n_{\sigma}\ll 1$) since
the fermion model del Valle et al. (2009) gives the same results. The system
is however almost decoupled due to the large detuning. The emission is thus
basically that of the bare cavity (Lorentzian). Figure 2: Converged spectra
with the full fermion model del Valle et al. (2009) [in solid black] for the
choice of pumping reservoirs and parameters of Yao _et al._ , along with the
approximate spectra proposed by these authors [in dashed red]. Their
approximation is correct only at the smallest values of pumping, where it also
recovers the boson results of Refs. Laussy et al. (2008, 2009). Figure 3:
Converged populations [in solid black] for the cavity,
$\langle{a^{\dagger}}a\rangle$ [thin] and the QD,
$\langle\sigma^{+}\sigma^{-}\rangle$ [thick], for the choice of pumping
reservoirs and parameters of Yao _et al._ , along with the approximate values
proposed by these authors [in dashed red]. Again, their approximation is
correct only at the smallest values of pumping, where it also recovers the
boson results of Refs. Laussy et al. (2008, 2009). The breakdown of their
approximation is further manifest by the inversion of population of the QD
(indicated by the arrow), which is notoriously impossible with thermal
reservoirs. Our model shows the expected saturation bounded by 1/2.
We now turn to the approach of Yao _et al._ Yao et al. (2009). They miss a
factor $\Gamma_{c}$ in the second term of the denominator of their Eq. (4),
although in their case this is a misprint only since their figures match the
formulas that follow from their approximations. These approximations, however,
are incorrect.
As we discussed above, it is straightforward in our approach to constrain the
coefficients, following a given choice of the reservoirs of excitations. If we
choose the thermal baths advocated by Yao _et al._ , we are able with our
approach del Valle et al. (2009) to apply the full fermion model with no
truncation in the number of excitations, for the parameters of these authors.
We find that their approximation is a poor one, as seen in Fig. 2, where we
superimpose the exact (numerical) result, in solid black, to their approximate
(analytical) formula, in dashed red. As could be expected, the agreement is
good only at very low pumping (linear regime) and very high pumping (uncoupled
regime). It is incorrect in the most relevant region of the transition where
the doublet collapses (as has been reported before, e.g., Fig. 13 of our Ref.
del Valle et al. (2009)).
Their approximation is also basically flawed in that it allows an inversion of
population for the two-level system, although it is excited by thermal
reservoirs, as seen in the magnified version of their Fig. 3(a), that we
reproduce in dashed lines in our Fig. 3, along with the converged solutions
from our model del Valle et al. (2009) (with their parameters and choices of
reservoirs). Beyond the poor quantitative agreement when pumping is non-
vanishing, at the point indicated by the arrow and above, the QD is inverted,
which indicates a pathology of their approximation.
Their implication that cavity pumping is determinant to achieve lasing is in
contradiction with well-known and established facts of the one-atom laser
theory. See for instance our text Ref. del Valle et al. (2009) where lasing is
achieved without cavity pumping, thanks to the gain-medium that our general
model allows. On the other hand, thermal reservoirs of Yao _et al._ forbids
lasing, regardless of the magnitude of cavity pumping (note that with their
parameters, very high cavity populations are already achieved, but they have
thermal statistics, with second-order correlator $g^{(2)}$ that increases
rapidly towards 2 with pumping).
Finally, we want to stress that their Fig. 2(b), that supposedly represents
our model, does not make any meaningful comparison, since, fitting some data
with their model [that we have just shown is wrong, but even if it was
correct], they proceed to plot our _boson_ model with _their_ fitting
parameters. It is obvious that, the two formulas being different, the best-
fitting parameters for one of them will yield poor agreement on the same data
for the other. Beside, they should have used our _fermion_ model, since they
consider a supposedly two-level emitter in a nonlinear regime. Fitting them
independently, on the one hand, and comparing them on statistical grounds on
the other hand, rather than settling for some aesthetic of the agreement, is
the correct course of action. Fitting with the nonlinear fermion model is not
a trivial task. With E. Cancellieri and A. Gonzalez-Tudela, we have recently
obtained results in this direction, to be published shortly.
In conclusion, we have shown that our work Laussy et al. (2008, 2009); del
Valle et al. (2009) is correct and that the critics addressed against it
Ridolfo et al. (2009); Yao et al. (2009) are unsubstantiated on the one hand,
and the proposed substitutes are incorrect on the other hand. These authors do
not derive any master equation. They settle for thermal reservoirs, which
derivation is a standard textbook material. This is a particular case of our
work that they apply incorrectly or beyond its limits of validity. We have
already provided the valid limit for the nonlinear regime del Valle et al.
(2009). There remain many open questions in the field. Some can be settled by
statistical analysis of experimental data with our model, Eq. (1), which
correlations between the fitting parameters can teach about underlying
microscopic mechanisms (such as the nature of the bath of excitations, among
other). To this intent, we invite experimentalists to make their raw data
available to everybody 444http://sciencecommons.org/about/towards.
## References
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|
arxiv-papers
| 2009-10-05T17:00:01 |
2024-09-04T02:49:05.684987
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "F.P. Laussy, E. del Valle, C. Tejedor",
"submitter": "Fabrice Laussy Dr.",
"url": "https://arxiv.org/abs/0910.0808"
}
|
0910.0932
|
Complete lists of low dimensional complex associative algebras
I.S. Rakhimov1, I.M. Rikhsiboev2 and W.Basri3
1,3Department of Mathematics, Faculty of Science,
1,2,3Laboratory of Theoretical studies, Institute for Mathematical Research
(INSPEM),
University Putra Malaysia
1 _risamiddin@gmail.com, 2ikromr@gmail.com, 3witri@science.upm.edu.my_
###### Abstract
In this paper we present a complete classification (isomorphism classes with
some isomorphism invariants) of complex associative algebras up to dimension
five (including both cases: unitary and non-unitary). In some symbolic
computations we used Maple software.
Keywords: Associative algebras, isomorphism, invariants, nilpotent, semi-
simple algebra.
## 1 Introduction
In algebra, there are three strongly related classical algebras: associative,
Lie and Jordan algebras. The objects of our attention in this paper are
associative algebras, exactly finite dimensional associative algebras over
complex numbers. The major theorems on associative algebras include the most
splendid results of the great heroes of algebra: Wedderburn, Artin, Noether,
Hasse, Brauer, Albert, Jacobson, and many others.
It is known that the universal enveloping algebra of a Lie algebra has the
structure of an associative algebra. Introduced by Loday [5], the notion of
Leibniz algebra is a generalization of the Lie algebra, where the skew-
symmetry in the bracket is dropped. Loday [6] also showed that the
relationship between Lie and associative algebras can be translated into an
analogous relationship between Leibniz and the so-called dialgebras (or
diassociative algebras), which are a generalization of associative algebra
possessing two operations $\dashv$ and $\vdash.$ In particular, it was shown
that any dialgebra becomes a Leibniz algebra under the bracket $[x,y]=x\dashv
y-y\vdash x.$
The classification of low dimensional complex Lie and Leibniz algebras can be
found in [4], [5], [2].
Our motivation to classify associative algebras comes out from the intention
to classify the diassociative algebras. A diassociative algebra is a
generalization of associative algebra, it can be obtained as a combination of
two associative algebras. In order to make up this combination we need the
list of all associative algebras (both unitary and non-unitary). The latest
lists of all unitary associative algebras in dimension two, three, four, and
five are available in [8], [1], [3] and [7], respectively.
In 1993 Loday [5] introduced a non-antisymmetric version of Lie algebras,
whose bracket satisfies the Leibniz identity and therefore this generalization
has been called Leibniz algebras. The Leibniz identity, combined with
antisymmetricity, is a variation of the Jacobi identity. Hence a Lie algebra
is antisymmetric Leibniz algebra. Loday also introduced an “associative”
version of Leibniz algebras, called diassociative algebras, equipped with two
binary operations, which satisfy certain identities. These identities are all
variations of the associative law. An associative algebra is a diassociative
algebra when the two operations coincide. The main motivation of Loday to
introduce this class of algebras was the search of an “obstruction” to the
periodicity of algebraic K-theory. Besides this purely algebraic motivation,
some relationships with classical geometry, non-commutative geometry and
physics have been recently discovered. We are interested in classification of
diassociative algebras. One of the approaches to describe a finite dimensional
diassociative law is to consider it as a combination of two associative
algebras. Therefore, we need a complete list of associative algebras.
Obviously, the classification problem of algebras (even associative algebras),
in general, is nearly unreal problem. All existence classifications of
associative algebras concern unitary ones. In order to use the classification
of associative algebras to classify the diassociative algebras we need both
lists of unitary and non unitary associative algebras. In the present, paper
we give complete lists of low-dimensional complex associative algebras.
## 2 Lists of low-dimensional associative algebras with some isomorphism
invariants
Now and what it follows, all algebras are assumed to be over the field of
complex numbers $\mathbb{C}$.
In the next section we give lists of all complex associative algebras in
dimensions 2– 4. Some remarks on the tables. In the tables $As_{p}^{q}$ stands
for $q^{th}$ algebra in dimension $p$. In the second column only nonzero
products are given. The third column describes the automorphism groups of the
algebras. For all algebras we test to be nilpotent or not, in the last case we
indicate nilpotent $N$ and semi-simple $S$ parts of $As_{p}^{q}$. By
$C(As_{p}^{q}),$ $L(As_{p}^{q})$, and $R(As_{p}^{q})$ the maximal commutative
subalgebra, the left annihilator, and the right annihilator of $As_{p}^{q}$,
respectively are denoted, and their dimensions are given in the last three
columns of the tables.
A proof, that any associative algebra of dimensions 2– 4 is included in the
lists, is available from the authors. Because of its length, it is omitted
from the paper. We tabulate only indecomposable algebras.
### 2.1 Two-dimensional associative algebras
| Table of multiplication | Automorphisms | Type of algebra | dim | dim | dim
---|---|---|---|---|---|---
| | | | C$(As_{p}^{q})$ | L$(As_{p}^{q})$ | R$(As_{p}^{q})$
$As_{2}^{1}:$ | ${e_{1}e_{1}=e_{2}}$ | $\,\left(\begin{array}[]{l}{a\,\,\,\,\,0}\\\ {b\,\,\,\,a^{2}}\end{array}\right)$ | commutative,
nilpotent | 2 | 1 | 1
$As_{2}^{2}:$ | $e_{1}e_{1}=e_{1},$ $e_{1}e_{2}=e_{2}$ | $\,\left(\begin{array}[]{l}{1\,\,\,\,0}\\\ {a\,\,\,\,b}\end{array}\right)$ | $A=N\dotplus S,$
$N=<e_{2}>$,
$S=<e_{1}>$ | 1 | 1 | 0
$As_{2}^{3}:$ | $e_{1}e_{1}=e_{1},$ $e_{2}e_{1}=e_{2}$ | $\,\left(\begin{array}[]{l}{1\,\,\,\,0}\\\ {a\,\,\,\,b}\end{array}\right)$ | $A=N\dotplus S,$
$N=<e_{2}>$,
$S=<e_{1}>$ | 1 | 0 | 1
$As_{2}^{4}:$ | $e_{1}e_{1}=e_{1},$ $e_{1}e_{2}=e_{2},$
$e_{2}e_{1}=e_{2}$ | $\left(\begin{array}[]{l}{1\,\,\,\,0}\\\ {0\,\,\,\,a}\end{array}\right)$ | commutative,
unitary,
$A=N\dotplus S,$
$N=<e_{2}>$,
$S=<e_{1}>$ | 2 | 0 | 0
### 2.2 Three-dimensional associative algebras
| Table of
multiplication | Automorphisms | Type of algebra | dim C$(As_{p}^{q})$ | dim L$(As_{p}^{q})$ | dim R$(As_{p}^{q})$
---|---|---|---|---|---|---
$As_{3}^{1}:$ | $e_{1}e_{3}=e_{2},$ $e_{3}e_{1}=e_{2}$ | $\left(\begin{array}[]{l}{a\,\,\,\,\,0\,\,\,\,\,0}\\\ {b\,\,\,\,ac\,\,\,d}\\\ {0\,\,\,\,\,0\,\,\,\,\,c}\end{array}\right)$, $\left(\begin{array}[]{l}{0\,\,\,\,\,0\,\,\,\,\,a}\\\ {b\,\,\,\,ac\,\,\,d}\\\ {c\,\,\,\,\,0\,\,\,\,\,0}\end{array}\right)$ | commutative,
nilpotent | 3 | 1 | 1
$As_{3}^{2}:$ | $e_{1}e_{3}=e_{2},$
$e_{3}e_{1}=\alpha e_{2},$
$\alpha$ $\in\mathbb{C}\backslash\ \\{1\\}$ | $\,\left(\begin{array}[]{l}{a\,\,\,\,\,0\,\,\,\,\,0}\\\ {b\,\,\,\,ac\,\,\,d}\\\ {0\,\,\,\,\,0\,\,\,\,\,c}\end{array}\right)$,$\,\left(\begin{array}[]{l}{0\,\,\,\,\,0\,\,\,\,\,a}\\\ {b\,\,\,\,ac\,\,\,d}\\\ {c\,\,\,\,\,0\,\,\,\,\,0}\end{array}\right)$,
$\left(\begin{array}[]{l}{a\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,b}\\\ {c\,\,\,\,af-be\,\,\,d}\\\ {e\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,f}\end{array}\right)$ | nilpotent | 2 | 1 | 1
$As_{3}^{3}:$ | $e_{1}e_{1}=e_{2},$ $e_{1}e_{2}=e_{3}$,
$e_{2}e_{1}=e_{3}$ | $\left(\begin{array}[]{l}{a\,\,\,\,\,\,\,0\,\,\,\,\,\,0}\\\ {b\,\,\,\,\,\,a^{2}\,\,\,\,\,0}\\\ {c\,\,\,2ab\,\,\,a^{3}}\end{array}\right)$ | commutative,
nilpotent | 3 | 1 | 1
$As_{3}^{4}:$ | $e_{1}e_{3}=e_{2},$ $e_{2}e_{3}=e_{2},$
$e_{3}e_{3}=e_{3}$ | $\,\left(\begin{array}[]{l}{a\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,0}\\\ {b\,\,\,\,a+b\,\,\,c}\\\ {0\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,1}\end{array}\right)$ | $A=N\dotplus S,$
$N=<e_{1},e_{2}>,$
$S=<e_{3}>$ | 2 | 0 | 2
---|---|---|---|---|---|---
$As_{3}^{5}:$ | $e_{2}e_{3}=e_{2},$ $e_{3}e_{1}=e_{1},$
$e_{3}e_{3}=e_{3}$ | $\,\left(\begin{array}[]{l}{a\,\,\,\,0\,\,\,b}\\\ {0\,\,\,\,c\,\,\,\,d}\\\ {0\,\,\,\,0\,\,\,\,1}\end{array}\right)$ | $A=N\dotplus S,$
$N=<e_{1},e_{2}>,$
$S=<e_{3}>$ | 2 | 1 | 1
$As_{3}^{6}:$ | $e_{3}e_{1}=e_{2},$ $e_{3}e_{2}=e_{2},$
$e_{3}e_{3}=e_{3}$ | $\,\left(\begin{array}[]{l}{a\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,0}\\\ {b\,\,\,\,a+b\,\,\,c}\\\ {0\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,1}\end{array}\right)$ | $A=N\dotplus S,$
$N=<e_{1},e_{2}>,$
$S=<e_{3}>$ | 2 | 2 | 0
$As_{3}^{7}:$ | $e_{1}e_{2}=e_{1},$ $e_{2}e_{2}=e_{2},$
$e_{3}e_{1}=e_{1},$ $e_{3}e_{3}=e_{3}$ | $\,\left(\begin{array}[]{l}{a\,\,\,\,b\,\,\,-b}\\\ {0\,\,\,\,\,1\,\,\,\,\,\,\,0}\\\ {0\,\,\,\,0\,\,\,\,\,\,\,\,1}\end{array}\right)$ | unitary,
$A=N\dotplus S,$
$N=<e_{1}>,$
$S=<e_{2},e_{3}>$ | 2 | 0 | 0
$As_{3}^{8}:$ | $e_{1}e_{3}=e_{1},$ $e_{2}e_{3}=e_{2},$
$e_{3}e_{1}=e_{1},$ $e_{3}e_{3}=e_{3}$ | $\,\left(\begin{array}[]{l}{a\,\,\,\,0\,\,\,\,0}\\\ {0\,\,\,\,b\,\,\,\,c}\\\ {0\,\,\,\,0\,\,\,\,1}\end{array}\right)$ | $A=N\dotplus S,$
$N=<e_{1},e_{2}>,$
$S=<e_{3}>$ | 2 | 0 | 1
$\,As_{3}^{9}:$ | $e_{2}e_{3}=e_{2},$ $e_{3}e_{1}=e_{1},$
$e_{3}e_{2}=e_{2},$ $e_{3}e_{3}=e_{3}$ | $\,\left(\begin{array}[]{l}{a\,\,\,\,0\,\,\,\,b}\\\ {0\,\,\,\,c\,\,\,\,0}\\\ {0\,\,\,\,0\,\,\,\,1}\end{array}\right)$ | $A=N\dotplus S,$
$N=<e_{1},e_{2}>,$
$S=<e_{3}>$ | 2 | 1 | 0
$\,As_{3}^{10}:\,$ | $e_{1}e_{3}=e_{1},$ $e_{2}e_{3}=e_{2},$
$e_{3}e_{1}=e_{1},$ $e_{3}e_{2}=e_{2},$
$e_{3}e_{3}=e_{3}$ | $\,\left(\begin{array}[]{l}{a\,\,\,\,b\,\,\,\,0}\\\ {c\,\,\,\,d\,\,\,\,0}\\\ {0\,\,\,\,0\,\,\,\,1}\end{array}\right)$ | commutative, unitary,
$A=N\dotplus S,$
$N=<e_{1},e_{2}>,$
$S=<e_{3}>$ | 3 | 0 | 0
$\,As_{3}^{11}:$ | $e_{1}e_{3}=e_{2},$ $e_{2}e_{3}=e_{2},$
$e_{3}e_{1}=e_{2},$ $e_{3}e_{2}=e_{2},$
$e_{3}e_{3}=e_{3}$ | $\,\left(\begin{array}[]{l}{a\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,0}\\\ {b\,\,\,\,a+b\,\,\,0}\\\ {0\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,1}\end{array}\right)$ | commutative,
$A=N\dotplus S,$
$N=<e_{1},e_{2}>,$
$S=<e_{3}>$ | 3 | 0 | 0
$\,As_{3}^{12}:\,$ | $e_{1}e_{1}=e_{2},$ $e_{1}e_{3}=e_{1},$
$e_{2}e_{3}=e_{2},$ $e_{3}e_{1}=e_{1},$
$e_{3}e_{2}=e_{2},$ $e_{3}e_{3}=e_{3}$ | $\,\left(\begin{array}[]{l}{a\,\,\,\,\,0\,\,\,\,\,0}\\\ {b\,\,\,\,\,a^{2}\,\,\,0}\\\ {0\,\,\,\,\,\,0\,\,\,\,\,\,1}\end{array}\right)$ | commutative, unitary,
$A=N\dotplus S,$
$N=<e_{1},e_{2}>,$
$S=<e_{3}>$ | 3 | 0 | 0
### 2.3 Four-dimensional associative algebras
| Table of
multiplication | Automorphisms | Type of algebra | dim C$(As_{p}^{q})$ | dim L$(As_{p}^{q})$ | dim R$(As_{p}^{q})$
---|---|---|---|---|---|---
$As_{4}^{1}:$ | $e_{1}e_{2}=e_{3},$ $e_{2}e_{1}=e_{4}$ | $\left(\begin{array}[]{l}{a\,\,\,\,0\,\,\,\,0\,\,\,\,\,0}\\\ {0\,\,\,\,a\,\,\,\,0\,\,\,\,\,0}\\\ {b\,\,\,\,c\,\,\,\,a^{2}\,\,\,0}\\\ {d\,\,\,\,e\,\,\,\,0\,\,\,\,a^{2}}\end{array}\right)$, $\left(\begin{array}[]{l}{0\,\,\,\,a\,\,\,\,0\,\,\,\,\,0}\\\ {b\,\,\,\,0\,\,\,\,0\,\,\,\,\,0}\\\ {c\,\,\,\,d\,\,\,\,0\,\,\,ab}\\\ {e\,\,\,\,f\,\,\,\,ab\,\,\,\,0}\end{array}\right)$ | nilpotent | 3 | 2 | 2
$As_{4}^{2}:$ | $e_{1}e_{2}=e_{4},$ $e_{3}e_{1}=e_{4}$ | $\left(\begin{array}[]{l}{\,\,a\,\,\,\,\,0\,\,\,\,\,0\,\,\,\,\,0}\\\ {\,\,c\,\,\,\,\,b\,\,\,\,\,0\,\,\,\,\,0}\\\ {-c\,\,\,0\,\,\,\,b\,\,\,\,\,0}\\\ {\,\,d\,\,\,\,e\,\,\,\,f\,\,\,\,ab}\end{array}\right)$ | nilpotent | 3 | 2 | 2
$As_{4}^{3}:$ | $e_{1}e_{2}=e_{3}$, $e_{2}e_{1}=e_{4}$,
$e_{2}e_{2}=-e_{3}$ | $\left(\begin{array}[]{l}{a\,\,\,\,0\,\,\,\,0\,\,\,\,\,0}\\\ {0\,\,\,\,a\,\,\,\,0\,\,\,\,\,0}\\\ {b\,\,\,\,c\,\,\,\,a^{2}\,\,\,0}\\\ {d\,\,\,\,e\,\,\,\,0\,\,\,\,a^{2}}\end{array}\right)$ | nilpotent | 3 | 2 | 2
$As_{4}^{4}:$ | $e_{1}e_{2}=e_{3},$ $e_{2}e_{2}=e_{4}$,
$e_{2}e_{1}=-e_{3}$ | $\left(\begin{array}[]{l}{a\,\,\,\,c\,\,\,\,0\,\,\,\,\,0}\\\ {0\,\,\,\,b\,\,\,\,0\,\,\,\,\,0}\\\ {d\,\,\,\,e\,\,\,\,ab\,\,\,0}\\\ {f\,\,\,\,g\,\,\,\,0\,\,\,\,b^{2}}\end{array}\right)$ | nilpotent | 3 | 2 | 2
$As_{4}^{5}:$ | $e_{1}e_{2}=e_{4},$ $e_{3}e_{3}=e_{4}$
$e_{2}e_{1}=-e_{4},$ | $\left(\begin{array}[]{l}{a\,\,-\frac{b^{2}}{c}\,\,\,\,0\,\,\,\,\,0}\\\ {c\,\,\,\,\,\,\,0\,\,\,\,\,\,\,0\,\,\,\,\,\,\,0}\\\ {0\,\,\,\,\,\,\,0\,\,\,\,\,\,\,b\,\,\,\,\,\,\,0}\\\ {d\,\,\,\,\,\,\,e\,\,\,\,\,\,\,f\,\,\,\,\,b^{2}}\end{array}\right)$, | nilpotent | 3 | 1 | 1
| | $\left(\begin{array}[]{l}{\frac{d^{2}+ab}{c}\,\,\,a\,\,\,\,0\,\,\,\,\,0}\\\ {\,\,\,\,b\,\,\,\,\,\,\,\,c\,\,\,\,\,\,\,0\,\,\,\,\,\,\,0}\\\ {\,\,\,\,0\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,d\,\,\,\,\,\,\,0}\\\ {\,\,\,\,e\,\,\,\,\,\,\,\,f\,\,\,\,\,\,\,h\,\,\,\,\,d^{2}}\end{array}\right)$ | | | |
$As_{4}^{6}(\alpha):$ | $e_{1}e_{2}=e_{4}$, $e_{2}e_{2}=e_{3},$
$e_{2}e_{1}=\frac{1+\alpha}{1-\alpha}e_{4}$, $\alpha\neq 1$ | $\left(\begin{array}[]{l}{a\,\,\,\,c\,\,\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0}\\\ {0\,\,\,\,b\,\,\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0}\\\ {d\,\,\,\,e\,\,\,\,\,\,\,\,\,\,\,\,b^{2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0}\\\ {f\,\,\,\,g\,\,\,\,bc(1+\alpha)\,\,\,\,ab}\end{array}\right)$ | nilpotent | 3 | 2 | 2
$As_{4}^{7}:$ | $e_{1}e_{1}=e_{1},$ $e_{1}e_{4}=e_{4},$
$e_{2}e_{1}=e_{2},$ $e_{2}e_{4}=e_{3}$ | $\left(\begin{array}[]{l}{1\,\,\,\,\,0\,\,\,\,\,0\,\,\,\,\,0}\\\ {a\,\,\,\,\,b\,\,\,\,\,0\,\,\,\,\,0}\\\ {ac\,\,bc\,\,bd\,\,ad}\\\ {c\,\,\,\,\,0\,\,\,\,\,0\,\,\,\,\,d}\end{array}\right)$ | $A=N\dotplus S,$
$N=<e_{2},e_{3},e_{4}>$,
$S=<e_{1}>$ | 2 | 2 | 2
$As_{4}^{8}:$ | $e_{1}e_{1}=e_{1},$ $e_{3}e_{1}=e_{3},$
$e_{4}e_{3}=e_{2},$ $e_{2}e_{1}=e_{2}$ | $\left(\begin{array}[]{l}{\,\,1\,\,\,\,\,\,\,\,0\,\,\,\,\,0\,\,\,\,0}\\\ {\,\,a\,\,\,\,\,\,bc\,\,\,\,d\,\,\,\,e}\\\ {-\frac{e}{c}\,\,\,0\,\,\,\,\,b\,\,\,\,\,0}\\\ {\,\,0\,\,\,\,\,\,0\,\,\,\,\,0\,\,\,\,c}\end{array}\right)$ | $A=N\dotplus S,$
$N=<e_{2},e_{3},e_{4}>$,
$S=<e_{1}>$ | 2 | 0 | 2
$As_{4}^{9}:$ | $e_{1}e_{1}=e_{1},$ $e_{1}e_{3}=e_{3},$
$e_{1}e_{2}=e_{2},$ $e_{3}e_{4}=e_{2}$ | $\left(\begin{array}[]{l}{1\,\,\,\,\,0\,\,\,\,\,0\,\,\,\,\,\,0}\\\ {a\,\,\,\,bc\,\,\,d\,\,\,-ce}\\\ {e\,\,\,\,\,0\,\,\,\,\,b\,\,\,\,\,\,0}\\\ {0\,\,\,\,\,0\,\,\,\,\,0\,\,\,\,\,\,c}\end{array}\right)$ | $A=N\dotplus S,$
$N=<e_{2},e_{3},e_{4}>$,
$S=<e_{1}>$ | 2 | 2 | 0
$As_{4}^{10}:$ | $e_{1}e_{1}=e_{3},$ $e_{1}e_{3}=e_{4},$
$e_{2}e_{2}=-e_{4},$
$e_{3}e_{1}=e_{4}$ | $\left(\begin{array}[]{l}{\sqrt[3]{a^{2}}\,\,0\,\,\,\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0}\\\ {\,b\,\,\,\,\,\,\,\,a\,\,\,\,\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0}\\\ {\,c\,\,\,\,\,\,b\sqrt[3]{a}\,\,\,\,\,\,\,\,\,\sqrt[3]{a^{2}}\,\,\,\,\,\,\,\,\,0}\\\ {d\,\,\,\,\,\,e\,\,\,\,\,\,2c\sqrt[3]{a^{2}}-b^{2}\,\,\,a^{2}}\end{array}\right)$ | commutative,
nilpotent | 4 | 1 | 1
$As_{4}^{11}:$ | $e_{1}e_{1}=e_{4},$ $e_{2}e_{1}=e_{3},$
$e_{1}e_{4}=-e_{3},$
$e_{4}e_{1}=-e_{3}$ | $\left(\begin{array}[]{l}{a\,\,\,\,\,0\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,\,0}\\\ {b\,\,\,\,a^{2}\,\,\,0\,\,\,\,\,\,\,\,\,\,\,0}\\\ {c\,\,\,\,\,d\,\,\,\,\,a^{3}\,\,a(b-2e)}\\\ {e\,\,\,\,\,0\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,\,a^{2}}\end{array}\right)$ | nilpotent | 3 | 1 | 2
$As_{4}^{12}:$ | $e_{1}e_{1}=e_{1},$ $e_{1}e_{2}=e_{2},$
$e_{3}e_{1}=e_{3},$ $e_{4}e_{1}=e_{4}$ | $\left(\begin{array}[]{l}{1\,\,\,\,\,0\,\,\,\,0\,\,\,\,\,0}\\\ {a\,\,\,\,b\,\,\,\,0\,\,\,\,\,0}\\\ {c\,\,\,\,\,0\,\,\,\,d\,\,\,\,e}\\\ {f\,\,\,\,0\,\,\,\,g\,\,\,\,h}\end{array}\right)$ | $A=N\dotplus S,$
$N=<e_{2},e_{3},e_{4}>$,
$S=<e_{1}>$ | 3 | 1 | 2
$As_{4}^{13}:$ | $e_{2}e_{2}=e_{2},$ $e_{2}e_{3}=e_{3},$
$e_{2}e_{4}=e_{4},$ $e_{1}e_{2}=e_{1}$ | $\left(\begin{array}[]{l}{a\,\,\,\,\,b\,\,\,\,0\,\,\,\,\,0}\\\ {0\,\,\,\,\,1\,\,\,\,0\,\,\,\,\,0}\\\ {0\,\,\,\,\,c\,\,\,\,d\,\,\,\,e}\\\ {0\,\,\,\,f\,\,\,\,g\,\,\,\,h}\end{array}\right)$ | $A=N\dotplus S,$
$N=<e_{1},e_{3},e_{4}>$,
$S=<e_{2}>$ | 3 | 2 | 1
$As_{4}^{14}:$ | $e_{1}e_{1}=e_{1},$ $e_{3}e_{1}=e_{3},$
$e_{4}e_{1}=e_{4},$ $e_{2}e_{1}=e_{2}$ | $\left(\begin{array}[]{l}{1\,\,\,\,\,0\,\,\,\,0\,\,\,\,\,0}\\\ {a\,\,\,\,\,b\,\,\,\,c\,\,\,\,\,d}\\\ {e\,\,\,\,f\,\,\,g\,\,\,\,h}\\\ {i\,\,\,\,\,j\,\,\,\,k\,\,\,\,l}\end{array}\right)$ | $A=N\dotplus S,$
$N=<e_{2},e_{3},e_{4}>,$
$S=<e_{1}>$ | 3 | 0 | 3
---|---|---|---|---|---|---
$As_{4}^{15}:$ | $e_{2}e_{2}=e_{2},$ $e_{2}e_{3}=e_{3},$
$e_{2}e_{4}=e_{4},$ $e_{2}e_{1}=e_{1}$ | $\left(\begin{array}[]{l}{a\,\,\,\,\,b\,\,\,\,c\,\,\,\,\,d}\\\ {0\,\,\,\,\,1\,\,\,\,0\,\,\,\,\,0}\\\ {e\,\,\,\,f\,\,\,g\,\,\,\,h}\\\ {i\,\,\,\,\,j\,\,\,\,k\,\,\,\,l}\end{array}\right)$ | $A=N\dotplus S,$
$N=<e_{1},e_{3},e_{4}>$,
$S=<e_{2}>$ | 3 | 3 | 0
$As_{4}^{16}(\alpha):$ | $e_{1}e_{1}=e_{4},$ $e_{1}e_{2}=e_{4},$
$e_{2}e_{1}=\alpha e_{4},$
$e_{3}e_{3}=e_{4}$ | $\left(\begin{array}[]{l}{a\,\,\,0\,\,\,0\,\,\,\,0}\\\ {0\,\,\,a\,\,\,0\,\,\,\,0}\\\ {0\,\,\,0\,\,\,a\,\,\,0}\\\ {b\,\,\,c\,\,\,d\,\,\,a^{2}}\end{array}\right)$, $\left(\begin{array}[]{l}{a\,\,\,\,0\,\,\,\,0\,\,\,\,0}\\\ {b\,\,\,\frac{c^{2}}{a}\,\,\,0\,\,\,\,0}\\\ {0\,\,\,\,0\,\,\,c\,\,\,\,0}\\\ {d\,\,\,\,e\,\,\,f\,\,\,c^{2}}\end{array}\right)$ | nilpotent | 3 | 1 | 1
$As_{4}^{17}:$ | $e_{1}e_{1}=e_{4},$ $e_{1}e_{2}=e_{3},$
$e_{2}e_{1}=-e_{3},$
$e_{2}e_{2}=-2e_{3}+e_{4}$ | $\left(\begin{array}[]{l}{a\,\,\,\,0\,\,\,\,0\,\,\,\,\,0}\\\ {0\,\,\,\,a\,\,\,\,0\,\,\,\,\,0}\\\ {b\,\,\,\,c\,\,\,\,a^{2}\,\,\,0}\\\ {d\,\,\,\,e\,\,\,\,0\,\,\,\,a^{2}}\end{array}\right)$, $\left(\begin{array}[]{l}{0\,\,\,\,a\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,0}\\\ {a\,\,\,\,0\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,0}\\\ {b\,\,\,\,c\,\,-a^{2}\,-2a^{2}}\\\ {d\,\,\,\,e\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,a^{2}}\end{array}\right)$ | nilpotent | 3 | 2 | 2
$As_{4}^{18}(\alpha):$ | $e_{1}e_{1}=e_{4},$ $e_{1}e_{2}=e_{3},$
$e_{2}e_{1}=-\alpha e_{4},$
$e_{2}e_{2}=-e_{3}$ | $\left(\begin{array}[]{l}{a\,\,\,\,0\,\,\,\,0\,\,\,\,\,0}\\\ {0\,\,\,\,a\,\,\,\,0\,\,\,\,\,0}\\\ {b\,\,\,\,c\,\,\,\,a^{2}\,\,\,0}\\\ {d\,\,\,\,e\,\,\,\,0\,\,\,\,a^{2}}\end{array}\right)$, $\left(\begin{array}[]{l}{0\,\,\,\,a\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,0}\\\ {a\,\,\,\,0\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,0}\\\ {b\,\,\,\,c\,\,-a^{2}\,-2a^{2}}\\\ {d\,\,\,\,e\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,a^{2}}\end{array}\right)$ | nilpotent | 3 | 2 | 2
$As_{4}^{19}:$ | $e_{1}e_{1}=e_{1},$ $e_{2}e_{2}=e_{2},$
$e_{2}e_{3}=e_{3},$ $e_{3}e_{1}=e_{3},$
$e_{4}e_{1}=e_{4}$ | $\,\left(\begin{array}[]{l}{1\,\,\,\,\,\,0\,\,\,\,\,0\,\,\,\,\,0}\\\ {0\,\,\,\,\,\,1\,\,\,\,\,0\,\,\,\,\,0}\\\ {a\,\,\,-a\,\,\,\,b\,\,\,\,0}\\\ {c\,\,\,\,\,\,0\,\,\,\,\,0\,\,\,\,d}\end{array}\right)\,\,\,$ | $A=N\dotplus S,$
$N=<e_{3},e_{4}>,$
$S=<e_{1},e_{2}>$ | 2 | 0 | 1
$As_{4}^{20}:$ | $e_{1}e_{1}=e_{1},$ $e_{1}e_{3}=e_{3},$
$e_{1}e_{4}=e_{4},$ $e_{2}e_{2}=e_{2},$
$e_{3}e_{2}=e_{3}$ | $\left(\begin{array}[]{l}{1\,\,\,\,\,\,0\,\,\,\,\,0\,\,\,\,\,0}\\\ {0\,\,\,\,\,\,1\,\,\,\,\,0\,\,\,\,\,0}\\\ {a\,\,\,-a\,\,\,\,b\,\,\,\,0}\\\ {c\,\,\,\,\,\,0\,\,\,\,\,0\,\,\,\,d}\end{array}\right)\,\,\,$ | $A=N\dotplus S,$
$N=<e_{3},e_{4}>,$
$S=<e_{1},e_{2}>$ | 2 | 1 | 0
$As_{4}^{21}:$ | $e_{1}e_{1}=e_{1},$ $e_{2}e_{2}=e_{2},$
$e_{2}e_{4}=e_{4},$ $e_{4}e_{1}=e_{4},$
$e_{3}e_{2}=e_{3}$ | $\left(\begin{array}[]{l}{1\,\,\,\,\,\,0\,\,\,\,\,0\,\,\,\,\,0}\\\ {0\,\,\,\,\,\,1\,\,\,\,\,0\,\,\,\,\,\,0}\\\ {0\,\,\,\,\,a\,\,\,\,\,\,b\,\,\,\,\,0}\\\ {c\,\,-c\,\,\,\,0\,\,\,\,d}\end{array}\right)\,\,\,$ | $A=N\dotplus S,$
$N=<e_{3},e_{4}>,$
$S=<e_{1},e_{2}>$ | 2 | 0 | 1
$As_{4}^{22}:$ | $e_{2}e_{2}=e_{2},$ $e_{2}e_{4}=e_{4},$
$e_{3}e_{3}=e_{3},$ $e_{3}e_{1}=e_{1},$
$e_{1}e_{2}=e_{1}$ | $\left(\begin{array}[]{l}{a\,\,\,\,\,\,b\,\,-b\,\,\,\,0}\\\ {0\,\,\,\,\,\,1\,\,\,\,\,\,\,0\,\,\,\,\,0}\\\ {0\,\,\,\,\,0\,\,\,\,\,\,\,\,1\,\,\,\,\,0}\\\ {0\,\,\,\,\,\,c\,\,\,\,\,\,0\,\,\,\,\,d}\end{array}\right)\,\,\,$ | $A=N\dotplus S,$
$N=<e_{1},e_{4}>,$
$S=<e_{2},e_{3}>$ | 2 | 1 | 0
$As_{4}^{23}:$ | $e_{1}e_{1}=e_{4},$ $e_{1}e_{4}=-e_{3},$
$e_{2}e_{1}=e_{3},$ $e_{2}e_{2}=e_{3},$
$e_{4}e_{1}=-e_{3}$ | $\left(\begin{array}[]{l}{1\,\,\,\,\,\,0\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,\,\,\,0}\\\ {a\,\,\,\,\,\,1\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,\,\,\,\,0}\\\ {b\,\,\,\,\,c\,\,\,\,\,1\,\,\,\,a(a+1)-2d}\\\ {d\,\,\,\,\,a\,\,\,\,\,o\,\,\,\,\,\,\,\,\,\,\,\,\,\,1}\end{array}\right)\,\,\,$ | nilpotent | 3 | 1 | 1
$As_{4}^{24}:$ | $e_{2}e_{2}=e_{2},$ $e_{2}e_{3}=e_{3},$
$e_{2}e_{1}=e_{1},$ $e_{4}e_{2}=e_{4},$
$e_{1}e_{2}=e_{1}$ | $\left(\begin{array}[]{l}{a\,\,\,\,0\,\,\,\,0\,\,\,\,0}\\\ {0\,\,\,\,1\,\,\,\,0\,\,\,\,0}\\\ {0\,\,\,\,b\,\,\,\,c\,\,\,\,0}\\\ {0\,\,\,\,d\,\,\,\,o\,\,\,\,e}\end{array}\right)\,\,\,$ | $A=N\dotplus S,$
$N=<e_{1},e_{3},e_{4}>,$
$S=<e_{2}>$ | 3 | 1 | 1
$As_{4}^{25}:$ | $e_{1}e_{2}=e_{4},$ $e_{1}e_{3}=e_{4},$
$e_{2}e_{1}=-e_{4},$ $e_{2}e_{2}=e_{4},$
$e_{3}e_{1}=e_{4}$ | $\left(\begin{array}[]{l}{\,\,\,a\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,0\,\,\,\,\,0}\\\ {\,\,\,b\,\,\,\,\,\,\,\,\,\,a\,\,\,\,\,\,0\,\,\,\,\,0}\\\ {-\frac{b^{2}}{2a}\,\,-b\,\,\,\,a\,\,\,\,0}\\\ {\,\,\,\,c\,\,\,\,\,\,\,\,\,d\,\,\,\,\,\,e\,\,\,\,a^{2}}\end{array}\right)\,\,\,$ | nilpotent | 3 | 1 | 1
$As_{4}^{26}:$ | $e_{1}e_{1}=e_{1},$ $e_{1}e_{2}=e_{2},$
$e_{2}e_{1}=e_{2},$ $e_{4}e_{1}=e_{4},$
$e_{3}e_{1}=e_{3}$ | $\left(\begin{array}[]{l}{1\,\,\,\,\,0\,\,\,\,\,0\,\,\,\,\,0}\\\ {0\,\,\,\,a\,\,\,\,\,0\,\,\,\,\,0}\\\ {b\,\,\,\,0\,\,\,\,\,c\,\,\,\,\,d}\\\ {e\,\,\,\,\,0\,\,\,\,\,f\,\,\,g}\end{array}\right)\,\,\,$ | $A=N\dotplus S,$
$N=<e_{2},e_{3},e_{4}>,$
$S=<e_{1}>$ | 3 | 0 | 2
$As_{4}^{27}:$ | $e_{1}e_{1}=e_{1},$ $e_{1}e_{2}=e_{2},$
$e_{1}e_{4}=e_{4},$ $e_{1}e_{3}=e_{3},$
$e_{2}e_{1}=e_{2}$ | $\left(\begin{array}[]{l}{1\,\,\,\,\,0\,\,\,\,\,0\,\,\,\,\,0}\\\ {0\,\,\,\,a\,\,\,\,\,0\,\,\,\,\,0}\\\ {b\,\,\,\,0\,\,\,\,\,c\,\,\,\,\,d}\\\ {e\,\,\,\,\,0\,\,\,\,\,f\,\,\,g}\end{array}\right)\,\,\,$ | $A=N\dotplus S,$
$N=<e_{2},e_{3},e_{4}>,$
$S=<e_{1}>$ | 3 | 2 | 0
---|---|---|---|---|---|---
$As_{4}^{28}(\alpha):$ | $e_{1}e_{1}=e_{4},$ $e_{1}e_{2}=\alpha e_{4},$
$e_{2}e_{1}=-\alpha e_{4},$
$e_{2}e_{2}=e_{4},$ $e_{3}e_{3}=e_{4},$ | $\left(\begin{array}[]{l}{a\,\,\,b\,\,\,\,c\,\,\,\,\,\,\,\,\,d}\\\ {e\,\,\,f\,\,\,g\,\,\,\,\,\,\,\,\,h}\\\ {i\,\,\,j\,\,\,\,k\,\,\,\,\,\,\,\,\,\,l}\\\ {0\,\,\,0\,\,\,0\,\,af-be}\end{array}\right)$ | nilpotent | 3 | 1 | 1
$As_{4}^{29}:$ | $e_{1}e_{1}=e_{1},$ $e_{1}e_{3}=e_{3},$
$e_{2}e_{2}=e_{2},$ $e_{2}e_{4}=e_{4},$
$e_{3}e_{1}=e_{3},$ $e_{4}e_{1}=e_{4}$ | $\left(\begin{array}[]{l}{1\,\,\,\,\,\,0\,\,\,\,\,0\,\,\,\,\,0}\\\ {0\,\,\,\,\,1\,\,\,\,\,\,0\,\,\,\,\,0}\\\ {0\,\,\,\,\,0\,\,\,\,\,a\,\,\,\,\,0}\\\ {b\,\,-b\,\,\,0\,\,\,\,c}\end{array}\right)\,\,\,$ | unitary,
$A=N\dotplus S,$
$N=<e_{3},e_{4}>,$
$S=<e_{1},e_{2}>$ | 3 | 0 | 0
$As_{4}^{30}:$ | $e_{1}e_{1}=e_{1},$ $e_{1}e_{3}=e_{3},$
$e_{1}e_{4}=e_{4},$ $e_{2}e_{2}=e_{2},$
$e_{3}e_{1}=e_{3},$ $e_{4}e_{2}=e_{4}$ | $\left(\begin{array}[]{l}{1\,\,\,\,\,\,0\,\,\,\,\,0\,\,\,\,\,0}\\\ {0\,\,\,\,\,1\,\,\,\,\,\,0\,\,\,\,\,0}\\\ {0\,\,\,\,\,0\,\,\,\,\,a\,\,\,\,\,0}\\\ {b\,\,-b\,\,\,0\,\,\,\,c}\end{array}\right)\,\,\,$ | unitary,
$A=N\dotplus S,$
$N=<e_{3},e_{4}>,$
$S=<e_{1},e_{2}>$ | 3 | 0 | 0
$As_{4}^{31}:$ | $e_{1}e_{1}=e_{1},$ $e_{1}e_{4}=e_{4},$
$e_{2}e_{2}=e_{2},$ $e_{2}e_{3}=e_{3},$
$e_{3}e_{1}=e_{3},$ $e_{4}e_{2}=e_{4}$ | $\left(\begin{array}[]{l}{0\,\,\,\,\,1\,\,\,\,\,\,0\,\,\,\,\,0}\\\ {1\,\,\,\,\,\,0\,\,\,\,\,\,0\,\,\,\,\,0}\\\ {a\,\,-a\,\,\,\,0\,\,\,\,b}\\\ {c\,\,-c\,\,\,\,d\,\,\,\,0}\end{array}\right)$, $\left(\begin{array}[]{l}{1\,\,\,\,\,0\,\,\,\,\,\,0\,\,\,\,\,0}\\\ {0\,\,\,\,\,\,1\,\,\,\,\,\,0\,\,\,\,\,0}\\\ {a\,\,-a\,\,\,\,b\,\,\,\,0}\\\ {c\,\,-c\,\,\,\,0\,\,\,\,d}\end{array}\right)$ | unitary,
$A=N\dotplus S,$
$N=<e_{3},e_{4}>,$
$S=<e_{1},e_{2}>$ | 2 | 0 | 0
$As_{4}^{32}:$ | $e_{2}e_{1}=e_{3},$ $e_{3}e_{4}=e_{3}$,
$e_{4}e_{2}=e_{2},$ $e_{4}e_{3}=e_{3},$
$e_{4}e_{4}=e_{4},$ $e_{1}e_{4}=e_{1}$ | $\left(\begin{array}[]{l}{\,\,a\,\,\,\,\,\,0\,\,\,\,\,\,0\,\,\,-\frac{d}{b}}\\\ {\,\,0\,\,\,\,\,\,b\,\,\,\,\,0\,\,\,\,\,\,\,\,c}\\\ {-ac\,\,d\,\,\,\,ab\,\,\,\,\frac{cd}{b}}\\\ {\,\,0\,\,\,\,\,\,0\,\,\,\,\,\,0\,\,\,\,\,\,\,1}\end{array}\right)\,\,\,$ | $A=N\dotplus S,$
$N=<e_{1},e_{2},e_{3}>,$
$S=<e_{4}>$ | 2 | 0 | 0
$As_{4}^{33}:$ | $e_{1}e_{1}=e_{2},$ $e_{1}e_{2}=e_{3},$
$e_{1}e_{3}=e_{4},$ $e_{2}e_{1}=e_{3},$
$e_{2}e_{2}=e_{4},$ $e_{3}e_{1}=e_{4}$ | $\left(\begin{array}[]{l}{a\,\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,0}\\\ {b\,\,\,\,\,\,\,\,\,\,a^{2}\,\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,0}\\\ {c\,\,\,\,\,\,\,\,\,2ab\,\,\,\,\,\,\,\,\,a^{3}\,\,\,\,\,\,\,\,0}\\\ {d\,\,\,2ac+b^{2}\,\,3a^{2}b\,\,\,\,a^{4}}\end{array}\right)\,\,$ | commutative,
nilpotent | 4 | 1 | 1
$As_{4}^{34}:$ | $e_{2}e_{2}=e_{2},$ $e_{2}e_{3}=e_{3},$
$e_{2}e_{4}=e_{4},$ $e_{2}e_{1}=e_{1},$
$e_{4}e_{2}=e_{4},$ $e_{4}e_{3}=e_{1}$ | $\left(\begin{array}[]{l}{ab\,\,\,\,c\,\,\,\,d\,\,\,\,ae}\\\ {0\,\,\,\,\,1\,\,\,\,\,0\,\,\,\,\,0}\\\ {0\,\,\,\,\,e\,\,\,\,\,b\,\,\,\,\,0}\\\ {0\,\,\,\,\,0\,\,\,\,0\,\,\,\,\,a}\end{array}\right)\,\,\,$ | $A=N\dotplus S,$
$N=<e_{1},e_{3},e_{4}>,$
$S=<e_{2}>$ | 2 | 2 | 0
$As_{4}^{35}:$ | $e_{1}e_{2}=e_{1},$ $e_{3}e_{2}=e_{3},$
$e_{2}e_{2}=e_{2},$ $e_{2}e_{4}=e_{4},$
$e_{3}e_{4}=e_{1},$ $e_{4}e_{2}=e_{4}$ | $\left(\begin{array}[]{l}{ab\,\,\,\,c\,\,\,\,d\,\,\,\,ae}\\\ {0\,\,\,\,\,1\,\,\,\,\,0\,\,\,\,\,0}\\\ {0\,\,\,\,\,e\,\,\,\,\,b\,\,\,\,\,0}\\\ {0\,\,\,\,\,0\,\,\,\,0\,\,\,\,\,a}\end{array}\right)\,\,\,$ | $A=N\dotplus S,$
$N=<e_{1},e_{3},e_{4}>,$
$S=<e_{2}>$ | 2 | 0 | 2
$As_{4}^{36}:$ | $e_{1}e_{1}=e_{1},$ $e_{2}e_{2}=e_{2},$
$e_{2}e_{3}=e_{3},$ $e_{2}e_{4}=e_{4},$
$e_{3}e_{1}=e_{3},$ $e_{4}e_{1}=e_{4}$ | $\left(\begin{array}[]{l}{1\,\,\,\,\,0\,\,\,\,\,0\,\,\,\,\,0}\\\ {0\,\,\,\,\,1\,\,\,\,\,0\,\,\,\,\,0}\\\ {a\,\,-a\,\,\,\,b\,\,\,\,\,c}\\\ {d\,\,-d\,\,\,\,e\,\,\,f}\end{array}\right)\,\,\,$ | unitary,
$A=N\dotplus S,$
$N=<e_{3},e_{4}>,$
$S=<e_{1},e_{2}>$ | 2 | 0 | 0
$As_{4}^{37}:$ | $e_{1}e_{1}=e_{1},$ $e_{1}e_{2}=e_{2},$
$e_{1}e_{3}=e_{3},$ $e_{2}e_{1}=e_{2},$
$e_{3}e_{1}=e_{3},$ $e_{4}e_{1}=e_{4}$ | $\left(\begin{array}[]{l}{1\,\,\,\,0\,\,\,\,\,0\,\,\,\,\,0}\\\ {0\,\,\,\,a\,\,\,\,b\,\,\,\,\,0}\\\ {0\,\,\,\,\,c\,\,\,\,d\,\,\,\,0}\\\ {e\,\,\,\,\,0\,\,\,\,0\,\,\,\,f}\end{array}\right)\,\,\,$ | $A=N\dotplus S,$
$N=<e_{2},e_{3},e_{4}>,$
$S=<e_{1}>$ | 3 | 0 | 1
$As_{4}^{38}:$ | $e_{1}e_{1}=e_{1},$ $e_{1}e_{2}=e_{2},$
$e_{1}e_{3}=e_{3},$ $e_{1}e_{4}=e_{4},$
$e_{2}e_{1}=e_{2},$ $e_{3}e_{1}=e_{3}$ | $\left(\begin{array}[]{l}{1\,\,\,\,0\,\,\,\,\,0\,\,\,\,\,0}\\\ {0\,\,\,\,a\,\,\,\,b\,\,\,\,\,0}\\\ {0\,\,\,\,\,c\,\,\,\,d\,\,\,\,0}\\\ {e\,\,\,\,\,0\,\,\,\,0\,\,\,\,f}\end{array}\right)\,\,\,$ | $A=N\dotplus S,$
$N=<e_{2},e_{3},e_{4}>,$
$S=<e_{1}>$ | 3 | 1 | 0
$As_{4}^{39}:$ | $e_{1}e_{1}=e_{1},$ $e_{1}e_{2}=e_{2},$
$e_{1}e_{3}=e_{3},$ $e_{2}e_{1}=e_{2},$
$e_{2}e_{2}=e_{3},$ $e_{4}e_{1}=e_{4},$
$e_{3}e_{1}=e_{3}$ | $\left(\begin{array}[]{l}{1\,\,\,\,0\,\,\,\,\,0\,\,\,\,\,0}\\\ {0\,\,\,\,a\,\,\,\,0\,\,\,\,\,0}\\\ {0\,\,\,\,b\,\,\,\,a^{2}\,\,0}\\\ {c\,\,\,\,\,0\,\,\,\,0\,\,\,\,d}\end{array}\right)\,\,\,$ | $A=N\dotplus S,$
$N=<e_{2},e_{3},e_{4}>,$
$S=<e_{1}>$ | 3 | 0 | 1
$As_{4}^{40}:$ | $e_{1}e_{1}=e_{1},$ $e_{1}e_{2}=e_{2},$
$e_{1}e_{3}=e_{3},$ $e_{1}e_{4}=e_{4},$
$e_{2}e_{1}=e_{2},$ $e_{3}e_{1}=e_{3},$
$e_{4}e_{1}=e_{4}$ | $\left(\begin{array}[]{l}{1\,\,\,\,0\,\,\,\,\,0\,\,\,\,\,0}\\\ {0\,\,\,\,a\,\,\,\,b\,\,\,\,\,c}\\\ {0\,\,\,\,\,d\,\,\,\,e\,\,\,\,f}\\\ {0\,\,\,\,\,i\,\,\,\,j\,\,\,\,k}\end{array}\right)\,\,\,$ | commutative,
unitary,
$A=N\dotplus S,$
$N=<e_{2},e_{3},e_{4}>,$
$S=<e_{1}>$ | 4 | 0 | 0
---|---|---|---|---|---|---
$As_{4}^{41}:$ | $e_{1}e_{1}=e_{1},$ $e_{1}e_{2}=e_{2},$
$e_{1}e_{4}=e_{4},$ $e_{1}e_{3}=e_{3},$
$e_{2}e_{1}=e_{2},$ $e_{2}e_{2}=e_{3},$
$e_{3}e_{1}=e_{3}$ | $\left(\begin{array}[]{l}{1\,\,\,\,0\,\,\,\,\,0\,\,\,\,\,0}\\\ {0\,\,\,\,a\,\,\,\,0\,\,\,\,\,0}\\\ {0\,\,\,\,b\,\,\,\,a^{2}\,\,0}\\\ {c\,\,\,\,\,0\,\,\,\,0\,\,\,\,d}\end{array}\right)\,\,$ | $A=N\dotplus S,$
$N=<e_{2},e_{3},e_{4}>,$
$S=<e_{1}>$ | 3 | 1 | 0
$As_{4}^{42}:$ | $e_{1}e_{1}=e_{1},$ $e_{1}e_{2}=e_{2},$
$e_{2}e_{3}=e_{1},$ $e_{2}e_{4}=e_{2},$
$e_{3}e_{1}=e_{3},$ $e_{3}e_{2}=e_{4},$
$e_{4}e_{3}=e_{3},$ $e_{4}e_{4}=e_{4}$ | $\left(\begin{array}[]{l}{\,\,\,\,1\,\,\,\,\,\,\,\,\,\,\,a\,\,\,\,\,\,\,\,0\,\,\,\,\,0}\\\ {\,\,\,\,0\,\,\,\,\,\,\,\,\,\,\frac{\,1}{b}\,\,\,\,\,\,\,\,0\,\,\,\,\,0}\\\ {-ab\,\,-a^{2}b\,\,\,b\,\,\,\,ab}\\\ {\,\,\,\,0\,\,\,\,\,\,\,-a\,\,\,\,\,\,0\,\,\,\,\,1}\end{array}\right)\,\,\,$ | unitary,
$A=N\dotplus S,$
$N=<e_{2},e_{3}>,$
$S=<e_{1},e_{4}>$ | 2 | 0 | 0
$As_{4}^{43}:$ | $e_{1}e_{1}=e_{1},$ $e_{1}e_{2}=e_{2},$
$e_{1}e_{3}=e_{3},$ $e_{1}e_{4}=e_{4},$
$e_{2}e_{1}=e_{2},$ $e_{2}e_{2}=e_{4},$
$e_{3}e_{1}=e_{3},$ $e_{4}e_{1}=e_{4}$ | $\left(\begin{array}[]{l}{1\,\,\,\,0\,\,\,\,\,0\,\,\,\,\,0}\\\ {0\,\,\,\,a\,\,\,\,0\,\,\,\,\,0}\\\ {0\,\,\,\,b\,\,\,\,c\,\,\,\,\,0}\\\ {0\,\,\,\,d\,\,\,\,e\,\,\,\,a^{2}}\end{array}\right)\,\,\,$ | commutative,
unitary,
$A=N\dotplus S,$
$N=<e_{2},e_{3},e_{4}>,$
$S=<e_{1}>$ | 4 | 0 | 0
$As_{4}^{44}(\alpha):$ | $e_{1}e_{1}=e_{1},$ $e_{1}e_{2}=e_{2},$
$e_{1}e_{3}=e_{3},$ $e_{1}e_{4}=e_{4},$
$e_{2}e_{1}=e_{2},$ $e_{2}e_{3}=\alpha e_{4}$,
$e_{3}e_{1}=e_{3},$ $e_{3}e_{2}=e_{4},$
$e_{4}e_{1}=e_{4}$ | $\left(\begin{array}[]{l}{1\,\,\,\,0\,\,\,\,\,0\,\,\,\,\,\,0}\\\ {0\,\,\,\,0\,\,\,\,a\,\,\,\,\,\,\,0}\\\ {0\,\,\,\,b\,\,\,\,0\,\,\,\,\,\,\,0}\\\ {0\,\,\,\,c\,\,\,\,d\,\,\,\,\,ab}\end{array}\right)$, $\left(\begin{array}[]{l}{1\,\,\,\,0\,\,\,\,\,0\,\,\,\,0}\\\ {0\,\,\,\,a\,\,\,\,0\,\,\,\,\,0}\\\ {0\,\,\,\,0\,\,\,\,b\,\,\,\,\,0}\\\ {0\,\,\,\,c\,\,\,\,d\,\,\,\,\,ab}\end{array}\right)$, $\left(\begin{array}[]{l}{1\,\,\,0\,\,\,0\,\,\,\,\,\,\,0}\\\ {0\,\,\,a\,\,\,b\,\,\,\,\,\,\,0}\\\ {0\,\,\,c\,\,\,d\,\,\,\,\,\,\,0}\\\ {0\,\,\,e\,\,\,f\,\,ad-bc}\end{array}\right)$ | unitary,
$A=N\dotplus S,$
$N=<e_{2},e_{3},e_{4}>,$
$S=<e_{1}>$ | 3 | 0 | 0
$As_{4}^{45}:$ | $e_{1}e_{1}=e_{1},$ $e_{1}e_{2}=e_{2},$
$e_{1}e_{3}=e_{3},$ $e_{1}e_{4}=e_{4},$
$e_{2}e_{1}=e_{2},$ $e_{2}e_{2}=e_{3},$
$e_{2}e_{3}=e_{4},$ $e_{3}e_{1}=e_{3},$
$e_{3}e_{2}=e_{4},$ $e_{4}e_{1}=e_{4}$ | $\left(\begin{array}[]{l}{1\,\,\,\,0\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,0}\\\ {0\,\,\,\,a\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,0}\\\ {0\,\,\,\,b\,\,\,\,\,\,\,a^{2}\,\,\,\,\,\,0}\\\ {0\,\,\,\,c\,\,\,\,2a^{2}b\,\,\,a^{3}}\end{array}\right)\,\,\,$ | commutative,
unitary,
$A=N\dotplus S,$
$N=<e_{2},e_{3},e_{4}>,$
$S=<e_{1}>$ | 4 | 0 | 0
$As_{4}^{46}:$ | $e_{1}e_{1}=e_{1},$ $e_{1}e_{2}=e_{2},$
$e_{1}e_{3}=e_{3},$ $e_{1}e_{4}=e_{4},$
$e_{2}e_{1}=e_{2},$ $e_{2}e_{2}=-e_{4},$
$e_{2}e_{3}=-e_{4},$ $e_{3}e_{1}=e_{3},$
$e_{3}e_{2}=e_{4},$ $e_{4}e_{1}=e_{4}$ | $\left(\begin{array}[]{l}{1\,\,\,\,0\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,0}\\\ {0\,\,\,\,a\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,0}\\\ {0\,\,\,\,b\,\,\,\,\,\,\,a^{2}\,\,\,\,\,\,0}\\\ {0\,\,\,\,c\,\,\,\,2a^{2}b\,\,\,a^{3}}\end{array}\right)\,\,\,$ | unitary,
$A=N\dotplus S,$
$N=<e_{2},e_{3},e_{4}>,$
$S=<e_{1}>$ | 3 | 0 | 0
where $a,b,c,d,e,f,g,h,i,j,k,l,\alpha\in\mathbb{C}.$
## References
* [1] Azarina P., Caldwell S., Davis J., Frederick B., Three-dimensional Associative Unital Algebras, Journal of PGSS, www.pgss.mcs.cmu.edu/home/Publications.html, 227 - 237.
* [2] Ayupov Sh.A., Omirov B.A., On 3-dimensional Leibniz algebras, Uzbek Math. Jour., No.1,(1999), 9-14.
* [3] Gabriel R. Finite representation type is open. Lecture Notes in Math., No. 488 (1974).
* [4] Jacobson N., Lie algebras. Interscience Tracts in Pure and Applied Mathematics, No.10,(1962), New York.
* [5] Loday J.-L., Une version non commutative des algebras de Lie: les algebras de Leibniz, Enseign. Math, No. 39, (1993), 269-293.
* [6] Loday J.-L., Frabetti A., Chapoton F., Goichot F., Dialgebras and Related Operads, Lecture Notes in Math., No. 1763, (2001), Springer, Berlin.
* [7] Mazolla G. The algebraic and geometric classification associative algebras of dimension five. _Manuscripta math._ Vol.27, (1979), 1-21.
* [8] Peirce B. Linear associative algebra. Amer. J. Math., No. 4, (1881), 97-221.
* [9] Rakhimov I.S., Rikhsiboev I.M. A simple classification of three-dimensional complex associative algebras. _Math Digest. Research Bulletin of INSPEM _(_ Institute for Mathematical Research, UPM _)__ , (to appear).
|
arxiv-papers
| 2009-10-06T07:28:26 |
2024-09-04T02:49:05.693106
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "I.S. Rakhimov, I.M. Rikhsiboev and W.Basri",
"submitter": "Ikrom Rikhsiboev Dr",
"url": "https://arxiv.org/abs/0910.0932"
}
|
0910.0988
|
# The combined effect of chemical and electrical synapses in small Hindmarsh-
Rose neural networks on synchronisation and on the rate of information
M. S. Baptista1, F. M. Moukam Kakmeni2, C. Grebogi1 1 Institute for Complex
Systems and Mathematical Biology, King’s College, University of Aberdeen, AB24
3UE Aberdeen, United Kingdom
2 Laboratory of Research on Advanced Materials and Nonlinear Science
(LaRAMaNS), Department of Physics, Faculty of sciences, University of Buea, P.
O. Box 63 Buea, Cameroon
###### Abstract
In this work we studied the combined action of chemical and electrical
synapses in small networks of Hindmarsh-Rose (HR) neurons on the synchronous
behaviour and on the rate of information produced (per time unit) by the
networks. We show that if the chemical synapse is excitatory, the larger the
chemical synapse strength used the smaller the electrical synapse strength
needed to achieve complete synchronisation, and for moderate synaptic
strengths one should expect to find desynchronous behaviour. Otherwise, if the
chemical synapse is inhibitory, the larger the chemical synapse strength used
the larger the electrical synapse strength needed to achieve complete
synchronisation, and for moderate synaptic strengths one should expect to find
synchronous behaviours. Finally, we show how to calculate semi-analytically an
upper bound for the rate of information produced per time unit (Kolmogorov-
Sinai entropy) in larger networks. As an application, we show that this upper
bound is linearly proportional to the number of neurons in a network whose
neurons are highly connected.
###### pacs:
05.45.-a; 05.45.Gg; 05.45.Pq; 05.45.Xt
## I Introduction
Intercellular communication is one of the most important characteristics of
all animal species because it makes the many components of such complex
systems operate together. Among the many types of intercellular communication,
we are interested in the communication among brain cells, the neurons, that
exchange information mediated by chemical and electrical synapses sudhof .
The uncovering of the essence of behaviour and perception in animals and human
beings is one of the main challenges in brain research. While the behaviour is
believed to be linked to the way neurons are connected (the topology of the
neural network and the physical connections among the neurons), the perception
is believed to be linked to synchronisation. This comes from the binding
hypothesis malsburg1 , which states that synchronisation functionally binds
neural networks coding the same feature or objects. This hypothesis raised one
of the most important contemporary debates in neurobiology pareti because
desynchronisation seems to play an important role in perception as well. The
binding hypothesis is mainly supported by the belief that a convenient
environment for neurons to exchange information appears when they become more
synchronous.
Despite the explosive growth in the field of complex networks, it is still
unclear for which conditions synchronisation implies information transmission
and it is still unclear which topology favours the flowing of information.
Additionally, most of the models being currently studied in complex networks
consider networks whose nodes (such as neurons) are either linearly or non-
linearly connected. But, recent works have shown that neurons that were
believed to make only non-linear (chemical) synapses make also simultaneously
linear (electrical) synapses Gibson ; Connors ; Galarreta1999 ; Hestrin ;
Galarreta2001 . To make the scenario even more complicated, neurons connect
chemically in an excitatory and/or an inhibitory way. In this work, we aim to
study the relationship between synchronisation and information transmission in
such neural networks, whose neurons are simultaneously connected by chemical
and electrical synapses.
The electrical synapse is the result of the potential difference between the
neurons and causes an immediate physiological response of the latter one,
linearly proportional to the potential difference. The chemical synapse is
mediated by the exchange of neurotransmitters from the pre to the postsynaptic
neuron and can only be released once the presynaptic neuron membrane achieves
a certain action potential. The chemical interaction is described by a
nonlinear function Greengard .
While the electrical synapses between neurons is localised in the neuron cell
and therefore it is a local connection, the chemical synapse is in the neuron
axon and is therefore mainly responsible for the non-local nature of the
synapses.
Chemical synapses can be inhibitory and excitatory. When an inhibitory neuron
spikes (the pre-synaptic neuron), a neuron connected to it (the post-synaptic
neuron) is prevented from spiking. As shown in Ref. vree , inhibition promotes
synchronisation. When an excitatory neuron spikes, it induces the post-
synaptic neuron to spike. Several types of synchronisation were found in
networks of chaotic neurons coupled with only electrical synapses. One can
have complete synchronisation, generalised synchronisation and phase
synchronisation, the latter appearing for small synapse strength
baptista_PRE2008 . Complete synchrony strongly depends on the network
structure and the number of cells. In networks of chemically coupled neurons
Baptista1 , the net input a neuron receives from synaptic neurons emitting
synchronised spikes is proportional to the number of connected units. Hence,
for chemical synapses, if all the nodes in the network have the same degree,
synchronisation will be enhanced; if different nodes have different degrees,
synchronisation will be hampered Cosenza . In fact, Ref. Hasler has shown
analytically that the stability of the completely synchronous state in such
networks only depends on the number of signals each neuron receives,
independent of all other details of the network topology.
The most obvious possible role of electrical synapses within networks of
inhibitory neurons is to couple the membrane potential of connected cells,
leading to an increase in the probability of synchronised action potentials.
This synchronous firing could coordinate the activity of other cortical cell
populations. For example, it has been reported that the introduction of
electrical synapses among GABAergic neurons that are also chemically connected
can promote oscillatory rhythmic activity Hestrin . These possibilities have
been addressed experimentally by several investigators and have been reviewed
recentlyGalarreta2001 ; Connors ; Bennett .
Motivated by these observations and also by the fact that the behaviour of
micro-circuitry in the cerebral cortex is not well understood, we analyse the
combined effect of these two types of synapses on the stability of the
synchronous behaviour and on the information transmission in small neural
networks. In order to deal with this problem analytically we consider
idealistic networks, composed of equal neurons with mutual connections of
equal strengths (see Sec. II). A basic assumption characterising most of the
early works on synchronisation in neural networks is that, by adding a
relatively small amount of electrical synapse to the inhibitory synapse, one
can increase the degree of synchronisation far more than a much larger
increase in inhibitory conductance Kopell ; Pfeuty .
Our results agree with this finding in the sense that for larger inhibitory
synaptic strengths complete synchronisation can only be achieved if the
electrical synapse strength is larger than a certain amount. But in contrast,
we found that for moderate inhibitory synaptic strengths, the larger the
chemical synapse strength is the larger the electrical synapse strength needs
to be to achieve complete synchronisation. Additionally, we introduce in this
work analytical approaches to understand when complete synchronisation should
be expected to be found and what is the relation of that with the amount of
information produced by the network.
Information is an important concept shannon . It measures how much uncertainty
one has about an event before it happens. It is a measure of how complex a
system is. Very complicated and higher dimensional systems might be actually
very predictable, and as a consequence the content of information of such a
system might be very limited. But measuring the amount of information is
something difficult to accomplish. Normally, there is always some bias or
error on the calculation of it paninski , and one has to rely on alternative
approaches. Measuring the Shannon entropy of a chaotic trajectory is extremely
difficult because one has to calculate an integral of the probability density
of a fractal chaotic set. But for chaotic systems that have absolutely
continuous conditional measures, one can calculate Shannon’s entropy per unit
of time, a quantity known as Kolmogorov-Sinai (KS) entropy kolmogorov , by
summing all the positive Lyapunov exponents LS . A system that has absolutely
continuous conditional measures is a system whose trajectory continuously
distribute along unstable directions. More precisely, systems whose
trajectories continuously distribute along unstable manifolds at points that
have positive probability measure. These systems form a large class of
nonuniformly hyperbolic systems young : the Hénon family; Hénon-like attractor
arising from homoclinic bifurcations; strange attractors arising from Hopf
Bifurcations (e.g. Rössler oscillator); some classes of mechanical models with
periodic forcing. The result in Ref. LS extends a previous result by Pesin
pesin that demonstrated that for hyperbolic maps, the KS entropy is equal to
the sum of the positive Lyapunov exponents. We are not aware of any rigorous
result proving the equivalence of the KS entropy and the sum of Lyapunov
exponent for the Hindmarsh-Rose neural model neither to a network constructed
with them. But the chaotic attractors arising in this neuron model are similar
to the ones appearing from Homoclinic bifurcations. Additionally, for two
coupled neurons, we show in Sec. VII (using the non-rigorous methods described
in Appendix XI) that a lower bound estimation of the KS entropy is indeed
close to the sum of all the positive Lyapunov exponents. Despite the lack of a
rigorous proof, we will assume that the results in Refs. LS ; young apply in
here in the sense that the sum of the positive Lyapunov exponents provide a
good estimation for the KS entropy.
The KS entropy for chaotic networks has another important meaning. It provides
one the so called network capacity baptista_PRE2008 , the maximal amount of
information that all the neurons in the network can simultaneously process
(per unit of time). A network that produces information at a higher rate is
more unpredictable and more complex. Arguably, the network capacity is an
upper bound for the amount of information that the network is capable of
processing from external stimuli. In Ref. baptista_PRE2008 we discuss a
situation were that is indeed the case.
To understand the scope of this paper and the methods used, we first justify
the chosen network topologies in Sec. II. Then, in Sec. III, we describe the
dynamical system of our network and derive the variational equations of it in
the eigenmode form, a necessary analytical tool in order to be able to study
the onset of complete synchronisation (CS) and to calculate the rate of
information produced by the network. Complete synchronisation happens when the
trajectories of all neurons are equal.
Our main results can be summarised as in the following:
* •
We show (Secs. IV and V) how one can calculate the synaptic strengths
(chemical and electrical) necessary for a network of $N$ neurons to achieve
complete synchronisation when one knows the strengths for which two mutually
coupled neurons become completely synchronous.
* •
We show numerically (Sec. VI) parameter space diagrams indicating the
electrical and chemical synapse strengths responsible to make complete
synchronisation to appear in different networks. The analytical derivation
from Sec. V are found to be sufficiently accurate. There are two scenarios for
the appearance of complete synchronisation for inhibitory networks. If the
chemical synapse strength is small, the larger the chemical synapse strength
used the larger the electrical synapse strength needed to be to achieve
complete synchronisation. Otherwise, if the chemical synapse strength is
large, complete synchronisation appears if the electrical synapse strength is
lager than a certain value. In excitatory networks both synapses work in a
constructive way to promote complete synchronisation: the larger the chemical
synapse strength is the smaller the electrical synapse strength needs to be to
achieve complete synchronisation.
* •
We show (Secs. VII) that the sum of the positive Lyapunov exponents provides a
good estimation for the KS entropy. Additionally, we show that there are
optimal ranges of values for the chemical and electrical strengths for which
the amount of information is large.
* •
If complete synchronisation is absent, we show (Sec. VIII) that while in
inhibitory networks one can typically expect to find high levels of
synchronous behaviour, in excitatory networks one is likely to expect
desynchronous behaviour.
* •
We calculate (Sec. IX) an upper bound for the rate of information produced per
time unit (Kolmogorov-Sinai entropy) by larger networks using the rate at
which information is produced by two mutually coupled neurons.
## II The topology of the studied networks
In order to consider the combined action of these two different types of
synapses, we need to consider in our theoretical approach idealistic networks,
constructed by nodes possessing equal dynamics and particular coupling
topologies such that a synchronisation manifold exists and CS is possible. If
we had studied networks whose neurons were exclusively connected by electrical
means, we could have considered networks with arbitrary topologies. On the
other hand, if we had studied networks whose neurons are exclusively connected
by chemical means, we would have considered networks whose neurons receive the
same number of chemical connections. These conditions are the same ones being
usually made to study complete synchronisation in complex networks Pecora ;
Hasler .
In order to analytically study networks formed by neurons that make
simultaneously chemical and electrical connections, we have not only to assume
that the neurons have equal dynamics and that every neuron receives the same
number of chemical connections coming from other neurons, but also that the
Laplacian matrix for the electrical synapses (that provides topology of the
electrical connections) and the Laplacian matrix for the chemical synapses
commute, as we clarify later in this paper. Naturally, there is a large number
of Laplacian matrices that commute. In this work we construct networks that
are biologically plausible. Since the electrical connection is local, we
consider that neurons connect electrically only to their nearest neighbours.
Since neurons connected chemically make a large number of connections (of the
order of 1000), it is reasonable to consider that for small networks the
neurons that are chemically connected are fully connected, i.e., every neuron
connects to all the other neurons. Notice however that while reciprocal
connections are commonly found in electrically coupled neurons, that is not
typical for chemically connected neurons.
Since our small networks are composed of no more than 8 neurons, we make an
abstract assumption and admit another possible type of network in which
neurons that are connected electrically can also make non-local connections,
allowing them to become fully connected to the other neurons. Notice, however,
that our theoretical approach remains valid for larger networks that admit a
synchronisation manifold.
## III The networks of coupled neurons and master stability analysis
The dynamics of the Hindmarsh-Rose (HR) model for neurons is described by
$\displaystyle\dot{p}$ $\displaystyle=$ $\displaystyle
q-ap^{3}+bp^{2}-n+I_{ext}$ $\displaystyle\dot{q}$ $\displaystyle=$
$\displaystyle c-dp^{2}-q$ (1) $\displaystyle\dot{n}$ $\displaystyle=$
$\displaystyle r[s(p-p_{0})-n]$
where $p$ is the membrane potential, $q$ is associated with the fast current,
$Na^{+}$ or $K^{+}$, and $n$ with the slow current, for example, $Ca^{2+}$.
The parameters are defined as $a=1,b=3,c=1,d=5,s=4,r=0.005,p_{0}=-1.60$ and
$I_{ext}=3.2$ where the system exhibits a multi-time-scale chaotic behaviour
characterised as spike-bursting.
The dynamics of a neural networks of $N$ neurons connected simultaneously by
electrical (a linear coupling) and chemical (a non-linear coupling) synapses
is described by
$\displaystyle\dot{p_{i}}$ $\displaystyle=$ $\displaystyle
q_{i}-ap_{i}^{3}+bp_{i}^{2}-n_{i}+I_{ext}$ $\displaystyle-
g_{n}(p_{i}-V_{syn})\sum_{j=1}^{N}\mathbf{C}_{ij}S(p_{j})+g_{l}\sum_{j=1}^{N}\mathbf{G}_{ij}\mathbf{H}(p_{j})$
$\displaystyle\dot{q_{i}}$ $\displaystyle=$ $\displaystyle c-dp_{i}^{2}-q_{i}$
(2) $\displaystyle\dot{n_{i}}$ $\displaystyle=$ $\displaystyle
r[s(p_{i}-p_{0})-n_{i}]$
$(i,j)=1,\ldots,N$, where $N$ is the number of neurons.
In this work we consider that $\mathbf{H}(p_{i})=p_{i}$. But we preserve the
function $\mathbf{H}(p_{i})$ in our remaining analytical derivation to
maintain generality.
The chemical synapse function is modelled by the sigmoidal function
$S(p_{j})=\displaystyle\frac{1}{1+e^{-\lambda(p_{j}-\Theta_{syn})}},$ (3)
with $\Theta_{syn}=-0.25$, $\lambda=10$ and $V_{syn}=2.0$ for excitatory and
$V_{syn}=-2.0$ for inhibitory. For the chosen parameters and all the networks
that we have worked $|p_{i}|<2$ and the term $(p_{i}-V_{syn})$ is always
negative for excitatory networks and positive for inhibitory networks. If two
neurons are connected under an inhibitory (excitatory) synapse then, when the
presynaptic neuron spikes, it induces the postsynaptic neuron not to spike (to
spike).
The matrix $\mathbf{G}_{ij}$ describes the way neurons are electrically
connected. It is a Laplacian matrix and therefore $\sum_{j}\mathbf{G}_{ij}=0$.
The matrix $\mathbf{C}_{ij}$ describes the way neurons are chemically
connected and it is an adjacent matrix, therefore $\sum_{j}\mathbf{C}_{ij}=k$,
for all $i$. For both matrices, a positive off-diagonal term placed in the
line $i$ and column $j$ means that neuron $i$ perturbs neuron $j$ with an
intensity given by $g_{l}\mathbf{G}_{ij}$ (or by $g_{n}\mathbf{C}_{ij}$).
Since the diagonal elements of the adjacent matrix are zero, $k$ represents
the number of connections that neuron $i$ receives from all the other neurons
$j$ in the network. This is a necessary condition for the existence of the
synchronous solution Hasler by the subspace
$P=P_{1}=P_{2}=..=P_{N},P_{i}=(p_{i},q_{i},n_{i})$.
Under these assumptions and, as previously explained, we consider networks
with three topologies: topology I, when all the neurons are mutually fully
(all-to-all) connected with chemical synapses and mutually diffusively
(nearest neighbours) connected with electrical synapses; topology II, when all
the neurons are mutually fully connected with chemical synapses and mutually
fully connected with electrical synapses; topology III, when all the neurons
are mutually diffusively (nearest neighbours) connected with chemical and
electrical synapses. We consider networks with 2, 4 and 8 neurons. By nearest
neighbours, we consider that the neurons are forming a closed ring.
The synchronous solutions $P=(p,q,n)$ take the form
$\displaystyle\dot{p}$ $\displaystyle=$ $\displaystyle
q-ap^{3}+bp^{2}-n+I_{ext}-g_{n}k(p-V_{syn})S(p)$ $\displaystyle\dot{q}$
$\displaystyle=$ $\displaystyle c-dp^{2}-q$ (4) $\displaystyle\dot{n}$
$\displaystyle=$ $\displaystyle r[s(p-p_{0})-n]$
The variational equation of the network in (III) [calculated around the
synchronisation manifold (III)] is given by
$\displaystyle\dot{\delta p_{i}}$ $\displaystyle=$ $\displaystyle\delta
q_{i}-3ap_{i}^{2}\delta p_{i}+2bp_{i}\delta p_{i}-\delta n_{i}$
$\displaystyle-g_{n}(p_{i}-V_{syn})S^{{}^{\prime}}(p)\left(k\delta
p_{i}+\sum_{j=1}^{N}\mathbf{\tilde{G}}_{ij}\delta p_{j}\right)$
$\displaystyle-kg_{n}S(p)\delta
p_{i}+g_{l}\sum_{j=1}^{N}\mathbf{G}_{ij}D\mathbf{H}(p)\delta p_{i}$
$\displaystyle\dot{\delta q_{i}}$ $\displaystyle=$ $\displaystyle 2d\delta
p_{i}-\delta q_{i}$ (5) $\displaystyle\dot{\delta n_{i}}$ $\displaystyle=$
$\displaystyle r(s\delta p_{i}-\delta n_{i})$
The matrix $\mathbf{C}_{ij}$ has been transformed to a Laplacian matrix by
$\mathbf{\tilde{G}}=\mathbf{C}_{ij}-k\mathbb{I}$. $D\mathbf{H}(p)$ represents
the derivative of $\mathbf{H}$ with respect to $p$, which in this work equals
1.
The term $S^{\prime}(p)$ refers to the spatial derivative $\frac{dS(p)}{dp}$
and equals
$S^{\prime}(p)=\frac{\lambda\exp{{}^{-\lambda(p-\Theta_{syn})}}}{[1+\exp{{}^{-\lambda(p-\Theta_{syn})}}]^{2}}.$
(6)
Notice that if $S(p)=1$ (what happens for $p>>\Theta_{syn}$), then
$S^{\prime}(p)=0$ and if $S(p)=0$ ($p<<\Theta_{syn}$), then $S^{\prime}(p)=0$.
$S^{\prime}(p)$ is not zero when the value of $S(p)$ changes from 1 to 0 (and
vice-versa) and $p\approxeq\Theta_{syn}$.
Equation (III) is referred to as the variational equation and is often the
starting point for determining whether the synchronisation manifold is stable.
This equation is rather complicated since, given arbitrary synapses $g_{n}$
and $g_{l}$, it can become quite higher dimensional. Also the coupling
matrices $\mathbf{G}$ and $\mathbf{\tilde{G}}$ can be arbitrary making the
situation to become even more complicated. However, assuming that whenever
there is a chemical synapse (and $g_{n}>0$), the matrices $\mathbf{G}$ and
$\mathbf{\tilde{G}}$ commute, then the problem can be simplified by noticing
that the arbitrary state $\delta X$ (where $\delta X=(\delta p_{i},\delta
q_{i},\delta n_{i})$ is the deviation of the $i$th vector state from the
synchronisation manifold) can be written as $\delta
X=\sum_{i=1}^{N}\textbf{v}_{i}\bigotimes\kappa_{i}(t)$, with
$\kappa_{i}(t)=(\eta_{i},\psi_{i},\varphi_{i})$. The $\textbf{v}_{i}$ be the
eigenvector and $\gamma_{i}$ and $\tilde{\gamma}_{i}$ the corresponding
eigenvalues for the matrices $\mathbf{G}$ and $\mathbf{\tilde{G}}$
respectively. So, if that is the case, by applying $\textbf{v}_{j}^{T}(t)$
(with
$\textbf{v}_{j}^{T}(t)\cdot\textbf{v}_{i}=\delta_{ij}\>\>\text{where}\>\>\delta_{ij}\>\>\text{is
the Kronecker delta}$), to the left (right) side of each term in Eq. (III) one
finally obtains the following set of N variational equations in the eigenmode
$\displaystyle\dot{\eta_{j}}$ $\displaystyle=$
$\displaystyle(2bp-3ap^{2})\eta_{j}-\varphi_{j}+\psi_{j}-\Gamma(p)\eta_{j}$
$\displaystyle\dot{\psi_{j}}$ $\displaystyle=$ $\displaystyle
2d\eta_{j}-\psi_{j}$ $\displaystyle\dot{\varphi_{j}}$ $\displaystyle=$
$\displaystyle r(s\eta_{j}-\varphi_{j})$ (7) $\displaystyle j$
$\displaystyle=$ $\displaystyle 1,2,3,...N$
where the term $\Gamma(p)$ is given by
$\Gamma(p)=kg_{n}S(p)-g_{n}(V_{syn}-p)S^{{}^{\prime}}(p)\left(k+\tilde{\gamma}_{j}\right)-g_{l}\gamma_{j}$
(8)
in which $\gamma_{j}$ (with $\gamma_{1}$=0, and $\gamma_{j}<$0, $j\geq 2$) are
the eigenvalues of $\mathbf{G}$ and $\tilde{\gamma_{j}}$ are the eigenvalues
of $\mathbf{\tilde{G}}$. The eigenvalues $\gamma_{j}$ are negative because the
off-diagonal elements of $\mathbf{G}$ are positive.
For networks with $N=2$ we have that $|\gamma_{2}|=2$ and $k=1$, meaning that
the neurons are connected in an all-to-all fashion. For networks with $N=4$,
if the neurons are connected in an all-to-all fashion, we have that
$|\gamma_{2}|=4$ and $k=3$ or if the neurons are connected with their nearest
neighbours we have that $|\gamma_{2}|=2$ and $k=2$. For $N=8$,
$|\gamma_{2}|=8$ and $k=7$ (all-to-all) and $|\gamma_{2}|=0.585786402$ and
$k=2$ (nearest-neighbour). These values are placed in Table 1 for further
reference.
Table 1: Values of $\gamma_{2}$ in absolute value and $k$ for the considered networks. | all-to-all | nearest-neighbour
---|---|---
$N=2$ | $\gamma_{2}=2$, $k$=1 | $\gamma_{2}=2$, $k$=1
$N=4$ | $\gamma_{2}=4$, $k$=3 | $\gamma_{2}=2$, $k$=2
$N=8$ | $\gamma_{2}=8$, $k$=7 | $\gamma_{2}=0.585786402$, $k$=2
The previous equations are integrated using the 4th-order Range-Kutta method
with a step size of 0.001. The calculations of the Lyapunov exponents are
performed considering a time interval of 600 [sufficient for a neuron to
produce approximately 600 spikes ($p>0$)]. We discard a transient time of 300,
corresponding to 300,000 integrations.
## IV Stability analysis
The stability of the synchronisation manifold can be seen from the perspective
of control Hasler ; Femat ; Moukam ; Bowong by imagining that the term
$\Gamma(p)$ stabilises Eq. (7) at the origin. This term can be interpreted as
the main gain of a feedback control law $u(t)=\Gamma(p)\eta_{j}$ such that
$\eta_{j}$ (resp. $\psi_{j}$ and $\varphi_{j}$ ) tends to $0$ as $t$ tends to
infinity. In fact, the controlling force $u(t)=\Gamma(p)\eta_{j}$ could be
designed with no previous knowledge of the system under consideration assuming
that it has a parametric dependence. A drawback of such a general control
approach is that it leads to non-feedback control strategy, which have not
guaranteed stability margins. More robust approaches for determining the
structural stability of the synchronisation manifold of systems whose
equations of motion are partially unknown have been recently developed Femat ;
Moukam ; Bowong .
In this work, however, we determine the stability of the synchronisation
manifold from the master stability analysis of Refs. Hasler ; Pecora . A
necessary condition for the linear stability of the synchronised state is that
all Lyapunov exponents associated with $\gamma_{j}$ and/or
$\tilde{\gamma_{j}}$ for each $j=2,3,...,N$ (the directions transverse to the
synchronisation manifold) are negative. This criterion is a necessary
condition for complete synchronisation only locally, i.e. close to the
synchronisation manifold.
## V Rescaling of Eqs. (III) and (7)
When working with networks formed by nodes possessing equal dynamical rules,
we wish to predict the behaviour of a large network from the behaviour of two
coupled nodes. That can always be done whenever the equations of motion of the
network can be rescaled into the form of the equations describing the two
coupled nodes. That means that, given that two mutually coupled neurons
completely synchronise for the electrical and chemical synapse strengths
$g_{l}^{*}(N=2)$ and $g_{n}^{*}(N=2)$, respectively, then it is possible to
calculate the synapse strengths $g_{l}^{*}(N)$ and $g_{n}^{*}(N)$ for which a
network composed by $N$ nodes completely synchronises.
In order to rescale the equations for the synchronisation manifold and for its
stability, Eqs. (III) and (7), respectively, we need to preserve the form of
these equations as we consider different networks.
Concerning Eq. (7), we need to show under which conditions it is possible to
have $\Gamma(p,N=2)=\Gamma(p,N)$, where $\Gamma$ is the term responsible to
make the stability of the synchronisation manifold to depend among other
things on the topology of the network and on the coupling function $S(p)$.
Notice that $S(p)$ assumes for most of the time either the value 0 or 1\. For
some short time interval $S(p)$ changes its value from 0 to 1 (and vice-versa)
and at this time $S^{\prime}(p)$ is different from zero [see Eqs. (3) and
(6)]. For that reason we will treat $S^{\prime}(p)$ as a small perturbation in
our further calculations and will ignore it, most of the times. That leave us
with two relevant terms in both Eqs. (III) and (7) that need to be taken into
consideration in our rescaling analyses. These terms are $g_{l}\gamma_{j}$ and
$kg_{n}S(p)$. While the first term comes from the electrical synapse, the
second term comes from the chemical synapse.
The first term depends on the eigenvalues of $\mathbf{G}_{ij}$ (which varies
according to the number of nodes and the topology of the network) and on the
synapse strength $g_{l}$. If this term assumes a particular value for a given
network, for another network one can suitably vary $g_{l}$ in order for the
whole term to assume this same value in the other network. So, the term
$g_{l}\gamma_{j}$ can always be rescaled by finding an appropriate value of
$g_{l}$.
The rescaling of the second term, $kg_{n}S(p)$ is more complicated because it
depends on the trajectory $(p)$ of the attractor. Naturally, we wish to find a
proper rescaling for the function $S(p)$, which implies that the attractors
appearing as solutions on the synchronisation manifold should present some
kind of invariant property.
In order to find such an invariant property, we study the time average
$\langle S(p)\rangle$ of the function $S(p)$ for attractors appearing as
solutions of Eq. (III) for 5 network topologies. In Fig. 1 we show in the
boxes (A-E) the values of $N,$$|\gamma_{2}|$, $k$ and the type of topology
considered in the networks of Figs. 2, 3, 4, and 5.
Figure 1: The topology of the networks considered in Figs. 2, 3, 4 and 5 and
the values of $N$, $|\gamma_{2}|$ and $k$.
The result for excitatory networks can be seen in Fig. 2(A-E), which shows
this value as a function of $kg_{n}$. Apart from some small differences, the
function $\langle S(p)\rangle$ remains invariant for the different networks
considered. We identify two relevant values for $\langle S(p)\rangle$. Either
$\langle S(p)\rangle\approxeq 0.9$, for $g_{n}<g_{n}^{(c)}$ or $\langle
S(p)\rangle=0$, for $g_{n}\geq g_{n}^{(c)}$. $g_{n}^{(c)}\approx 1.67$.
We also find an invariant curve of $\langle S(p)\rangle$ for inhibitory
networks. In Fig. 3(A-E) we show this curve for the same networks of Fig. 2.
For these networks, we define $g_{n}^{(c)}\approx 1.5$ as the value of $g_{n}$
for which the curve of $\langle S(p)\rangle$ reaches its maximum. In the
considered inhibitory networks, $\langle S(p)\rangle=1$ is a consequence of
the fact that the neurons loose their chaotic behaviour and become a stable
limit cycle. Notice that the value of $\langle S(p)\rangle$ does not depend on
the value of the electrical synapse strength $g_{l}$. This is due to the fact
that $g_{l}$ is not present in the equations for the synchronisation manifold
[Eq. (III)].
Figure 2: (A-E) The value of $\langle S(p)\rangle$ with respect to a rescaled
chemical synapse strength $kg_{n}$ for excitatory networks with a
configuration shown in Figs. 1(A-E). Initial conditions of the neurons are set
to be equal (and $g_{l}$=0).
Figure 3: (A-E) The value of $\langle S(p)\rangle$ with respect to a rescaled
chemical synapse strength $kg_{n}$ for inhibitory networks with a
configuration shown in Figs. 1(A-E). Initial conditions of the neurons are set
to be equal (and $g_{l}$=0).
Let us rescale Eq. (III). First notice that the average
$\langle(p-V_{syn})\rangle$ has the same invariant properties of the average
$\langle S(p)\rangle$. Then, we assume that both $S(p)$ and $(p-V_{syn})$ make
small oscillations around their average value. That implies that
$S(p)(p-V_{syn})\approxeq\langle S(p)(p-V_{syn})\rangle$. From Figs. 2 and 3
we have that the average $\langle S(p,N)\rangle$ can be written as a function
of $g_{n}(N)$, as well as $\langle(p-V_{syn})\rangle$. Therefore, we can write
$\langle S(p)(p-V_{syn})\rangle$ as a function of $g_{n}(N)$. It is clear that
the value of this average obtained for $g_{n}(N=2)$ should be approximately
equal to the value obtained for $kg_{n}(N)$, and so this average function can
be rescaled by $kg_{n}(N)\approxeq g_{n}(N=2)$. Therefore, Eq. (III)
describing a large network can be rescaled into this same equation describing
two mutually coupled neurons by
$\displaystyle g_{n}(N)=\displaystyle\frac{g_{n}(N=2)}{k}$ (9)
Now, we need to show that it is also possible to do the same to Eq. (7), the
equation responsible for the stability of the synchronous solution.
Assuming again that $S(p)$ make small oscillations around its average value
allows us to write $\Gamma(p,N)$ as a function of $\langle S(p)\rangle$ as in
$\Gamma(p,N)\cong kg_{n}(N)\langle S(p,N)\rangle-g_{l}(N)\gamma_{j}$. Notice
from Figs. 2 and 3 that the average $\langle S(p,N)\rangle$ can be written as
a function of $g_{n}(N)$. In order to rescale Eq. (7), describing a network of
$N$ nodes in terms of a network of 2 nodes, we need to have that
$\Gamma(p,N)=\Gamma(p,N=2)$ leading to
$\displaystyle kg_{n}(N)\langle S[g_{n}(N)]\rangle-\gamma_{2}g_{l}(N)$
$\displaystyle=$ $\displaystyle g_{n}(N=2)\langle
S[g_{n}(N=2)]\rangle+2g_{l}(N)$ (10)
where we have considered only the second largest eigenvalue $\gamma_{2}$, the
one responsible for the stability of the synchronisation manifold; we have
ignored terms that appear together with $S^{\prime}$ in $\Gamma$.
We make now a reasonable hypothesis that if a stable synchronous solutions for
Eq. (III) exists for $g_{n}(N=2)=g_{n}^{*}(N=2)$ (for a two mutually coupled
neurons), then this same stable synchronous solution exists for
$kg_{n}^{*}(N)$ (for a network composed by $N$ neurons mutually connected).
This hypothesis is constructed from the observation that equivalent attractors
can be found in different networks if the rescaling in Eq. (9) is employed. We
are assuming that if $g_{n}^{*}(N=2)$ represents the chemical synapse strength
for which complete synchronisation appears in two mutually coupled neurons,
then complete synchronisation would appear in a network of $N$ nodes if
$\displaystyle g^{*}_{n}(N)=\displaystyle\frac{g^{*}_{n}(N=2)}{k}$ (11)
If the previous hypothesis is satisfied, i.e. Eq. (11) is satisfied, we see
from Figs. 2 and 3 that $\langle S[g_{n}(N)]\rangle\approxeq\langle
S[g_{n}(N=2)]\rangle$ and assuming that these two averages are equal, then Eq.
(10) takes us to
$\displaystyle
g^{*}_{l}(N)=\displaystyle\frac{2g^{*}_{l}{(N=2)}}{|\gamma_{2}(N)|}$ (12)
where $g^{*}_{l}(N)$ represents the electrical synapse strength for which
complete synchronisation occurs in a network composed by $N$ neurons.
In the following, we analyse two special cases of Eq. (10) when the function
$S(p)$ is constant and the previous approximations (expanding $\Gamma$ around
its average and that $\langle S[g_{n}(N)]\rangle=\langle
S[g_{n}(N=2)]\rangle$) to arrive to Eqs. (11) and (12) are exact.
### V.1 Rescaling in excitatory networks ($V_{syn}=2.0$)
Case 1: A large chemical synapse strength, $kg_{n}(N)>g_{n}^{(c)}$, with
$g_{n}^{(c)}\approxeq$1.67, makes for all the time $p<\Theta$, leading to
$S(p)=0$ and $S^{\prime}(p)$=0 (see Fig. 2). The neurons become completely
synchronous to a stable equilibrium point.
### V.2 Rescaling in inhibitory networks ($V_{syn}=-2.0$)
Case 2: a large chemical synapse strength, $kg_{n}(N)>g_{n}^{(c)}$, with
$g_{n}^{(c)}\approx 1.50$, makes for all the time $p>\Theta$ and as a
consequence $S(p)=1$ and $S(p)^{\prime}=0$ (see Fig. 3). The neurons become
completely synchronous to a limit cycle.
## VI Combined effect of the chemical and electrical synapses on the
synchronous behaviour
The analytical derivations done in the previous section are approximations,
except for some special values of the synaptic strengths (case 1 and 2).
However, as we show in this section, our calculations provide a good
estimation of what to expect from parameter spaces of larger networks when the
parameter space of two mutually coupled neurons is known. The parameter space
is constructed by considering the synapses $(g_{l},g_{n})$ and they identify
the regions where the state of complete synchronisation is stable.
The stability is determined from Eqs. (7), by verifying whether there are no
lyapunov exponents associated with transversal directions to the
synchronisation manifold. These exponents are numerically obtained, without
any approximation.
In Fig. 4, we show in black the synchronous regions (all transversal
conditional exponents are negative) for the excitatory networks and in Fig. 5
the same network topologies but for inhibitory networks. To simplify the
understanding of these two figures, in Fig. 1 we show in boxes (A-E) the
values of $N$, $|\gamma_{2}|$, $k$ and the type of topology considered in the
networks of Figs. 4(A-E) and 5(A-E). The values of $g_{l}$ and $g_{n}$ were
rescaled by using Eqs. (11) and (12). As expected, in excitatory networks our
rescaling works very well and roughly in inhibitory networks. So, the vertical
axis of Figs. 4(B-E) and 5(B-E) show the quantity $kg_{n}(N)$ and the
horizontal axis of these same figures show the quantity
$\frac{|\gamma_{2}|g_{l}(N)}{2}$.
To assist the analysis of the parameter spaces, imagine a curve $\Sigma$ that
is the border between the regions defining parameters for which the
synchronisation manifold is unstable (white regions) and regions defining
parameters for which the synchronisation manifold is stable (black regions).
There are four main characteristics in these two types (excitatory and
inhibitory) of networks concerning the occurrence of complete synchronisation.
Figure 4: Excitatory networks. Black points represent values of the synapse
strengths for which all transversal conditional exponents are negative. In
(B-E) the horizontal axis represent $g_{l}(N)|\gamma_{2}(N)|/2$ and the
vertical axis $kg_{n}$. Initial conditions of the neurons are set to be equal.
Figure 5: Inhibitory networks. Black points represent values of the synapse
strengths for which all transversal conditional exponents are negative. In
(B-E) the horizontal axis represent $g_{l}(N)|\gamma_{2}(N)|/2$ and the
vertical axis $kg_{n}$. Initial conditions of the neurons are set to be equal.
* $\bullet$
In excitatory networks, the electrical and the chemical synapses act in a
combined way to foster synchronisation. The neurons become completely
synchronous to a stable equilibrium point. The asynchronous neurons (white
regions) are chaotic. The curve $\Sigma$ would look like a diagonal line with
a negative slope. Such a curve could be defined by an equation similar to
$kg(N)+\gamma_{2}g_{l}\approx C$, $C$ being a function that is approximately
constant (see Fig. 4).
* $\bullet$
In excitatory networks, with $kg_{n}(N)>$1.67, Neurons are completely
synchronous to a stable equilibrium point (see Fig. 4).
* $\bullet$
In inhibitory networks, with $kg_{n}(N)<$5, the larger the chemical synapse
strength is the larger the electrical synapse strength needs to be to achieve
complete synchronisation. Neurons become completely synchronous to either a
limit cycle (large chemical synapse strength) or to a chaotic attractor (small
chemical synapse strength). The curve $\Sigma$ would look like a diagonal line
with a positive slope. Such a curve could be defined by an equation similar to
$kg(N)-\gamma_{2}g_{l}\approx C$, $C$ being a function that is approximately
constant (see Fig. 5).
* $\bullet$
In inhibitory networks, for large values of $kg_{n}(N)$, complete
synchronisation appears for $\gamma_{2}g_{l}>C$ and neurons become completely
synchronous to a stable limit cycle, which is unstable if $\gamma_{2}g_{l}<C$.
The curve $\Sigma$ would look like a straight vertical line. Such a curve
could be defined by an equation similar to $\gamma_{2}g_{l}\approx C$. $C$
being a function that is approximately constant (see Fig. 5).
If the neurons are set with different initial conditions, but sufficiently
close, complete synchronisation is found for similar synaptic strengths for
which the synchronisation manifold is stable.
If the neurons are set with sufficiently different initial conditions, and we
construct parameter spaces that represent synaptic strengths for which CS
takes place, we would have obtained parameter spaces with similar structure as
the one observed in Figs. 4 and 5. However, the network can become completely
synchronous to other synchronous solutions of Eq. (III), different from the
synchronous solutions observed for the parameters used to make Figs. 4 and 5.
In other words, parameter spaces that show CS in networks whose neurons are
set with different initial conditions constructed for the same synaptic
strengths and networks considered in Figs. 4 and 5 would present additional
black points in the white areas of Figs. 4 and 5.
## VII Combined effect of the chemical and electrical synapses on the amount
of information
Figure 6: [Color Online] We show the value of the sum of all the positive
Lyapunov exponents $H_{L}$ in black line and an estimation of the lower bound
for the KS entropy in filled squares (red line online) for two mutually
chemically coupled neurons under an excitatory synapse (A) and an inhibitory
synapse (B), as we vary the chemical synapse strength. We consider a constant
electrical synapse of strength $g_{l}$=0.1. Initial conditions are not equal.
First, we calculate the sum of all the positive Lyapunov exponents of the
attractor obtained from integrating the neural network [Eq. (III)] and
represent it by $H_{L}$. The Lyapunov exponents are calculated from the
variational equation of the network in Eq. (III). As previously discussed, it
is reasonable to assume that $H_{L}\approx H_{KS}$, where $H_{KS}$ represents
the KS entropy kolmogorov , which measures the amount of information
(Shannon’s entropy) produced per time unit.
In Figs. 6(A-B) we show in the thin line $H_{L}$ for two mutually chemically
and electrically coupled neurons ($g_{l}$=0.1) for excitatory synapse (A) and
for inhibitory synapse (B). To confirm that the sum of the positive Lyapunov
exponents have an entropic meaning for the studied Hindmarsh-Rose neuron
model, we have estimated a lower bound for the KS entropy, represented by the
tick line with filled squares (red online) in Fig. 6(A-B).
We see that for both cases, as one increases the synaptic strength, $H_{L}$
decreases. For the excitatory case, for $g_{n}>1.52$, the neurons trajectories
go to an equilibrium point and we obtain $H_{L}=0$. If $H_{L}=0$, that means
that there are no positive Lyapunov exponents and therefore no chaos. The
maximal value of $H_{L}$, calculated varying the synaptic strengths, is almost
equal for both types of synapses. One sees that there is a range of strength
values in both figures within which $H_{L}$ is large. For example, in (A)
$H_{L}$ is large for $g_{n}\in[0.7,1.2]$ and in (B) $H_{L}$ is large for
$g_{n}\in[0.3,0.7]$. This was also observed in 3D parameter space diagrams
(not shown in here) that show the value of $H_{L}$ versus $g_{n}$ and $g_{l}$.
These diagrams indicate that there is an optimal range of values for $g_{n}$
and $g_{l}$ for which $H_{L}$ remains large.
The reason we have shown results for two coupled neurons is because for such a
configuration a lower bound estimation of the KS entropy can be calculated by
encoding the trajectory into a binary symbolic sequence. Since the sequence is
binary, this method is only capable of measuring an information rate that is
less or equal than 1bit/symbol or 1bit/unit of time. Since that for two
coupled neurons, $H_{L}<1$bit/unit of time, and assuming that $H_{L}$ is a
good estimation for $H_{KS}$, then the employed method to calculate a lower
bound of the KS entropy is appropriate. The details of this estimation can be
seen in Appendix XI.
Notice that in Fig. 6(A-B) for $g_{n}\approx$0 (as well as in (B) for
$g_{n}\approx 2$) the estimations of $H_{KS}$ are larger than $H_{L}$. That is
the result of a known problem in the estimation of entropic quantities which
prevents the estimation to be small. The problem arises because the symbolic
sequences considered are not infinitely long for one to realise that there
exists a few or only one symbolic sequence encoding the trajectory. For
example, a long periodic orbit would be encoded by a series of short symbolic
sequences making the estimation of $H_{KS}$ to be positive instead of zero as
it should be.
## VIII Synchronisation (and desynchronisation) versus inhibition (and
excitation) versus Information
To understand the relation between synchronisation (desynchronisation) and
inhibition (excitability), when complete synchronisation is absent we do the
following. But notice that the following results are based on a conjecture
that is currently not demonstrated.
We calculate the Lyapunov exponents along the synchronisation manifold, which
are just the Lyapunov exponents of the network by assuming that all neurons
are completely synchronous. We call these exponents conditional Lyapunov
exponents and the sum of all the positive ones is denoted by $H_{C}$. There
are two ways for calculating them, either using Eq. (III) or (7), Eq. (7)
being simpler because of the dimensionality of the orthogonal vectors employed
to calculate the Lyapunov exponents. While the use of Eq. (III) requires 3N
vectors, each one with dimensionality 3N, the use of Eq. (7) requires N
vectors each one with dimensionality 3. Additionally, once the function that
relates the conditional exponents of two mutually coupled neurons with $g_{n}$
and $g_{l}$ is known, then one can calculate this function for all the
conditional exponents of larger networks as long as Eqs. (III) and (7) can be
rescaled.
We can then classify these neural networks into 2 types. The types UPPER or
LOWER. More specifically,
$\displaystyle H_{C}(N,g_{n},g_{l})$ $\displaystyle>$ $\displaystyle
H_{L}(N,g_{n},g_{l}),\mbox{\ \ \ \ \ UPPER}$ (13) $\displaystyle
H_{C}(N,g_{n},g_{l})$ $\displaystyle<$ $\displaystyle
H_{L}(N,g_{n},g_{l}),\mbox{\ \ \ \ \ LOWER}$ (14)
To understand what $H_{C}$ and $H_{L}$ exactly mean and the reason for such a
classification, notice that the networks here considered admit a synchronous
solution. This synchronous solution might be unstable (an unstable saddle) and
typical initial conditions depart from the neighbourhood of the synchronous
solution and asymptotically tend towards a stable solution, the chaotic
attractor. This attractor describes a network whose nodes are not synchronous.
In such a situation, the network admits at least two relevant solutions: a
stable desynchronous one (the chaotic attractor) and an unstable synchronous
one (the synchronisation manifold). While $H_{C}$ can be associated with the
amount of information produced by the unstable synchronous solution, $H_{L}$
can be associated with the amount of information produced by the desynchronous
chaotic attractor. If the complete synchronous state is stable, then,
$H_{C}=H_{L}$, and the network in Eq. (III) possesses only one stable
synchronous solution, for typical initial conditions. The nomenclature in Eqs.
(13) and (14) comes from the fact that if
$H_{C}(N,g_{n},g_{l})>H_{L}(N,g_{n},g_{l})$ then, $H_{C}$ is an upper bound
for $H_{L}$, otherwise it is a lower bound baptista_NJP2008 .
Assume now that the more information a network produces, the more
desynchronisation is observed among pair of neurons baptista_NJP2008 ;
baptista_PLOS2008 . If $H_{C}(N,g_{n},g_{l})>H_{L}(N,g_{n},g_{l})$ (UPPER),
then $H_{L}(N,g_{n},g_{l})$ is limited. As a consequence, the production of
information in the network is limited and therefore the level of
desynchronisation is small. On the other hand, if
$H_{C}(N,g_{n},g_{l})<H_{L}(N,g_{n},g_{l})$ (LOWER), then
$H_{L}(N,g_{n},g_{l})$ can be large implying a large level of
desynchronisation. Another way of understanding the relationship between
synchronisation and information is by using a result from Ref.
baptista_NJP2008 , which shows that for two coupled maps (but this result is
trivially extended to networks), the largest transversal conditional exponent,
when the maps have a LOWER character, is larger than this exponent for when
they have an UPPER character. Since this exponent provides a necessary
condition for the stability of the synchronisation manifold, it can be
interpreted as a measure of the level of desynchronisation in the network. The
larger this exponent is, the more desynchronous the network is. Therefore,
UPPER networks should have neurons more synchronous than LOWER networks.
If $H_{C}(N,g_{n},g_{l})>H_{L}(N,g_{n},g_{l})$ (UPPER), the synapse forces the
trajectory to approach the synchronisation manifold and, as a consequence,
there is a high level of synchronisation in the network. On the other hand, if
$H_{C}(N,g_{n},g_{l})<H_{L}(N,g_{n},g_{l})$ (LOWER), the synapse forces the
trajectory to depart from the synchronisation manifold and, as a consequence,
there is a high level of desynchronisation in the network.
Figure 7: [Color online] Gray regions (green online) indicate ($g_{n},g_{l}$)
values for which $H_{C}>H_{L}$ (UPPER) and black regions indicate
($g_{n},g_{l}$) values for which the complete synchronisation state is stable,
in excitatory networks (A-D) and inhibitory networks (E-H). The networks
considered in (A-D) as well as in (E-H) have the parameters shown in Fig.
1(A-D). In (B-D) and (F-H) the horizontal axis represent
$g_{l}(N)|\gamma_{2}(N)|/2$ and the vertical axis $kg_{n}$. Gray points (green
online) appearing on black regions represent synaptic strengths for which in
fact one has $H_{C}=H_{L}$, but numerically we obtain that
$H_{C}=H_{L}+\epsilon$, with $\epsilon$ being a very small positive constant.
One can check that in Fig. 7, which shows as gray, the parameter regions for
which $H_{C}>H_{L}$ and as black the parameter regions for which the
synchronisation manifold is stable and there is complete synchronisation (and
therefore, $H_{C}=H_{L}$) for typical initial conditions. Gray points
appearing on black regions represent synaptic strengths for which in fact one
has $H_{C}=H_{L}$, but numerically we obtain that $H_{C}=H_{L}+\epsilon$, with
$\epsilon$ being a very small positive constant. Typically, neurons coupled
via an excitatory synapse [(A-D)] present a LOWER character while via an
inhibitory synapse [(E-H)] present an UPPER character.
This classification is also important because as it was shown in Ref.
baptista_NJP2008 , once two coupled neurons are UPPER (or LOWER) there is
always a synaptic strength range for which a large network is UPPER (or
LOWER). And these synaptic strength ranges can be calculated using the
rescalings in Eqs. (11) and (12).
In Figs. 7(B-C) and 7(F-H), we show that the UPPER and LOWER character of two
mutually coupled neurons is preserved in networks composed by a number of
neurons larger than 2, if one considers the rescalings of Eqs. (11) and (12).
This result is of fundamental importance, specially for synaptic strengths
that promote the network to have an UPPER character because it allows us to
calculate an upper bound for the KS entropy of larger networks by knowing the
value of $H_{C}$ for two mutually coupled neurons. Such a situation arises for
inhibitory networks for a large range of both synaptic strengths. One finds an
UPPER character in excitatory networks for a small value of the chemical
synapse strength.
The electrical synapse favours the neurons to synchronise. As a consequence,
it is expected that networks with neurons connected exclusively by electrical
synapses are of the UPPER type. This can be checked in all figures for when
$g_{n}\approxeq$0.
We are currently trying to prove the conjecture in Ref. baptista_NJP2008 by
studying the relationship between the stability of unstable periodic orbits
paulo embedded in the attractors appearing in complex networks and the
stability of the equilibrium points. All the equilibrium points of a
polynomial network can be calculated by the methods in Refs. ra1 ; ra2 ; ra3 .
## IX Upper bound for the rate of information
According to Ruelle ruelle , the sum of all the positive Lyapunov exponents is
an upper bound for the Kolmogorov-Sinai entropy kolmogorov . Therefore,
whenever $H_{C}(N)>H_{L}(N)$ (UPPER) it is valid to write that
$H_{C}(N)>H_{KS}(N)$ (15)
where $H_{KS}(N)$ denotes the Kolmogorov-Sinai entropy of a network composed
of $N$ neurons.
As we have previously seen, the UPPER character of two mutually coupled
neurons is preserved in the special larger networks here studied. In addition
to this, if the positive conditional exponents of two mutually coupled neurons
are known for a given $g_{n}$ and $g_{l}$, allowing us to calculate
$H_{C}[N=2,g_{n}(N=2),g_{l}(N=2)]$, then one can calculate the positive
conditional exponents of a network with $N$ neurons,
$H_{C}[N,g_{n}(N),g_{l}(N)]$. In other words, if the ratio of information
production of two mutually coupled neurons that have equal trajectories,
$H_{C}(N=2)$, is known and the neurons have an UPPER character, one can
calculate the upper bound for the ratio of information production in larger
networks, as long as Eqs. (III) and (7) can be rescaled. Therefore, in UPPER
networks connected simultaneously with electrical and inhibitory chemical
synapses we can always calculate an upper bound for the rate of information
production in terms of this quantity in two mutually coupled inhibitory
neurons.
Consider two mutually coupled neurons. Denote $\lambda_{1}(N=2,g_{n})$ as the
sum for the positive Lyapunov conditional exponents associated with the
synchronisation manifold for a chemical synapse strength $g_{n}$ and
$\lambda_{2}(N=2,g_{n},g_{l})$ as the sum of the positive Lyapunov exponents
associated with the only one transversal direction for a chemical synapse
strength $g_{n}$ and an electrical synapse strength $g_{l}$. Remind that
$\lambda_{1}$ and $\lambda_{2}$ are calculated using Eq. (7) for the index
$j=1$ and $j=2$, respectively.
Now, consider a network formed by N neurons. Using similar arguments than the
ones presented in Sec. V and based on the conjecture proposed in
baptista_NJP2008 , the value of the synapse strengths $g_{l}(N),g_{n}(N)$ for
which the exponent $\lambda_{1}(N)$ has the same value of $\lambda_{1}(N=2)$
can be calculated by
$g_{n}(N)=\frac{g_{n}(N=2)}{k}$ (16)
and the value of the synapse strengths $g_{l}(N),g_{n}(N)$ for which the sum
of the positive conditional exponent $\lambda_{w}(N,g_{n},g_{l})$ (for $w\geq
2$) has the same value of $\lambda_{2}(N=2,g_{n},g_{l})$ can be calculated by
$\displaystyle g_{n}(N)$ $\displaystyle=$ $\displaystyle\frac{g_{n}(N=2)}{k}$
(17) $\displaystyle g_{l}(N)$ $\displaystyle=$
$\displaystyle\frac{g_{l}(N=2)|\gamma_{2}(N=2)|}{|\gamma_{w}(N)|}$ (18)
Denote $\lambda^{max}_{1}(N=2)$ and $\lambda^{max}_{2}(N=2)$ as the maximal
values of $\lambda_{1}(N=2,g_{n})$ and $\lambda_{2}(N=2,g_{n},g_{l})$ with
respect to $g_{n}$ and $g_{l}$.
As an example of how to use Eqs. (16), (17) and (18) in order to calculate the
upper bound for the rate of information produced in the network, we consider
that the neurons in the network with $N$ nodes are coupled via electrical and
excitatory chemical synapses in an all-to-all configuration (topology II),
then $k=N-1$, $|\gamma_{w}(N)|=N$ and $|\gamma_{2}(N=2)|=2$.
Now, we search for a synapse strength range for which two mutually coupled
neurons have an UPPER character. For example, let us say the range
$g_{l}(N=2)\in[0,1]$ and $g_{n}(N=2)\in[2,10]$, in Fig. 7(E), for two
inhibitory mutually coupled neurons.
From Eqs. (17) and (18), as long as the network with $N$ nodes has
$g_{n}(N)\leq\frac{1}{2(N-1)}$ and $\frac{0.3}{k}\leq
g_{l}(N)\leq\frac{1}{N}$, then $\lambda^{max}_{1}(N)=\lambda^{max}_{1}(N=2)$
and $\lambda^{max}_{w}(N)=\lambda^{max}_{2}(N=2)$, and therefore for this
synapse range, the maximum of $H_{C}$ is
$\max_{g_{n},g_{l}}{[H_{C}(N,g_{n},g_{l})]}=\lambda^{max}_{1}(N=2)+(N-1)\lambda^{max}_{2}(N=2)$
(19)
Notice that Eq. (19) is valid to any network topology as long as Eqs. (III)
and (7) can be rescaled.
For very large networks that are very well connected, $g_{l}(N)$ and
$g_{n}(N)$ will be very small, since $k$ and $N$ are large. As a consequence,
$\lambda^{max}_{1}\approxeq\lambda^{max}_{2}$, since neurons are equal, and we
can write
$\max_{g_{n},g_{l}}{[H_{C}(N,g_{n},g_{l})]}=N\lambda^{max}_{2}(N=2)$ (20)
which means that the rate of information produced by large UPPER neural
networks whose neurons are highly connected has an upper bound that increases
linearly with the number of neurons. A similar result is obtained when the
neurons are connected with only electrical synapses baptista_NJP2008 .
## X Conclusion
We have studied the combined action of chemical and electrical synapses in
small networks of Hindmarsh-Rose (HR) neurons in the process of
synchronisation and on the rate of information production.
There are mainly two scenarios for the appearance of complete synchronisation
for the studied inhibitory networks. If the chemical synapse strength is
small, the larger the chemical synapse strength used the larger the electrical
synapse strength needs to be to achieve complete synchronisation. Otherwise,
if the chemical synapse strength is large, complete synchronisation appears if
the electrical synapse strength is larger than a certain value. In the studied
excitatory networks both synapses work in a constructive way to promote
complete synchronisation: the larger the chemical synapse strength is the
smaller the electrical synapse strength needs to be to achieve complete
synchronisation.
When neurons connect simultaneously by electrical and chemical ways, there is
an optimal range of synaptic strengths for which the production of information
is large. For strengths larger than values within this optimal range, the
larger the electrical and chemical synaptic strengths are the smaller the
production of information of coupled neurons.
In the absence of complete synchronisation, it is intuitive to expect that
excitatory networks have neurons that are more desynchronous while inhibitory
networks have neurons that are more synchronous. This intuitive idea can be
better formalised by understanding the relationship between excitation
(inhibition), synchronisation (desynchronisation) and the rate of information
production. For that we classify the network as having an UPPER or a LOWER
character. In a UPPER (LOWER) network, the sum of all the positive Lyapunov
exponents, denoted by $H_{L}$, is bounded from above (below) by the sum of all
the positive conditional Lyapunov exponents, denoted by $H_{C}$, the Lyapunov
exponents of the synchronisation manifold and the transversal directions.
Networks that have neurons connected simultaneously by inhibitory chemical
synapses and electrical synapses can be expected to have an UPPER character.
In such networks, one should expect to find synchronous behaviour, since the
synapses force the trajectory to approach the synchronisation manifold. On the
other hand, networks whose chemical synapse are of the excitatory type might
likely have a LOWER character. In such networks one should expect to find
desynchronous behaviour since the synapses force the trajectory to depart from
the synchronisation manifold.
Notice that $H_{L}(N)$ can only be numerically obtained whereas $H_{C}(N)$ can
be calculated from the conditional exponents numerically obtained for two
mutually coupled neurons that have equal trajectories. For UPPER networks,
$H_{C}(N)>H_{L}(N)$, and by Ruelle ruelle $H_{L}(N)\geq H_{KS}(N)$, where
$H_{KS}$ is the Kolmogorov-Sinai entropy, the amount of information (Shannon’s
entropy) produced by time unit; we have then that $H_{C}$ is an upper bound
for $H_{KS}(N)$. That can be advantageously used in order to calculate the
rate of information produced by a large network, composed of $N$ neurons by
using only the rate at which information is produced in two mutually coupled
neurons that are completely synchronous and have equal trajectories.
We have worked with idealistic networks. However, our results can be extended
to more realistic networks Baptista1 . For UPPER networks, our numerical
results show that more realistic networks constructed with non-equal nodes (or
networks of equal nodes but with random synapse strengths baptista_PLOS2008 )
have $H_{L}$ smaller than the networks with equal nodes. Therefore, even
though networks with equal nodes might not be realistic, their entropy
production per time unit is an upper bound for the entropy production of more
realistic networks.
Acknowledgment MSB and FMMK thank the Max-Planck-Institut für Physik komplexer
Systeme (Dresden) for the partial support of this research. MSB acknowledges
the partial financial support of ”Fundação para a Ciência e Tecnologia (FCT),
Portugal” through the programmes POCTI and POSI, with Portuguese and European
Community structural funds. The authors are deeply grateful for the 4
anonimous referees for their important comments and suggestions that were
considered in this new version of the manuscript.
## XI Appendix
### XI.1 A lower bound for the KS entropy
Imagine a 2D chaotic system as the one studied in Ref. baptista_PRE2008 [Eqs.
(5) and (6)]. Following the same ideas from there, the KS entropy of two
coupled maps with variables $x^{\alpha}$ and $x^{\beta}$ can be estimated from
the Shannon’s entropy of the probabilities that a trajectory point makes a
given itinerary in the phase space $(x^{\alpha},x^{\beta})$, divided by the
time interval for the trajectory to make that itinerary.
In practice, calculating the Shannon’s entropy shannon for all possible
itineraries on the phase space ($x^{\alpha}$,$x^{\beta}$) of a chaotic
trajectory is equivalent to calculating the joint entropy between the
probabilities of finding a point following simultaneously an itinerary along
the variable the variable $x^{\alpha}$ and another itinerary along the
variable $x^{\beta}$.
Since we are unable to make a high resolution partition of the phase space
(nor we do not know the Markov partition) in the neural networks studied in
this work, we estimate a lower bound for the KS entropy by calculating the
joint entropy between symbolic sequences encoding the trajectory. Such
calculation of probabilities involve large matrix operations and for that
reason we restrain ourselves to the calculation of the joint entropy between
two neurons.
It is a lower bound due to two reasons. The first one is because the entropy
will be measured considering the probabilities of occupation of a projected
trajectory in a subspace of the network. The second one is because we
calculate the entropy considering the probabilities of binary symbolic
sequences and obviously a binary sequence may contain much less information
than the content of a continuous signal paninski .
In the following, we show in more details how this estimation is done. The way
we encode the trajectory is partially based on the time encoding proposed in
Ref. baptista_PLOS2008 .
Given two symbolic sequences $S_{1}$ and $S_{2}$, generated by neuron 1 and 2,
respectively, a lower bound for the KS entropy can be estimated by
$H_{low}=\frac{1}{\langle\tau\rangle}H(S_{1};S_{2})$ (21)
with $H(S_{1};S_{2})$ representing the joint entropy between the symbolic
sequences $S_{1}$ and $S_{2}$. To create the symbolic sequences, we represent
the time at which the $n$-th maxima happens in neuron 1 by $T_{1}^{n}$, and
the time interval between the n-th and the (n+1)-th maxima, by $\delta
T_{1}^{n}$. A maxima represents the moment when the action potential reaches
its maximal value. The quantity $\langle\tau\rangle$ represents the average
time between two spikes. We then encode the spiking events using the following
rule. The $i$-th symbol of the encoding is a “1” if a spike is found in the
time interval $[i\Delta,(i+1)\Delta[$, and “0” otherwise. We choose
$\Delta\in[\min{(\delta T_{1}^{n})},\max{(\delta T_{1}^{n})}]$ in order to
maximise $H_{low}$. Each neuron produces a symbolic sequence that is split
into small non-overlapping sequences of length $L$=8.
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|
arxiv-papers
| 2009-10-06T12:40:35 |
2024-09-04T02:49:05.701880
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M. S. Baptista, F. M. Moukam Kakmeni, C. Grebogi",
"submitter": "Murilo Baptista S.",
"url": "https://arxiv.org/abs/0910.0988"
}
|
0910.1198
|
11institutetext: Department of Physics, Eastern Mediterranean University,
Gazimagosa, North Cyprus, Mersin 10, Turkey 11email:
hale.pasaoglu@emu.edu.tr22institutetext: 22email: izzet.sakalli@emu.edu.tr
# Hawking Radiation of Linear Dilaton Black Holes in Various Theories
H.Pasaoglu 11 I.Sakalli 22 Address for offprint requests
###### Abstract
Using the Damour-Ruffini-Sannan, the Parikh-Wilczek and the thin film brick-
wall models, we investigate the Hawking radiation of uncharged massive
particles from 4-dimensional linear dilaton black holes, which are the
solutions to Einstein-Maxwell-Dilaton, Einstein-Yang-Mills-Dilaton and
Einstein-Yang-Mills-Born-Infeld-Dilaton theories. Our results show that the
tunneling rate is related to the change of Bekenstein-Hawking entropy.
Contrary to the many studies in the literature, here the emission spectrum is
precisely thermal. This implies that the derived emission spectrum is not
consistent with the unitarity of the quantum theory, which would possibly lead
to the information loss.
###### Keywords:
Entropy, Linear dilaton black holes, Tunneling effect, Thin film brick-wall
model
††journal: Int J Theor Phys††: Head note that is usually deleted††offprints:
Offprints Assistant
## 1 Introduction
Obeying the laws of black hole mechanics Bardeen , Hawking Nature ; Commun
proved that a stationary black hole can emit particles from its event horizon
with a temperature proportional to the surface gravity. According to this
idea, the vacuum fluctuations near the horizon would produce a virtual
particle pair, similar to electron-positron pair creation in a constant
electric field. When a virtual particle pair is created just inside or outside
the horizon, the sign of its energy changes as it crosses the horizon. So
after one member of the pair has tunneled to the opposite side, the pair can
materialize with zero total energy. This discovery also announced the relation
between the triple subjects – the quantum mechanics, thermodynamics and the
gravitation. After this pioneering study of Hawking, many methods have been
proposed to calculate the Hawking radiation for the last three decades.
One of the commonly used methods is known as Damour-Ruffini-Sannan (DRS)
Damour ; Sannan method. This method is applicable to any Hawking temperature
problem in which the asymptotic behaviors of the wave equation near the event
horizon are known.
In 2000, Parikh and Wilczek ParikhWilczek proposed a method based on null
geodesics in order to clarify more the Hawking radiation via tunneling across
the event horizon. Namely, they treated the Hawking radiation as a tunneling
process, and used the WKB approximation to determine the correction spectrum
for the black hole’s Hawking radiation. In their study, it is supposed that
the barrier depends on the tunneling particle itself. The crucial point of
this method is not to violate the energy conservation during the process of
particle emission and to pass to an appropriate coordinate system at horizon.
In general, the tunneling process is not precisely a thermal effect and it
explains the modification of the black hole radiation spectrum in which it
leads to the unitarity in the quantum theory INTParikh ; GRGParikh ;
ArxivParikh .
Another possible method to study the statistical origin of the black hole
entropy is the brick-wall model initially proposed by t’Hooft Hooft . The
brick-wall model identifies the black hole entropy by the entropy of a thermal
gas of quantum field excitations outside the event horizon. Since then, this
method has been satisfactorily applied to many black hole geometries (see for
instance Liu , and the references therein). Although t’ Hooft made significant
contribution to clarify the understanding and calculating the entopy of the
black holes, there were some drawbacks in his model. Those drawbacks are
overcome by the improved form of the original brick-wall model, which is
called as thin film brick-wall model PRDLiZhao . The thin film brick-wall
model gives us acceptable and net physical meaning of the entropy calculation.
In summary, since the entropy calculated by the thin film brick-wall model is
just from a small region (thin film) near the horizon, this improved version
of the brick-wall model represents explicitly the correlation between the
horizon and the entropy. In this study, we obtain the ultraviolet cut-off
distance as 90$\beta,$ where $\beta$ is the Boltzmann factor.
Hawking described the black hole radiation as tunneling triggered by the
vacuum fluctuations near the horizon. His discovery, which treats the black
hole radiation as being pure thermal gave also rise to a new paradox in the
black hole physics – the information loss paradox. Although, Parikh and
Wilczek’s tunneling process ParikhWilczek is a way to overcome the
information loss paradox in the Hawking radiation, the information might not
be conserved in some black hole geometries. For instance, if only the
tunneling process of the outer horizon of the Reissner-Nordström black hole is
considered QJiang ; Ren , it can be shown that the information loss is
possible. The similar violation in the conservation of information happens in
the $4$-dimensional linear dilaton black holes (LDBHs) in various theories,
and we will explain its reason by using the differences in entropies of the
black holes before and after the emission.
The paper is organized as follows: In section 2, a brief overview of the
$4$-dimensional LDBHs in Einstein-Maxwell-Dilaton (EMD), Einstein-Yang-Mills-
Dilaton (EYMD) and Einstein-Yang-Mills-Born-Infeld-Dilaton (EYMBID) theories,
which they have been recently employed in Sakalli for calculating the Hawking
radiation via the method of semi-classical radiation spectrum is given. Next,
we apply the DRS method to find the temperature of the LDBHs and the tunneling
rate of the chargeless particles crossing the event horizon. Section 3 is
devoted to the calculation of the entropy of the horizon by using all those
methods mentioned above. As it is expected, they all conclude with the same
result. Finally, we draw our conclusions and discussions.
Throughout the paper, the units $G=c=$h $=k_{B}$=1 are used.
## 2 LDBHs, Calculation of Their Temperature and Tunneling Rate
The line-element of $N$-dimensional ($N\geq 4$) LDBHs, which are static
spherically symmetric solutions in various theories (EMD, EYMD and EYMBID)
have been recently summarized by Sakalli . However, throughout this paper we
restrict ourselves to the $4$-dimensional LDBHs and follow the notations of
Sakalli .
Consider a general class of static, spherically symmetric spacetime for the
LDBHs as
$ds^{2}=-fdt^{2}+\frac{dr^{2}}{f}+A^{2}rd\Omega^{2},$ (1)
where $d\Omega^{2}=d\theta^{2}+\sin^{2}\theta d\phi^{2}.$ Here, the metric
function $f$ is given by Sakalli
$f=\Sigma r(1-\frac{r_{+}}{r}),$ (2)
where $r_{+}$ is the radius of the event horizon. The coefficients $\Sigma$
and $A$ in the metric (1) take different values according to the concerned
theory.
Since the present form of the metric represent asymptotically non-flat
solutions, one should consider the quasi-local mass definition $M$ of the
metric (1). In Sakalli , the relationship between the horizon $r_{+}$ and the
mass $M$ is explicitly given as
$r_{+}=\frac{4M}{\Sigma A^{2}},$ (3)
In the EMD theory Sakalli ; Chan ; Clement , the coefficients $\Sigma$ and $A$
are found as
$\Sigma\rightarrow\Sigma_{EMD}=\frac{1}{\gamma^{2}}\text{ \ and \ \ \
}A\rightarrow A_{EMD}=\gamma\text{,}$ (4)
where $\gamma$ is a constant related to the electric charge of the black hole.
Meanwhile, one can match the metric (1) to the LDBH’s metric of Clément et.al.
Clement by setting $\gamma\equiv r_{0}$. Next, if one considers the EYMD and
EYMBID theories Mazhari1 ; Mazhari2 , the coefficients in the metric (1)
become
$\Sigma\rightarrow\Sigma_{EYMD}=\frac{1}{2Q^{2}}\text{ \ and \ \ \
}A\rightarrow A_{EYMD}=\sqrt{2}Q,$ (5)
and
$\Sigma\rightarrow\Sigma_{EYMBID}=\frac{1}{Q_{C}^{2}}\left[1-\sqrt{1-\frac{Q_{C}^{2}}{Q^{2}}}\right]\text{
\ and \ }A\rightarrow
A_{EYMBID}=\sqrt{2}Q\left(1-\frac{Q_{C}^{2}}{Q^{2}}\right)^{\frac{1}{4}},$ (6)
where $Q$ and $Q_{C}$ are YM charge and the critical value of YM charge,
respectively. The existence of the metric (1) in EYMBID theory depends
strictly on the condition Mazhari2
$Q^{2}>Q_{C}^{2}=\frac{1}{4\tilde{\beta}^{2}},$ (7)
where $\tilde{\beta}$ is the Born-Infeld parameter. Meanwhile, it is not
necessary to say that values of $\Sigma$ in equations (4), (5) and (6) are
always positive.
By using the definition of the surface gravity Wald , we get
$\kappa=\lim_{r\rightarrow r_{+}}\frac{f^{\prime}(r)}{2}=\frac{\Sigma}{2}.$
(8)
Since the surface gravity (8) is positive, one can deduce that it is directed
towards the singularity. As a consequence, it is attractive and the matter can
only fall into the black hole. This horizon is a future horizon to an
observer, who is located outside of it.
In curved spacetime, a massive test scalar field $\Phi$ with mass $\mu$ obeys
the covariant Klein-Gordon (KG) equation, which is given by
$\frac{1}{\sqrt{-\det g}}\partial_{\mu}\left(\sqrt{-\det
g}g^{\mu\nu}\partial_{\nu}\Phi\right)-\mu^{2}\Phi=0,$ (9)
The massive scalar wave equation $\Phi$ in metric (1) can be separated as
$\Phi=Y(\theta,\varphi)\psi(t,r)$ in which the radical KG equation (9)
satisfies the following equation:
$\frac{\partial^{2}\psi}{\partial
t^{2}}+f(\frac{f}{r}+\Sigma)\frac{\partial\psi}{\partial
r}+f^{2}\frac{\partial^{2}\psi}{\partial
r^{2}}-f(\mu^{2}-\frac{l(l+1)}{r})\psi=0,$ (10)
where $l$ is the angular quantum number. In order to change equation (10) into
a standard wave equation at the horizon, we introduce the tortoise coordinate
transformation, which is obtained from
$dr_{\ast}=\frac{dr}{f},$ (11)
After making the straightforward calculation, we find an appropriate
$r_{\ast}$ as
$r_{\ast}=\frac{1}{2\kappa}\ln(r-r_{+}),$ (12)
Thus, one can transform the radical equation (10) into the following form
$\frac{\partial^{2}\psi}{\partial
t^{2}}-\frac{f}{r}\frac{\partial\psi}{\partial
r_{\ast}}-\Sigma\frac{\partial\psi}{\partial
r_{\ast}}+\Sigma\frac{\partial\psi}{\partial
r_{\ast}}-\frac{\partial^{2}\psi}{\partial
r_{\ast}^{2}}+f[\mu^{2}-\frac{l(l+1)}{r}]\psi=0,$ (13)
While $r\rightarrow r_{+}$ in which $f\rightarrow 0$, the transformed radical
equation (13) can be reduced to the following standard form of the wave
equation as
$\frac{\partial^{2}\psi}{\partial t^{2}}-\frac{\partial^{2}\psi}{\partial
r_{\ast}^{2}}=0,$ (14)
This form of the wave equation reveals that there are propagating waves near
the horizon. The solutions of equation (14), which give us the ingoing and
outgoing waves at the black hole horizon surface $r_{+}$ are
$\psi_{out}=\exp(-i\omega t+i\omega r_{\ast}),$ (15)
$\psi_{in}=\exp(-i\omega t-i\omega r_{\ast}),$ (16)
When we introduce the ingoing Eddington-Finkelstein coordinate,
$v=t+r_{\ast}$, the line-element (1) of the LDBHs becomes
$ds^{2}=-fdv^{2}+2dvdr+A^{2}rd\Omega^{2},$ (17)
The present form of the metric does not attribute a singularity to the
horizon, so that the ingoing wave equation behaves regularly at the horizon.
This yields the solutions of ingoing and outgoing waves at the horizon $r_{+}$
as follows
$\psi_{out}=e^{-i\omega v}e^{2i\omega r_{\ast}},$ (18)
$\psi_{in}=e^{-i\omega v},$ (19)
Now, we consider only the outgoing waves. Namely,
$\psi_{out}(r>r_{+})=e^{-i\omega v}(r-r_{+})^{\frac{{}^{i\omega}}{\kappa}},$
(20)
which has a singularity at the horizon $r_{+}$. Therefore, equation (20) can
only describe the outgoing particles outside the horizon and strictly cannot
describe the particles, which are inside the horizon. In other words, the
description of the particles’ behavior inside horizon has to be made as well.
To this end, the outgoing wave $\psi_{out}$ should be analytically extended
from outside to the interior of the black hole by the lower half complex
$r$-plane
$(r-r_{+})\rightarrow\left|r-r_{+}\right|e^{-i\pi}=(r_{+}-r)e^{-i\pi},$ (21)
We can derive the solution of outgoing wave inside the horizon as follows
$\psi_{out}(r<r_{+})=\psi_{out}^{{}^{\prime}}\left(r<r_{+}\right)e^{\frac{{}^{\omega\pi}}{\kappa}},$
(22)
where
$\psi_{out}^{{}^{\prime}}\left(r<r_{+}\right)=e^{-i\omega
v}(r_{+}-r)^{\frac{{}^{i\omega}}{\kappa}},$ (23)
According to the Damour-Ruffini-Sannan (DRS) Damour ; Sannan method, it is
possible to calculate the emission rate. The total outgoing wave function can
be written in a uniform form
$\psi=N_{\omega}[\Theta(r-r_{+})\psi_{out}\left(r>r_{+}\right)+e^{\frac{\omega\pi}{\kappa}}\Theta(r_{+}-r)\psi_{out}^{{}^{\prime}}\left(r<r_{+}\right)],$
(24)
where $\Theta$ is the Heaviside step function and $N_{\omega}$ represents the
normalization factor. From the normalization condition
$\left(\psi,\psi\right)=\pm 1,$ (25)
we can obtain the resulting radiation spectrum of scalar particles
$N_{\omega}^{2}=\frac{\Gamma}{1-\Gamma}=\frac{1}{e^{\frac{\omega}{T}}-1},$
(26)
and read the temperature of the horizon as
$T=\frac{\kappa}{2\pi},$ (27)
In equation (26) $\Gamma$ symbolizes the emission or tunneling rate, which is
found by the following ratio
$\Gamma=\left|\frac{\psi_{out}\left(r>r_{+}\right)}{\psi_{out}\left(r<r_{+}\right)}\right|^{2}=e^{\frac{-^{2\pi\omega}}{\kappa}}.$
(28)
One can remark for this section that the resulting temperature (27) obtained
from the DRS method is in agreement with the statistical Hawking temperature
Wald computed as usual by dividing the surface gravity by $2\pi$.
## 3 Entropy of the Horizon
In this section, we shall use three different methods in order to show that
they all lead to the same entropy result. We first employ the DRS method,
which is worked in detail and obtained remarkable results in the previous
section. The second method will be the Parikh-Wilczek method ParikhWilczek
describing the Hawking radiation as a tunneling process. Last method that is
also going to be used in the calculation of the entropy is the thin film
brick-wall model PRDLiZhao .
In the DRS method, the emission rate of outgoing particles is found as in
equation (28). Accordingly, the probability of emission can be modified into
QJiang ; Jiang (and references therein)
$\Gamma=e^{{}^{-2\pi\int_{0}^{\omega}\frac{d\omega^{\prime}}{\kappa}}}=e^{-\int_{0}^{\omega}\frac{d\omega^{\prime}}{T}}=e^{\Delta
S_{BH}},$ (29)
where $\Delta S_{BH}$ is the difference of Bekenstein-Hawking entropies of the
LDBHs before and after the emission of the particle.
On the other hand, the novel study on the tunneling effect is designated by
Parikh-Wilczek method ParikhWilczek , which proposes an approach for
calculating the tunneling rate at which particles tunnel across the event
horizon. They treated Hawking radiation as a tunneling process, and used the
WKB method ArxivParikh . In classical limit, we can also find the tunneling
rate by applying WKB approximation. This relates the tunneling amplitude to
the imaginary part of the particle action at stationary phase and the
Boltzmann factor for emission at the Hawking temperature.
In the WKB approximation, the imaginary part of the amplitude for outgoing
positive energy particle which crosses the horizon outwards from initial
radius of the horizon $r_{in}$ to the final radius of the horizon $r_{out}$
could be expressed by
$\mathop{\mathrm{I}m}I=\mathop{\mathrm{I}m}\int_{r_{in}}^{r_{out}}p_{r}dr=\mathop{\mathrm{I}m}\int_{r_{in}}^{r_{out}}\int_{0}^{p_{r}}dp_{r}^{\prime}dr,$
(30)
By using the standard quantum mechanics, the tunneling rate $\Gamma$ is given
in the WKB approximation as KrausWilczek1 ; KrausWilczek2 ,
$\Gamma\sim\exp(-2\mathop{\mathrm{I}m}I),$ (31)
Here we can consider the particle with energy $\omega$ as a shell of energy
and fix the total mass $M$ (quasi-local mass) and allow the hole mass to
fluctuate. Then the Hamilton’s equation of motion can be used to write
$dp_{r}=\frac{dH}{\dot{r}},$ and it can be noted that the horizon moves
inwards from $M$ to $M-\omega$ while a particle emits. Introducing
$H=M-\omega$ and inserting the value of the
$\dot{r}\equiv\frac{dr}{dv}=\frac{f}{2}$ obtained from the null geodesic
equation into (30), we obtain
$\mathop{\mathrm{I}m}\int_{r_{in}}^{r_{out}}\int_{0}^{p_{r}}dp_{r}^{\prime}dr=\mathop{\mathrm{I}m}\int_{r_{in}}^{r_{out}}\int_{M}^{M-\omega}\frac{dr}{\dot{r}}dH=\mathop{\mathrm{I}m}\int_{0}^{\omega}\int_{r_{in}}^{r_{out}}\frac{2dr}{\Sigma(r-r_{+})}\left(-d\omega^{\prime}\right),$
(32)
The $r$-integral can be done by deforming the contour. The deformation of the
integral is based on an assumption that the contour semicircles the residue in
a clockwise fashion. In this way, one can obtain
$\mathop{\mathrm{I}m}I=2\pi\int_{0}^{\omega}\frac{d\omega^{\prime}}{\Sigma},$
(33)
So, the tunneling rate (31) is
$\Gamma\sim\exp(-2\mathop{\mathrm{I}m}I)=\exp\left(-4\pi\int_{0}^{\omega}\frac{d\omega^{\prime}}{\Sigma}\right)=\exp\left(\Delta
S_{BH}\right),$ (34)
Our result (34) is consistent with the results of the other works
ParikhWilczek ; Kerner ; Zhang ; Chen ; Li .
Now, we come to the stage to apply the thin film brick-wall model PRDLiZhao ,
which was based on the brick wall model proposed firstly by t’Hooft Hooft .
According to this model, the considered field outside the horizon is assumed
to be non-zero only in a thin film, which exists in a small region bordered by
$r_{+}+\varepsilon$ and $r_{+}+\varepsilon+\delta$. Here, $\varepsilon$ is the
ultraviolet cut-off distance and $\delta$ is the thickness of the thin film.
In summary, both $\varepsilon$ and $\delta$ are positive infinitesimal
parameters. This model treats the entropy as being associated with the field
in the considered small region in which the local thermal equilibrium and the
statistical laws are valid Tian . That is why one can work out the entropy of
the horizon by using this model.
If one redefines the massive test scalar field $\Phi$ as being
$\Phi=e^{-i\omega t}\psi(r)Y_{lm}(\theta,\phi)$ in the KG equation (9) and
considers its radial part only, the wave vector is found with the help of WKB
approximation as
$k^{2}=\frac{1}{\Sigma r(1-\frac{r_{+}}{r})}[\frac{\omega^{2}}{\Sigma
r(1-\frac{r_{+}}{r})}-(\mu^{2}+\frac{l(l+1)}{A^{2}r})],$ (35)
Using the quantum statistical mechanics, we calculate the free energy from
$F=\frac{-1}{\pi}\int_{0}^{\infty}d\omega\int_{r}dr\int_{l}(2l+1)\frac{k}{e^{\beta\omega}-1}dl,$
(36)
While integrating equation (36) with respect to $l,$ one should consider the
upper limit of integration such that $k^{2}$ remains positive, and the lower
limit becomes zero. Briefly, we get
$F\cong\frac{-2A^{2}}{3\pi\Sigma^{2}}\int_{0}^{\infty}\frac{d\omega}{e^{\beta\omega}-1}\int_{r}\frac{r}{(r-r_{+})^{2}}[\omega^{2}-\mu^{2}\Sigma(r-r_{+})]^{\frac{3}{2}}dr,$
(37)
where $\beta$ denotes the inverse of the temperature. In equation (37), the
integration with respect to $r$ is quite difficult. On the other hand, the
thin film brick-wall model imposes us to take only the free energy of a thin
layer near horizon of a black hole, and the integration with respect to $r$
must be limited in the region $r_{+}+\varepsilon\leq r\leq
r_{+}+\varepsilon+\delta.$ The natural result of this choice sets the
coefficient of $\mu^{2}$ to zero, and whence the integration of equation (37)
with respect to $\omega$ becomes very simple such that it can be easily found
as $\pi^{4}/15\beta^{4}$. Finally, the equation (37) reduces to
$F\cong\frac{-2\pi^{3}A^{2}}{45\beta^{4}\Sigma^{2}}\int_{r_{+}+\varepsilon}^{r_{+}+\varepsilon+\delta}\frac{r}{(r-r_{+})^{2}}dr,$
(38)
$\cong\frac{-2\pi^{3}A^{2}r_{+}}{45\beta^{4}\Sigma^{2}}\int_{r_{+}+\varepsilon}^{r_{+}+\varepsilon+\delta}\frac{dr}{(r-r_{+})^{2}},$
(39)
$F\cong\frac{-2\pi^{3}A^{2}r_{+}}{45\beta^{4}\Sigma^{2}}\frac{\delta}{\varepsilon(\delta+\varepsilon)},$
(40)
and we can get the entropy
$S_{BH}=\beta^{2}\frac{\partial
F}{\partial\beta}=[\frac{8\pi^{3}A^{2}r_{+}}{45\beta^{3}\Sigma^{2}}]\frac{\delta}{\varepsilon(\delta+\varepsilon)},$
(41)
Since the beta is the inverse of the temperature
$\beta=\frac{1}{T}=\frac{4\pi}{\Sigma},$ (42)
and if we select an appropriate cut-off distance $\varepsilon$ and thickness
of thin film $\delta$ to satisfy
$\frac{\delta}{\varepsilon(\delta+\varepsilon)}=90\beta,$ (43)
the total entropy of the horizon becomes
$S_{BH}=\frac{1}{4}A_{h},$ (44)
where $A_{h}$ is the area of the the black hole horizon, i.e. $A_{h}=4\pi
A^{2}r_{+}.$ The derivative of the entropy (44) with respect to $M$ is
$\frac{\partial S_{BH}}{\partial M}=\pi A^{2}\frac{\partial r_{+}}{\partial
M}=\frac{4\pi}{\Sigma},$ (45)
Getting the integral of $M$, equation (45) becomes to
$\Delta S_{BH}=\int_{M}^{M-\omega}\frac{\partial S_{BH}}{\partial
M^{\prime}}dM^{\prime}=4\pi\int_{M}^{M-\omega}\frac{dM^{\prime}}{\Sigma},$
(46)
After substituting $M^{\prime}=M-\omega^{\prime}$ into the above equation, we
obtain
$\Delta S_{BH}=-\int_{0}^{\omega}\frac{\partial S_{BH}}{\partial
M^{\prime}}d\omega^{\prime}=-4\pi\int_{0}^{\omega}\frac{d\omega^{\prime}}{\Sigma},$
(47)
One can easily see that equation (47) is nothing but the results obtained both
from Parikh-Wilczek method (34) and the DRS method (29).
On the other hand, for the LDBHs the change of the entropy before and after
the radiation is
$\Delta S_{BH}=S(M-\omega)-S(M)=-\frac{2\pi\omega}{\kappa}.$ (48)
Since equation (48) contains only $\omega$, we deduce that the spectrum is
precisely thermal. In other words, the thermal spectrum does not suggest the
underlying unitary theory, and whence we can understand that the conservation
of information is violated.
## 4 Discussion and Conclusion
In this paper, we have effectively utilized three different methods (the DRS
model, the Parikh-Wilczek model and the thin film brick wall model) to
investigate the Hawking radiation for massive 4-dimensional LDBHs in the EMD,
EYMD and EYMBID theories. By considering the DRS method, the tunneling
probability for an outgoing positive energy particle or simply the tunneling
rate is neatly found. Later on, it is shown that the tunneling rate found from
the DRS method can be expressed in terms of the difference of Bekenstein-
Hawking entropies $\Delta S_{BH}$ of the black holes. Beside this, the other
two methods i.e. the Parikh-Wilczek method and the thin film brick-wall model
attribute also the same $\Delta S_{BH}$ result. In the thin film brick-wall
model, the cut-off factor is found to be 90$\beta$, which is exactly same as
in the calculation of the entropy for the Schwarzschild black hole Liancheng .
On the other hand, the obtained$\ \Delta S_{BH}$ result shows us that the
emission spectrum is nothing but a pure thermal spectrum. This result is not
consistent with the unitarity principle of quantum mechanics. It also implies
the violation of the conservation of information in the LDBHs.
Finally, further application of the Hawking radiation of the charged massive
particles via different methods to the case of LDBHs in higher dimensions
Sakalli may reveal more information compared to the present case. This will
be our next problem in the near future.
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|
arxiv-papers
| 2009-10-07T09:28:38 |
2024-09-04T02:49:05.713038
|
{
"license": "Public Domain",
"authors": "H.Pasaoglu and I.Sakalli",
"submitter": "Izzet Sakalli",
"url": "https://arxiv.org/abs/0910.1198"
}
|
0910.1286
|
# The variable 6307Å emission line in the spectrum of Eta Carinae: blueshifted
[S III] $\lambda$6313 from the interacting winds
T. R. Gull11affiliation: Laboratory for Extrasolar Planets and Stellar
Astrophysics, Exploration of the Universe Division, Code 667, Goddard Space
Flight Center, Greenbelt, MD 20771
###### Abstract
The 6307Å emission line in the spectrum of $\eta$ Car (Martin et al., 2006) is
blue-shifted [S iii] $\lambda$6313 emission originating from the outer wind
structures of the massive binary system. We realized the identification while
analyzing multiple forbidden emission lines not normally seen in the spectra
of massive stars. The high spatial and moderate spectral resolutions of
HST/STIS resolve forbidden lines of Fe+, N+, Fe+2, S+2, Ne+2 and Ar+2 into
spatially and velocity-resolved rope-like features originating from
collisionally-excited ions photo-ionized by UV photons or collisions. While
the [Fe ii] emission extends across a velocity range of $\pm$500 km s-1 out to
0$\farcs$7, more highly ionized forbidden emissions ( [N ii], [Fe iii], [S
iii], [Ar iii], and [Ne iii]) range in velocity from $-$500 to $+$200 km s-1,
but spatially extend outward to only 0$\farcs$4\. The [Fe ii] defines the
outer regions of the massive primary wind. The [N ii], [Fe iii] emission
define the the outer wind interaction regions directly photo-ionized by far-UV
radiation. Variations in emission of [S iii] $\lambda$$\lambda$9533, 9071 and
6313 suggest density ranges of 106 \- 1010 cm-3 for electron temperatures
ranging from 8,000 to 13,000° K. Mapping the temporal changes of the emission
structure at critical phases of the 5.54-year period will provide important
diagnostics of the interacting winds.
stars: binaries:spectroscopic, stars: individual: Eta Carinae, stars:winds
## 1 Introduction
Martin et al. (2006) found a variable emission line centered at 6307Å in
multiple spectra of $\eta$ Car, recorded by HST/STIS and by VLT/UVES. They
were unable to identify the origin of the emission line, but demonstrated that
the line was present across the high state (defined by presence of forbidden
lines of doubly-ionized elements, see Damineli et al. (2008) and references
therein) and disappeared during the low state.
Recently Gull et al. (2009), using the same spectra, focused on the spatially-
extended forbidden line emission both from high-ionization (herein defined as
$>$14 eV) and low-ionization (8-13 eV). We found that the forbidden emission
originated from 1) the Weigelt condensations (Weigelt & Ebersberger, 1986), as
narrow lines centered on $-$43 km s-1, 2) the boundaries of the primary wind
($\eta$ Car A) as rope-like [Fe ii], photo-ionized by mid-UV and collisionally
excited at densities around Nc=107cm-3, and 3) the wind interaction region, as
high-ionization emission from [N ii], [Fe iii], [Ar iii], [Ne iii] and [S
iii], photo-ionized by far-UV and collisionally excited for densities, nc
ranging from 105 to 108cm-3. The high-ionization emission lines are present
both for the Weigelt condensations and the wind interaction region during the
5 year high state, but every 5.5 years, disappear during the low state. X-ray
models (Pittard & Corcoran, 2002; Parkin et al., 2009) place the binary
periastron event near the onset of the low state with a three to six month
recovery.
We applied a 3D SPH (smoothed particle hydrodynamics) model (Okazaki et al.,
2008) extended out to 1700 AU (0$\farcs$67) to match the spatial structure
seen in these forbidden lines and realized that the bulk of the high-
ionization emission structure originated from the wind interaction region in
the outer portion of the massive wind structure. We found that portions of the
wind interaction structure, moving ballistically outward, are directly
illuminated by the far-UV radiation of the hot secondary, $\eta$ Car B,
leading to highly-ionized, collisionally excited gas and hence the high-
ionization extended emission.
Further examination of the HST/STIS longslit spectra showed that the
previously unidentified emission at 6307Å is blue-shifted [S iii]
$\lambda$6313 emission from the interacting wind structure. The evidence is
subtle, but convincing. We summarize the observations in Section 2. A
description of how the spectro-images were produced is in Section 3. The
forbidden emission structures are described in Section 4. Discussion in
Section 5 provides insight on the ionization and excitation leading to [S iii]
emission and the potential for monitoring changes with orbital phase,
including mapping temperature with density dependence. We conclude with a
summary in Section 6. Throughout this paper, all wavelengths are in vacuum,
the velocities are heliocentric, directions are compass points (N=north, NNW=
north by northwest and the phase of the binary orbit is referenced to the
X-ray minimum beginning at 1997.9604 (Corcoran, 2005).
## 2 The HST/STIS Observations
Figure 1: HST/STIS aperture positions: Left: Centered on $\eta$ Car. Right:
Centered on Weigelt D. The 2″$\times$2″ Field of View is extracted from an
HST/ACS image recorded in February 2003 through the 550M filter. Weigelt
condensations B, C and D are indicated by the black dots. The projection of
the 52″$\times$0$\farcs$1 aperture is indicated. Note the field is rotated by
69° placing the aperture vertical. North is indicated by the compass.
The spectra discussed here are a portion of the Eta Carinae Treasury
observations accessible through the STScI archives (
http://archive.stsci.edu/prepds/etacar) as reduced by a special reduction tool
developed by K. Ishibashi and K. Davidson. For brevity we focus on two sets of
observations, recorded in July 2002 ($\phi$=0.820) and July 2003
($\phi$=1.001).
Observations were recorded with the HST/STIS moderate dispersion gratings and
CCD detector through the 52″$\times$0$\farcs$1 aperture. We wanted to monitor
the change in both $\eta$ Car and the Weigelt condensations (Weigelt &
Ebersberger, 1986), which drop in excitation during the low state. However the
range in aperture position angle (PA) is limited by the required orientation
of HST solar panels and changes throughout the year. can prevent inclusion of
any one of the three Weigelt condensations within the aperture when centered
on $\eta$ Car (Figure 1). For observations centered around the X-ray minimum,
predicted to be around 1 July, 2003, we scheduled a visit one year earlier,
July 2, 2002 (orbital phase, $\phi$=0.820), at a pre-selected PA=69°, that
would be accessible just before($\phi$=0.995) and after the X-ray minimum
($\phi$=1.001). During all three visits, separate observations centered on
$\eta$ Car and Weigelt D were obtained with the aperture placed as shown in
Figure 1. Additional information on the observations are presented in Martin
et al. (2006) and Gull et al. (2009).
## 3 Spatially-resolved Emission
The HST/STIS spatial-resolution (0$\farcs$1 at H$\alpha$) separates the
spectrum of $\eta$ Car’s core from extended structures, especially the narrow
line emission that originates from the Weigelt condensations (Davidson et al.,
1995), located between 0$\farcs$1 to 0$\farcs$3 in the NW quadrant relative to
$\eta$ Car (see Figure 1). As described by Gull et al. (2009), we found
considerable differences between many broad forbidden emission line profiles
of $\eta$ Car as recorded by the VLT/UVES and the HST/STIS. Extractions with a
0$\farcs$127-high slice of the STIS spectra (five half rows in the reduced
spectro-images) yielded broad wind line profiles for H i, He I and Fe ii lines
that compared favorably with those recorded by VLT/UVES. In contrast, broad
profiles of forbidden lines recorded by VLT/UVES were nearly absent in the
HST/STIS extractions centered on $\eta$ Car. Examination of the HST/STIS long
aperture spectra revealed faint structure in these emission lines extending
out to 0$\farcs$7\. However, the observed emission was highly variable both
with aperture PA and orbital phase, $\phi$. Clearly the forbidden line
emission is spatially extended on scales resolved by HST/STIS but not by
VLT/UVES.
We examined individual lines in more detail and attempted to enhance
visibility of the extended line emission by several reduction procedures.
While various spatial filters were tried, the best results were obtained by
subtraction of measured continuum on a spatial row-by-row basis. We used
spectral plots of the Weigelt condensations (Zethson, 2001) to identify 10 to
20Å intervals with no obvious presence of narrow or broad line emission. At
each position along the aperture, we measured and subtracted the average
continuum in that spectral interval. Examples of the resulting spectro-images
(intensity images with x$=$velocity and y$=$angular size) are presented in
Figure 2.
Figure 2: Spectro-images centered on $\eta$ Car (Columns 1 and 2) and on
Weigelt D (Columns 3 and 4). [Fe II] $\lambda$4815 (Row 1) [Fe III]
$\lambda$4703 (Row 2) [N II] $\lambda$5756 (Row 3) and [S III]
$\lambda\lambda$6313, 9071, 9533 (Rows 4$-$6) Note: Continuum in regions free
of narrow or broad-line emission has been subtracted on a spatial line-by-line
basis to display the extended emission structure. All plots are with a grey
level proportional to $\sqrt{Intensity}$. All observations were recorded with
PA=69°.
These spectro-images provide only a qualitative view of the line profile. We
caution the reader that quantitative measures require much more precision for
the following reasons:
1. 1.
The STIS utilized a three-axis mechanism to select a grating and to set the
correct tilt angle for the spectral interval of choice. While return to that
grating position is within a few CCD pixels, variations in the tilt are not
fully reproducible. A tilt of 1/20 pixel along the 1024 element row led to
significant photometric variation when attempting to extract spectra at the
0$\farcs$1 spatial resolution.
2. 2.
The standard calibration for the STIS photometry is properly referenced to
extractions of a stellar spectrum with a 2″-wide aperture, allowing for full
capture of flux from a point source. Apertures with widths comparable to the
diffraction limit of HST sample the point spread function of the telescope,
which changes dynamically even within an orbit.
3. 3.
Charge transfer inefficiency (CTI) leads to a trail in the direction of
columns and, with on-orbit time, increases. For complex sources like extended
structures, a proper extraction is not available.
These problems complicate attempts to show extended structure in the vicinity
of a bright star, which is exactly the situation with $\eta$ Car. This leads
to the obvious linear striations at the star position in the spectrally-
dispersed (velocity) coordinate. These variations do affect measures of the
extended emission closest to the stellar position. However, for offsets to
Weigelt D, the stellar flux is blocked by the aperture, and quantitative
measures are then possible. We note that the stellar spectrum scattered from
the direction of Weigelt D is very different from the direct spectrum of
$\eta$ Car. The lack of P Cygni absorption in H$\alpha$ across the high state
indicates fully-ionized hydrogen in the region spatially located between
$\eta$ Car and Weigelt D (Gull et al., 2009). On the side of caution, we limit
this discussion to a description of the spatial and velocity structure of the
lines. Even with qualitative descriptions, we gain much insight on the
spatially resolved wind interactions and the source of the 6307Å emission.
## 4 Description of the emission structures
We refer the reader to Figure 2 for the following descriptions of the
forbidden line emission. The first two columns of spectro-images are extracted
from spectra centered on $\eta$ Car in the high state ($\phi$=0.820) and early
in the low state ($\phi$=1.001). Likewise, columns 3 and 4 are centered on
Weigelt D (A more complete summary on variation of the emission structures
with ionization potential and orbital phase is presented by Gull et al.
(2009)).
Four basic structures contribute to these spectro-images:
1. 1.
The central core of $\eta$ Car, not resolved by HST at 0$\farcs$1, which
contributes the bulk of the continuum and P Cygni wind lines, notably of H i
and Fe ii.
2. 2.
Weigelt D and other, lesser condensations that contribute many narrow emission
lines centered at $-$40 km s-1.
3. 3.
Rope-like structures of high-ionization forbidden emission lines with velocity
components extending from $+$200 to $-$500 km s-1.
4. 4.
Noticeably more-diffuse, rope-like structures of low-ionization forbidden
emission lines, specifically [Fe ii].
Spectro-images of [Fe ii] $\lambda$4815 (Row 1) show narrow rope-like features
extending to 0$\farcs$7 at $-$500 km s-1and other, more diffuse structures
closer to the star extending $\pm$500 km s-1. The narrow emission at $-$40 km
s-1 originates from extended structure WSW of $\eta$ Car, not noted by Weigelt
& Ebersberger (1986), but present throughout the observational period from
1999 to 2004 whenever the STIS aperture sampled this position. The narrow [Fe
ii] emission centered on Weigelt D extends 0$\farcs$5 E and W of Weigelt D,
but at about 0$\farcs$25 E of D, a diffuse emission extends to $-$400 km
s-1and away from $\eta$ Car. During the low state, the outer [Fe ii] emission
drops, becomes more diffuse and is located closer to $\eta$ Car.
The [Fe iii] $\lambda$4703 (Row 2) is interior to the rope-like [Fe ii]
$\lambda$4815\. A series of highly filamentary loops extend from $\eta$ Car to
the east at velocities from $-$40 to $-$500 km s-1. No red-shifted velocity
components are seen at this PA. However, Gull et al. (2009) find that at
PA=$-$28°, observed six times from 1998.0 to 2004.3 ($\phi$=0.000 to 1.122),
red-shifted components extend to $+$200 km s-1 at early phases, but fade late
in the high state. All [Fe iii] $\lambda$4703 disappears during the low state.
The [N ii] $\lambda$5756 (Row 3) has very similar structure to that of [Fe
iii] $\lambda$4703 with higher S/N. A narrow emission line, [Fe ii]
$\lambda$5748, appears at the $-$550 km s-1 position and persists in the low
state, along with weak [N ii] $\lambda$5756\. The structure of [N ii]
$\lambda$5756 extends from $-$40 to $-500$ km s-1 in the spectro-image
centered on Weigelt D during the high state, but also disappears in the low
state.
Three [S iii] lines are shown in Rows 4$-$6 as each is important in accounting
for the 6307Å emission. Unfortunately, the [S iii] $\lambda$6313 (Row 4) was
recorded only at $\phi$=0.820, but the other two lines were observed at both
phases. Most noticeable in the spectro-image of $\eta$ Car at $\phi$=0.820 is
a knot of emission, centered on the stellar position at $-$400 km s-1. The
effective wavelength is 6307Å. That spectral interval was recorded at other
phases, but at other position angles, during the 2003.5 minimum with no [S
iii] $\lambda$6313 present either at the positions of $\eta$ Car or Weigelt D.
The extended structure is less apparent in the spectro-image of $\eta$ Car at
$\phi$=0.820, but is well-defined in the spectro-image centered on Weigelt D.
Narrow lines of [O i] $\lambda$6302, Fe ii $\lambda$$\lambda$6307, 6309 and
6319 contaminate the spectro-image and persist during the minimum.
The [S iii] $\lambda$ 9071 (Row 5) weak emission extends off of $\eta$ Car,
but several narrow lines (N i $\lambda$9063, Fe ii $\lambda$9073) also
contribute to the spectro-image. The high-velocity arc of [S iii] $\lambda$
9071 extends blueward from Weigelt D.
The [S iii] $\lambda$9533 emission (Row 6) is quite similar to that of [Fe ii]
$\lambda$4703 (Row 2) and confirms that [S iii] emission extends to $-$500 km
s-1. The bright emission to the red of [S iii] $\lambda$9533 is H i Pa 8
$\lambda$9548, which originates primarily from the central core.
## 5 Discussion
We associate the low-ionization structure with the massive, slow-moving wind
of $\eta$ Car A. The high-ionization emission is from the interacting wind
region piled up by the fast-moving, less-massive wind of $\eta$ Car B (Pittard
& Corcoran, 2002). The bulk of the interacting wind, by its velocity, appears
to be mostly ionized wind of $\eta$ Car A. The higher velocity side of the
shock is likely less dense and more highly ionized by $\eta$ Car B.
Using the 3D SPH models of Okazaki et al. (2008) and simple geometric models,
we determined that the high-ionization emission originates from a distorted
paraboloidal structure lying in the skirt of the Homunculus. Based upon the
blue-shifted velocities and near symmetry for PAs ranging from $+$22 to
$+$38°, the paraboloid points in our general direction with axis of rotation
projecting onto the sky at PA$\approx-$25°.
Martin et al. (2006) performed a very complete analysis on the unidentified
6307Å line in both the HST/STIS and VLT/UVES spectra, finding very similar
behavior with orbital phase. Their search of possible line identifications
focused primarily on singly-ionized species such as Fe+, V+ and S+, although
they do list [S iii] as a narrow nebular line identified by Zethson (2001) in
the spectrum of the Weigelt condensations. Their candidate of greatest
interest appeared to be Fe iii] $\lambda$6306.43 with unknown atomic data for
the transition.
Most important in their analysis was the tracking of the strength of the
emission throughout the 5.5-year orbit. They found that the line disappeared
during the low state, but might be anti-correlated with Fe ii $\lambda$5529\.
Both suggest a high-ionization source. Nielsen et al. (2007) analyzed the
behavior of the He i absorption, finding an anti-correlation with Fe ii
absorption.
Salient are three facts:
1. 1.
On the star, both HST/STIS and VLT/UVES see the same emission bump with
similar strengths.
2. 2.
The emission correlates with high-ionization variations, not the behavior of
the low-ionization emission of Fe ii.
3. 3.
The extended emission of the extended emission correlated with [S iii]
correlates very well with the extended emission identified with [S iii]
$\lambda\lambda$9071 and 9533.
## 6 Conclusions
We have presented conclusive evidence that the emission line at 6307Å, noted
in the spectra of $\eta$ Car by Martin et al. (2006) is blue-shifted emission
of [S iii] $\lambda$6313 originating from the distorted paraboloidal,
interaction region located between the massive binary members. While the
massive primary, $\eta$ Car A, provides the dominant wind ejecta, 10-3
$M_{\odot}$/y at 500 km s-1, the hotter secondary provides a less massive,
faster wind, 10-5 $M_{\odot}$/y at 3000 km s-1, and far-UV photons that ionize
iron, neon, argon and sulfur to doubly ionized states. Thermal collisions,
mid-UV photons and possible charge exchange excite the doubly-ionized species
to upper states with forbidden transitions leading to forbidden line emission
in regions with densities close to nc. Specifically, [S iii]
$\lambda$$\lambda$9533, 9071 and 6313 have extended spatial structures. The
intensity ratio (Flux ($\lambda$9533) + Flux
($\lambda$9071)/Flux($\lambda$6313) leads to density estimates ranging from
107 \- 108 cm-3on the scale of 0$\farcs$1, the limit of HST/STIS spatial
capabilities. Mapping in these and other doubly-ionized lines will provide
powerful measures for models of wind interactions using various 3-D
hydrodynamical codes.
The observations were accomplished with the NASA/ESA Hubble Space Telescope.
Support for Program numbers 7302, 8036, 8483, 8619, 9083, 9337, 9420, 9973,
10957 and 11273 was provided by NASA directly to the Space Telescope Imaging
Spectrograph Science Team and through grants from the Space Telescope Science
Institute, which is operated by the Association of Universities for Research
in Astronomy, Incorporated, under NASA contract NAS5-26555. All analysis was
done using STIS IDT software tools on data available through the HST $\eta$
Car Treasury public archive.
## References
* Corcoran (2005) Corcoran, M. F. 2005, AJ, 129, 2018
* Damineli et al. (2008) Damineli, A., Hillier, D. J., Corcoran, M. F., & et al. 2008, MNRAS, 384, 1649
* Davidson et al. (1995) Davidson, K., Ebbets, D., Weigelt, G., Humphreys, R. M., Hajian, A. R., Walborn, N. R., & Rosa, M. 1995, AJ, 109, 1784
* Gull et al. (2009) Gull, T. R., Nielsen, K. E., Corcoran, M. F., Madura, T. I., Owocki, S. P., Russell, C. M. P., Hillier, D. J., Hamaguchi, K., Kober, G. V., Weis, K., Stahl, O., & Okazaki, A. T. 2009, MNRAS, accepted
* Martin et al. (2006) Martin, J. C., Davidson, K., Humphreys, R. M., & et al. 2006, ApJ, 640, 474
* Nielsen et al. (2007) Nielsen, K. E., Corcoran, M. F., Gull, T. R., & et al. 2007, ApJ, 660, 669
* Okazaki et al. (2008) Okazaki, A. T., Owocki, S. P., Russell, C. M. P., & Corcoran, M. F. 2008, MNRAS, 388, L39
* Parkin et al. (2009) Parkin, E. R., Pittard, J. M., Corcoran, M. F., Hamaguchi, K., & Stevens, I. R. 2009, MNRAS, in press
* Pittard & Corcoran (2002) Pittard, J. M. & Corcoran, M. F. 2002, A&A, 383, 636
* Weigelt & Ebersberger (1986) Weigelt, G. & Ebersberger, J. 1986, A&A, 163, L5
* Zethson (2001) Zethson, T. 2001, PhD thesis, Lund University
|
arxiv-papers
| 2009-10-07T15:14:50 |
2024-09-04T02:49:05.720016
|
{
"license": "Public Domain",
"authors": "T. R. Gull",
"submitter": "Theodore Gull",
"url": "https://arxiv.org/abs/0910.1286"
}
|
0910.1623
|
# Modified Basis Pursuit Denoising(Modified-BPDN) for noisy compressive
sensing with partially known support
Wei Lu and Namrata Vaswani
Department of Electrical and Computer Engineering, Iowa State University,
Ames, IA
{luwei,namrata}@iastate.edu
###### Abstract
In this work, we study the problem of reconstructing a sparse signal from a
limited number of linear ‘incoherent’ noisy measurements, when a part of its
support is known. The known part of the support may be available from prior
knowledge or from the previous time instant (in applications requiring
recursive reconstruction of a time sequence of sparse signals, e.g. dynamic
MRI). We study a modification of Basis Pursuit Denoising (BPDN) and bound its
reconstruction error. A key feature of our work is that the bounds that we
obtain are computable. Hence, we are able to use Monte Carlo to study their
average behavior as the size of the unknown support increases. We also
demonstrate that when the unknown support size is small, modified-BPDN bounds
are much tighter than those for BPDN, and hold under much weaker sufficient
conditions (require fewer measurements).
###### Index Terms:
Compressive sensing, Sparse reconstruction
## I Introduction
In this work, we study the problem of reconstructing a sparse signal from a
limited number of linear ‘incoherent’ noisy measurements, when a part of its
support is known. In practical applications, this may be obtained from prior
knowledge, e.g. it can be the lowest subband of wavelet coefficients for
medical images which are sparse in the wavelet basis. Alternatively when
reconstructing time sequences of sparse signals, e.g. in a real-time dynamic
MRI application, it could be the support estimate from the previous time
instant.
In [3], we introduced modified-CS for the noiseless measurements’ case.
Sufficient conditions for exact reconstruction were derived and it was argued
that these are much weaker than those needed for CS. Modified-CS-residual,
which combines the modified-CS idea with CS on LS residual (LS-CS) [5], was
introduced for noisy measurements in [4] for a real-time dynamic MRI
reconstruction application. In this paper, we bound the recosntruction error
of a simpler special case of modified-CS-residual, which we call modified-
BPDN. We use a strategy similar to the results of [2] to bound the
reconstruction error and hence, just like in [2], the bounds we obtain are
computable. We are thus able to use Monte Carlo to study the average behavior
of the reconstruction error bound as the size of the unknown support,
$\Delta$, increases or as the size of the support itself, $N$, increases. We
also demonstrate that modified-BPDN bounds are much smaller than those for
BPDN (which corresponds to $|\Delta|=|N|$) and hold under much weaker
sufficient conditions (require fewer measurements).
In parallel and independent work recently posted on Arxiv, [7] also proposed
an approach related to modified-BPDN and bounded its error. Their bounds are
based on Candes’ results and hence are not computable. Other related work
includes [8] (which focusses on the time series case and mostly studies the
time-invariant support case) and [9] (studies the noiseless measurements’ case
and assumes probabilistic prior knowledge).
### I-A Problem definition
We obtain an $n$-length measurement vector $y$ by
$y=Ax+w$ (1)
Our problem is to reconstruct the $m$-length sparse signal $x$ from the
measurement $y$ with $m>n$. The measurement is obtained from an $n\times m$
measurement matrix $A$ and corrupted by a $n$-length vector noise $w$. The
support of $x$ denoted as $N$ consists of three parts: $N\triangleq
T\cup\Delta\setminus\Delta_{e}$ where $\Delta$ and $T$ are disjoint and
$\Delta_{e}\subseteq T$. $T$ is the known part of support while $\Delta_{e}$
is the error in the known part of support and $\Delta$ is the unknown part. We
also define $N_{e}\triangleq T\cup\Delta=N\cup\Delta_{e}$.
Notation: We use ′ for conjugate transpose. For any set $T$ and vector $b$, we
have $(b)_{T}$ to denote a sub-vector containing the elements of $b$ with
indices in $T$. $\|b\|_{k}$ means the $l_{k}$ norm of the vector $b$. $T^{c}$
denotes the complement of set $T$ and $\emptyset$ is the empty set. For the
matrix $A$, $A_{T}$ denotes the sub-matrix by extracting columns of $A$ with
indices in $T$. The matrix norm $\|A\|_{p}$, is defined as
$\|A\|_{p}\triangleq\max_{x\neq 0}\frac{\|Ax\|_{p}}{\|x\|_{p}}$
We also define $\delta_{S}$ to be the $S$-restricted isometry constant and
$\theta_{S,S^{\prime}}$ to be the $S,S^{\prime}$ restricted orthogonality
constant as in [6].
## II Bounding modified-BPDN
In this section, we introduce modified-BPDN and derive the bound for its
reconstruction error.
### II-A Modified-BPDN
In [3], equation (5) gives the modified-CS algorithm under noiseless
measurements. We relax the equality constraint of this equation to propose
modified-BPDN algorithm using a modification of the BPDN idea[1]. We solve
$\min_{b}\ \frac{1}{2}\|y-Ab\|_{2}^{2}+\gamma\|b_{T^{c}}\|_{1}$ (2)
Then the solution to this convex optimization problem $\hat{x}$ will be the
reconstructed signal of the problem. In the following two subsections, we
bound the reconstruction error.
### II-B Bound of reconstruction error
We now bound the reconstruction error. We use a strategy similar to [2]. We
define the function
$\ \ L(b)=\frac{1}{2}\|y-Ab\|_{2}^{2}+\gamma\|b_{T^{c}}\|_{1}$ (3)
Look at the solution of the problem (2) over all vectors supported on $N_{e}$.
If $A_{N_{e}}$ has full column rank, the function $L(b)$ is strictly convex
when minimizing it over all $b$ supported on $N_{e}$ and then it will have a
unique minimizer. We denote the unique minimizer of function $L(b)$ over all
$b$ supported on $N_{e}$ as
$\tilde{b}=[\tilde{b}^{\prime}_{N_{e}}\ \ \textbf{0}^{\prime}_{N_{e}^{c}}]$
(4)
Also, we denote the genie-aided least square estimate supported on $N_{e}$ as
$c:=[c^{\prime}_{N_{e}}\ \ \textbf{0}^{\prime}_{N_{e}^{c}}]\text{ where
}c_{N_{e}}:=(A^{\prime}_{N_{e}}A_{N_{e}})^{-1}A^{\prime}_{N_{e}}y$ (5)
Since $\|c-x\|_{2}\leq\frac{\|w\|}{\sqrt{1-\delta_{|N_{e}|}}}$ is quite small
if noise is small and $\delta_{|N_{e}|}$ is small, we just give the error
bound for $\tilde{b}$ with respect to $c$ in the following lemma and will
prove that it is also the global unique minimizer under some sufficient
condition.
###### Lemma 1
Suppose that $A_{N_{e}}$ has full column rank, and let $\tilde{b}$ minimize
the function $L(b)$ over all vectors supported on $N_{e}$. We have the
following conclusions:
1. 1.
A necessary and sufficient condition for $\tilde{b}$ to be the unique
minimizer is that
$c_{N_{e}}-\tilde{b}_{N_{e}}=\left[\begin{array}[]{c}-\gamma(A_{T}^{\prime}A_{T})^{-1}A^{\prime}_{T}A_{\Delta}(A^{\prime}_{\Delta}MA_{\Delta})^{-1}g_{\Delta}\\\
\gamma(A^{\prime}_{\Delta}MA_{\Delta})^{-1}g_{\Delta})\end{array}\right]$
where $M\triangleq I-A_{T}(A^{\prime}_{T}A_{T})^{-1}A^{\prime}_{T}$ and
$g\in\partial(\|b_{T^{c}}\|_{1})|_{b=\tilde{b}}$.
$\partial(\|b_{T^{c}}\|_{1})$ is the subgradient set of $\|b_{T^{c}}\|_{1}$.
Thus, $g_{T}=0$ and $\|g_{\Delta}\|_{\infty}=1$.
2. 2.
Error bound in $l_{\infty}$ norm
$\displaystyle\|\tilde{b}-c\|_{\infty}\leq\gamma\max(\|(A^{\prime}_{T}A_{T})^{-1}A^{\prime}_{T}A_{\Delta}(A^{\prime}_{\Delta}MA_{\Delta})^{-1}\|_{\infty}$
$\displaystyle,\|(A^{\prime}_{\Delta}MA_{\Delta})^{-1}\|_{\infty})\hskip
5.69054pt$ (6)
3. 3.
Error bound in $l_{2}$ norm
$\displaystyle\|\tilde{b}-c\|_{2}\leq\gamma\sqrt{|\Delta|}\cdot\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$
$\displaystyle\sqrt{\|(A_{T}^{\prime}A_{T})^{-1}A_{T}^{\prime}A_{\Delta}(A_{\Delta}^{\prime}MA_{\Delta}^{-1})\|_{2}^{2}+\|(A_{\Delta}^{\prime}MA_{\Delta})^{-1}\|_{2}^{2}}$
$\displaystyle\leq\gamma\sqrt{|\Delta|}\sqrt{\frac{\theta_{|T|,|\Delta|}^{2}}{(1-\delta_{|T|})^{2}}+1}\cdot\frac{1}{1-\delta_{|\Delta|}-\frac{\theta_{|\Delta|,|T|}^{2}}{1-\delta_{|T|}}}$
The proof is given in the Appendix.
Next, we obtain sufficient condition under which $\tilde{b}$ is also the
unique global minimizer of $L(b)$.
###### Lemma 2
If the following condition is satisfied, then the problem (2) has a unique
minimizer which is equal to $\tilde{b}$ defined in (4).
$\|A^{\prime}(y-A_{N_{e}}c_{N_{e}})\|_{\infty}<\gamma\big{[}1-\max_{\omega\notin
N_{e}}\|(A^{\prime}_{\Delta}MA_{\Delta})^{-1}A^{\prime}_{\Delta}MA_{\omega}\|_{1}\big{]}$
The proof of Lemma 2 is in the appendix.
Combining Lemma 1 and 2 and bounding $\|c-x\|$,we get the following Theorem:
###### Theorem 1
If $A_{N_{e}}$ has full column rank and the following condition is satisfied
$\|A^{\prime}(y-A_{N_{e}}c_{N_{e}})\|_{\infty}<\gamma\big{[}1-\max_{\omega\notin
N_{e}}\|(A^{\prime}_{\Delta}MA_{\Delta})^{-1}A^{\prime}_{\Delta}MA_{\omega}\|_{1}\big{]}$
(7)
then,
1. 1.
Problem (2) has a unique minimizer $\tilde{b}$ and it is supported on $N_{e}$.
2. 2.
The unique minimizer $\tilde{b}$ satisfies
$\displaystyle\|\tilde{b}-x\|_{\infty}\leq\gamma\max(\|(A^{\prime}_{T}A_{T})^{-1}A^{\prime}_{T}A_{\Delta}(A^{\prime}_{\Delta}MA_{\Delta})^{-1}\|_{\infty}$
$\displaystyle,\|(A^{\prime}_{\Delta}MA_{\Delta})^{-1}\|_{\infty})+\|(A_{N_{e}}^{\prime}A_{N_{e}})^{-1}A_{N_{e}}^{\prime}\|_{\infty}\|w\|_{\infty}$
(8)
and
$\displaystyle\|\tilde{b}-x\|_{2}\leq\|(A_{N_{e}}^{\prime}A_{N_{e}})^{-1}A_{N_{e}}^{\prime}\|_{2}\|w||_{2}+\gamma\sqrt{|\Delta|}\cdot$
$\displaystyle\sqrt{\|(A_{T}^{\prime}A_{T})^{-1}A_{T}^{\prime}A_{\Delta}(A_{\Delta}^{\prime}MA_{\Delta}^{-1})\|_{2}^{2}+\|(A_{\Delta}^{\prime}MA_{\Delta})^{-1}\|_{2}^{2}}$
(9)
$\displaystyle\leq\gamma\sqrt{|\Delta|}\sqrt{\frac{\theta_{|T|,|\Delta|}^{2}}{(1-\delta_{|T|})^{2}}+1}\cdot\frac{1}{1-\delta_{|\Delta|}-\frac{\theta_{|\Delta|,|T|}^{2}}{1-\delta_{|T|}}}+\frac{\|w\|_{2}}{\sqrt{1-\delta_{|N_{e}|}}}$
(10)
Now consider BPDN. From theorem 8 of [2](the same thing also follows by
setting $T=\emptyset$ in our result), if $A_{N}$ has full rank and if
$\|A^{\prime}(y-A_{N}(A_{N}^{\prime}A_{N})^{-1}A_{N}^{\prime}y)\|_{\infty}<\gamma[1-\max_{\omega\notin
N}\|(A_{N}^{\prime}A_{N})^{-1}A_{N}^{\prime}A_{\omega}\|_{1}]$ (11)
then $\tilde{b}_{BPDN}$
$\|\tilde{b}_{BPDN}-x\|_{\infty}\leq\gamma\|(A_{N}^{\prime}A_{N})^{-1}\|_{\infty}+\|(A_{N}^{\prime}A_{N})^{-1}A_{N}^{\prime}\|_{\infty}\|w\|_{\infty}$
(12)
Similarly, we can have the $l_{2}$ norm bound of BPDN is
$\|\tilde{b}_{BPDN}-x\|_{2}\leq\gamma\sqrt{|N|}\frac{1}{1-\delta_{|N|}}+\frac{\|w\|_{2}}{\sqrt{1-\delta_{|N|}}}$
(13)
Compare (10) and (13) for the case when
$|\Delta|=|\Delta_{e}|=\frac{|N|}{10}$(follows from [4]), the second terms are
mostly equal. Consider an example assuming that $\delta_{|N|}=0.5$,
$\delta_{|\Delta|}=0.1$, $\theta_{|T|,|\Delta|}=0.2$ and
$|\Delta|=\frac{1}{10}|N|$ which is practical in real data. Then the bound for
BPDN is $2\gamma_{BPDN}\sqrt{|N|}+0.7||w||_{2}$ and the bound for modified-
BPDN approximates to $1.3\gamma_{modBPDN}|\Delta|+0.7||w||_{2}$. Using a
similar argument, $\gamma_{modBPDN}$ which is the smallest $\gamma$ satisfying
(7), will be smaller than $\gamma_{BPDN}$ which is the smallest $\gamma$
satisfying (11). Since $|\Delta|=\frac{1}{10}|N|$ and $\gamma_{BPDN}$ will be
larger than $\gamma_{modBPDN}$, the bound for modified-BPDN will be much
smaller than that of BPDN. This is one example, but we do a detailed
simulation comparison in the next section using the computable version of the
bounds given in (8) and (9).
## III Simulation Results
In this section, we compare both the computable $l_{\infty}$ and $l_{2}$ norm
bounds for modified-BPDN with those of BPDN using Monte Carlo simulation. Note
that, BPDN is a special case of modified-BPDN when $\Delta=N$ and
$\Delta_{e}=\emptyset$. Therefore, we do the following simulation to check the
change of error bound when $|\Delta|$ increases and compare the bounds of
modified-BPDN with those of BPDN.
We do the simulation as follows:
1. 1.
Fix $m=1024$ and size of support $|N|$.
2. 2.
Select $n$, $|\Delta|$ and $|\Delta_{e}|$.
3. 3.
Generate the $n\times m$ random-Gaussian matrix, $A$ (generate an $n\times m$
matrix with i.i.d. zero mean Gaussian entries and normalize each column to
unit $\ell_{2}$ norm).
4. 4.
Repeat the following $\text{tot}=50$ times
1. (a)
Generate the support, $N$, of size $|N|$, uniformly at random from $[1:m]$.
2. (b)
Generate the nonzero elements of the sparse signal $x$ on the support $N$ with
i.i.d Gaussian distributed entries with zero mean and variance 100. Then
generate a random i.i.d Gaussian noise $w$ with zero mean and variance
$\sigma_{w}^{2}$. Compute $y:=Ax+w$.
3. (c)
Generate the unknown part of support, $\Delta$, of size $|\Delta|$ uniformly
at random from the elements of $N$.
4. (d)
Generate the error in known part of support, $\Delta_{e}$, of size
$|\Delta_{e}|$, uniformly at random from $[1:m]\setminus N$
5. (e)
Use $T=N\cup\Delta_{e}\setminus\Delta$ to compute $\gamma^{*}$ by
$\gamma^{*}=\frac{\|A^{\prime}(y-A_{N_{e}}c_{N_{e}})\|_{\infty}}{1-\max_{\omega\notin
N_{e}}\|(A^{\prime}_{\Delta}MA_{\Delta})^{-1}A^{\prime}_{\Delta}MA_{\omega}\|_{1}}$
and do reconstruction with $\gamma=\gamma^{*}$ using modified-BPDN to obtain
$\hat{x}_{modBPDN}$.
6. (f)
Compute the reconstruction error $\|\hat{x}_{modBPDN}-c\|_{\infty}$
7. (g)
Compute the $l_{\infty}$ norm bound from (6) and the $l_{2}$ norm bound from
(9).
5. 5.
Compute the average bounds and average error for the given $n$, $|\Delta|$,
$|\Delta_{e}|$.
6. 6.
Repeat for various values of $n$,$|\Delta|$ and $|\Delta_{e}|$.
Fig.1 shows the average bound(RHS of (9)) for different $|\Delta|$ when
$|N|=100\approx 10\%m$ which is practical for real data as in [3, 4]. The
noise variance is $\sigma_{w}^{2}=0.001$. We show plots for different choice
of $n$. The case $\frac{|\Delta|}{|N|}=1$ in Fig. 1 corresponds to BPDN. From
the figures, we can observe that when $|\Delta|$ increases, the bounds are
increasing. One thing needed to be mentioned is that for
BPDN($\Delta=N,\Delta_{e}=\emptyset$) in this case, the RHS of (7) is negative
and the bound can only hold when number of measurements $n\geq 0.95m$.
Therefore, BPDN is difficult to meet the unique minimizer condition when $|N|$
increases to $0.1m$. However, when $|\Delta|$ is small, modified-BPDN can
easily satisfy the condition, even with very few measurements($n=0.2m$ when
$|\Delta|=0.05|N|$). Hence, the sufficient conditions for modified-BPDN
require much fewer measurements than those for BPDN when $|\Delta|$ is small.
(a) $|N|=100,\Delta_{e}=\emptyset$
(b) $|N|=100,|\Delta_{e}|=\frac{1}{10}|N|$
Figure 1: The average bound(9) on $||\tilde{b}-x||_{2}$ is plotted. Signal
length $m=1024$ and support size $|N|=100$. For fixed $n$ and $|\Delta_{e}|$,
the bound increases when $|\Delta|$ increases. When number of measurements $n$
increases, the bound decreases. When $n=0.2m$ and $|\Delta|\geq 0.05|N|$, the
RHS of (7) is negative and thus the bound does not hold. We do not plot the
case of BPDN($\Delta=N,\Delta_{e}=\emptyset$) since it requires $n\geq 0.95m$
measurements to make RHS of (7) positive.
Fig.2 gives another group of results showing average bound(RHS of (9)) for
different $|\Delta|$ when $|N|=15\approx 1.5\%m$. The noise variance is
$\sigma_{w}^{2}=0.0003$ and $\Delta_{e}=\emptyset$. We can also obtain the
same conclusions as Fig.1. Note that we do not plot the average error and
bound for $|\Delta|\geq\frac{2}{3}|N|$ when $n=0.2m$ since the RHS of (7) is
negative and thus the bound does not hold. Hence, the more we know the
support, the fewer measurements modified-BPDN requires.
In this case, we also compute the average error and the bound (6) on
$\|\tilde{b}-c\|_{\infty}$. Since $|N_{e}|=15$ is small and noise is small
$\|c-x\|_{\infty}$ will be small and equal for any choice of $|\Delta|$. Thus
we just compare $\|\tilde{b}-c\|_{\infty}$ with its upper bound given in (6).
For the error and bound on $\|\tilde{b}-c\|_{\infty}$, when we fix $n=0.3m$
and $\Delta_{e}=\emptyset$, the error and the bound are both 0 for
$|\Delta|=0$ which verifies that the unique minimizer is equal to the genie-
aided least square estimation on support $N$ in this case. For
$|\Delta|=\frac{1}{3}|N|$, the error is $0.08$ and the bound is $0.09$. For
$|\Delta|=\frac{2}{3}|N|$, the error is $0.21$ and the bound is $0.27$. When
$|\Delta|=|N|$ which corresponds to BPDN in this case, the error increases to
$3.3$ and the bound increases to $9$. Therefore, we can observe that when
$|\Delta|$ increases, both the error and the bound are increasing. Also, we
can see the gap between error and bound(gap=bound-error) increases with
$|\Delta|$.
Figure 2: The average bound(9) on $||\tilde{b}-x||_{2}$ is plotted. Signal
length $m=1024$, support size $|N|=15$ and $|\Delta_{e}|=0$. For fixed $n$,
the bound on $||\tilde{b}-x||_{2}$ increases when $|\Delta|$ increases. When
number of measurements $n$ increases, the bound decreases. When $n=0.2m$ and
$|\Delta|\geq\frac{2}{3}|N|$, the RHS of (7) is negative and thus the bound
does not hold.
From the simulation results, we conclude as follows:
1. 1.
The error and bound increase as $|\Delta|$ increases.
2. 2.
The error and bound increase as $|N|$ increases.
3. 3.
The gap between the error and bound increases as $|\Delta|$ increases.
4. 4.
The error and bound decrease as $n$ increases.
5. 5.
For real data, $|N|\approx 0.1m$. In this case, BPDN needs $n\geq 0.95m$ to
apply the bound while modified-BPDN can much easily to apply its bound under
very small $n$.
6. 6.
When $n$ is large enough, e.g. $n=0.5m$ for $|N|=15=15\%m$, the bounds are
almost equal for all values of $|\Delta|$ (the black plot of Fig. 2) including
$|\Delta|=|N|$ (BPDN).
## IV Conclusions
We proposed a modification of the BPDN idea, called modified-BPDN, for sparse
reconstruction from noisy measurements when a part of the support is known,
and bounded its reconstruction error. A key feature of our work is that the
bounds that we obtain are computable. Hence we are able to use Monte Carlo to
show that the average value of the bound increases as the unknown support size
or the size of the error in the known support increases. We are also able to
compare with the BPDN bound and show that (a) for practical support sizes
(equal to 10% of signal size it holds under very strong assumptions (require
more than 95% random Gaussian measurements for the bound to hold) and (b) for
smaller support sizes (e.g. 1.5% of signal size), the BPDN bound is much
larger than the modified-BPDN bound.
## V appendix
### V-A Proof of Lemma 1
Suppose $\text{supp(b)}\subseteq N_{e}$. We know the vectors
$y-Ac=y-A_{N_{e}}c_{N_{e}}$ and $Ac-Ab=A_{N_{e}}(b_{N_{e}}-c_{N_{e}})$ are
orthogonal because $A_{N_{e}}^{\prime}(y-A_{N_{e}}c_{N_{e}})=0$ using (5).
Thus we minimize function $L(b)$ over all vectors supported on set $N_{e}$ by
minimizing:
$F(b)=\frac{1}{2}\|A_{N_{e}}c_{N_{e}}-A_{N_{e}}b_{N_{e}}\|_{2}^{2}+\gamma\|b_{T^{c}}\|_{1}$
(14)
Since this function is strictly convex, then $0\in\partial F(\tilde{b})$.
Hence,
$A^{\prime}_{N_{e}}A_{N_{e}}\tilde{b}_{N_{e}}-A^{\prime}_{N_{e}}A_{N_{e}}c_{N_{e}}+\gamma
g_{N_{e}}=0$ (15)
Then, we have
$c_{N_{e}}-\tilde{b}_{N_{e}}=\gamma(A^{\prime}_{N_{e}}A_{N_{e}})^{-1}g_{N_{e}}$
(16)
Since
$A^{\prime}_{N_{e}}A_{N_{e}}=\left[\begin{array}[]{cc}A^{\prime}_{T}A_{T}&A^{\prime}_{T}A_{\Delta}\\\
A^{\prime}_{\Delta}A_{T}&A^{\prime}_{\Delta}A_{\Delta}\end{array}\right]$
By using the block matrix inversion and $g_{T}=0$, we get
$c_{N_{e}}-\tilde{b}_{N_{e}}=\left[\begin{array}[]{c}-\gamma(A^{\prime}_{T}A_{T})^{-1}A^{\prime}_{T}A_{\Delta}(A^{\prime}_{\Delta}MA_{\Delta})^{-1}g_{\Delta}\\\
\gamma(A^{\prime}_{\Delta}MA_{\Delta})^{-1}g_{\Delta})\end{array}\right]$
Thus, we can obtain the $l_{\infty}$ norm bound of error as below:
$\displaystyle\|\tilde{b}_{N_{e}}-c_{N_{e}}\|_{\infty}=\gamma\|(A^{\prime}_{N_{e}}A_{N_{e}})^{-1}g_{N_{e}}\|_{\infty}$
$\displaystyle\leq$
$\displaystyle\gamma\max(\|(A^{\prime}_{T}A_{T})^{-1}A^{\prime}_{T}A_{\Delta}(A^{\prime}_{\Delta}MA_{\Delta})^{-1}g_{\Delta}\|_{\infty},$
$\displaystyle\quad\quad\quad\quad\quad\|(A^{\prime}_{\Delta}MA_{\Delta})^{-1}g_{\Delta}\|_{\infty})$
$\displaystyle\leq$
$\displaystyle\gamma\max(\|(A^{\prime}_{T}A_{T})^{-1}A^{\prime}_{T}A_{\Delta}(A^{\prime}_{\Delta}MA_{\Delta})^{-1}\|_{\infty},$
$\displaystyle\quad\quad\quad\quad\quad\|(A^{\prime}_{\Delta}MA_{\Delta})^{-1}\|_{\infty})$
This follows using $\|g_{\Delta}\|_{\infty}=1$. Also, using
$\|g_{\Delta}\|_{2}\leq\sqrt{|\Delta|}$, we get the $l_{2}$ norm bound of
$\tilde{b}-c$.
Using $\|(A_{T}^{\prime}A_{T})^{-1}\|_{2}\leq\frac{1}{1-\delta_{|T|}}$,
$\|A_{\Delta}^{\prime}A_{\Delta}\|_{2}\geq 1-\delta_{|\Delta|}$ and
$\|A_{T}^{\prime}A_{\Delta}\|_{2}\leq\theta_{|T|,|\Delta|}$, we get (10).
### V-B Proof of Lemma 2
Suppose that $A_{N_{e}}$ has full column rank, and let $\tilde{b}$ minimize
the function $L(b)$ over all $b$ supported on $N_{e}=T\cup\Delta$. We need to
prove under this condition, $\tilde{b}$ is the unique global minimizer of
$L(b)$.
The idea is to prove under the given condition, any small perturbation $h$ on
$\tilde{b}$ will increase function $L(\tilde{b})$,i.e.
$L(\tilde{b}+h)-L(\tilde{b})>0,\forall||h||_{\infty}\leq\delta$ for $\delta$
small enough. Since $L(b)$ is a convex function, $\tilde{b}$ should be the
unique global minimizer.
Similar to [2], we first split the perturbation into two parts $h=u+v$ where
$supp(u)=N_{e}$ and $supp(v)=N_{e}^{c}$. Clearly
$||u||_{\infty}\leq||h||_{\infty}\leq\delta$. Then we have
$L(\tilde{b}+h)=\frac{1}{2}||y-A(\tilde{b}+u)-Av||_{2}^{2}+\gamma||(\tilde{b}+u)_{T^{c}}+v_{T^{c}}||_{1}$
(17)
Then expand the first term, we can obtain
$\displaystyle\|y-A(\tilde{b}+u)-Av\|_{2}^{2}=\|y-A(\tilde{b}+u)\|_{2}^{2}+\|Av\|_{2}^{2}$
$\displaystyle-2Re\langle y-A\tilde{b},Av\rangle+2Re\langle Au,Av\rangle$ (18)
The second term of (17) becomes
$\|(\tilde{b}+u)_{T^{c}}+v_{T^{c}}\|_{1}=\|(\tilde{b}+u)_{T^{c}}\|_{1}+\|v_{T^{c}}\|_{1}$
(19)
Then we have
$\displaystyle
L(\tilde{b}+h)-L(\tilde{b})=L(\tilde{b}+u)-L(\tilde{b})+\frac{1}{2}\|Av\|_{2}^{2}$
$\displaystyle-Re\langle y-A\tilde{b},Av\rangle+Re\langle
Au,Av\rangle+\gamma\|v_{T^{c}}\|_{1}$ (20)
Since $\tilde{b}$ minimizes $L(b)$ over all vectors supported on $N_{e}$,
$L(\tilde{b}+u)-L(\tilde{b})\geq 0$. Then since
$L(\tilde{b}+u)-L(\tilde{b})\geq 0$ and $\|Av\|_{2}^{2}\geq 0$, we need to
prove that the rest are non-negative:$\gamma\|v_{T^{c}}\|_{1}-Re\langle
y-A\tilde{b},Av\rangle+Re\langle Au,Av\rangle\geq 0$. Instead, we can prove
this by proving a stronger one $\gamma\|v_{T^{c}}\|_{1}-|\langle
y-A\tilde{b},Av\rangle|-|\langle Au,Av\rangle|\geq 0$.
Since $\langle y-A\tilde{b},Av\rangle=v^{\prime}A^{\prime}(y-A\tilde{b})$ and
$supp(v)=N_{e}^{c}$,
$|\langle
y-A\tilde{b},Av\rangle|=|v_{N_{e}^{c}}^{\prime}A_{N_{e}^{c}}^{\prime}(y-A\tilde{b})|\leq\|v\|_{1}\|A_{N_{e}^{c}}(y-A\tilde{b})\|_{\infty}$
Thus,
$\displaystyle|\langle y-A\tilde{b},Av\rangle|\leq\max_{\omega\notin
N_{e}}|\langle y-A\tilde{b},A_{\omega}\rangle|||v||_{1}$ (21)
The third term of (17) can be written as
$|\langle
Au,Av\rangle|\leq\|A^{\prime}Au\|_{\infty}||v||_{1}\leq\delta\|A^{\prime}A\|_{\infty}||v||_{1}$
(22)
And $\|v\|_{1}=\|v_{T^{c}}\|_{1}$ since $supp(v)=N_{e}^{c}\subseteq T^{c}$.
Therefore,
$L(\tilde{b}+h)-L(\tilde{b})\geq\big{[}\gamma-\max_{\omega\notin
N_{e}}|\langle
y-A\tilde{b},A_{\omega}\rangle|-\delta||A^{\prime}A||_{\infty}\big{]}||v||_{1}$
(23)
Since we can select $\delta>0$ as small as possible, then we just need to have
$\gamma-\max_{\omega\notin N_{e}}|\langle y-A\tilde{b},A_{\omega}\rangle|>0$
(24)
Invoke Lemma 1, we have $A_{N_{e}}(c_{N_{e}}-\tilde{b}_{N_{e}})=\gamma
MA_{\Delta}(A^{\prime}_{\Delta}MA_{\Delta})^{-1}g_{\Delta}$. Since
$y-A\tilde{b}=(y-A_{N_{e}}c_{N_{e}})+A_{N_{e}}(c_{N_{e}}-\tilde{b}_{N_{e}})$,
therefore,
$\displaystyle|\langle y-A\tilde{b},A_{\omega}\rangle|\leq|\langle
y-A_{N_{e}}c_{N_{e}},A_{\omega}\rangle|\quad\quad$
$\displaystyle+\gamma|\langle(A^{\prime}_{\Delta}MA_{\Delta})^{-1}A^{\prime}_{\Delta}MA_{\omega},g_{\Delta}\rangle|$
(25)
Then we only need to have the condition
$\displaystyle\gamma-\max_{\omega\notin
N_{e}}\big{[}\gamma|\langle(A^{\prime}_{\Delta}MA_{\Delta})^{-1}A^{\prime}_{\Delta}MA_{\omega},g_{\Delta}\rangle|+$
$\displaystyle|\langle y-A_{N_{e}}c_{N_{e}},A_{\omega}\rangle|\big{]}>0$ (26)
Since $y-A_{N_{e}}c_{N_{e}}$ is orthogonal to $A_{w}$ for each $\omega\in
N_{e}$, then $\max_{\omega\notin N_{e}}|\langle
y-A_{N_{e}}c_{N_{e}},A_{\omega}\rangle|=\|A^{\prime}(y-A_{N_{e}}c_{N_{e}})\|_{\infty}$.
Also, we know that $\max_{\omega\notin
N_{e}}|\langle(A^{\prime}_{\Delta}MA_{\Delta})^{-1}A^{\prime}_{\Delta}MA_{\omega},g_{\Delta}\rangle|\leq\max_{\omega\notin
N_{e}}\|(A^{\prime}_{\Delta}MA_{\Delta})^{-1}A^{\prime}_{\Delta}MA_{\omega}\|_{1}\big{]}$.
Thus, (26) holds if the following condition holds
$||A^{\prime}(y-A_{N_{e}}c_{N_{e}})||_{\infty}<\gamma\big{[}1-\max_{\omega\notin
N_{e}}||(A^{\prime}_{\Delta}MA_{\Delta})^{-1}A^{\prime}_{\Delta}MA_{\omega}||_{1}\big{]}$
(27)
i.e. $\tilde{b}$ is the unique global minimizer if (27) holds.
## References
* [1] S. S. Chen, D. L. Donoho, and M. A. Saunders,Atomic decomposition by basis pursuit, SIAM J. Sci. Comput., vol. 20, no. 1, pp. 33-61, 1999.
* [2] Joel A. Tropp, Just Relax: Convex Programming Methods for Identifying Sparse Signals in Noise, IEEE Trans. on Information Theory, 52(3), pp. 1030 - 1051, March 2006.
* [3] Namrata Vaswani and Wei Lu, Modified-CS: Modifying Compressive Sensing for Problems with Partially Known Support, IEEE Intl. Symp. Info. Theory (ISIT), 2009
* [4] Wei Lu and Namrata Vaswani,Modified Compressive Sensing for Real-time Dynamic MR Imaging, IEEE Intl. Conf. Image Proc (ICIP), 2009
* [5] Namrata Vaswani,Analyzing Least Squares and Kalman Filtered Compressed Sensing, IEEE Intl. Conf. Acous. Speech. Sig. Proc. (ICASSP), 2009.
* [6] E. Candes and T. Tao. Decoding by Linear Programming, IEEE Trans. Info. Th., 51(12):4203 - 4215, Dec. 2005.
* [7] L. Jacques, A short Note on Compressed Sensing with Partially Known Signal Support, Arxiv preprint arXiv:0908.0660v1, 2009.
* [8] D. Angelosante, E. Grossi, G. B. Giannakis,Compressed Sensing of time-varying signals, DSP 2009
* [9] A. Khajehnejad, W. Xu, A. Avestimehr, B. Hassibi, Weighted l1 Minimization for Sparse Recovery with Prior Information, IEEE Intl. Symp. Info. Theory(ISIT),2009
|
arxiv-papers
| 2009-10-08T22:00:19 |
2024-09-04T02:49:05.730339
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Wei Lu, Namrata Vaswani",
"submitter": "Wei Lu",
"url": "https://arxiv.org/abs/0910.1623"
}
|
0910.1692
|
∎
11institutetext: M. Beckett and C. Maynard 22institutetext: The University of
Edinburgh, Edinburgh, United Kingdom
Tel.: +44-131-650-5030
Fax: +44-131-650-6555
22email: george.beckett@ed.ac.uk c.maynard@ed.ac.uk 33institutetext: B. Joo
44institutetext: Scientific Computing Group, Jefferson Lab,
12000 Jefferson Avenue, Newport News,
VA 23606, U.S.A 44email: bjoo@jlab.org 55institutetext: D. Pleiter
66institutetext: Deutsches Elektronen-Synchrotron DESY, 15738 Zeuthen, Germany
Tel.: +49-33762-77-381
Fax: +49-33762-77-216
66email: dirk.pleiter@desy.de 77institutetext: T. Yoshie 88institutetext:
Center for Computational Sciences, University of Tsukuba, Tsukuba 305-8577,
Japan
Tel.: +81-29-853-6492
Fax: +81-29-853-6406
88email: yoshie@ccs.tsukuba.ac.jp
# Building the International Lattice Data Grid
Mark G. Beckett Bálint Joó Chris M. Maynard Dirk Pleiter Osamu Tatebe
Tomoteru Yoshie
(Received: date / Accepted: date)
###### Abstract
We present the International Lattice Data Grid (ILDG), a loosely federated
grid of grids for sharing data from Lattice Quantum Chromodynamics (LQCD)
simulations. The ILDG comprises of metadata, file format and web-service
standards, which can be used to wrap regional data-grid interfaces, allowing
seamless access to catalogues and data in a diverse set of collaborating
regional grids. We discuss the technological underpinnings of the ILDG,
primarily the metadata and the middleware, and offer a critique of its various
aspects with the hindsight of the design work and the first full year of
production.
###### Keywords:
ILDG data grids lattice QCD
††journal: Journal of Grid Computing
## 1 Introduction
In this paper, we present the International Lattice Data Grid (ILDG). The ILDG
project is a mostly volunteer effort within the Lattice Quantum Chromodynamics
(LQCD) community, to share data worldwide, and to thus amortise the very high
computational cost of producing the data. In terms of organisation it is a
data-grid, but it is also a loosely federated grid of grids.
Large data sets require significant scientific endeavour to amass them. This
may represent intellectual property, as well as physical resources. In the
case of LQCD, the resources are both intellectual – such as the scientific
ideas and algorithmic development – as well as other resources, such as the
manpower required to write the computer code and the resources to
procure/develop and operate a large supercomputer. Why then do scientists wish
to share this valuable data? It is precisely because this data is so valuable
that scientists make it available for others to use. A mechanism is required
whereby those who generate shared data can receive credit for doing so.
For the LQCD community there are two compelling reasons to share data. First,
fully exploiting the data requires computing and manpower resource. A
particular group may generate a dataset to compute a target physical quantity
with sufficient precision to have an impact on experimental results, and yet
not have sufficient resources or even the expertise to calculate many other
possible quantities on that dataset. At this stage, rather than waste some of
the scientific potential of the data, a group may give the data away freely
provided some basic use conditions are met such as citing a certain paper in
any resulting publication. Second, the resources required to generate ever
more potent data sets require ever greater resources, outstripping Moore’s Law
and scientific innovation. This forces different groups to collaborate:
jointly baring the cost of data generation.
Quantum Chromodynamics (QCD) is a theory of sub-atomic particles
(specifically, quarks and gluons) and their interactions. Lattice QCD (LQCD)
is a version of QCD where space-time is discretized, making the theory
amenable to calculation by computers. LQCD computations are of utility in a
variety of theoretical particle physics contexts including Nuclear Physics and
High Energy Particle Physics, and have historically consumed a large fraction
of available computing cycles worldwide. The interested reader can find
several excellent books and review articles on LQCD in the literature, for
example Creutz:1984mg ; Montvay:1994cy and Gupta:1997nd .
LQCD Computations are based on Markov Chain Monte Carlo methods (see
Peardon:2003fv for a recent review) and typically the primary data from such
calculations are samples of the QCD vacuum known as gauge configurations. The
Monte Carlo process will generate an ensemble of such configurations for each
set of physical and algorithmic input parameters. At the time of writing, the
typical cost of generating an ensemble is $O(1)-O(10)$ Teraflop years
depending on the precise formulation employed, and this cost is expected to
grow to the Exaflop-year scale as one simulates lattices with finer lattice
spacings, larger physical volumes and physically light quarks.
A set of ensembles is amenable to many different type of secondary analysis.
One can, for example, perform calculations of nuclear structure on the same
configurations one also uses to perform calculations of fundamental parameters
of the Standard Model of Particle Interactions. Alternatively, an ensemble
generated to measure Nuclear Energy spectra may also be useful in the study of
the nuclear strong force binding together nucleons into atomic nuclei.
Since the generation of ensembles is very demanding in terms of effort, and
since the ensembles can facilitate multiple uses, it makes sense to share them
amongst the LQCD community to get maximum value out of a particular generation
project. The ILDG infrastructure discussed in this paper, is designed to
promote and facilitate such data sharing.
The LQCD community has a history of sharing data before the formation of the
ILDG. The MILC collaboration MILC has pioneered the approach of freely giving
away the data, after publishing results for their target quantities. This
conservative approach is necessary for scientific prudence. The data has been
very widely used, and the MILC collaboration policy of data release is seen as
successful and beneficial to the collaboration. There are many examples of
different groups collaborating together to share the burden of generating the
data. In some sense the ILDG is similar to other kinds of data archives and
Science Gateways, of which there are now many throughout the world. However,
it does present some particularly unique aspects, which stem from it being a
grid of grids.
In 2002, different groups were starting to make use of grid technologies to
store and retrieve data, primarily within their own collaboration. A proposal
by Richard Kenway Davies:2002mu , at the annual lattice QCD conference, to use
grid technologies to store and share data, was well received and supported.
The ILDG was formed from interested groups that were willing to share data.
There is no central authority forcing policy on the member collaborations:
rather the ILDG is a collaboration of groups that are prepared to commit some
resource to a central service. This idea of an aggregation or grid-of-grids is
a powerful one, which allows each group to retain control of its own resources
whilst making them available to the greater whole.
This paper is organised as follows: we outline the basic requirements needed
for such an infrastructure in section 2. The ILDG development has been split
into two broad overlapping groups, the metadata working group (MWG) and the
middleware working group (MWWG). In sections 3 and 4, we consider aspects of
metadata and middleware, respectively. Finally, in section 5, we review
aspects of the ILDG project from over several years of activity and over one
year of production. We present operational details of the infrastructure as
well as criticism of various aspects. We also have a chance in section 5 to
compare and contrast the ILDG with some related or similar efforts. Finally,
we summarise and discuss the potential for future work in section 6.
## 2 Requirements
Put in simple terms, the goal of the ILDG project is to allow scientists to
share their data, across the different research collaborations within the
project.
In order to attain the goal, the team have had to translate it into a set of
concrete requirements, which has then been used to guide the development of
the ILDG infrastructure (that is, the technologies, policies, and processes).
These requirements are summarised in this section.
### 2.1 Data management
The team started by quantifying the data to be shared. This data is file-based
and – as noted above – represents lattice gauge configurations, which are
collected together into ensembles pertaining to Monte Carlo simulations.
The nature of the simulations implies that a configuration is only meaningful
as a member of the ensemble. Thus, scientists almost always want to access the
whole ensemble (or a significant part thereof): this equates to Terabytes of
data. Also, scientists typically need to have access to local copies of data,
in order to complete the required analysis processes. Thus, it is clear that
sharing data involves copying multi-Terabyte file sets from the storage
facility of one research group to a remote scientist’s local system.
All file copy operations are intended to be undertaken over the Internet.
Thus, even with good bandwidth, it is clear that multi-Terabyte transfers
represent time-consuming operations, requiring a reliable, high performance
bulk data transfer mechanism.
Ensembles of gauge configurations that pre-date ILDG are typically identified
using locally agreed naming conventions. For example, a particular
configuration might be identified by a combination of the Unix path to the
file and the hostname of the server on which it resides. While this approach
may be suitable for a small group of researchers working in a particular
collaboration, it is inadequate for a community like ILDG that is loosely
coupled and distributed across multiple research groups.
What is required is a method for assigning a unique and persistent identifier
to each file (that is, gauge configuration) that is to be held within the
infrastructure. In addition, there needs to be an equivalent method for
identifying each ensemble.
### 2.2 Data Curation
For a configuration (or an ensemble) to be useful to a researcher, it must be
apparent what it represents in scientific terms. This information is provided
by metadata – literally, data about data. Metadata may be captured in a number
of different ways. For example, a widely used approach is based on descriptive
filenames that follow an agreed naming format. For Lattice QCD, the detail
required to describe a dataset is too great to be realistically encoded in its
filename, especially considering the various different formulations of QCD
available, all with different parameters. The process of scientific annotation
has warranted a more sophisticated approach.
ILDG researchers require a scientific annotation that thoroughly and
unambiguously describes a configuration (or ensemble of) for other members of
the community. The annotation should be extensible: that is, it should support
the introduction of new descriptive elements. This may be required — for
example — to accommodate new science.
A user should easily be able to search the catalogue of scientific annotations
and, complementing this, the generation of metadata should be a lightweight
and straightforward process. Where possible, elements of the description
should be populated automatically.
As well as having an agreed mechanism for describing data, one must also be
able to read the binary files that hold the data. This motivates convergence
to a common file format (for gauge configurations, at least). At the inception
of ILDG, a number of different file formats existed, based on the conventions
used in the most popular LQCD codes. Alongside the formalisation of the
scientific metadata, it has been decided that a community-wide, flexible,
extensible binary file format is required.
### 2.3 Infrastructure
Pre-dating the formation of ILDG, the five collaborations that make up the
core of the consortium have procured or developed storage facilities to host
the ensembles of data that they each generate. These systems are all
accessible, in principle, over the Internet, but via different and
incompatible access protocols and access control systems targeted at local
(that is, institution-based) users.
To work around this issue, two specific requirements need to be fulfilled.
First, a layer of software is required on top of the local infrastructures, to
provide a uniform interface to an end-user. Second, an access control
mechanism needs to be established that permits ILDG members from different
collaborations to access designated data at partner institutions/storage
facilities.
### 2.4 Operation and monitoring
To be useful, the ILDG infrastructure must achieve high levels of
availability. High availability must be attained in spite of the decentralised
and heterogeneous nature of the component elements, and should efficiently
exploit the support effort available at the regional grids. It has therefore
been decided that an automated monitoring service should be set up within the
infrastructure, fulfilling the following specific attributes. The monitoring
service needs to:
* •
be reliable – since it is the primary means in which problems and failures are
identified.
* •
be flexible – in order that the diversity of ILDG components can be
represented and monitored.
* •
produce accurate and informative alarms, which will allow regional-grid
support teams to quickly and effectively diagnose and resolve issues.
* •
post alarms using email – as this is the primary medium over which regional-
grid teams communicate.
* •
maintain a record of system performance, to inform coordinators as to overall
reliability and to highlight any specific weaknesses.
For easy access to all ILDG resources, a centrally coordinated user management
system is required: making all globally registered users known to all local-
resource providers. To this end, we have adopted the concept of a Virtual
Organisation (VO), with membership being managed by the VO itself. With ILDG
consisting of several regional grids, a setup is however needed that allows
the decision – as to whether an application for VO membership is to be
approved or declined – to be delegated to the regional grids. For users to
access ILDG resources only a single sign-on should be required: that is, a
single trust domain has to be defined. This domain should include a
sufficiently large set of trusted Certificate Authorities that every potential
users can be provided with a certificate that is acceptable to any of the
resource providers.
While it is not envisaged that the regional-grid make-up of ILDG will change
in a particularly dynamic manner, it is expected that new collaborations will
wish to join the infrastructure, either independently or as part of an
existing group. With this in mind, it is important that the infrastructure
evolves in a way that does not prevent expansion. Specifically:
* •
ILDG specifications (for example, service definitions) are thoroughly
documented in a manner intended to facilitate the creation of new
implementations.
* •
the technology layer is supported by a test suite, which allows new
implementations of ILDG services to be validated against the specification.
* •
where possible, ILDG uses open (or at least widely adopted) technologies and
standards, aiming to increase coverage of user groups and to reduce the risk
of systems becoming obsoleted.
* •
the technological aspect of the infrastructure is specified as a thin layer
(that is, focused on a baseline set of functionalities), which can easily be
incorporated into existing infrastructures with low levels of development
effort.
## 3 Metadata
To motivate the need for metadata, consider an example where there is no
metadata. Configurations from different ensembles are all stored in a single
directory with potentially random strings for names. Clearly this data is now
not accessible. A scheme is required to describe the data. As noted above,
many groups have in the past constructed ad-hoc schemes for describing the
data based on filenames and directory structures. Whilst this approach is not
without merit, it does not scale when many groups are sharing data. In
constructing this scheme there are likely to be several assumptions which are
specific to the group which uses the scheme. Another group may well find these
assumptions are not valid for their data, and hence their data will not fit
into the scheme. Modifying the scheme is only possible where the assumptions
used in its construction are still valid. To accommodate several potential
different formulations of LQCD, and the needs of different groups a different
approach is required.
Extensibility is a critical requirement of any annotation scheme. Any new data
will need new metadata to describe it and the scheme will have to be modified.
In an extensible scheme this can be done without breaking the original scheme.
That is, the new scheme is an extension of the old one. Furthermore, any
document which was valid in the old scheme is valid in the new one, so that
the old documents don’t have to be updated to be valid in the new scheme.
Data provenance is likewise an important requirement. Can the data be
recreated from the metadata? Taken to the limit this question is extremely
challenging. In principle the code used to generate the data and its inputs
should allow the data to be regenerated. However, this doesn’t include any
machine, compiler or library information. Moreover, in the context of sharing
data, the application belonging to one group may not be able to parse and
process the input parameters of the application belonging to a different
group. Hence while a full archival of a statically linked code, its inputs
should allow recreation of the data if the original producing machine were to
be available, archiving to this level of detail is not practical.
Correspondingly some of the data provenance requirements may need to be
softened in practise.
Lattice QCD metadata is hierarchical in nature and the annotation scheme
should reflect this. Markup languages combine text and information about the
text, and thus are perhaps a natural choice for a language in which to
construct the scheme. Semantic or descriptive languages don’t mandate
presentational or any other interpretation of the markup. XML was chosen as it
is the most widely used and best supported markup language. Similarly XML
schema was chosen as the schema language to define the scheme or set of rules
for the metadata.
In order to make sharing lattice QCD data useful and effective, lattice QCD
metadata should be recorded uniformly throughout the grid. The metadata
working group designed an XML schema called QCDml for the metadata. The
primary use case being data discovery via the metadata.
As described above a key concept for Lattice QCD data is the organisation of
the data as configurations and ensembles to which the configurations belong.
The metadata is divided into two linked XML schemata, one for the
configurations, and one for the ensemble. The two schemata are linked together
by a unique Uniform Resource Identifier (URI), called the markovChainURI,
which lives in the name-space of the ILDG and which appears in the XML
instance documents (IDs) of the configuration and the ensemble to which it
belongs. There is no formal mechanism for ensuring uniqueness, but a simple
convention has been adopted whereby the name of the group who generated the
data appears in the URI, and responsibility for uniqueness is thereafter
delegated to that group.
The separation of the metadata into two pieces, besides reflecting the nature
of lattice QCD data, has two advantages. First, metadata capture is
potentially simplified, as only the configuration-specific information has to
be recorded for each configuration, and the information specific to an
ensemble has to be recorded only once. Second, the performance of searches on
the data may be improved since the split represents a factoring of the
original more complicated schemata.
The metadata scheme is encoded as a set of XML schemata XMLSchemata and
whilst this does not mandate how the metadata is stored and accessed, for
simplicity it is often stored in native XML databases such as eXist eXist . It
is well known that the speed of access of hierarchical databases, such as
native XML database is vastly inferior to that of relational databases.
Scientists are almost always interested in finding an ensemble rather than
finding an individual configuration. Therefore, for most cases, the separation
of ensemble and configuration XML reduces the number of documents to be
searched by ${\mathcal{O}}(100-1000)$.
In each configuration ID the logical file name (LFN) of the data file is
recorded. The LFN is a unique and persistent identifier of the file in the
ILDG name-space. The ILDG and local grid services then map the LFN to actual
file instances.
The data itself is stored in a file format known as LIME LIME . LIME is short
for Limited Internet Message Encapsulation, and is a simplified and
generalised version of the DIME (Direct Internet Message Encapsulation) DIME
Internet standard, which was proposed as an Internet Standard and which is now
part of the Microsoft .NET framework. LIME is a record-oriented message format
which simplifies and extends the original DIME framework by introducing 64-bit
length records instead of the original 32-bit ones, and correspondingly it
eliminates the need for continuation records. LIME thus allows the packing of
descriptive text records and binary data records in the same file. This format
itself is very flexible and extensible since the types and sequence of records
are not mandated in the file format itself. The ILDG however, specifies and
requires a set of LIME records, including: a record containing some XML file
format metadata describing the size of the space-time lattice and data
precision; a record containing the data itself in a specified data ordering;
and a record of the LFN for the data, to allow the linking of ILDG data files
to their metadata catalogue entries. LIME was developed by the USQCD
collaboration through the SciDAC software initiative and a C-code to read and
write lime files on a serial machine (C-LIME) can be downloaded from the USQCD
web-site LIME . The QIO package also developed by the USQCD collaboration has
facilities for reading ILDG formatted data files on both serial and parallel
machines QIO
The scientific core of the metadata is contained within the ensemble schema.
The most important section from the data discovery viewpoint is the action
which contains the details of the physics. Here, the object-oriented ideas of
inheritance are used to build an inheritance tree of actions based on the XML
schema concepts of extension and restriction as appropriate. This enables
users to make both very specific searches and more general searches on the
names, types and/or parameters of the actions. The exact details of the
physics can only be encoded in mathematics, which is not suited to an XML
description. A reference to a paper, and the URL of an external _g_ lossary
document which contain the mathematical descriptions of the physics are
included. Clearly an application cannot parse this information, but it is
included to avoid ambiguity in the names used in the inheritance tree.
QCDml uses a namespace defined by an URI. This URI includes the version
number. Backward compatible, extensible updates to the schema don’t change the
URI of the namespace, so XML IDs don’t need to be modified. Clearly the XML ID
of the schema itself is modified, so a new URL for the extended XML ID of the
schema is needed. All versions persist on the web, but with incremental URLs.
Non-extensible updates to the schema require a change of namespace and the URI
which identifies it.
Lattice QCD algorithms are very complex with many different algorithmic
components. They are also an active area of research, and changes and
improvements are common. This makes designing a scheme, especially an
extensible one rather difficult. The defined names and inheritance tree ideas
used for the action would be too cumbersome for describing the algorithms.
QCDml has only small scope for the algorithms limited to name, value pairs for
the parameters. Algorithmic details can be expressed mathematically in an
external, non-parseble glossary document. This approach further limits the
data provenance of QCDml. However, individual groups can import their own
namespace with as much detail and structure as they see fit, which can help
ameliorate the data provenance issue even if the metadata is no longer
universal.
Before leaving this general discussion of the metadata schema we note that the
current set of schemata may be found at EnsembleSchema (ensembles) and
ConfigurationSchema (configurations). More Physics-oriented information about
the metadata can be found in Coddington:2007gz .
## 4 Middleware
As described above a grid-of-grids concept has been adopted for ILDG. Each of
the regional grids has to provide the following services: a metadata catalogue
(MDC) for metadata-based file discovery, a file catalogue (FC) for data file
location and one or more storage elements (SE) which can then serve the data
to the user.
The user can discover available datasets by sending a query to the MDC of each
of the regional grids. On input this search requires an XPath expression. On
output the services will return the list of Logical Files Names (LFNs) of
those documents for which the XPath expression identifies a non-zero set of
nodes.
To identify all copies of a particular file a query to the file catalogue has
to be performed, which takes an LFN on input and returns a source URL (SURL)
for each replica of the file. The scheme part of the SURL tells the client
whether it can either directly download the file using the transfer protocols
HTTP or GridFTP or whether it has to connect to a Storage Resource Manager
(SRM) interface.
The SRM protocol srm is evolving to an open standard for grid middleware to
communicate with site-specific storage fabrics. The ILDG, at the time of
writing, requires the SRM to be adhere to version 2 of the SRM specification.
A particularly appealing feature of the SRM is that implementations of the
service are typically provided with a standard Web-Service interface, allowing
the SRM component to fit in easily with the comparatively less complex, ILDG-
defined MDC and FC.
SRM is a system to manage local storage fabrics comprised of several data
servers and possibly long-term backup storage. From the point of view of an
ILDG transaction, the SRM may be required to: stage a file to some transfer
location on a server, negotiate a transfer protocol between the server and the
client, and to then arrange for the transfer to occur. Once the file is ready
for transfer, the location and the transfer protocol are returned to the
client by the SRM service in a transfer URL (TURL). The client and the data
server can then carry out the transfer independently of the rest of the SRM
system. Typically GridFTP is used as a transport protocol.
By adopting a grid-of-grids concept with different middleware stacks being
used by the different regional grids, interoperability becomes a challenge.
Interoperability is required to provide standardised interfaces towards the
application layer. While there are clear similarities between the different
grid architectures, there are crucial conceptional differences and
incompatibilities of the interfaces.
For all services, which have not been specifically designed for ILDG, two
strategies have been applied to overcome this interoperability issue. Firstly,
wherever possible, common grid standards supported by all the used middleware
stacks have been adopted. One example is the transfer protocol GridFTP.
Secondly, interface services have been defined and implemented. Instead of
accessing a service directly, the user connects to the interface service which
will process the request on behalf of the user.
If a service requires authentication the corresponding interface service has
to provide a credential delegation service. Within ILDG we use an
implementation of such a service that has been developed within the GridSite
project gridsite and is now part of the gLite middleware stack. On request of
the client the server returns a proxy certificate request, which is signed on
the client side and returned back to the server. Since the proxy certificate
has only a limited lifetime, the risk due to a compromised server hosting the
interface service is considered to be acceptable small.
To standardise a web-service within ILDG a WSDL description is implemented and
additionally a behavioural specification is provided. The WSDL description
specifies the structure of the service’s input and output data structure,
while a functional description of the service is provided by the behavioural
specification. Additionally, test suites have been defined and implemented
which can be used to verify whether a service conforms to the ILDG standard.
Access to most services is restricted to members of the Virtual Organisation
(VO) ILDG. For the management of this VO we use a VOMRS service vomrs . Each
user which wants to join the VO has to submit an application and nominate one
of their regional grid’s representatives. For each regional grid at least two
representatives have been assigned which can accept or reject the request. For
each regional grid an individual group has been created. Information on group
membership may be used by the regional grids as input for authorisation
services.
The only other global service which is used within ILDG is the monitoring
service. This monitoring service has been implemented using INCA inca . In the
framework of INCA a set of so-called reporter managers regularly execute test
scripts accessing grid resources. The information returned by the reporter
managers is collected in a repository. In case of failures a notification
email is generated and sent to the regional grid which is responsible for a
particular service. Data in the repository can later be used to check the
service’s availability.
## 5 Review
### 5.1 General Status
At the time of writing, the ILDG has been in production use for a little over
a year. It is comprised of five main-partner regional grids. These are: The
Center for the Structure of Subnuclear Matter (CSSM) in Australia; The Japan
Lattice Data Grid (JLDG); Latfor Data Grid (LDG) for continental Europe
(primarily Germany, Italy and France); the regional grid of the UKQCD
Collaboration in the UK; and the regional grid of the USQCD collaboration in
the United States. The ILDG VO has 113 registered users and the combined ILDG
hosts some 207 gauge configuration ensembles, corresponding to various lattice
volumes, gauge and fermion actions. Each single ensemble represents
significant portions – potentially years – of human and supercomputing
resources. Thus these archives are immensely valuable.
On the management side, the Middleware working group hosts monthly
teleconferences to discuss operational exceptions, experiences and future
development efforts while at the higher level, the ILDG holds bi-annual video
conferences, so that regional partners can discuss more general progress.
### 5.2 Benefits of Sharing
Hosting such a wealth of data has had great benefit on computational lattice
QCD worldwide. In the case of some regional grids, the regional grid itself
has become the primary means of data distribution, for multi-site projects,
prime examples of which are the LDG and UKQCD collaborations Beckett:2009 .
A number of research activities have been enabled, thanks to the ILDG
infrastructure. Scientists in Japan have been using data produced by the MILC
collaboration (in United States) as part of their research Furui:2008um and,
complementing this, a team at $\chi$-QCD (University of Kentucky) has accessed
data from CP-PACS (Japan), in the ILDG community Draper:2008tp ; Doi:2009sq .
Other examples of ILDG use can be found in Ehmann:2007hj and
Ilgenfritz:2006gp where two groups made use of lattices generated by the
German QCDSF Collaboration.
Both inter-collaboration and intra-collaboration activities are enabled by
ILDG. In Stuben:2005uf , a number of ILDG-enabled activities are noted
relating to data sharing across LDG sites. The fact that the ILDG is making a
serious impact in international collaboration can also be seen in the fact
that Physics workshops are being held within the community that focus, not
only on the generation of QCD data, but also on accomplishing calculations by
sharing the data via the ILDG TsukubaWorkshop .
### 5.3 Criticism of the ILDG
While ILDG appears to be operating successfully, there are some aspects of it
that could be improved. Using the ILDG to locate and share data is relatively
straightforward especially with the easy to install client tools ILDGTools as
is described in Yoshie:2008aw . However, contributing to the ILDG potentially
involves a lot of effort. Depending on the level of involvement, one may need
to maintain storage and database resources as well as having to mark up
configuration and ensemble metadata.
In order to create ensemble metadata markup, one needs to get a unique key to
identify it (MarkovChainURI). There is no service which can supply one or
necessarily check that a manually chosen key is in fact unique. Further,
ensemble metadata markup is not straightforward to automate and may need to be
done by hand. If a new collaboration wishes to extend the XML Schemata to mark
up data for which no QCDml exists, the process of standardisation of the
markup may take a substantial amount of time.
Marking up configurations may be more straightforward, and may be automated.
However, it too involves some amount of post-processing. The checksum needed
in the configuration metadata document is not easy to compute in a parallel
program and likewise a unique key; the configuration LFN; needs to be known in
order to create both the configuration metadata and in order to write a fully
ILDG compliant configuration file as described previously. However, the LFN
may not be known at the time of production. Thus typically configuration
metadata is generated post-production, and the configurations typically do not
contain the LFN on creation. This has to be added on insertion to the ILDG.
While much of this activity can be automated, the initial goal of the
computation producing the configuration metadata and the ILDG compliant
configuration at the same time has been sacrificed in order to agree on other
aspects. There is thus scope in the data production workflow, for data to lay
idle for quite some time before being added to the ILDG with the consequent
loss of history and provenance information. Hopefully future software tools
can alleviate this problem.
Although it was thought that these difficulties will be a major stumbling
block to ILDG participation, in practice metadata creation proved to be less
of a stumbling block than initially expected. The ensemble metadata typically
needs to be created only once, making it worth the effort and as mentioned
previously the workflow for configuration metadata markup and publication can
be substantially automated. Hence while the in principle issues discussed
above remain, at a practical level the bar for participation in ILDG came not
from the metadata, but rather from maintaining the middleware stack of the
participating organisations such as managing grid security certificate
infrastructure.
One aspect of the ILDG to remark upon is that it is most definitely a
volunteer, and altruistic activity. It receives very little in the way of
funding for itself and is usually piggybacked discretely onto other grid
related projects or to regional grid activities. Correspondingly, it can
become difficult to maintain effort focused on the ILDG, which limits large
scale development and essentially forces simple solutions.
We can contrast the ILDG with some other related work. Other non-ILDG lattice
archives include the Gauge Connection (at NERSC) GaugeConnection and the
QCDOC Configuration download site QCDOCSite (at the Brookhaven National
Laboratory) which is very similar in structure to the Gauge Connection and we
shall treat the two identically below. The Gauge Connection was created before
the era of Web Services and Grid services. It hosts files on a single
filesystem and one can download all the configurations over HTTP. The file
format used is an ASCII header followed by a binary data segment. The header
contains rudimentary metadata (e.g. information about the creators, a
checksum, and some derived measurement). Hence there is no separation between
the configuration files and their metadata like there is in our case.
Ensembles are not marked up in terms of XML at all, but there is some human-
readable description for each one. Authentication and authorisation is done at
the Web-Server level and one needs to register with the site to gain access.
This setup, though very simple has worked very robustly and well. On the other
hand, it becomes harder to search this archive, since there is no actual
metadata catalogue as such. A human must read through a list of available
ensembles until he finds the one he wants from the description. The Gauge
Connection served as a guide to the ILDG effort. In particular the layout of
the data in the binary part of the Gauge Connection format has been kept in
the ILDG data record.
We should also mention in this section the LQCD Archive (LQA) LQA which is
maintained at the Center for Computational Sciences at the University of
Tsukuba in Japan. The LQA began development prior to the ILDG to distribute
the data of the CP-PACS collaboration as a configuration download service
similar to the Gauge Connection. However, upon inception of the ILDG, the LQA
was re-developed to be the front end portal to the data available on the JLDG.
It currently provides metadata search facilities as well as HTTP based
download which may be useful to users who do not wish to set up a full grid
client infrastructure on their machine. The JLDG data is of course also
available through the usual ILDG client tools independent of this portal. To
use this service, one is required to register. The portal post a list of
publications to which citations should be made on publication of results that
come from the downloaded datasets.
Download services have proved useful to the community however they have
several shortcomings. They allow downloading primarily through HTTP which may
encounter performance limitations when one considers downloading entire
ensembles, especially since the size of configurations is expected to
increase. There has been no attempt to provide a common file format. The
individual architectures do not lend themselves to data replication and lack a
common security infrastructure (each requiring separate registrations). That
having been said, historically the Gauge Connection share their file format
while the LQA as noted above has been extensively redeveloped to complement
rather than contrast with the ILDG.
One can also compare the ILDG to the concept of a Science Gateway. Quoting
from the definition of Science Gateways on the TeraGrid
TeraGridScienceGateways , “A Science Gateway is a community developed set of
tools, applications and data that is integrated via a portal, or suite of
applications, usually in a graphic interface that is customised to meet the
needs of a target community.” In this sense the gateways have a broader scope
than the ILDG, they can offer codes, grid services, as well as access to data
collections. As an example we consider the “Massive Pulsar Surveys Using the
Arecibo L-band Feed Array (ALFA)” TeraGrid Science Gateway which allows one to
brows data on pulsars and is similar in scope to the ILDG. One can browse
pulsar information, and can download associated data-products. On the other
hand, the SCEC Earthworks Gateway actually allows the running of earthquake
simulations on TeraGrid resources. Both these gateways can be found at
GatewayList .
One unique feature of the ILDG, in contrast to a Science Gateway, is that the
ILDG is the result of a collaboration of collaborations. A single Science
Gateway would typically consist of a single portal maintained by a group on
behalf of a larger community. This group then has some freedom (within
community limits) in defining internal formats, markup and can settle on a
single set of software tools. The ILDG instead is a loose federation of
existing grids, some of which at the inception of the ILDG had no grid
infrastructure and some of whom were already heavily invested in their own
systems. The worldwide community had to therefore come together in order to
define metadata standards, middleware operation and thin, easy to implement
interfaces that could then wrap any potentially existing, underlying
infrastructure. Another difference between the ILDG and Science Gateways may
be their philosophy.
## 6 Summary and Future Work
In summary, the ILDG is a loosely federated grid-of-grids to facilitate the
sharing of LQCD data worldwide. The technology allows it to operate across
regional grid boundaries, relies on a simple and thin layer of middleware
standard definitions, and a standardised metadata markup.
In six years of design and a little over one year of operation, the ILDG
effort has brought together the lattice QCD community and has fostered QCD
research and collaboration.
Potential future work focuses on several areas including but not limited to
data replication, and the storage and mark up of secondary large data such as
quark propagators.
###### Acknowledgements.
Notice: Authored in part by Jefferson Science Associates, LLC under U.S. DOE
Contract No. DE-AC05-06OR23177. The U.S. Government retains a non-exclusive,
paid-up, irrevocable, world-wide license to publish or reproduce this
manuscript for U.S. Government purposes.
## References
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* (3) R. Gupta, arXiv:hep-lat/9807028.
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* (5) http://www.physics.indiana.edu/~sg/milc.html
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* (7) http://www.w3.org/XML/Schema
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* (12) http://www.lqcd.org/ildg/QCDml/ensemble1.4/QCDmlEnsemble1.4.4.xsd
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* (14) P. Coddington, B. Joo, C. M. Maynard, D. Pleiter and T. Yoshie, PoS LAT2007 (2007) 048 [arXiv:0710.0230 [hep-lat]].
* (15) http://sdm.lbl.gov/srm-wg/doc/SRM.v2.2.pdf.
* (16) A. McNab, S. Kaushal, “The GridSite Proxy Delegation Service,” UK e-Science All Hands Conference, Nottingham, September 2006.
* (17) L. Bauerdick et al., “The virtual organisation management registration service,” CHEP 2006 proceedings. Proc. International Conference on Computing in High Energy Physics (CHEP06), Mumbai (India), February 2006.
* (18) S. Smallen et al., “User-level grid monitoring with Inca 2,” Proceedings of the 2007 workshop on Grid monitoring, 2007.
* (19) M. G. Beckett et al., Phil. Trans. R. Soc. A 367 (2009).
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* (21) T. Draper, T. Doi, K. F. Liu, D. Mankame, N. Mathur and X. f. Meng, arXiv:0810.5512 [hep-lat].
* (22) T. Doi et al., arXiv:0903.3232 [hep-ph].
* (23) C. Ehmann and G. Bali, PoS LAT2007 (2007) 094 [arXiv:0710.0256 [hep-lat]].
* (24) E. M. Ilgenfritz, M. Muller-Preussker, A. Sternbeck and A. Schiller, arXiv:hep-lat/0601027.
* (25) H. Stuben and S. Wollny, Nucl. Phys. Proc. Suppl. 153, 300 (2006) [arXiv:hep-lat/0512008].
* (26) http://www.ccs.tsukuba.ac.jp/workshop/EP09
* (27) http://www-zeuthen.desy.de/latfor/ldg/doc/swinstall.html
* (28) T. Yoshie, PoS LATTICE2008 (2008) 019 [arXiv:0812.0849 [hep-lat]].
* (29) http://qcd.nersc.gov
* (30) http://lattices.qcdoc.bnl.gov
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* (34) http://www.globus.org
|
arxiv-papers
| 2009-10-09T09:29:06 |
2024-09-04T02:49:05.736710
|
{
"license": "Public Domain",
"authors": "G. Beckett, B. Joo, C.M. Maynard, D. Pleiter, O. Tatebe, T. Yoshie",
"submitter": "Chris Maynard",
"url": "https://arxiv.org/abs/0910.1692"
}
|
0910.1713
|
# Isomorphisms and automorphisms of quantum groups
Li-Bin Li and Jie-Tai Yu School of Mathematics, Yangzhou University, Yangzhou
225002, China lbli@yzu.edu.cn Department of Mathematics, The University of
Hong Kong, Pokfulam, Hong Kong SAR, China yujt@hkucc.hku.hk
yujietai@yahoo.com
###### Abstract.
We consider isomorphisms and automorphisms of quantum groups. Let $k$ be a
field and suppose $p,\ q\in k^{*}$ are not roots of unity. We prove a new
result that the two quantum groups $U_{q}(\mathfrak{sl}_{2})$ and
$U_{p}(\mathfrak{sl}_{2})$ over a field $k$ are isomorphic as $k$-algebras if
and only if $p=q^{\pm 1}$. We also rediscover the description of the group of
all $k$-automorphisms of $U_{q}(\mathfrak{sl}_{2})$ of Alev and Chamarie, and
that $\text{Aut}_{k}(U_{q}(\mathfrak{sl}_{2}))$ is isomorphic to
$\text{Aut}_{k}(U_{p}(\mathfrak{sl}_{2}))$.
###### Key words and phrases:
Quantum groups, isomorphisms, automorphisms, center, polynomial algebras,
simple $U_{q}(\mathfrak{sl}_{2})$-modules, Casimir elements, symmetry, $PBW$
type basis, degree function, graded algebra structure.
###### 2000 Mathematics Subject Classification:
16G10, 16S10, 16W20, 16Z05, 17B10, 20C30
The research of Li-Bin Li was partially supported by NSFC Grant No.10771182.
The research of Jie-Tai Yu was partially supported by an RGC-GRF grant.
## 1\. Introduction and the main results
The Drinfeld-Jimbo quantum group $U_{q}(\mathfrak{g})$ over a field $k$ (see
[D1, D2, J, Ja]), associated with a simple finite dimensional Lie algebra
$\mathfrak{g}$, plays a crucial role in the study of the quantum Yang-Baxter
equations, two dimensional solvable lattice models, the invariants of
3-manifolds, the fusion rules of conformal field theory, and the modular
representations (see, for instance, [K, L, LZ, RT]). It is natural to raise
###### Problem 1.1.
When are the two quantum groups $U_{q}(\mathfrak{g})$ and
$U_{p}(\mathfrak{g})$ over a field $k$ isomorphic as $k$-algebras?
It is closely related to
###### Problem 1.2.
Describe the structure of $\text{Aut}_{k}(U_{q}(\mathfrak{g}))$ for the
quantum group $U_{q}(\mathfrak{g})$ over a field $k$.
See, for instance, Alev and Chamarie [AC] for a description of
$\text{Aut}_{k}(U_{q}(\mathfrak{sl}_{2}))$. See also Launois [La1, La2], and
Launois and Lopes [LL] and references therein for related description of
$\text{Aut}_{k}(U_{q}^{+}(\mathfrak{g}))$.
In particular, we may formulate
###### Problem 1.3.
When are the two quantum groups $U_{q}(\mathfrak{sl}_{n})$ and
$U_{p}(\mathfrak{sl}_{n})$ over a field $k$ isomorphic as $k$-algebras?
To the authors, the above problems are also motivated by the similar questions
regarding the isomorphisms and automorphisms of affine Hecke algebras
$\mathbb{H}_{q}$ and $\mathbb{H}_{p}$ over a field $k$ recently considered by
Nanhua Xi and Jie-Tai Yu [XY]. See also Rong Yan [Y].
In this paper, we fully classify the quantum groups $U_{q}(\mathfrak{sl}_{2})$
by $q$ provided $q$ is not a root of unity.
###### Theorem 1.4.
Suppose $q\in k^{*}$ is not a root of unity in a field $k$, then
$U_{q}(\mathfrak{sl}_{2})$ and $U_{p}(\mathfrak{sl}_{2})$ are isomorphic as
$k$-algebras if and only if $p=q^{\pm 1}$. Moreover, any such $k$-isomorphism
must take the generator $c_{q}$ of the center $Z(U_{q}(\mathfrak{sl}_{2}))$ of
$U_{q}(\mathfrak{sl}_{2})$ to $c_{p}$ or $-c_{p}$, where $c_{p}$ is the
generator of the center $Z(U_{p}(\mathfrak{sl}_{2}))$ of
$U_{p}(\mathfrak{sl}_{2})$.
In case $q$ is not a root of unity, we also rediscover the description of
$\text{Aut}_{k}(U_{q}(\mathfrak{sl}_{2}))$ of Alev and Chamarie [AC] by a
different method.
###### Proposition 1.5.
Suppose $q\in k^{*}$ is not a root of unity in a field $k$, then
$\alpha\in\text{Aut}_{k}(U_{q}(\mathfrak{sl}_{2}))$ if and only if
(1) $\alpha(K)=K,\ \alpha(E)=\lambda EK^{r},\ \alpha(F)=\lambda^{-1}K^{-r}F;\
$
or
(2) $\alpha(K)=-K,\ \alpha(E)=\lambda EK^{r},\
\alpha(F)=-\lambda^{-1}K^{-r}F;$
or
(3) $\alpha(K)=K^{-1},\ \alpha(E)=\lambda K^{r}F,\
\alpha(F)=\lambda^{-1}EK^{-r};$
or
(4) $\alpha(K)=-K^{-1},\ \alpha(E)=\lambda K^{r}F,\
\alpha(F)=-\lambda^{-1}EK^{-r}$
for some $r\in\mathbb{Z}$ and some $\lambda\in K^{*}$.
The techniques used here depend on the description of the center of the
quantum group $U_{q}(\mathfrak{sl}_{2})$ as a polynomial algebra in one
indeterminate over $k$ and its $k$-automorphisms, the classification of finite
dimensional simple $U_{q}(\mathfrak{sl_{2}})$-modules, and in particular, the
‘symmetry’ of the Casimir element action on finite-dimensional simple
$U_{q}(\mathfrak{sl}_{2})$-module. We also use the well-known $PBW$ type
basis, the degree function, and the graded algebra structure of
$U_{q}(\mathfrak{sl}_{2})$.
As a consequence of Proposition 1.5, we obtain that the two groups of
$k$-automorphisms of $U_{q}(\mathfrak{sl}_{2})$ and $U_{p}(\mathfrak{sl}_{2})$
are isomorphic provided both $q$ and $p$ are not roots of unity.
###### Proposition 1.6.
Suppose both $q,p\in k^{*}$ are not roots of unity in a field $k$, then the
two groups $\text{Aut}_{k}(U_{q}(\mathfrak{sl}_{2}))$ and
$\text{Aut}_{k}(U_{p}(\mathfrak{sl}_{2}))$ are isomorphic.
Based on the main results of this paper and some more involved methodology, we
will treat the general cases of Problems 1.1, 1.2 and 1.3 in a forthcoming
paper [LY]. In particular, in [LY] we completely solve Problem 1.3 and get the
condition $p=q^{\pm 1}$ as Theorem 1.4 in this paper.
## 2\. Preliminaries
In this section, we first recall some fundamental facts about the quantum
group $U_{q}(\mathfrak{sl}_{2})$ over a field $k$, where $q\in K^{*}$ is not a
root of unity in $k$ (see, for instance, Jantzen [Ja], or Kassel [K]). We also
prove a technical lemma, which classifies the unit elements in
$U_{q}(\mathfrak{sl}_{2})$. Finally, we recall an elementary lemma about
automorphisms of polynomial algebras. All of these will be used in the proof
of the main results in the next section.
Recall that for given $q\in k^{*}$ and $q^{2}\neq 1$, the quantum group
$U_{q}(\mathfrak{sl}_{2})$, introduced by Kulish and Reshetikhin[KR],
Reshetikhin and Turaev [RT] (see Takeuchi [T] for notations used in this
paper), is the associative algebra over $k$ generated by $K$, $K^{-1}$, $E$,
$F$ subject to the following defining relations:
$\displaystyle KK^{-1}=K^{-1}K=1,\ \ KEK^{-1}=q^{2}E,$ $\displaystyle
KFK^{-1}=q^{-2}F,\ \ \ EF-FE=\frac{K-K^{-1}}{q-q^{-1}}.$
It is well-known that the algebra $U_{q}(\mathfrak{sl}_{2})$ is an iterated
Ore extension and a Noetherian domain and has a $PBW$ type basis
$\\{E^{i}F^{j}K^{s}|$ $i,j\in\mathbb{N}$, $s\in\mathbb{Z}\\}$ as a $k$-vector
space. If $q$ is not a root of unity, then the center
$Z(U_{q}(\mathfrak{sl}_{2}))$ of $U_{q}(\mathfrak{sl}_{2})$ is the subalgebra
generated by the Casimir element
$\displaystyle
c_{q}=EF+\frac{q^{-1}K+qK^{-1}}{{(q-q^{-1})}^{2}}=FE+\frac{qK+q^{-1}K^{-1}}{{(q-q^{-1})}^{2}},$
hence $Z(U)=k[c_{q}]$ is a polynomial algebra in one indeterminate over $k$.
For $\varepsilon\in\\{-1,1\\}$ and each $n\in\mathbb{N}$, define an
$(n+1)$-dimensional $U$-module $V_{q}^{\varepsilon}(n)$ with a basis
$\\{v_{0}^{\varepsilon},v_{1}^{\varepsilon},\cdots,v_{n}^{\varepsilon}\\}$,
and the actions of the generators of $U$ on the basis vectors are given by the
following rules:
$Kv_{i}^{\varepsilon}=\varepsilon q^{n-2i}v_{i}^{\varepsilon}$
$Ev_{i}^{\varepsilon}=\varepsilon[n-i+1]v_{i-1}^{\varepsilon}$
$Fv_{i}^{\varepsilon}=[i+1]v_{i+1}^{\varepsilon},$
where
$i=0,1,\cdots,n,v_{-1}^{\varepsilon}=v_{n+1}^{\varepsilon}=0,[n]=\frac{q^{n}-q^{-n}}{q-q^{-1}}$,
$[n]!=[n][n-1]\cdots[2][1].$
It is well-known that {$V_{p}^{\varepsilon}(n)|\,\
\varepsilon\in\\{{-1,1}\\},n\in\mathbb{N}$} forms a complete-non-redundant
list of finite dimensional simple $U_{q}(\mathfrak{s}l(2))$-module. Note that
the Casimir element $c_{q}$ acts on $V_{q}^{\varepsilon}(n)$ via the following
scalar
$\varepsilon\frac{{{q^{n+1}+q^{-(n+1)}}}}{(q-q^{-1})^{2}}.$
The following lemma describe the unit elements in $U_{q}(\mathfrak{sl}_{2})$.
###### Lemma 2.1.
An element $u\in U_{q}(\mathfrak{sl}_{2})$ is multiplicative invertible if and
only if there exist $\lambda\in k^{*}$, $m\in\mathbb{Z}$ such that $u=\lambda
K^{m}$.
###### Proof.
The ‘if’ part is clear. Suppose $u\in U_{q}(\mathfrak{sl}_{2})$ is invertible,
then based on the $PBW$ type basis, $u$ can be written uniquely as a sum of
the terms $E^{r}h_{rs}F^{s}$ with non-negative integers $r,\ s$ and $h_{rs}\in
k[K,K^{-1}]-\\{0\\}$. Let $E^{m}h_{mn}F^{n}$ be the leading term of $u$
determined by the lexicographic order of $\\{r,s\\}$ by
$\\{r,s\\}>\\{r_{1},s_{1}\\}$ if $r>r_{1}$, or $r=r_{1}$ and $s>s_{1}$. Let
$v$ be the inverse of $u$ with the leading term
$E^{m_{1}}h_{m_{1}n_{1}}F^{n_{1}}$. Then by Lemma 1.1.7 and Proposition 1.1.8
in [Ja], $1=uv$ has the leading term of the form $E^{m+m_{1}}hF^{n+n_{1}}=1$
with some $h\in k[K,K^{-1}]-\\{0\\}$. It forces that $m=n=0=n_{1}=n_{1}$.
Hence $u\in k[K,K^{-1}]$. Now if $u$ is not a monomial, then based on
expansion of $u^{-1}\in k(K,K^{-1})$ as power series, $u^{-1}$ must contain
infinite many terms, hence not in $k[K,K^{-1}]$. Therefore $u$ must be a
monomial. ∎
We also need
###### Lemma 2.2.
Let $k[x]$ be the polynomial algebra in one indeterminate $x$ over a field
$k$. The the only $k$-automorphisms $\alpha$ of $k[x]$ are fully determined by
$\alpha(x)=ax+b$, where $a\in k^{*},$ $b\in k$.
###### Proof.
This is well-known. The proof is elementary and direct.∎
## 3\. Proof of the main results
Proof of Theorem 1.4.
The ‘if’ part is trivial. Suppose there exists an isomorphism $\Phi$ sending
$U_{q}(\mathfrak{sl}_{2})$ onto $U_{p}(\mathfrak{sl}_{2})$. Then $\Phi$
induces an isomorphism sending the center $k[c_{q}]$ of $U_{q}(sl_{2})$ onto
the center $k[c_{p}]$ of $U_{p}(\mathfrak{sl}_{2})$. Hence the center of
$U_{p}(\mathfrak{sl}_{2})$ is also a polynomial algebra in one indeterminate
over $k$. By [Ja], it forces $q$ is also not a root of unity in $k$ and the
center of $U_{p}(\mathfrak{sl}_{2})$ is $k[c_{p}]$. The isomorphism $\Phi$
induces an automorphism of $k[c_{p}]$ taking $\Phi(c_{q})$ to $c_{p}$ and its
inverse takes $c_{p}$ to $\Phi(c_{q})$. By Lemma 2.2, $\Phi(c_{q})=ac_{p}+b$,
for some $a\in k^{*}$ and $b\in k$. Therefore, under the isomorphism $\Phi$,
the $(n+1)$-dimensional simple $U_{p}(\mathfrak{sl}_{2})$-module
$V_{p}^{1}(n)$ becomes an $(n+1)$-dimensional simple $U_{q}$-module
$V_{q}^{\varepsilon}(n)$ for some $\varepsilon\in\\{-1,1\\}$. That is,
$V_{q}^{\varepsilon}(n)$= $V_{p}^{1}(n)$ as a vector space, and the action on
$V_{p}^{1}(n)$ of $x\in U_{q}(\mathfrak{sl}_{2})$ is given by $x\cdot
v:=\Phi(x)v$. Note that the Casimir elements $c_{q},\ c_{p}$ act on
$V_{q}^{\varepsilon}(n)$ and $V_{p}^{1}(n)$ via the scalars
$\varepsilon\frac{q^{n+1}+q^{-(n+1)}}{{(q-q^{-1})}^{2}}$
and
$\frac{p^{n+1}+p^{-(n+1)}}{(p-p^{-1})^{2}},$
respectively. Hence
(5)
$\varepsilon\frac{{q^{n+1}+q^{-(n+1)}}}{(q-q^{-1})^{2}}=a\frac{p^{n+1}+p^{-(n+1)}}{(p-p^{-1})^{2}}+b.$
Set $e=q+q^{-1}$, $f=p+p^{-1}$ and $n=0,1,2,3,4$, by (3.1), we get
(6) $\frac{\varepsilon e}{e^{2}-4}=\frac{fa}{f^{2}-4}+b,$ (7)
$\frac{\varepsilon(e^{2}-2)}{e^{2}-4}=\frac{a(f^{2}-2)}{f^{2}-4}+b,$ (8)
$\frac{\varepsilon(e^{3}-3e)}{e^{2}-4}=\frac{a(f^{3}-3f)}{f^{2}-4}+b,$ (9)
$\frac{\varepsilon(e^{4}-4e^{2}+2)}{e^{2}-4}=\frac{a(f^{4}-4f^{2}+2)}{f^{2}-4}+b,$
(10)
$\frac{\varepsilon(e^{5}-5e^{3}+5e)}{e^{2}-4}=\frac{a(f^{5}-5f^{3}+5f)}{f^{2}-4}+b.$
Performing (4)-(2), we obtain
(11) $\varepsilon e=af.$
Performing (5)-(3), we get
(12) $\varepsilon(e^{2}-1)=a(f^{2}-1).$
Performing (6)-(4), we obtain
(13) $\varepsilon e(e^{2}-2)=af(f^{2}-2).$
By (7) and (9), we get
(14) $e^{2}=f^{2}.$
By (8) and (10), we obtain
(15) $\varepsilon=a.$
By (7) and (11), we get
(16) $e=f.$
Thus $q+q^{-1}=p+p^{-1}$, therefore $(q-p)(1-qp)=0$, it forces that
$p=q^{\pm 1}.$
It is clear now $\Phi(c_{q})=\varepsilon c_{p}=\pm c_{p}$ as $a=\varepsilon$.∎
Proof of Proposition 1.5.
The ‘if’ part is obvious. Let
$\alpha\in\text{Aut}_{k}(U_{q}(\mathfrak{sl}_{2}))$. By Lemma 2.1,
$\alpha(K)=\lambda K^{m}$ for some $m\in\mathbb{Z}$. Under the automorphism
$\alpha$, the $(n+1)$-dimensional simple $U_{q}(\mathfrak{sl}_{2})$-module
$V_{q}^{1}(n)$ becomes an $(n+1)$-dimensional simple
$U_{q}(\mathfrak{sl}_{2})$-module $V_{q}^{\varepsilon}(n)$ for some
$\varepsilon\in\\{-1,1\\}$ via the action
$x\cdot v_{i}=\alpha(x)v_{i},$
where $\\{v_{0},\dots,v_{n}\\}$ is the standard basis of
$V_{q}^{\varepsilon}(n)$ as in Section 2. It follows that
$K\cdot v_{i}=\lambda K^{m}v_{i}=\lambda q^{(n-2i)m}v_{i},$
and the action of $K$ on $V_{q}^{\varepsilon}(n)$ is diagonalizable with the
eigenvalue set
$\\{\lambda q^{nm},\ \lambda q^{(n-2)m},\dots,\lambda
q^{-nm}\\}=\\{\varepsilon q^{n},\ \varepsilon q^{n-2},\dots,\varepsilon
q^{-n}\\},$
it forces that $m=\pm 1$ and $\lambda=\varepsilon=\pm 1$. Therefore
$\alpha(K)=\varepsilon K=\pm K^{m}=\pm K^{\pm 1}$.
In the sequel we will only give a detailed proof for the case $m=1$, as the
proof for the case $m=-1$ is similar. As $m=1$, $\alpha(K)=\varepsilon K$,
$K\cdot v_{0}=\varepsilon q^{n}v_{0}$ and $K\cdot v_{i}=\varepsilon
q^{n-2i}v_{i}$. Note that $E\cdot v_{i}$ is an eigenvector with corresponding
eigenvalue $\varepsilon q^{n-2i+2}$. It follows that
a) $E\cdot v_{i}=\lambda_{i}v_{i-1}$ for some $\lambda_{i}\in k$.
Similarly
b) $F\cdot v_{i}=\theta_{i}v_{i+1}$ for some $\theta_{i}\in k$.
Since $V_{q}^{\varepsilon}(n)$ is simple,
c) $\lambda_{0}=\theta_{n}=0$, $\lambda_{i}\neq 0$ for $0<i\leq n$, and
$\theta_{j}\neq 0$ for $0\leq j<n$.
As $KEK^{-1}=q^{2}E$, we get
$K\alpha(E)K^{-1}=(\varepsilon K)\alpha(E)(\varepsilon
K)^{-1}=\alpha(KEK^{-1})=q^{2}\alpha(E),$
hence $\alpha(E)$ is homogeneous with degree $1$ by [Ja]. Thus we may express
uniquely
$\alpha(E)=\sum_{i\geq 0}E^{i+1}h_{i}F^{i},\ h_{i}\in k[K,K^{-1}]-\\{0\\}.$
If there exists an index $i>0$ in the above sum, we may choose a positive
integer $i_{0}$ such that $n\geq i_{0}>0$ and $i\geq i_{0}$ for all index $i$
in the sum, then by the formulas a), b) and c) above,
$0\neq\lambda_{n-i_{0}+1}v_{n-i_{0}}=E\cdot v_{n-i_{0}+1}=\alpha(E)\cdot
v_{n-i_{0}+1}$ $=\sum_{i\geq 0}[(E^{i+1}h_{i})\cdot(F^{i}\cdot
v_{n-i_{0}+1})]=\sum_{i\geq 0}[(E^{i+1}h_{i})\cdot 0]=0,$
a contradiction, as by repeatly applying the action of $F$,
$F^{i}\cdot v_{n-i_{0}+1}=F^{i-i_{0}}\cdot(F^{i_{0}}\cdot
v_{n-i_{0}+1})=F^{i-i_{0}}\cdot 0=0.$
It follows that $\alpha(E)=Eh$, where $h\in k[K,K^{-1}]-\\{0\\}$. Similarly
$\alpha(F)=gF$, where $g\in k[K,K^{-1}]-\\{0\\}$.
But by the proof of Theorem 1.4, $\alpha(c_{q})=\varepsilon c_{q}$, that is,
$\alpha(EF+\frac{q^{-1}K+qK^{-1}}{(q-q^{-1})^{2}})=\varepsilon
EF+\varepsilon\frac{q^{-1}K+qK^{-1}}{(q-q^{-1})^{2}}$
$=EhgF+\varepsilon\frac{q^{-1}K+qK^{-1}}{(q-q^{-1})^{2}}.$
The uniqueness of expression, due to the $PBW$ type basis, forces that
$\alpha(EF)=EhgF=\varepsilon EF=\pm EF$. It follows that in the case
$\varepsilon=1$, $hg=1$, hence by Lemma 2.1, $h=\lambda K^{r}$,
$g=\lambda^{-1}K^{-r}$ for some $\lambda\in K^{*}$, $m\in\mathbb{Z}$; and in
the case $\varepsilon=-1$, $hg=-1$, hence by Lemma 2.1, $h=\lambda K^{r}$,
$g=-\lambda^{-1}K^{-r}$ for some $\lambda\in K^{*}$, $r\in\mathbb{Z}$.∎
Proof of Proposition 1.6. Denote the $k$-automorphisms of
$U_{q}(\mathfrak{sl}_{2})$ in Theorem 1.5 (1) by $\alpha_{q}(1,1,r)$, in
Theorem 1.5 (2) by $\alpha_{q}(-1,1,r)$, in Theorem 1.5 (3) by
$\alpha_{q}(1,-1,r)$, in Theorem 1.5 (4) by $\alpha_{q}(-1,-1,r)$. Define a
map
$\phi:\
\text{Aut}(U_{q}(\mathfrak{sl}_{2}))\to\text{Aut}(U_{p}(\mathfrak{sl}_{2}))$
by $\phi(\alpha_{q}(a,b,c))=\alpha_{p}(a,b,c)$. One readily checks that $\phi$
is a bijective group homomorphism, hence an isomorphism.∎
## 4\. Acknowledgements
Jie-Tai Yu is grateful to Yangzhou University, Yunnan Normal University,
Shanghai University, Osaka University, Kwansei Gakuin University, Saitama
University, Beijing International Center for Mathematical Research (BICMR) and
Chinese Academy of Sciences for warm hospitality and stimulating environment
during his visits, when this work was carry out. The authors thank Stephane
Launois for valuable references and comments, in particular for pointing out
the references [AC, La1, La2, LL]. The authors also thank I-Chiau Huang and
Shigeru Kuroda for providing the reference [J].
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* [D2] V.G.Drinfeld, Quantum Groups, Proc. ICM, Berkeley, 1986, 798-820.
* [J] M.Jimbo, A $q$-difference analogue of $U(\mathfrak{g})$ and the Yang-Baxter equation, Lett.Math. Phys. 10 (1985) 63-69.
* [Ja] J.C.Jantzen, Lectures on quantum groups, Graduate Studies in Mathematics, Volumn 6, American Mathematical Society, Providence, RI, 1995.
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* [LY] L.-B. Li and J.-T.Yu, Isomorphisms between quantum groups of type $A$, Preprint 2009.
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|
arxiv-papers
| 2009-10-09T11:35:06 |
2024-09-04T02:49:05.744007
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Li-Bin Li and Jie-Tai Yu",
"submitter": "Jie-Tai Yu",
"url": "https://arxiv.org/abs/0910.1713"
}
|
0910.1874
|
# Solutions of the Maxwell equations and photon wave functions
Peter J. Mohr National Institute of Standards and Technology, Gaithersburg,
MD 20899-8420, USA
###### Abstract
Properties of six-component electromagnetic field solutions of a matrix form
of the Maxwell equations, analogous to the four-component solutions of the
Dirac equation, are described. It is shown that the six-component equation,
including sources, is invariant under Lorentz transformations. Complete sets
of eigenfunctions of the Hamiltonian for the electromagnetic fields, which may
be interpreted as photon wave functions, are given both for plane waves and
for angular-momentum eigenstates. Rotationally invariant projection operators
are used to identify transverse or longitudinal electric and magnetic fields.
For plane waves, the velocity transformed transverse wave functions are also
transverse, and the velocity transformed longitudinal wave functions include
both longitudinal and transverse components. A suitable sum over these
eigenfunctions provides a Green function for the matrix Maxwell equation,
which can be expressed in the same covariant form as the Green function for
the Dirac equation. Radiation from a dipole source and from a Dirac atomic
transition current are calculated to illustrate applications of the Maxwell
Green function.
††journal: Annals of Physics
## 1 Introduction
For quantum mechanics to provide a complete description of nature, it is
necessary to have a wave function for something as important as
electromagnetic radiation or photons. This has become increasingly relevant as
the number of experiments on single photon production and detection, motivated
by interest in the fields of quantum computation and quantum cryptography, has
grown rapidly over the past two decades [1]. The history of theoretical
efforts to define photon wave functions dates back to the early days of
quantum mechanics and is still unfolding. Overviews have been given in [2, 3,
4]. However, there is not yet a consensus on the form a photon wave function
should take or the properties it should have. Further investigation of these
questions is warranted, and possible answers are given in this paper.
Quantum electrodynamics (QED) accurately describes the interaction of
radiation with free electrons and electrons bound in atoms, but as it is
formulated in terms of an $S$ matrix, asymptotic states, and Feynman diagrams,
it does not readily lend itself to the description of the time evolution of
radiation. In particular, interference effects or the space-time behavior of a
photon wave packet would be more naturally described in the framework of wave
mechanics with a wave function for a single photon.
There are a number of requirements that need to be be imposed on a formalism
for a quantum mechanical description of photons. First, the predicted behavior
of radiation should be consistent with the Maxwell equations. Time-dependent
solutions of the Maxwell equations provide the basis for both classical
electromagnetic theory and QED, and it can be expected that a photon wave
function should also be based on solutions of the Maxwell equations. This
means that the wave function is simultaneously the solution of both of the
first-order Maxwell equations with time derivatives and not just a solution of
a second-order scalar wave equation.
A second requirement is that the wave functions obey the quantum mechanical
principle of linear superposition. Simply stated, this means that if two wave
functions describe possible states of radiation, then a linear combination of
these wave functions also describes a possible state. For example, a wave
function for circularly polarized radiation can be written as a linear
combination of two wave functions for linearly polarized radiation, all of
which must be solutions of the same wave equation.
Another requirement is that the formalism be Lorentz invariant in order to
properly describe the space-time behavior of radiation. An approximation
scheme like the reduction of the Dirac equation to obtain the Schrödinger
equation for electron velocities that are small compared to the speed of light
is not an option for radiative photons.
Finally, it is necessary for the formalism to provide the tools for methods
associated with quantum mechanics. This includes a wave equation with a
Hamiltonian that describes the time development of states, wave functions that
comprise a complete set of eigenfunctions of the Hamiltonian, normalizable
states with a probability distribution that corresponds to the location of the
photon, a law of conservation of probability, operators with expectation
values for observables, and wave packets that realistically describe the
propagation of photons in space and time.
To arrive at a wave equation that addresses these requirements, we examine an
approach in which the four-component matrix Dirac equation for a spin one-half
electron is adapted to a six-component form of the Maxwell equations for a
spin-one photon. This version of the Maxwell equations is a direct extension
of the Dirac equation for the electron in which two-by-two Pauli matrices are
replaced by analogous three-by-three matrices. Since the quantum mechanical
properties of the Dirac equation, Hamiltonian, and wave functions are well
understood and tested experimentally, it is natural to consider the analogous
Maxwell equation, Hamiltonian, and wave functions as a quantum mechanical
description of photons.
There are fundamental differences between the Dirac equation and the matrix
Maxwell equation, so the extension requires a detailed analysis. The most
prominent difference is the fact that there is a possible source term in the
Maxwell equation which has no analog for the Dirac equation [5]. Also, some
properties of the three-by-three spin matrices differ from those of the Pauli
matrices, even though they have the same commutation relations.
Linear operators that are representations of the inhomogeneous Lorentz group
can replace the wave equation of a system for free electrons or transverse
photons with no sources [6, 7, 8], but the source terms and longitudinal
solutions of the Maxwell equation fall outside this framework. Taking the view
that the Maxwell equation with a source is the most direct contact with
experiment, our approach is to start from the matrix Maxwell equation with a
source term and explicitly work out the Lorentz transformations of the
solutions. It is shown that the six-component equation is invariant under
Lorentz transformations, as it should be, but this is not self-evident, since
the source term is essentially the three-vector current density.
Next, six-component solutions are constructed and shown to be complete sets of
orthogonal coordinate-space eigenfunctions of the Maxwell Hamiltonian,
parameterized by physical properties, such as linear momentum, angular
momentum, and parity. These properties are associated with operators that
commute with the Hamiltonian. Complete sets of both plane-wave solutions and
angular-momentum eigenfunctions are given. Bilinear products of normalizable
linear combinations of these functions provide expressions for the probability
density and flux. The eigenfunctions are further classified according to
whether they represent transverse or longitudinal states. These properties are
associated with the electric and magnetic fields, with the result that under a
velocity boost, the transformed transverse solutions are also transverse,
unlike solutions corresponding to a transverse vector potential. Moreover, by
summing over both transverse and longitudinal solutions, we obtain a covariant
Green function for the Maxwell equation, which is of the same form as the
Green function for the Dirac equation.
Solutions are obtained directly from the Maxwell equation, with no recourse to
a vector potential. This avoids problems such as extra polarization components
and ambiguities associated with gauge transformations [9, 10, 11]. Although
integrals over a closed path of the potential may be observables, as discussed
in [12], such integrals can be expressed in terms of the magnetic flux through
the loop [13], so it is expected that the fields alone provide a complete
description of electrodynamics. It is also clear that photon wave functions
are closely aligned with electric and magnetic fields, and an approach that
starts with classical electrodynamics expressed in terms of fields only
provides a natural framework for the transition to the wave mechanics of
photons. A possible advantage of using a vector potential is that it is the
solution of a scalar wave equation, which has a well-known Green function.
However, this advantage is offset by the fact that we provide a covariant
Green function for the Maxwell equation.
This paper is organized as follows. In Sec. 2 the vector Maxwell equations and
the Dirac equation are stated to define notation. The algebra of three-
component spin matrices is reviewed in Sec. 3, where both a spherical basis,
which is the direct extension of the Pauli matrices to three components, and a
Cartesian basis with real components, are defined. The Maxwell equations are
written in terms of the spherical spin matrices and combined into the Dirac
equation form in Sec. 4. In Sec. 5, transverse and longitudinal projection
operators are defined and used to separate the Maxwell equations and solutions
into the corresponding disjoint sectors. Lorentz invariance is addressed in
Sec. 6, where transformations of the coordinates and derivatives,
transformations of the Maxwell equation, and transformations of the solutions
are explicitly written. In Sec. 7, plane-wave solutions, which are
eigenfunctions of the momentum operator as well as the Hamiltonian, are given
for both transverse and longitudinal states, and the set of solutions is shown
to be complete. The explicit action of Lorentz transformations on the plane-
wave solutions is described. Properties of normalizable wave packets formed
from the plane-wave solutions are illustrated. The angular-momentum operator
and the corresponding eigenfunctions are given and shown to be complete in
Sec. 8. In Sec. 9, the Maxwell Green function is written as an integral over
the plane-wave solutions in a form analogous to the Dirac Green function. As
examples of applications of the Maxwell Green function, formulas for radiation
from a point dipole source and from a Dirac current source are derived in Sec.
10. A summary of the main points of the paper is in Sec. 11 and brief
concluding remarks are made in Sec. 12.
The relation of the present study to earlier work is indicated in the sections
where the particular topics are discussed.
## 2 Three-vector Maxwell equations and the Dirac equation
The Maxwell equations in vacuum, in the International System of Units (SI),
are
$\displaystyle\bm{\nabla\cdot E}$ $\displaystyle=$
$\displaystyle\frac{\rho}{\epsilon_{0}},{}$ (1) $\displaystyle\bm{\nabla\times
B}-\frac{1}{c^{2}}\frac{\partial\bm{E}}{\partial t}$ $\displaystyle=$
$\displaystyle\mu_{0}\bm{J},{}$ (2) $\displaystyle\bm{\nabla\times
E}+\frac{\partial\bm{B}}{\partial t}$ $\displaystyle=$ $\displaystyle 0,{}$
(3) $\displaystyle\bm{\nabla\cdot B}$ $\displaystyle=$ $\displaystyle 0,{}$
(4)
where $\bm{E}$ and $\bm{B}$ are the electric and magnetic fields, $\rho$ and
$\bm{J}$ are the charge and current densities, $\epsilon_{0}$ and $\mu_{0}$
are the electric and magnetic constants, and $c=(\epsilon_{0}\mu_{0})^{-1/2}$
is the speed of light. The continuity equation
$\displaystyle\frac{\partial\rho}{\partial t}+\bm{\nabla\cdot J}$
$\displaystyle=$ $\displaystyle 0{}$ (5)
follows from Eqs. (1) and (2).
The form of the Maxwell equations considered here is analogous to the Dirac
equation for the electron. The Dirac wave function $\phi(x)$ is a four-
component column matrix that is a function of the four-vector $x$. For a free
electron, the Dirac equation is
$\displaystyle\left(\,{\rm i}\,\hbar\gamma^{\mu}\partial_{\mu}-m_{\rm
e}c\right)\phi(x)=0,{}$ (6)
where $\hbar$ is the Planck constant divided by $2\pi$, $m_{\rm e}$ is the
mass of the electron, $\gamma^{\mu}$, $\mu=0,1,2,3$, are the $4\times 4$ Dirac
gamma matrices, given by
$\displaystyle\gamma^{0}=\left(\begin{array}[]{rrr}I&&0\\\
0&&-I\end{array}\right)\\!;\quad\gamma^{i}=\left(\begin{array}[]{rrr}0&&\sigma^{i}\\\
-\sigma^{i}&&0\end{array}\right)\\!,\ i=1,2,3,{}$ (11)
$I$ and $0$ are the $2\times 2$ identity and zero matrices, $\sigma^{i}$,
$i=1,2,3$, are the Pauli spin matrices
$\displaystyle\sigma^{1}=\left(\begin{array}[]{rrr}0&&1\\\
1&&0\end{array}\right)\\!,\ \sigma^{2}=\left(\begin{array}[]{rrr}0&&-{\rm
i}\\\ {\rm i}&&0\end{array}\right)\\!,\
\sigma^{3}=\left(\begin{array}[]{rrr}1&&0\\\ 0&&-1\end{array}\right)\\!,\ {}$
(18)
and the derivatives $\partial_{\mu}$ are
$\displaystyle\partial_{0}=\frac{\partial}{\partial
ct};\quad\partial_{i}=\frac{\partial}{\partial x^{i}},\ i=1,2,3.$ (19)
We take the metric tensor $g^{\mu\nu}$ to be
$\displaystyle g^{00}=1;\quad g^{ii}=-1,\ i=1,2,3;\quad g^{\mu\nu}=0,\
\mu\neq\nu.\quad$ (20)
In terms of the spin matrices, the derivative term in Eq. (6) can be written
as
$\displaystyle\gamma^{\mu}\partial_{\mu}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}I\,{\frac{\textstyle\partial}{\textstyle\partial
ct}}&&\bm{\sigma}\cdot\bm{\nabla}\\\
-\bm{\sigma}\cdot\bm{\nabla}&&-I\,{\frac{\textstyle\partial}{\textstyle\partial
ct}}\end{array}\right)\\!.$ (23)
The Pauli spin matrices act on two-component spin matrices in the electron
wave function. Oppenheimer has suggested that since the Maxwell equations
involve three-vectors, three-component matrices should be considered for
constructing a photon wave function [5]. Here we implement such an extension
by replacing the Pauli spin matrices in the Dirac equation by the analogous
$3\times 3$ matrices described in the next section.
## 3 Three-component spin matrices
As is well known, three-vectors and operations among them are interchangeable
with three-component matrices and matrix operations. In this section, formulas
for these matrices relevant to subsequent work are given. Some of these
formulas have been given in [14]. It is useful to define both Cartesian and
spherical matrices to represent three-vectors.
The Cartesian matrix representing a vector $\bm{a}$ may be written as
$\displaystyle\bm{a}_{\rm c}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}a^{1}\\\ a^{2}\\\
a^{3}\end{array}\right)$ (27)
where $a^{1},~{}a^{2},~{}a^{3}$ are the rectangular components of the vector
$\bm{a}$, and a spherical representation is denoted by
$\displaystyle\bm{a}_{\rm s}$ $\displaystyle=$ $\displaystyle\bm{M}\bm{a}_{\rm
c},{}$ (28)
where $\bm{M}$ is a $3\times 3$ unitary matrix specified in the following. The
dot product of two vectors is
$\displaystyle\bm{a}\cdot\bm{b}$ $\displaystyle=$ $\displaystyle\bm{a}_{\rm
c}^{\dagger}\bm{b}_{\rm c}=\bm{a}_{\rm s}^{\dagger}\bm{b}_{\rm s},{}$ (29)
where $\dagger$ denotes the combined operations of matrix transposition and
complex conjugation.
Explicit Hermitian $\bm{\tau}$ matrices ($\bm{\tau}^{\dagger}=\bm{\tau}$),
which are $3\times 3$ versions of the Pauli matrices, are obtained by taking
$\tau^{3}$ to be diagonal
$\displaystyle\tau^{3}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{rrrrr}1&&0&&0\\\ 0&&0&&0\\\
0&&0&&-1\end{array}\right){}$ (33)
and applying appropriate rotation matrices to obtain $\tau^{1}$ and
$\tau^{2}$:
$\displaystyle{}\tau^{1}$ $\displaystyle=$
$\displaystyle\bm{\mathfrak{D}}^{(1)}(\\{0,{\textstyle{\frac{\pi}{2}}},0\\})\,\tau^{3}\,\bm{\mathfrak{D}}^{(1)}(\\{0,-{\textstyle{\frac{\pi}{2}}},0\\})=\frac{1}{\sqrt{2}}\left(\begin{array}[]{rrrrr}0&&1&&0\\\
1&&0&&1\\\ 0&&1&&0\end{array}\right)\\!,$ (37) $\displaystyle{}\tau^{2}$
$\displaystyle=$
$\displaystyle\bm{\mathfrak{D}}^{(1)}(\\{0,0,{\textstyle{\frac{\pi}{2}}}\\})\,\tau^{1}\,\bm{\mathfrak{D}}^{(1)}(\\{0,0,-{\textstyle{\frac{\pi}{2}}}\\})=\frac{{\rm
i}}{\sqrt{2}}\left(\begin{array}[]{rrrrr}0&&-1&&0\\\ 1&&0&&-1\\\
0&&1&&0\end{array}\right)\\!,$ (41)
where $\bm{\mathfrak{D}}^{(1)}(\\{\alpha,\beta,\gamma\\})$ is the $j=1$
representation of the rotation group, parameterized by the Euler angles
$\alpha,\beta,\gamma$ [15]. In particular,
$\bm{\mathfrak{D}}^{(1)}(\\{0,{\textstyle{\frac{\pi}{2}}},0\\})$ represents
the rotation about the 2 axis by the angle $\pi/2$ and
$\bm{\mathfrak{D}}^{(1)}(\\{0,0,{\textstyle{\frac{\pi}{2}}}\\})$ represents
the rotation about the 3 axis by the angle $\pi/2$. The same rotations
starting from $\sigma^{3}$, with the $j={\textstyle{\frac{1}{2}}}$
representation, reproduce $\sigma^{1}$ and $\sigma^{2}$. The $\bm{\tau}$
matrices are related by
$\displaystyle\left[\tau^{i},\tau^{j}\right]={\rm
i}\,\epsilon_{ijk}\,\tau^{k},$ (42)
where $\epsilon_{ijk}$ is the Levi-Civita symbol.111The tau matrices defined
by Oppenheimer [5] are $\bm{N}^{\dagger}\tau^{i}\bm{N}$, where
$\bm{N}=\left(\begin{array}[]{ccc}1&0&0\\\ 0&{\rm i}&0\\\ 0&0&-1\\\
\end{array}\right)$. [The minus sign in $\tau^{2}$ in Eq. (10) of that paper
apparently is a typographical error, as indicated by inspection of Eq. (11).]
The matrices defined by Majorana [16] are $\bm{M}^{\dagger}\tau^{i}\bm{M}$,
where $\bm{M}$ is given in Eq. (50).
The cross product of two vectors $\bm{a}$ and $\bm{b}$ can be written in terms
of the scalar product of the tau matrices with the vector $\bm{a}$
$\displaystyle\bm{\tau}\cdot\bm{a}$ $\displaystyle=$
$\displaystyle\tau^{i}\,a^{i}$ (43)
acting on the spherical matrix for the vector $\bm{b}$ as
$\displaystyle\bm{\tau}\cdot\bm{a}\ \bm{b}_{\rm s}$ $\displaystyle=$
$\displaystyle{\rm i}\left(\bm{a}\times\bm{b}\right)_{\rm s},{}$ (44)
provided the matrix $\bm{M}$ in Eq. (28) is suitably chosen. To determine
$\bm{M}$, we take the Cartesian definition
$\displaystyle(\bm{a}\times\bm{b})^{i}$ $\displaystyle=$
$\displaystyle\epsilon_{ijk}a^{j}b^{k}$ (45)
and write Eq. (44) as
$\displaystyle\bm{\tau}\cdot\bm{a}\,\bm{M}\,\bm{b}_{\rm c}$ $\displaystyle=$
$\displaystyle{\rm i}\bm{M}(\bm{a}\times\bm{b})_{\rm c}.$ (46)
Imposing the requirement that this equation be valid for any vectors $\bm{a}$
and $\bm{b}$ fixes $\bm{M}$, up to a phase factor, to be
$\displaystyle\bm{M}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\begin{array}[]{ccc}-1&{\rm i}&0\\\
0&0&\sqrt{2}\\\ 1&{\rm i}&0\end{array}\right),{}$ (50)
which yields
$\displaystyle\bm{a}_{\rm s}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}-{1\over\sqrt{2}}(a^{1}-{\rm
i}\,a^{2})\\\ a^{3}\\\ {1\over\sqrt{2}}(a^{1}+{\rm
i}\,a^{2})\end{array}\right).{}$ (54)
Consequences of Eq. (44) are
$\displaystyle\bm{\tau}\cdot\bm{a}\ \bm{a}_{\rm s}$ $\displaystyle=$
$\displaystyle 0,{}$ (55) $\displaystyle\bm{\tau}\cdot\bm{a}\ \bm{b}_{\rm
s}+\bm{\tau}\cdot\bm{b}\ \bm{a}_{\rm s}$ $\displaystyle=$ $\displaystyle 0,{}$
(56) $\displaystyle\bm{a}_{\rm s}^{\dagger}\,\bm{\tau}\cdot\bm{b}\ \bm{c}_{\rm
s}$ $\displaystyle=$ $\displaystyle{\rm i}\,\bm{a}\cdot(\bm{b}\times\bm{c}),$
(57) $\displaystyle(\bm{\tau}\cdot\bm{a}\,\bm{c}_{\rm
s})\cdot(\bm{\tau}\cdot\bm{b}\,\bm{d}_{\rm s})$ $\displaystyle=$
$\displaystyle(\bm{a}\times\bm{c})\cdot(\bm{b}\times\bm{d})$ (58)
$\displaystyle=$ $\displaystyle\bm{a}\cdot\bm{b}\
\bm{c}\cdot\bm{d}-\bm{a}\cdot\bm{d}\ \bm{c}\cdot\bm{b}.{}\quad$
Equation (58) can be written as
$\displaystyle\bm{c}_{\rm
s}^{\dagger}(\bm{\tau}\cdot\bm{a})^{\dagger}\bm{\tau}\cdot\bm{b}\,\bm{d}_{\rm
s}$ $\displaystyle=$ $\displaystyle\bm{c}_{\rm
s}^{\dagger}(\bm{a}\cdot\bm{b}-\bm{b}_{\rm s}\,\bm{a}_{\rm
s}^{\dagger})\bm{d}_{\rm s}\quad$ (59)
for any vectors $\bm{c}$ and $\bm{d}$, which yields the relation
$\displaystyle(\bm{\tau}\cdot\bm{a})^{\dagger}\bm{\tau}\cdot\bm{b}$
$\displaystyle=$ $\displaystyle\bm{a}\cdot\bm{b}-\bm{b}_{\rm s}\bm{a}_{\rm
s}^{\dagger},{}$ (60)
where it is understood that the first term on the right includes the $3\times
3$ identity matrix as a factor and the second term is also a $3\times 3$
matrix. If $\bm{a}_{\rm c}$ has real components, then
$\displaystyle(\bm{\tau}\cdot\bm{a})^{\dagger}$ $\displaystyle=$
$\displaystyle\bm{\tau}\cdot\bm{a},$ (61)
$\displaystyle(\bm{\tau}\cdot\bm{a})^{3}$ $\displaystyle=$
$\displaystyle\bm{a}^{2}\ \bm{\tau}\cdot\bm{a},$ (62)
where $\bm{a}^{2}=\bm{a}\cdot\bm{a}$ is the ordinary real vector scalar
product.
Real Cartesian tau matrices $\bm{\tilde{\tau}}$ may be defined so that
$\displaystyle\bm{\tilde{\tau}}\cdot\bm{a}\ \bm{b}_{\rm c}$ $\displaystyle=$
$\displaystyle\left(\bm{a}\times\bm{b}\right)_{\rm c}.{}$ (63)
This relation follows from Eqs. (44) and (28) with the definition
$\displaystyle\tilde{\tau}^{i}$ $\displaystyle=$ $\displaystyle-{\rm
i}\bm{M}^{\dagger}\,\tau^{i}\,\bm{M},\quad i=1,2,3.$ (64)
These matrices are antisymmetric
$\bm{\tilde{\tau}}^{\top}=-\bm{\tilde{\tau}}$, where $\top$ denotes matrix
transposition, in contrast to $\bm{\tau}^{\dagger}=\bm{\tau}$. For vectors
$\bm{a}$ and $\bm{b}$ with real Cartesian components, we have
$\displaystyle\bm{\tilde{\tau}}\cdot\bm{a}\ \bm{\tilde{\tau}}\cdot\bm{b}$
$\displaystyle=$ $\displaystyle\bm{b}_{\rm c}\bm{a}_{\rm
c}^{\top}-\bm{a}\cdot\bm{b},$ (65)
$\displaystyle(\bm{\tilde{\tau}}\cdot\bm{a})^{3}$ $\displaystyle=$
$\displaystyle-\bm{a}^{2}\,(\bm{\tilde{\tau}}\cdot\bm{a}),$ (66)
$\displaystyle(\bm{\tilde{\tau}}\cdot\bm{a})^{ij}$ $\displaystyle=$
$\displaystyle-\epsilon_{ijk}a^{k}.$ (67)
The matrix
$\displaystyle\bm{\tilde{\tau}}\cdot\bm{a}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}0&-a^{3}&a^{2}\\\ a^{3}&0&-a^{1}\\\
-a^{2}&\ a^{1}&0\end{array}\right){}$ (71)
has the form of the lower right portion of the electromagnetic field-strength
tensor $F^{\mu\nu}$, as given in [17] for example.
## 4 Matrix Maxwell equation
In terms of the notation of the previous section, the matrix forms of the
Maxwell equations in (2) and (3), for the source-free case ($\bm{J}=0$), are
$\displaystyle{\rm i}\,\bm{\tau}\cdot\bm{\nabla}\bm{B}_{\rm
s}+\frac{1}{c}\frac{\partial\bm{E}_{\rm s}}{\partial ct}$ $\displaystyle=$
$\displaystyle 0,{}$ (72) $\displaystyle{\rm
i}\,\bm{\tau}\cdot\bm{\nabla}\bm{E}_{\rm s}-c\frac{\partial\bm{B}_{\rm
s}}{\partial ct}$ $\displaystyle=$ $\displaystyle 0.{}$ (73)
These equations may be written as two uncoupled equations
$\displaystyle\left(\bm{I}\frac{\partial}{\partial
ct}+\bm{\tau}\cdot\bm{\nabla}\right)\left(\bm{E}_{\rm s}+{\rm i}\,c\bm{B}_{\rm
s}\right)=0,{}$ (74) $\displaystyle\left(\bm{I}\frac{\partial}{\partial
ct}-\bm{\tau}\cdot\bm{\nabla}\right)\left(\bm{E}_{\rm s}-{\rm i}\,c\bm{B}_{\rm
s}\right)=0,{}$ (75)
where $\bm{I}$ is the $3\times 3$ identity matrix. In the Cartesian basis,
Eqs. (74) and (75) are
$\displaystyle\left(\bm{I}\frac{\partial}{\partial ct}+{\rm
i}\bm{\tilde{\tau}}\cdot\bm{\nabla}\right)\left(\bm{E}_{\rm c}+{\rm
i}\,c\bm{B}_{\rm c}\right)=0,{}$ (76)
$\displaystyle\left(\bm{I}\frac{\partial}{\partial ct}-{\rm
i}\bm{\tilde{\tau}}\cdot\bm{\nabla}\right)\left(\bm{E}_{\rm c}-{\rm
i}\,c\bm{B}_{\rm c}\right)=0.{}$ (77)
In these expressions, it is evident that for real electric and magnetic
fields, Eqs. (76) and (77) are complex conjugates of each other and reduce to
a single complex equation. It was recognized in lectures by Riemann in the
nineteenth century that this complex combination of $\bm{E}$ and $\bm{B}$ is a
solution of a single equation [18]. This fact was also discussed in [19, 20]
and is included in many works up to the present. Equations (76) and (77) may
be interpreted as Maxwell equations for right- and left-circularly polarized
radiation, analogous to the Weyl equations for right- and left-handed neutrino
fields [8, 14].
However, in this paper, we consider the more restrictive case of complex
electric and magnetic fields that are simultaneously solutions of both Eqs
(76) and (77), or equivalently both Eqs. (72) and (73), for any polarization
of radiation. The question of whether such solutions can be found is answered
by their explicit construction in subsequent sections of the paper. To
formulate this approach, we follow the Dirac equation and write
$\displaystyle\left(\begin{array}[]{ccc}\bm{I}\,{\frac{\textstyle\partial}{\textstyle\partial
ct}}&&\bm{\tau}\cdot\bm{\nabla}\\\
-\bm{\tau}\cdot\bm{\nabla}&&-\bm{I}\,{\frac{\textstyle\partial}{\textstyle\partial
ct}}\end{array}\right)\left(\begin{array}[]{c}\bm{E}_{\rm s}\\\ {\rm
i}\,c\bm{B}_{\rm s}\vbox to15.0pt{}\end{array}\right)=0,{}$ (82)
which is a restatement of Eqs. (72) and (73) in the form of the Dirac equation
for an electron wave function. It is a matrix equation with six components
that may be viewed as a single equation equivalent to Eqs. (72) and (73) for
any polarization of the fields. Any complex solution of Eq. (82) is a solution
of both Eqs. (74) and (75). Similar wave functions have been discussed in [21,
22, 23]. It should be noted that this formulation is different from the six-
component form considered by Oppenheimer in which the upper-three components
and lower-three components represent opposite helicity states [5].
If we define $6\times 6$ gamma matrices by
$\displaystyle\gamma^{0}=\left(\begin{array}[]{rrr}\bm{I}&&{\bm{0}}\\\
{\bm{0}}&&-\bm{I}\end{array}\right)\\!;\quad\gamma^{i}=\left(\begin{array}[]{rrr}{\bm{0}}&&\tau^{i}\\\
-\tau^{i}&&{\bm{0}}\end{array}\right)\\!,\ i=1,2,3,\quad{}$ (87)
where ${\bm{0}}$ is the $3\times 3$ zero matrix, and write
$\displaystyle{\it\Psi}(x)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}\bm{E}_{\rm s}(x)\\\ {\rm
i}\,c\bm{B}_{\rm s}(x)\vbox to15.0pt{}\end{array}\right),{}$ (90)
then Eq. (82) takes the covariant Dirac equation form
$\displaystyle\gamma^{\mu}\partial_{\mu}{\it\Psi}(x)$ $\displaystyle=$
$\displaystyle 0,{}$ (91)
which provides a concise expression for two of the Maxwell equations. We can
also write this as
$\displaystyle\overline{{\it\Psi}}(x)\overleftarrow{\partial}_{\mu}{\gamma^{\mu}}$
$\displaystyle=$ $\displaystyle 0,{}$ (92)
where $\overline{{\it\Psi}}(x)={\it\Psi}^{\dagger}(x)\,\gamma^{0}$ and
$\overleftarrow{\partial}_{\mu}$ denotes differentiation of the function to
the left. Although these equations are simply algebraic rearrangements of the
two Maxwell equations, the resemblance to the Dirac equation and wave function
is suggestive of a form that photon wave functions might take.
It is of interest to note that for solutions of the Dirac equation for the
hydrogen atom, the lower two components are small and approach zero in the
nonrelativistic limit, i.e., as the velocity of the bound electron approaches
zero. Similarly, for local electromagnetic fields generated by moving charges,
the magnetic field, given by the lower three components of ${\it\Psi}$, also
approaches zero in the limit as the velocity of the charges approaches zero.
To take source currents into account, Eq. (2) is written as
$\displaystyle{\rm i}\,\bm{\tau}\cdot\bm{\nabla}\bm{B}_{\rm
s}+\frac{1}{c}\frac{\partial\bm{E}_{\rm s}}{\partial ct}$ $\displaystyle=$
$\displaystyle-\mu_{0}\bm{J}_{\rm s},{}$ (93)
and a source term ${\it\Xi}$ is defined to be
$\displaystyle{\it\Xi}(x)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}-\mu_{0}c\bm{J}_{\rm s}(x)\\\
{\bm{0}\vbox to15.0pt{}}\end{array}\right),{}$ (96)
where ${\bm{0}}$ is a $3\times 1$ matrix of zeros. This yields the expressions
$\displaystyle\gamma^{\mu}\partial_{\mu}{\it\Psi}(x)$ $\displaystyle=$
$\displaystyle{\it\Xi}(x){}$ (97)
and
$\displaystyle\overline{{\it\Psi}}(x)\overleftarrow{\partial}_{\mu}{\gamma^{\mu}}$
$\displaystyle=$ $\displaystyle\overline{{\it\Xi}}(x),{}$ (98)
either of which is referred to as the Maxwell equation here. The source term
in Eq. (97) or (98) represents a fundamental difference between the Dirac
equation and the Maxwell equation, as mentioned in Sec. 1 [5].
In this framework, an energy-momentum density operator is
$\displaystyle p^{\mu}$ $\displaystyle=$
$\displaystyle\frac{\epsilon_{0}}{2c}\,\gamma^{\mu},{}$ (99)
which gives
$\displaystyle\overline{{\it\Psi}}\,cp^{0}{\it\Psi}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\epsilon_{0}|\bm{E}|^{2}+\frac{1}{\mu_{0}}|\bm{B}|^{2}\right)=u,$
(100) $\displaystyle\overline{{\it\Psi}}\,\bm{p}{\it\Psi}$ $\displaystyle=$
$\displaystyle\frac{{\rm i}\,\epsilon_{0}}{2}\left(\bm{E}_{\rm
s}^{\dagger}\bm{\tau}\bm{B}_{\rm s}-\bm{B}_{\rm
s}^{\dagger}\bm{\tau}\bm{E}_{\rm s}\right)$ (101) $\displaystyle=$
$\displaystyle\frac{1}{c^{2}\mu_{0}}\,{\rm
Re}\,\bm{E}\\!\times\bm{B}^{*}=\bm{g}.{}$
Eqs. (97) and (98) imply that
$\displaystyle\partial_{\mu}\overline{{\it\Psi}}(x){\gamma^{\mu}}{\it\Psi}(x)$
$\displaystyle=$
$\displaystyle\overline{{\it\Xi}}(x){\it\Psi}(x)+\overline{{\it\Psi}}(x)\,{\it\Xi}(x),\quad{}$
(102)
which is a complex form of the Poynting theorem [see Eq. (29)]
$\displaystyle\frac{\partial u}{\partial t}+\bm{\nabla}\cdot\bm{S}=-{\rm
Re}\,\bm{E}\cdot\bm{J},$ (103)
where
$\displaystyle\bm{S}=c^{2}\bm{g},{}$ (104)
which gives the conventional result if the fields and current are real [17].
## 5 Transverse and longitudinal fields
To make a Helmholtz decomposition of electromagnetic fields expressed in
matrix form into transverse and longitudinal components, we define $3\times 3$
matrix transverse and longitudinal Hermitian projection operators
$\bm{{\it\Pi}}_{\rm s}^{\rm T}(\bm{a})$ and $\bm{{\it\Pi}}_{\rm s}^{\rm
L}(\bm{a})$ to be
$\displaystyle\bm{{\it\Pi}}_{\rm s}^{\rm T}(\bm{a})$ $\displaystyle=$
$\displaystyle\frac{(\bm{\tau}\cdot\bm{a})^{\dagger}(\bm{\tau}\cdot\bm{a})}{\bm{a}\cdot\bm{a}},$
(105) $\displaystyle\bm{{\it\Pi}}_{\rm s}^{\rm L}(\bm{a})$ $\displaystyle=$
$\displaystyle\frac{\bm{a}_{\rm s}\bm{a}_{\rm
s}^{\dagger}}{\bm{a}\cdot\bm{a}}.$ (106)
Based on identities in Sec. 3, these operators have the following properties:
$\displaystyle\left[\bm{{\it\Pi}}_{\rm s}^{\rm T}(\bm{a})\right]^{2}$
$\displaystyle=$ $\displaystyle\bm{{\it\Pi}}_{\rm s}^{\rm T}(\bm{a}),$ (107)
$\displaystyle\left[\bm{{\it\Pi}}_{\rm s}^{\rm L}(\bm{a})\right]^{2}$
$\displaystyle=$ $\displaystyle\bm{{\it\Pi}}_{\rm s}^{\rm L}(\bm{a}),$ (108)
$\displaystyle\bm{{\it\Pi}}_{\rm s}^{\rm T}(\bm{a})+\bm{{\it\Pi}}_{\rm s}^{\rm
L}(\bm{a})$ $\displaystyle=$ $\displaystyle\bm{I},$ (109)
$\displaystyle\bm{{\it\Pi}}_{\rm s}^{\rm T}(\bm{a})\,\bm{{\it\Pi}}_{\rm
s}^{\rm L}(\bm{a})$ $\displaystyle=$ $\displaystyle 0,$ (110)
$\displaystyle\bm{{\it\Pi}}_{\rm s}^{\rm T}(\bm{a})\,\bm{a}_{\rm s}$
$\displaystyle=$ $\displaystyle 0,$ (111) $\displaystyle\bm{{\it\Pi}}_{\rm
s}^{\rm L}(\bm{a})\,\bm{a}_{\rm s}$ $\displaystyle=$ $\displaystyle\bm{a}_{\rm
s}.$ (112)
Acting on the matrix of an arbitrary vector $\bm{b}$, the operators project
the components perpendicular to and parallel to the argument $\bm{a}$
$\displaystyle\bm{{\it\Pi}}_{\rm s}^{\rm T}(\bm{a})\,\bm{b}_{\rm s}$
$\displaystyle=$ $\displaystyle\bm{b}_{\rm
s}-\frac{\bm{a}\cdot\bm{b}}{\bm{a}\cdot\bm{a}}\,\bm{a}_{\rm s},$ (113)
$\displaystyle\bm{{\it\Pi}}_{\rm s}^{\rm L}(\bm{a})\,\bm{b}_{\rm s}$
$\displaystyle=$
$\displaystyle\frac{\bm{a}\cdot\bm{b}}{\bm{a}\cdot\bm{a}}\,\bm{a}_{\rm s}.$
(114)
In addition to these algebraic relations, usefulness of the projection
operators arises from an extension to include differential and integral
operations acting on coordinate-space functions. Formally, we write
$\displaystyle\bm{{\it\Pi}}_{\rm s}^{\rm T}(\bm{\nabla})$ $\displaystyle=$
$\displaystyle\frac{(\bm{\tau}\cdot\bm{\nabla})^{2}}{\bm{\nabla}^{2}}\,,{}$
(115) $\displaystyle\bm{{\it\Pi}}_{\rm s}^{\rm L}(\bm{\nabla})$
$\displaystyle=$ $\displaystyle\frac{\bm{\nabla}_{\rm s}\bm{\nabla}_{\rm
s}^{\dagger}}{\bm{\nabla}^{2}}\,,{}$ (116)
which takes into account that fact that $\bm{\nabla}$ has real Cartesian
components (in the sense that they give real values when acting on a real
function). The inverse Laplacian is defined by the relation
$\displaystyle\frac{1}{\bm{\nabla}^{2}}\,f(\bm{x})$ $\displaystyle=$
$\displaystyle-\frac{1}{4\pi}\int{\rm
d}\,\bm{x}^{\prime}\,\frac{1}{|\bm{x}-\bm{x}^{\prime}|}\,f(\bm{x}^{\prime}),{}$
(117)
which yields
$\displaystyle\bm{\nabla}^{2}\,\frac{1}{\bm{\nabla}^{2}}\,f(\bm{x})$
$\displaystyle=$ $\displaystyle-\frac{1}{4\pi}\int{\rm
d}\,\bm{x}^{\prime}\,\bm{\nabla}^{2}\frac{1}{|\bm{x}-\bm{x}^{\prime}|}\,f(\bm{x}^{\prime})=f(\bm{x}),{}$
(118)
based on
$\displaystyle\bm{\nabla}^{2}\,\frac{1}{|\bm{x}-\bm{x}^{\prime}|}$
$\displaystyle=$ $\displaystyle-4\pi\,\delta(\bm{x}-\bm{x}^{\prime}),$ (119)
where
$\displaystyle\delta(\bm{x}-\bm{x}^{\prime})$ $\displaystyle=$
$\displaystyle\delta(x^{1}-x^{\prime 1})\,\delta(x^{2}-x^{\prime
2})\,\delta(x^{3}-x^{\prime 3}).\qquad$ (120)
Equation (118) indicates that the Laplacian operator follows analogs of the
rules of algebra in this context. For example, we have
$\displaystyle\left[\bm{{\it\Pi}}_{\rm s}^{\rm T}(\bm{\nabla})\right]^{2}$
$\displaystyle=$
$\displaystyle\left[\frac{(\bm{\tau}\cdot\bm{\nabla})^{2}}{\bm{\nabla}^{2}}\right]^{2}=\frac{\bm{\nabla}^{2}\,(\bm{\tau}\cdot\bm{\nabla})^{2}}{\bm{\nabla}^{2}\bm{\nabla}^{2}}$
(121) $\displaystyle=$
$\displaystyle\frac{(\bm{\tau}\cdot\bm{\nabla})^{2}}{\bm{\nabla}^{2}}=\bm{{\it\Pi}}_{\rm
s}^{\rm T}(\bm{\nabla})\,,$
where either simply canceling the $\bm{\nabla}^{2}$ factors in the numerator
and denominator or applying the definition in Eq. (117) for the operators
acting on a suitable function gives the same result. Transverse and
longitudinal components of the electric and magnetic fields and the current
density are identified by writing
$\displaystyle\bm{F}_{\rm s}$ $\displaystyle=$ $\displaystyle\bm{F}_{\rm
s}^{\rm T}+\bm{F}_{\rm s}^{\rm L},$ (122)
where
$\displaystyle\bm{F}_{\rm s}^{\rm T}$ $\displaystyle=$
$\displaystyle\bm{{\it\Pi}}_{\rm s}^{\rm T}(\bm{\nabla})\bm{F}_{\rm s},$ (123)
$\displaystyle\bm{F}_{\rm s}^{\rm L}$ $\displaystyle=$
$\displaystyle\bm{{\it\Pi}}_{\rm s}^{\rm L}(\bm{\nabla})\bm{F}_{\rm s},$ (124)
and $\bm{F}_{\rm s}$ may be any of $\bm{E}_{\rm s}$, $\bm{B}_{\rm s}$, or
$\bm{J}_{\rm s}$.
The separation of the Maxwell equations into two independent sets of equations
which involve either transverse components or longitudinal components takes
the following form. In terms of the spherical matrices, Eq. (1) is
$\displaystyle\bm{\nabla}_{\rm s}^{\dagger}\bm{E}_{\rm s}^{\rm L}$
$\displaystyle=$ $\displaystyle\frac{\rho}{\epsilon_{0}},{}$ (125)
where the transverse component of the electric field is absent, because
$\bm{\nabla}_{\rm s}^{\dagger}\bm{{\it\Pi}}_{\rm s}^{\rm T}(\bm{\nabla})=0$.
Equations (1) and (125) are equivalent, in the sense that each can be derived
from the other; Eq. (1) follows from Eq. (125) if the vanishing transverse
component is added to the latter equation. In the separated form, it is
evident that the equation neither contains information about or places any
constraint on the transverse component $\bm{E}_{\rm s}^{\rm T}$. The
transverse and longitudinal projection operators acting on Eq. (93), the
matrix form of Eq. (2), yield
$\displaystyle{\rm i}\,\bm{\tau}\cdot\bm{\nabla}\bm{B}_{\rm s}^{\rm
T}+\frac{1}{c}\frac{\partial\bm{E}_{\rm s}^{\rm T}}{\partial ct}$
$\displaystyle=$ $\displaystyle-\mu_{0}\bm{J}_{\rm s}^{\rm T},{}$ (126)
$\displaystyle\frac{1}{c}\frac{\partial\bm{E}_{\rm s}^{\rm L}}{\partial ct}$
$\displaystyle=$ $\displaystyle-\mu_{0}\bm{J}_{\rm s}^{\rm L}{}$ (127)
respectively, which take into account the commutation relation
$[\bm{{\it\Pi}}_{\rm s}^{\rm T}(\bm{\nabla}),\bm{\tau}\cdot\bm{\nabla}]=0$ and
the fact that $\bm{{\it\Pi}}_{\rm s}^{\rm
L}(\bm{\nabla})\,\bm{\tau}\cdot\bm{\nabla}=0$. Together, these equations are
equivalent to Eq. (93) which can be restored by writing the sum of Eq. (126)
and Eq. (127) and adding the term that vanishes. Evidently, this pair of
equations is independent of $\bm{B}_{\rm s}^{\rm L}$. Similarly, Eq. (3), or
equivalently Eq. (73), can be written as the pair
$\displaystyle{\rm i}\,\bm{\tau}\cdot\bm{\nabla}\bm{E}_{\rm s}^{\rm
T}-c\frac{\partial\bm{B}_{\rm s}^{\rm T}}{\partial ct}$ $\displaystyle=$
$\displaystyle 0,{}$ (128) $\displaystyle\frac{\partial\bm{B}_{\rm s}^{\rm
L}}{\partial ct}$ $\displaystyle=$ $\displaystyle 0,{}$ (129)
which are independent of $\bm{E}_{\rm s}^{\rm L}$. Equation (4) takes the form
$\displaystyle\bm{\nabla}_{\rm s}^{\dagger}\bm{B}_{\rm s}^{\rm L}$
$\displaystyle=$ $\displaystyle 0,{}$ (130)
independent of $\bm{B}_{\rm s}^{\rm T}$. The transverse and longitudinal
equations comprise two independent sets.
Six-dimensional transverse and longitudinal projection operators are defined
by
$\displaystyle{\it\Pi}^{\rm T}(\bm{\nabla})$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}\bm{{\it\Pi}}_{\rm s}^{\rm
T}(\bm{\nabla})&&{\bm{0}}\\\ {\bm{0}}&&\bm{{\it\Pi}}_{\rm s}^{\rm
T}(\bm{\nabla})\vbox to15.0pt{}\end{array}\right),{}$ (133)
$\displaystyle{\it\Pi}^{\rm L}(\bm{\nabla})$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}\bm{{\it\Pi}}_{\rm s}^{\rm
L}(\bm{\nabla})&&{\bm{0}}\\\ {\bm{0}}&&\bm{{\it\Pi}}_{\rm s}^{\rm
L}(\bm{\nabla})\vbox to15.0pt{}\end{array}\right),{}$ (136)
where ${\bm{0}}$ is the $3\times 3$ matrix of zeros, ${\it\Pi}^{\rm
T}(\bm{\nabla})+{\it\Pi}^{\rm L}(\bm{\nabla})={\cal I}$, and ${\cal I}$ is the
$6\times 6$ identity matrix. The transverse equations are summarized by
writing
$\displaystyle\gamma^{\mu}\partial_{\mu}{\it\Psi}^{\rm T}(x)$ $\displaystyle=$
$\displaystyle{\it\Xi}^{\rm T}(x),{}$ (137)
where
$\displaystyle{\it\Psi}^{\rm T}(x)$ $\displaystyle=$
$\displaystyle{\it\Pi}^{\rm T}(\bm{\nabla}){\it\Psi}(x),$ (138)
$\displaystyle{\it\Xi}^{\rm T}(x)$ $\displaystyle=$
$\displaystyle{\it\Pi}^{\rm T}(\bm{\nabla})\,{\it\Xi}(x).$ (139)
Equation (137) also follows directly from Eq. (97) and the fact that
$\left[{\it\Pi}^{\rm T}(\bm{\nabla}),\gamma^{\mu}\partial_{\mu}\right]=0$. The
longitudinal equations are Eqs. (125), (127), (129), and (130), together with
the continuity equation, Eq. (5), which can be expressed as
$\displaystyle\frac{\partial\rho}{\partial t}+\bm{\nabla}_{\rm
s}^{\dagger}\bm{J}_{\rm s}^{\rm L}$ $\displaystyle=$ $\displaystyle 0.{}$
(140)
Since the continuity equation follows from Eqs. (125) and (127), it is not
necessary to include it in an independent set of equations; it is listed here
only to show that it provides no restriction on $\bm{J}_{\rm s}^{\rm T}$.
Equations (129) and (130) are eliminated from consideration by taking
$\displaystyle\bm{B}_{\rm s}^{\rm L}=0.{}$ (141)
Constant fields are eliminated by the requirement that static fields vanish at
infinite distances for finite source distributions. However, a constant
magnetic field may be approximated by the field at the center of a current
loop with a radius that is large compared to the extent of the region of
interest. Such a steady-state current density is transverse, as shown by Eq.
(127), and so the magnetic field, given by Eq. (126), is also transverse,
which is consistent with Eq. (141). A complete set of equations, equivalent to
the set of Maxwell equations, is provided by Eqs. (97), (125), and (141), and
the transverse fields are completely described by Eq. (137).
## 6 Lorentz transformations
Lorentz transformations of the matrix Maxwell equation are examined here in
order to confirm that this form of the Maxwell equations is Lorentz invariant.
We adopt the convention that transformations apply to the physical system
rather than to the observer’s coordinates.
To represent four-vector coordinates, the Cartesian matrices are extended to
include a time component $x^{0}=ct$, so coordinate vectors take the form
$\displaystyle x=\left(\begin{array}[]{c}x^{0}\\\ x^{1}\\\ x^{2}\\\
x^{3}\end{array}\right)=\left(\begin{array}[]{c}ct\\\ \bm{x}_{\rm
c}\end{array}\right).$ (148)
We employ the Cartesian basis for coordinate and momentum vectors and the
spherical basis for fields and currents, with a few exceptions that will be
apparent. The notation $x$ represents either the four-coordinate argument of a
function or a column matrix, depending on the context. It is sufficient for
our purpose to consider only homogeneous Lorentz transformations and to
consider rotations and velocity transformations separately. These
transformations acting on four-vectors leave the scalar product
$\displaystyle x\cdot x$ $\displaystyle=$ $\displaystyle
x^{\top}g\,x=(ct)^{2}-\bm{x}^{2}{}$ (149)
invariant, where $g$ is the metric tensor given by
$\displaystyle g=\left(\begin{array}[]{ccc}1&&{\bm{0}}\\\
{\bm{0}}&&-\bm{I}\vbox to15.0pt{}\end{array}\right).{}$ (152)
A remark on notation is that a boldface ${\bm{0}}$ means either a $3\times 3$,
a $1\times 3$, or a $3\times 1$ rectangular array of zeros, as appropriate. We
take the liberty of using an ordinary zero on the right-hand side of equations
to mean whatever sort of zero matches the left-hand side.
### 6.1 Rotation of coordinates
Rotations are parameterized by a vector $\bm{u}=\theta\bm{\hat{u}}$, where
$\bm{\hat{u}}$ is a unit vector in the direction of the axis of the rotation
and $\theta$ is the angle of rotation. An infinitesimal rotation
$\delta\theta\,\bm{\hat{u}}$ changes the point at position $\bm{x}$ to the
point at position $\bm{x}^{\prime}$, where
$\displaystyle\bm{x}^{\prime}$ $\displaystyle=$
$\displaystyle\bm{x}+\delta\theta\,\bm{\hat{u}}\times\bm{x}+\dots\ ,$ (153)
or
$\displaystyle\bm{x}_{\rm c}^{\prime}$ $\displaystyle=$
$\displaystyle\left(\bm{I}+\delta\theta\,\bm{\tilde{\tau}}\cdot\bm{\hat{u}}\right)\bm{x}_{\rm
c}+\dots\ .$ (154)
For a finite rotation, the operation is exponentiated to give
$\displaystyle\bm{x}_{\rm c}^{\prime}$ $\displaystyle=$ $\displaystyle{\rm
e}^{\bm{\tilde{\tau}}\cdot\bm{u}}\bm{x}_{\rm c}=\bm{R}_{\rm
c}(\bm{u})\,\bm{x}_{\rm c}.{}$ (155)
Expansion of the exponential function in powers of $\theta$, taking into
account the fact that
$(\bm{\tilde{\tau}}\cdot\bm{\hat{u}})^{3}=-\bm{\tilde{\tau}}\cdot\bm{\hat{u}}$,
yields
$\displaystyle\bm{R}_{\rm c}(\bm{u})$ $\displaystyle=$
$\displaystyle\bm{I}+(\bm{\tilde{\tau}}\cdot\bm{\hat{u}})^{2}\left(1-\cos{\theta}\right)+\bm{\tilde{\tau}}\cdot\bm{\hat{u}}\,\sin{\theta}\quad$
(156) $\displaystyle=$ $\displaystyle\bm{\hat{u}}_{\rm c}\bm{\hat{u}}_{\rm
c}^{\top}-(\bm{\tilde{\tau}}\cdot\bm{\hat{u}})^{2}\,\cos{\theta}+\bm{\tilde{\tau}}\cdot\bm{\hat{u}}\,\sin{\theta}.$
Evidently, $\bm{R}_{\rm c}^{-1}(\bm{u})=\bm{R}_{\rm c}(-\bm{u})=\bm{R}_{\rm
c}^{\top}(\bm{u})$. It is confirmed that this operator has the appropriate
action on a vector by calculating
$\displaystyle\bm{x}^{\prime}$ $\displaystyle=$
$\displaystyle\bm{\hat{u}}\,\bm{\hat{u}}\cdot\bm{x}-\bm{\hat{u}}\times(\bm{\hat{u}}\times\bm{x})\cos{\theta}+\bm{\hat{u}}\times\bm{x}\sin{\theta}.\qquad{}$
(157)
We use the notation
$\displaystyle\bm{x}^{\prime}=\bm{R}(\bm{u})\bm{x}{}$ (158)
to represent the transformation in Eq. (157).
Rotations of a four-vector only change the spatial coordinates and are written
as
$\displaystyle x^{\prime}$ $\displaystyle=$ $\displaystyle
R(\bm{u})\,x=\left(\begin{array}[]{c}ct\\\ \bm{R}_{\rm c}(\bm{u})\,\bm{x}_{\rm
c}\vbox to15.0pt{}\end{array}\right),$ (161)
where
$\displaystyle R(\bm{u})$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}1&&{\bm{0}}\\\ {\bm{0}}&&\bm{R}_{\rm
c}(\bm{u})\vbox to15.0pt{}\end{array}\right).{}$ (164)
The scalar product $x\cdot x$ is invariant under rotations, since $\bm{x}^{2}$
is invariant.
The spatial coordinate rotation operator in the spherical basis, which follows
from
$\displaystyle\bm{R}_{\rm s}(\bm{u})$ $\displaystyle=$
$\displaystyle\bm{M}\bm{R}_{\rm c}(\bm{u})\bm{M}^{\dagger},{}$ (165)
is
$\displaystyle\bm{R}_{\rm s}(\bm{u})$ $\displaystyle=$ $\displaystyle{\rm
e}^{-{\rm i}\bm{\tau}\cdot\bm{u}}=\bm{\hat{u}}_{\rm s}\bm{\hat{u}}_{\rm
s}^{\dagger}+(\bm{\tau}\cdot\bm{\hat{u}})^{2}\,\cos{\theta}-{\rm
i}\,\bm{\tau}\cdot\bm{\hat{u}}\,\sin{\theta},\qquad{}$ (166)
and $\bm{R}_{\rm s}^{-1}(\bm{u})=\bm{R}_{\rm s}(-\bm{u})=\bm{R}_{\rm
s}^{\dagger}(\bm{u})$. Starting from the geometrical constraint that the
rotated cross product of two vectors is the cross product of the rotated
vectors, written as
$\displaystyle\bm{R}_{\rm s}(\bm{u})(\bm{a}\times\bm{b})_{\rm s}$
$\displaystyle=$ $\displaystyle(\bm{a}^{\prime}\times\bm{b}^{\prime})_{\rm
s},$ (167)
we have
$\displaystyle\bm{R}_{\rm s}(\bm{u})\,\bm{\tau}\cdot\bm{a}\,\bm{b}_{\rm s}$
$\displaystyle=$ $\displaystyle\bm{\tau}\cdot\bm{a}^{\prime}\,\bm{b}_{\rm
s}^{\prime}=\bm{\tau}\cdot\bm{a}^{\prime}\bm{R}_{\rm s}(\bm{u})\,\bm{b}_{\rm
s}.$ (168)
Since this relation holds for any vector $\bm{b}$, it yields
$\displaystyle\bm{R}_{\rm s}(\bm{u})\,\bm{\tau}\cdot\bm{a}$ $\displaystyle=$
$\displaystyle\bm{\tau}\cdot\bm{a}^{\prime}\bm{R}_{\rm s}(\bm{u}){}$ (169)
and
$\displaystyle\bm{R}_{\rm s}(\bm{u})\,\bm{\tau}\cdot\bm{a}\,\bm{R}^{-1}_{\rm
s}(\bm{u})$ $\displaystyle=$ $\displaystyle\bm{\tau}\cdot\bm{a}^{\prime}.{}$
(170)
A direct calculation provides the same result.
The relation between the rotated and the unrotated gradient operators is given
by
$\displaystyle{\nabla^{\prime}}^{i}$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial{x^{\prime}}^{i}}=\frac{\partial
x^{j}}{\partial{x^{\prime}}^{i}}\,\frac{\partial}{\partial
x^{j}}=\frac{\partial x^{j}}{\partial{x^{\prime}}^{i}}\,\nabla^{j},$ (171)
where, from Eq. (155), we have
$\displaystyle\frac{\partial x^{j}}{\partial{x^{\prime}}^{i}}$
$\displaystyle=$ $\displaystyle\bm{R}_{{\rm c}\,ji}^{-1}(\bm{u})=\bm{R}_{{\rm
c}\,ij}(\bm{u}),$ (172)
so that
$\displaystyle\bm{\nabla}_{\rm c}^{\prime}$ $\displaystyle=$
$\displaystyle\bm{R}_{\rm c}(\bm{u})\,\bm{\nabla}_{\rm c},$ (173)
and from Eq. (165),
$\displaystyle\bm{\nabla}_{\rm s}^{\prime}$ $\displaystyle=$
$\displaystyle\bm{R}_{\rm s}(\bm{u})\,\bm{\nabla}_{\rm s}.{}$ (174)
Since the spherical gradient operator transforms as a spherical vector, we
also have
$\displaystyle\bm{R}_{\rm
s}(\bm{u})\,\bm{\tau}\cdot\bm{\nabla}\bm{R}^{-1}_{\rm s}(\bm{u})$
$\displaystyle=$ $\displaystyle\bm{\tau}\cdot\bm{\nabla}^{\prime}{}$ (175)
from Eq. (170). Equations (174) and (175) imply that transverse and
longitudinal projection operators transform according to
$\displaystyle\bm{{\it\Pi}}_{\rm s}^{\rm T}(\bm{\nabla}^{\prime})$
$\displaystyle=$ $\displaystyle\bm{R}_{\rm s}(\bm{u})\,\bm{{\it\Pi}}_{\rm
s}^{\rm T}(\bm{\nabla})\bm{R}^{-1}_{\rm s}(\bm{u}),{}$ (176)
$\displaystyle\bm{{\it\Pi}}_{\rm s}^{\rm L}(\bm{\nabla}^{\prime})$
$\displaystyle=$ $\displaystyle\bm{R}_{\rm s}(\bm{u})\,\bm{{\it\Pi}}_{\rm
s}^{\rm L}(\bm{\nabla})\bm{R}^{-1}_{\rm s}(\bm{u}).{}$ (177)
The action of the inverse Laplacian in terms of the rotated coordinates is the
same as it is for unrotated coordinates, which follows either because
$\nabla^{\prime 2}=\nabla^{2}$ from Eq. (174) or by the definition in Eq.
(117), taking into account the fact that the Jacobian for a rotation is unity.
### 6.2 Velocity transformation of coordinates
Velocity transformations are parameterized by a velocity vector
$\bm{v}=c\,\tanh{\zeta}\,\bm{\hat{v}}$. If a space-time point is given an
infinitesimal velocity boost of $\delta\zeta\,c\,\bm{\hat{v}}$, its spatial
coordinate will change to
$\displaystyle\bm{x}^{\prime}$ $\displaystyle=$
$\displaystyle\bm{x}+\delta\zeta\,ct\,\bm{\hat{v}}+\dots\ ,$ (178)
and its time coordinate must transform in such a way that the scalar product
is invariant. In particular, we require $x^{\prime}\cdot x^{\prime}=x\cdot x$,
which yields
$\displaystyle ct^{\prime}$ $\displaystyle=$ $\displaystyle
ct+\delta\zeta\,\bm{\hat{v}}\cdot\bm{x}+\dots\ .$ (179)
The complete infinitesimal transformation is
$\displaystyle\left(\begin{array}[]{c}ct^{\prime}\\\ \bm{x}_{\rm
c}^{\prime}\end{array}\right)$ $\displaystyle=$
$\displaystyle\left[I+\delta\zeta\left(\begin{array}[]{ccc}0&&\bm{\hat{v}}_{\rm
c}^{\top}\\\ \bm{\hat{v}}_{\rm
c}&&{\bm{0}}\end{array}\right)\right]\left(\begin{array}[]{c}ct\\\ \bm{x}_{\rm
c}\end{array}\right)+\dots\ .\qquad$ (186)
This may be written in terms of a $4\times 4$ matrix valued function of the
velocity direction:
$\displaystyle K(\bm{\hat{v}})$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}0&\bm{\hat{v}}_{\rm c}^{\top}\\\
\bm{\hat{v}}_{\rm c}&{\bm{0}}\end{array}\right),$ (189)
for which
$\displaystyle K^{2}(\bm{\hat{v}})$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}1&{\bm{0}}\\\
{\bm{0}}&\bm{\hat{v}}_{\rm c}\bm{\hat{v}}_{\rm c}^{\top}\end{array}\right)$
(192)
and $K^{3}(\bm{\hat{v}})=K(\bm{\hat{v}})$. For a finite velocity, the
transformation is exponentiated to give
$\displaystyle x^{\prime}$ $\displaystyle=$ $\displaystyle{\rm e}^{\zeta
K(\bm{\hat{v}})}\,x=V(\bm{v})\,x.{}$ (193)
Expansion in powers of $\zeta$ yields
$\displaystyle V(\bm{v})$ $\displaystyle=$ $\displaystyle
I+K^{2}(\bm{\hat{v}})(\cosh{\zeta}-1)+K(\bm{\hat{v}})\sinh{\zeta}\qquad$ (196)
$\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}\cosh{\zeta}&&\bm{\hat{v}}_{\rm
c}^{\top}\sinh{\zeta}\\\ \bm{\hat{v}}_{\rm
c}\sinh{\zeta}&&\bm{I}+\bm{\hat{v}}_{\rm c}\bm{\hat{v}}_{\rm
c}^{\top}\left(\cosh{\zeta}-1\right)\vbox to15.0pt{}\end{array}\right).{}$
The relations $V^{\top}\\!(\bm{v})=V(\bm{v})$ and $gV(\bm{v})=V^{-1}(\bm{v})g$
confirm the invariance of the scalar product:
$\displaystyle x^{\prime}\cdot x^{\prime}$ $\displaystyle=$ $\displaystyle
x^{\top}V^{\top}\\!(\bm{v})\,g\,V(\bm{v})\,x=x\cdot x.$ (197)
The transformation yields
$\displaystyle ct^{\prime}$ $\displaystyle=$ $\displaystyle
ct\,\cosh{\zeta}+\bm{\hat{v}}\cdot\bm{x}\sinh{\zeta},$ (198)
$\displaystyle\bm{x}^{\prime}$ $\displaystyle=$
$\displaystyle\bm{x}+\bm{\hat{v}}\,\bm{\hat{v}}\cdot\bm{x}(\cosh{\zeta}-1)+ct\,\bm{\hat{v}}\,\sinh{\zeta}.$
(199)
A point with $\bm{x}=0$ has the boosted velocity
$\displaystyle\frac{\bm{x}^{\prime}}{t^{\prime}}$ $\displaystyle=$
$\displaystyle c\tanh{\zeta}\,\bm{\hat{v}}=\bm{v}.$ (200)
The spherical counterpart of the operator $V(\bm{v})$, in the velocity
transformation
$\displaystyle\left(\begin{array}[]{c}ct^{\prime}\\\ \bm{x}_{\rm
s}^{\prime}\end{array}\right)$ $\displaystyle=$ $\displaystyle V_{\rm
s}(\bm{v})\left(\begin{array}[]{c}ct\\\ \bm{x}_{\rm s}\end{array}\right),$
(205)
is
$\displaystyle V_{\rm s}(\bm{v})$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}1&{\bm{0}}\\\
{\bm{0}}&\bm{M}\end{array}\right)V(\bm{v})\left(\begin{array}[]{cc}1&{\bm{0}}\\\
{\bm{0}}&\bm{M}^{\dagger}\end{array}\right)$ (210) $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}\cosh{\zeta}&&\bm{\hat{v}}_{\rm
s}^{\dagger}\sinh{\zeta}\\\ \bm{\hat{v}}_{\rm
s}\sinh{\zeta}&&I+\bm{\hat{v}}_{\rm s}\bm{\hat{v}}_{\rm
s}^{\dagger}\left(\cosh{\zeta}-1\right)\vbox
to15.0pt{}\end{array}\right).\qquad{}$ (213)
For the four-gradient operator, we have
$\displaystyle\partial_{\mu}^{\prime}$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial{x^{\prime}}^{\mu}}=\frac{\partial
x^{\nu}}{\partial{x^{\prime}}^{\mu}}\,\frac{\partial}{\partial
x^{\nu}}=\frac{\partial
x^{\nu}}{\partial{x^{\prime}}^{\mu}}\,\partial_{\nu},{}$ (214)
and from Eq. (193), which can be written as
$\displaystyle x$ $\displaystyle=$ $\displaystyle
V^{-1}(\bm{v})\,x^{\prime}=V(-\bm{v})\,x^{\prime}$ (215)
or
$\displaystyle{x^{\nu}}$ $\displaystyle=$ $\displaystyle
V_{\nu\mu}(-\bm{v})\,{x^{\prime}}^{\mu},$ (216)
we also have
$\displaystyle\frac{\partial x^{\nu}}{\partial{x^{\prime}}^{\mu}}$
$\displaystyle=$ $\displaystyle V_{\nu\mu}(-\bm{v})=V_{\mu\nu}(-\bm{v}),$
(217)
which yields
$\displaystyle\partial_{\mu}^{\prime}$ $\displaystyle=$ $\displaystyle
V_{\mu\nu}(-\bm{v})\,\partial_{\nu}.{}$ (218)
If a Cartesian gradient operator is defined as
$\displaystyle\partial_{\rm c}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}{\frac{\textstyle\partial}{\textstyle\partial
ct}}\\\ \vbox to15.0pt{}-\bm{\nabla}_{\rm c}\end{array}\right),{}$ (221)
then Eq. (218) gives
$\displaystyle g\,\partial_{\rm c}^{\prime}$ $\displaystyle=$ $\displaystyle
V(-\bm{v})\,g\,\partial_{\rm c}$ (222)
or
$\displaystyle\partial_{\rm c}^{\prime}$ $\displaystyle=$ $\displaystyle
V(\bm{v})\,\partial_{\rm c},{}$ (223)
since $g\,V(-\bm{v})\,g=V(\bm{v})$.
### 6.3 Parity and time reversal of coordinates
Lorentz transformations that leave the scalar product in Eq. (149) invariant
include the parity transformation $P=g$, time reversal $T=-g$, and total
inversion $PT=-I$ operations. These transformations have the following
defining effects on the coordinate vectors:
$\displaystyle Px$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}ct\\\ -\bm{x}_{\rm
c}\end{array}\right),$ (226) $\displaystyle Tx$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}-ct\\\ \bm{x}_{\rm
c}\end{array}\right),$ (229) $\displaystyle PTx$ $\displaystyle=$
$\displaystyle-x.$ (230)
It is sufficient for the present purpose to consider only $P$ and $T$. The
coordinate derivatives transform as
$\displaystyle P\partial_{\rm c}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}{\frac{\textstyle\partial}{\textstyle\partial
ct}}\\\ \vbox to15.0pt{}\bm{\nabla}_{\rm c}\end{array}\right),{}$ (233)
$\displaystyle T\partial_{\rm c}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}-{\frac{\textstyle\partial}{\textstyle\partial
ct}}\\\ \vbox to15.0pt{}-\bm{\nabla}_{\rm c}\end{array}\right).{}$ (236)
Comparison of Eqs. (152) and (164) shows that parity transformations commute
with rotations. On the other hand, for velocity transformations, the relation
$\displaystyle PV(\bm{v})$ $\displaystyle=$ $\displaystyle V(-\bm{v})P{}$
(237)
applies as it should, because the space reflection of a point moving with a
velocity $\bm{v}$ is a point at the reflected position moving with a velocity
$-\bm{v}$. Similar conclusions follow for time-reversal transformations.
### 6.4 Rotation of ${\it\Psi}(x)$
The result of a rotation, parameterized by the vector $\bm{u}$, applied to the
field ${\it\Psi}(x)$ in Eq. (97) is the field ${\it\Psi}^{\prime}(x)$ given by
$\displaystyle{\it\Psi}^{\prime}(x)$ $\displaystyle=$ $\displaystyle{\cal
R}(\bm{u}){\it\Psi}\\!\big{(}R^{-1}(\bm{u})\,x\big{)},{}$ (238)
where ${\cal R}(\bm{u})$ is a $6\times 6$ matrix that gives the local
transformation of the field ${\it\Psi}(x)$ at any point $x$. The inverse
transformation of the argument on the right-hand-side takes into account the
fact that the transformed field at the point $x$ originated from the field at
the point that is mapped into $x$ by the transformation. Lorentz invariance is
confirmed by showing that the transformed field satisfies the same equation as
the original field. We expect the current to transform in the same way as
${\it\Psi}$ and write
$\displaystyle{\it\Xi}^{\prime}(x)$ $\displaystyle=$ $\displaystyle{\cal
R}(\bm{u})\,{\it\Xi}\\!\big{(}R^{-1}(\bm{u})\,x\big{)}.{}$ (239)
The objective is to show that
$\displaystyle\gamma^{\mu}\partial_{\mu}{\it\Psi}^{\prime}(x)$
$\displaystyle=$ $\displaystyle{\it\Xi}^{\prime}(x),{}$ (240)
for a suitable transformation ${\cal R}(\bm{u})$. In terms of the original
field and source, Eq. (240) is given by
$\displaystyle\gamma^{\mu}\partial_{\mu}{\cal
R}(\bm{u}){\it\Psi}\big{(}R^{-1}(\bm{u})\,x\big{)}$ $\displaystyle=$
$\displaystyle{\cal R}(\bm{u})\,{\it\Xi}\big{(}R^{-1}(\bm{u})\,x\big{)}\qquad$
(241)
or
$\displaystyle\gamma^{\mu}\partial_{\mu}^{\prime}{\cal R}(\bm{u}){\it\Psi}(x)$
$\displaystyle=$ $\displaystyle{\cal R}(\bm{u})\,{\it\Xi}(x),\qquad$ (242)
where the variable $x$ has been replaced by $x^{\prime}=R(\bm{u})\,x$. Thus
Eq. (240) will follow if
$\displaystyle{\cal R}^{-1}(\bm{u})\gamma^{\mu}\partial_{\mu}^{\prime}{\cal
R}(\bm{u})$ $\displaystyle=$ $\displaystyle\gamma^{\mu}\partial_{\mu}.{}$
(243)
We expect ${\cal R}(\bm{u})$ to be of the form
$\displaystyle{\cal R}(\bm{u})$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}\bm{R}_{\rm s}(\bm{u})&{\bm{0}}\\\
{\bm{0}}&\bm{R}_{\rm s}(\bm{u})\end{array}\right),{}$ (246)
which yields
$\displaystyle{\cal R}^{-1}(\bm{u})\gamma^{\mu}\partial_{\mu}^{\prime}{\cal
R}(\bm{u})$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}\bm{I}\,{\frac{\textstyle\partial}{\textstyle\partial
ct}}&\bm{R}^{-1}_{\rm
s}(\bm{u})\,\bm{\tau}\cdot\bm{\nabla}^{\prime}\bm{R}_{\rm s}(\bm{u})\\\
-\bm{R}^{-1}_{\rm s}(\bm{u})\,\bm{\tau}\cdot\bm{\nabla}^{\prime}\bm{R}_{\rm
s}(\bm{u})&-\bm{I}\,{\frac{\textstyle\partial}{\textstyle\partial ct}\vbox
to15.0pt{}}\end{array}\right),$ (249)
so Eq. (243) follows from
$\displaystyle\bm{R}^{-1}_{\rm
s}(\bm{u})\,\bm{\tau}\cdot\bm{\nabla}^{\prime}\bm{R}_{\rm s}(\bm{u})$
$\displaystyle=$ $\displaystyle\bm{\tau}\cdot\bm{\nabla},$ (250)
which, in turn, follows from Eq. (175). We conclude that, as expected, the
solution and source terms, transformed according to Eqs. (238) and (239),
where ${\cal R}(\bm{u})$ is given in Eq. (246), satisfy the same equation as
the original solution and source terms. The six-dimensional rotation operator
${\cal R}(\bm{u})$ may be written as
$\displaystyle{\cal R}(\bm{u})$ $\displaystyle=$ $\displaystyle{\rm e}^{-{\rm
i}\bm{{\cal S}}\cdot\bm{u}},{}$ (251)
where
$\displaystyle\bm{{\cal S}}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}\bm{\tau}&{\bm{0}}\\\
{\bm{0}}&\bm{\tau}\end{array}\right).{}$ (254)
Equations (238), (239), and (246) correspond to the separate equations
$\displaystyle\bm{E}_{\rm s}^{\prime}(x)$ $\displaystyle=$
$\displaystyle\bm{R}_{\rm s}(\bm{u})\,\bm{E}_{\rm
s}\big{(}R^{-1}(\bm{u})\,x\big{)},$ (255) $\displaystyle\bm{B}_{\rm
s}^{\prime}(x)$ $\displaystyle=$ $\displaystyle\bm{R}_{\rm
s}(\bm{u})\,\bm{B}_{\rm s}\big{(}R^{-1}(\bm{u})\,x\big{)},$ (256)
$\displaystyle\bm{J}_{\rm s}^{\prime}(x)$ $\displaystyle=$
$\displaystyle\bm{R}_{\rm s}(\bm{u})\,\bm{J}_{\rm
s}\big{(}R^{-1}(\bm{u})\,x\big{)}.$ (257)
It can be confirmed that Eqs. (1) and (4) in spherical form are invariant
under rotations. In particular, Eq. (1) for the rotated electric field and
charge density $\rho^{\prime}(x)=\rho\big{(}R^{-1}(\bm{u})x\big{)}$ is
$\displaystyle\bm{\nabla}_{\rm s}^{\dagger}\,\bm{E}^{\prime}_{\rm s}(x)$
$\displaystyle=$ $\displaystyle\frac{\rho^{\prime}(x)}{\epsilon_{0}}$ (258)
or
$\displaystyle\bm{\nabla}_{\rm s}^{\dagger}\,\bm{R}_{\rm s}(\bm{u})\bm{E}_{\rm
s}\big{(}R^{-1}(\bm{u})\,x\big{)}$ $\displaystyle=$
$\displaystyle\frac{\rho\big{(}R^{-1}(\bm{u})\,x\big{)}}{\epsilon_{0}}.$ (259)
The substitution $x\rightarrow R(\bm{u})\,x$ gives
$\displaystyle\bm{\nabla}_{\rm s}^{\prime\dagger}\,\bm{R}_{\rm
s}(\bm{u})\bm{E}_{\rm s}(x)$ $\displaystyle=$
$\displaystyle\frac{\rho(x)}{\epsilon_{0}},$ (260)
and Eq. (174) yields
$\displaystyle\bm{\nabla}_{\rm s}^{\dagger}\,\bm{E}_{\rm s}(x)$
$\displaystyle=$ $\displaystyle\frac{\rho(x)}{\epsilon_{0}}.$ (261)
Hence, the transformed field and charge density satisfy Eq. (1) if the
original field and charge density do. Similarly,
$\displaystyle\bm{\nabla}_{\rm s}^{\dagger}\,\bm{B}^{\prime}_{\rm s}(x)$
$\displaystyle=$ $\displaystyle\bm{\nabla}_{\rm s}^{\dagger}\,\bm{B}_{\rm
s}(x)=0.$ (262)
Thus, all of the Maxwell equations in matrix form are invariant under
rotations.
The separation into transverse and longitudinal components of the electric and
magnetic fields is also invariant under rotations. This can be seen by
considering the expression $\bm{{\it\Pi}}_{\rm s}(\bm{\nabla})\,\bm{F}_{\rm
s}(x)$, where $\bm{{\it\Pi}}_{\rm s}(\bm{\nabla})$ is either
$\bm{{\it\Pi}}^{\rm T}_{\rm s}(\bm{\nabla})$ or $\bm{{\it\Pi}}^{\rm L}_{\rm
s}(\bm{\nabla})$ and $\bm{F}_{\rm s}(x)$ is any of $\bm{E}_{\rm s}(x)$,
$\bm{B}_{\rm s}(x)$, or $\bm{J}_{\rm s}(x)$. We have
$\displaystyle\bm{{\it\Pi}}_{\rm s}(\bm{\nabla})\,\bm{F}^{\prime}_{\rm s}(x)$
$\displaystyle=$ $\displaystyle\bm{{\it\Pi}}_{\rm s}(\bm{\nabla})\,\bm{R}_{\rm
s}(\bm{u})\,\bm{F}_{\rm s}\big{(}R^{-1}(\bm{u})\,x\big{)}\qquad$ (263)
or
$\displaystyle\bm{{\it\Pi}}_{\rm
s}(\bm{\nabla}^{\prime})\,\bm{F}^{\prime}_{\rm s}(x^{\prime})$
$\displaystyle=$ $\displaystyle\bm{{\it\Pi}}_{\rm
s}(\bm{\nabla}^{\prime})\,\bm{R}_{\rm s}(\bm{u})\,\bm{F}_{\rm s}(x)$ (264)
$\displaystyle=$ $\displaystyle\bm{R}_{\rm s}(\bm{u})\,\bm{{\it\Pi}}_{\rm
s}(\bm{\nabla})\,\bm{F}_{\rm s}(x),{}$
where the last line follows from either Eq. (176) or Eq. (177). This means
that if the original field is transverse or longitudinal, then the rotated
field has the same character. These results extend directly to the six-
dimensional projection operators ${\it\Pi}(\bm{\nabla})$, solution
${\it\Psi}(x)$, and source ${\it\Xi}(x)$.
### 6.5 Velocity transformation of ${\it\Psi}(x)$
The result of the velocity transformation, by a velocity $\bm{v}$, applied to
the field ${\it\Psi}(x)$ in Eq. (97) is the function ${\it\Psi}^{\prime}(x)$
given by
$\displaystyle{\it\Psi}^{\prime}(x)$ $\displaystyle=$ $\displaystyle{\cal
V}(\bm{v}){\it\Psi}\\!\big{(}V^{-1}(\bm{v})\,x\big{)},{}$ (265)
where ${\cal V}(\bm{u})$ is a $6\times 6$ matrix that gives the local
transformation of the field ${\it\Psi}(x)$ at any point. The inverse
transformation of the argument on the right-hand-side plays the same role as
for rotations. Our objective is to establish the covariance of Eq. (97) by
showing that if ${\it\Psi}(x)$ is a solution of that equation with a source
${\it\Xi}(x)$, then
$\displaystyle\gamma^{\mu}\partial_{\mu}{\it\Psi}^{\prime}(x)$
$\displaystyle=$ $\displaystyle{\it\Xi}^{\prime}(x),{}$ (266)
where ${\it\Xi}^{\prime}(x)$ is a suitably transformed source term. Equation
(266) can be written as
$\displaystyle\gamma^{\mu}\partial_{\mu}{\cal
V}(\bm{v}){\it\Psi}\big{(}V^{-1}(\bm{v})\,x\big{)}$ $\displaystyle=$
$\displaystyle{\it\Xi}^{\prime}(x)\qquad$ (267)
or
$\displaystyle\gamma^{\mu}\partial_{\mu}^{\prime}{\cal V}(\bm{v}){\it\Psi}(x)$
$\displaystyle=$ $\displaystyle{\it\Xi}^{\prime}\left(V(\bm{v})\,x\right),{}$
(268)
where the variable $x$ has been replaced by $x^{\prime}=V(\bm{v})\,x$. The
$6\times 6$ matrix ${\cal V}(\bm{v})$ is based on the conventional local
velocity transformation of the electric and magnetic fields as discussed in A.
Here we write it as
$\displaystyle{\cal V}(\bm{v})$ $\displaystyle=$ $\displaystyle{\rm
e}^{\zeta\bm{{\cal K}}\cdot\bm{\hat{v}}},{}$ (269)
where
$\displaystyle\bm{{\cal K}}=\left(\begin{array}[]{cc}{\bm{0}}&\bm{\tau}\\\
\bm{\tau}&{\bm{0}}\end{array}\right).{}$ (272)
Expansion of the exponential function in Eq. (269) in powers of $\zeta$ yields
$\displaystyle{\cal V}(\bm{v})$ $\displaystyle=$ $\displaystyle{\cal
I}+\left(\bm{{\cal
K}}\cdot\bm{\hat{v}}\right)^{2}(\cosh{\zeta}-1)+\left(\bm{\bm{{\cal
K}}\cdot\hat{v}}\right)\,\sinh{\zeta}$ (275) $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}\bm{I}+(\bm{\tau}\cdot\bm{\hat{v}})^{2}\,(\cosh{\zeta}-1)&\bm{\tau}\cdot\bm{\hat{v}}\,\sinh{\zeta}\\\
\bm{\tau}\cdot\bm{\hat{v}}\,\sinh{\zeta}&\bm{I}+\left(\bm{\tau}\cdot\bm{\hat{v}}\right)^{2}(\cosh{\zeta}-1)\end{array}\right),{}$
where $(\bm{{\cal K}}\cdot\bm{\hat{v}})^{3}=\bm{{\cal K}}\cdot\bm{\hat{v}}$ is
taken into account, so that
$\displaystyle{\cal V}(\bm{v}){\it\Psi}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}\bm{E}_{\rm s}^{\prime}\\\ {\rm
i}\,c\bm{B}_{\rm s}^{\prime}\vbox
to15.0pt{}\end{array}\right)=\left(\begin{array}[]{c}\bm{E}_{\rm
s}+(\bm{\tau}\cdot\bm{\hat{v}})^{2}\bm{E}_{\rm s}(\cosh{\zeta}-1)+{\rm
i}\,\bm{\tau}\cdot\bm{\hat{v}}\,c\bm{B}_{\rm s}\sinh{\zeta}\\\ {\rm
i}\left[c\bm{B}_{\rm s}+(\bm{\tau}\cdot\bm{\hat{v}})^{2}c\bm{B}_{\rm
s}(\cosh{\zeta}-1)-{\rm i}\,\bm{\tau}\cdot\bm{\hat{v}}\,\bm{E}_{\rm
s}\sinh{\zeta}\,\right]\vbox to15.0pt{}\end{array}\right).\qquad$ (280)
In Eq. (268), we have
$\displaystyle\gamma^{\mu}\partial^{\,\prime}_{\mu}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}\bm{I}\,\frac{\textstyle\partial}{\textstyle\partial
ct^{\prime}}&&\bm{\tau}\cdot\bm{\nabla}^{\prime}\\\
-\bm{\tau}\cdot\bm{\nabla}^{\prime}&&-\bm{I}\,\frac{\textstyle\partial}{\textstyle\partial
ct^{\prime}}\end{array}\right),{}$ (283)
where, from Eqs. (221) and (223),
$\displaystyle\frac{\partial}{\partial ct^{\prime}}$ $\displaystyle=$
$\displaystyle\cosh{\zeta}\,\frac{\partial}{\partial
ct}-\sinh{\zeta}\,\bm{\hat{v}}\cdot\bm{\nabla},$ (284)
$\displaystyle\bm{\nabla}^{\prime}$ $\displaystyle=$
$\displaystyle\bm{\nabla}+(\cosh{\zeta}-1)\bm{\hat{v}}\,\bm{\hat{v}}\cdot\bm{\nabla}-\sinh{\zeta}\,\bm{\hat{v}}\,\frac{\partial}{\partial
ct}.\qquad$ (285)
Multiplication of Eq. (275) by Eq. (283) yields the identity
$\displaystyle\gamma^{\mu}\partial^{\,\prime}_{\mu}\,{\cal V}(\bm{v})$
$\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}\bm{I}+\bm{\hat{v}}_{\rm
s}\bm{\hat{v}}_{\rm s}^{\dagger}(\cosh{\zeta}-1)&&{\bm{0}}\\\
{\bm{0}}&&\bm{I}+\bm{\hat{v}}_{\rm s}\bm{\hat{v}}_{\rm
s}^{\dagger}(\cosh{\zeta}-1)\end{array}\right)\gamma^{\mu}\,\partial_{\mu}$
(291)
$\displaystyle+\left(\begin{array}[]{ccc}-\sinh{\zeta}\,\bm{\hat{v}}_{\rm
s}\bm{\nabla}_{\rm s}^{\dagger}&&{\bm{0}}\\\
{\bm{0}}&&\sinh{\zeta}\,\bm{\hat{v}}_{\rm s}\bm{\nabla}_{\rm
s}^{\dagger}\end{array}\right){}$
and hence
$\displaystyle\gamma^{\mu}\partial_{\mu}^{\prime}{\cal V}(\bm{v}){\it\Psi}(x)$
$\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}\bm{I}+\bm{\hat{v}}_{\rm
s}\bm{\hat{v}}_{\rm s}^{\dagger}(\cosh{\zeta}-1)&&{\bm{0}}\\\
{\bm{0}}&&\bm{I}+\bm{\hat{v}}_{\rm s}\bm{\hat{v}}_{\rm
s}^{\dagger}(\cosh{\zeta}-1)\end{array}\right){\it\Xi}(x)$ (297)
$\displaystyle+\left(\begin{array}[]{c}-\sinh{\zeta}\,\bm{\hat{v}}_{\rm
s}\,\bm{\nabla}\cdot\bm{E}(x)\\\ {\rm i}\sinh{\zeta}\,\bm{\hat{v}}_{\rm
s}\,c\,\bm{\nabla}\cdot\bm{B}(x)\end{array}\right)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}-\mu_{0}c\left[\bm{J}_{\rm
s}(x)+\bm{\hat{v}}_{\rm
s}\bm{\hat{v}}\cdot\bm{J}(x)(\cosh{\zeta}-1)+\sinh{\zeta}\,\bm{\hat{v}}_{\rm
s}c\rho(x)\right]\\\ {\bm{0}}\end{array}\right)$ (300) $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}-\mu_{0}c\bm{J}_{\rm s}^{\prime}(x)\\\
{\bm{0}}\end{array}\right),{}$ (303)
where $\bm{J}_{\rm s}^{\prime}(x)$ is the velocity transformed three-vector
source current, and
$\displaystyle{\it\Xi}^{\prime}(x)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}-\mu_{0}c\bm{J}_{\rm
s}^{\prime}\big{(}V^{-1}(\bm{v})\,x\big{)}\\\ {\bm{0}\vbox
to15.0pt{}}\end{array}\right).$ (306)
The result in Eq. (303) takes into account the additional two Maxwell
equations in (1) and (4), besides Eqs. (2) and (3) used to construct Eq. (97).
It also requires the conventional result that the current
$\displaystyle\left(\begin{array}[]{c}c\rho(x)\\\ \bm{J}_{\rm
s}(x)\end{array}\right)$ (309)
transforms as a four-vector under the velocity transformation given by Eq.
(213). Equation (303) establishes the validity of Eq. (266), provided the
transformed three-vector source current is the three-vector component of the
transformed four-vector current.
The covariance of Eq. (97), even though the charge density does not appear in
the source term, is linked to the fact that the current density satisfies the
continuity equation. Since the continuity equation follows from the Maxwell
equations, it cannot be expected that consistent solutions may be found for an
arbitrary four-vector current density. However, for valid sources, information
about the charge density may be obtained from the three-vector current density
and the continuity equation. For example, if the electric field is specified
at a particular time, then the charge density at that time is known from Eq.
(1) and may be determined at any other time from knowledge of the three-vector
current density by use of the continuity equation. Thus the time evolution of
the electromagnetic fields can be described relativistically with no reference
to the charge density. The Maxwell Green function, which provides the
solutions of Eq. (97), is discussed in Sec. 9.
Since ${\cal V}^{\dagger}\gamma^{0}{\cal V}=\gamma^{0}$ and ${\cal
V}^{\dagger}\gamma^{0}\eta{\cal V}=\gamma^{0}\eta$, where
$\displaystyle\eta=\left(\begin{array}[]{ccc}{\bm{0}}&&\bm{I}\\\
\bm{I}&&{\bm{0}}\end{array}\right),{}$ (312)
the invariance of the quantities
$\displaystyle\overline{{\it\Psi}}{\it\Psi}=|\bm{E}|^{2}-c^{2}|\bm{B}|^{2},$
(313) $\displaystyle\overline{{\it\Psi}}\eta{\it\Psi}=2\,{\rm i}\,c\,{\rm
Re}\,\bm{E}\cdot\bm{B}$ (314)
is evident.
### 6.6 Parity and time-reversal transformations of ${\it\Psi}(x)$
We expect the fields ${\it\Psi}(x)$ to transform under a parity change
according to
$\displaystyle{\it\Psi}^{\prime}(x)$ $\displaystyle=$ $\displaystyle{\cal
P}{\it\Psi}\\!\big{(}\,P^{-1}\,x\big{)},{}$ (315)
where ${\cal P}$ is a $6\times 6$ matrix, and we assume that the current
transforms in the same way, so that
$\displaystyle{\it\Xi}^{\prime}(x)$ $\displaystyle=$ $\displaystyle{\cal
P}{\it\Xi}\\!\big{(}\,P^{-1}\,x\big{)}.$ (316)
We can obtain
$\displaystyle\gamma^{\mu}\partial_{\mu}{\it\Psi}^{\prime}(x)$
$\displaystyle=$ $\displaystyle{\it\Xi}^{\prime}(x),{}$ (317)
by finding a suitable matrix ${\cal P}$. In terms of the original field and
source, Eq. (317) is given by
$\displaystyle\gamma^{\mu}\partial_{\mu}{\cal
P}{\it\Psi}\big{(}P^{-1}x\big{)}$ $\displaystyle=$ $\displaystyle{\cal
P}{\it\Xi}\big{(}P^{-1}x\big{)}\qquad$ (318)
or
$\displaystyle\gamma^{\mu}\partial_{\mu}^{\prime}{\cal P}{\it\Psi}(x)$
$\displaystyle=$ $\displaystyle{\cal P}{\it\Xi}(x),\qquad$ (319)
where $\partial_{\mu}^{\prime}$ is the parity transformed derivative given by
Eq. (233). Equation (317) will follow if
$\displaystyle{\cal P}^{-1}\gamma^{\mu}\partial_{\mu}^{\prime}{\cal P}$
$\displaystyle=$ $\displaystyle\gamma^{\mu}\partial_{\mu},$ (320)
which corresponds to
$\displaystyle{\cal P}^{-1}\gamma^{0}{\cal P}$ $\displaystyle=$
$\displaystyle\gamma^{0},$ (321) $\displaystyle{\cal P}^{-1}\gamma^{i}\,{\cal
P}$ $\displaystyle=$ $\displaystyle-\gamma^{i},\quad i=1,2,3.$ (322)
Solutions of these equations are provided by
$\displaystyle{\cal P}$ $\displaystyle=$
$\displaystyle\pm\left(\begin{array}[]{ccc}\bm{I}&&{\bm{0}}\\\
{\bm{0}}&&-\bm{I}\end{array}\right).{}$ (325)
The minus sign corresponds to the conventional choice of how classical
electric and magnetic fields transform under a parity change, that is, the
current and electric fields change sign and the magnetic fields do not [17].
To examine time-reversal invariance, we first consider ${\it\Psi}(x)$ as a
field which is real in the Cartesian basis. In this case, the conventional use
of an anti-unitary operator for time reversal is unnecessary, and the same can
be expected to be true for fields expressed in the spherical basis. We write
$\displaystyle{\it\Psi}^{\prime}(x)$ $\displaystyle=$ $\displaystyle{\cal
T}{\it\Psi}\\!\big{(}\,T^{-1}\,x\big{)}{}$ (326)
and
$\displaystyle{\it\Xi}^{\prime}(x)$ $\displaystyle=$ $\displaystyle-{\cal
T}\,{\it\Xi}\\!\big{(}\,T^{-1}\,x\big{)},$ (327)
where ${\cal T}$ is a suitable $6\times 6$ matrix. The minus sign for the
current provides the result that the electric field does not change sign under
time reversal and the current does. The objective is to find a matrix ${\cal
T}$ such that
$\displaystyle\gamma^{\mu}\partial_{\mu}{\it\Psi}^{\prime}(x)$
$\displaystyle=$ $\displaystyle{\it\Xi}^{\prime}(x)$ (328)
or
$\displaystyle\gamma^{\mu}\partial_{\mu}{\cal
T}{\it\Psi}\big{(}T^{-1}x\big{)}$ $\displaystyle=$ $\displaystyle-{\cal
T}{\it\Xi}\big{(}T^{-1}x\big{)},\qquad$ (329)
and so
$\displaystyle\gamma^{\mu}\partial_{\mu}^{\prime}{\cal T}{\it\Psi}(x)$
$\displaystyle=$ $\displaystyle-{\cal T}{\it\Xi}(x),\qquad$ (330)
where $\partial_{\mu}^{\prime}$ is given by Eq. (236). Such a matrix satisfies
$\displaystyle{\cal T}^{-1}\gamma^{\mu}\partial_{\mu}^{\prime}{\cal T}$
$\displaystyle=$ $\displaystyle-\gamma^{\mu}\partial_{\mu}$ (331)
or
$\displaystyle{\cal T}^{-1}\gamma^{0}{\cal T}$ $\displaystyle=$
$\displaystyle\gamma^{0},$ (332) $\displaystyle{\cal T}^{-1}\gamma^{i}{\cal
T}$ $\displaystyle=$ $\displaystyle-\gamma^{i},\quad i=1,2,3,$ (333)
with a solution provided by
$\displaystyle{\cal T}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}\bm{I}&&{\bm{0}}\\\
{\bm{0}}&&-\bm{I}\end{array}\right).$ (336)
The matrices ${\cal P}$ and ${\cal T}$ commute with the rotation matrix, as
they should, and we have the result that the matrix form of the Maxwell
equations is invariant under parity and time-reversal transformations.
For quantum-mechanical time reversal, the time-reversal operator is anti-
unitary and includes complex conjugation, or Hermitian conjugation in the case
of a matrix solution, which has the effect of interchanging initial and final
states. For an example where such an interchange corresponds to observable
consequences in QED, see [24, 25]. We thus write
$\displaystyle{\it\Psi}^{\prime}(x)$ $\displaystyle=$
$\displaystyle\mathfrak{T}{\it\Psi}\\!\big{(}\,T^{-1}\,x\big{)}={\it\Psi}^{\dagger}\big{(}\,T^{-1}\,x\big{)}\,{\cal
U}^{-1},$ (337) $\displaystyle{\it\Xi}^{\prime}(x)$ $\displaystyle=$
$\displaystyle-\mathfrak{T}{\it\Xi}\\!\big{(}\,T^{-1}\,x\big{)}=-{\it\Xi}^{\dagger}\big{(}\,T^{-1}\,x\big{)}\,{\cal
U}^{-1},$ (338)
where $\mathfrak{T}={\cal C}\,{\cal U}$ is the product of the Hermitian
conjugation operator ${\cal C}$, which has the action ${\cal
C}{\it\Psi}(x)={\it\Psi}^{\dagger}(x)$ and a unitary matrix ${\cal U}$. The
objective is to find a ${\cal U}$ such that
$\displaystyle{\it\Psi}^{\,\prime}(x)\gamma^{0}\overleftarrow{\partial}_{\\!\mu}{\gamma^{\mu}}$
$\displaystyle=$ $\displaystyle{\it\Xi}^{\,\prime}(x)\gamma^{0}{}$ (339)
if ${\it\Psi}(x)$ is a solution of Eq. (98). Equation (339) can be written as
$\displaystyle{\it\Psi}^{\dagger}(x)\,{\cal
U}^{-1}\gamma^{0}\overleftarrow{\partial^{\prime}}_{\\!\mu}{\gamma^{\mu}}$
$\displaystyle=$ $\displaystyle-{\it\Xi}^{\dagger}(x)\,{\cal
U}^{-1}\gamma^{0},{}$ (340)
where $\partial^{\prime}_{\mu}$ is given by Eq. (236). Equation (340) follows
from Eq. (98) provided
$\displaystyle\gamma^{0}\,{\cal
U}^{-1}\gamma^{0}\overleftarrow{\partial^{\prime}}_{\\!\mu}{\gamma^{\mu}}\gamma^{0}\,{\cal
U}\gamma^{0}$ $\displaystyle=$
$\displaystyle-\overleftarrow{\partial}_{\\!\mu}{\gamma^{\mu}},$ (341)
which has as a solution
$\displaystyle{\cal U}$ $\displaystyle=$ $\displaystyle{\cal T}$ (342)
and
$\displaystyle\mathfrak{T}$ $\displaystyle=$ $\displaystyle{\cal C}\,{\cal
T}.$ (343)
## 7 Plane-wave eigenfunctions
Following the analogy with the Dirac equation, the Hamiltonian for the Maxwell
equation is
$\displaystyle{\cal H}$ $\displaystyle=$ $\displaystyle
c\,\bm{\alpha}\cdot\bm{p}=-{\rm i}\,\hbar c\,\bm{\alpha}\cdot\bm{\nabla},{}$
(344)
where $\alpha^{i}=\gamma^{0}\,\gamma^{i}$, and the wave functions for the
photon may be identified as the complete set of eigenfunctions of ${\cal H}$.
The solutions considered here are coordinate-space plane waves characterized
by a wave vector $\bm{k}$ and a polarization vector
$\bm{\hat{\epsilon}}_{\lambda}$; both positive- and negative-energy solutions,
as well as zero-energy solutions are included to form a complete set. These
solutions are also eigenfunctions of the momentum operator
$\displaystyle\bm{{\cal P}}$ $\displaystyle=$ $\displaystyle{\cal
I}\bm{p}=-{\rm i}\,\hbar\,{\cal I}\,\bm{\nabla},{}$ (345)
which commutes with ${\cal H}$.
The plane-wave solutions are not normalizable, because their modulus squared
is independent of $\bm{x}$ and the integral over all space does not exist. As
a result, the solutions include an arbitrary multiplicative factor, that could
be a function of $\bm{k}$. Here a factor is chosen to provide the simple
result in Eq. (381).
### 7.1 Transverse plane-wave photons
We first consider transverse photons, i.,e., photons for which the electric
and magnetic fields are perpendicular to the wave vector. The polarization
vector is a unit vector proportional to the electric or magnetic fields,
represented by a three component, possibly complex, vector in the spherical
basis. As such, the polarization vector does not transform as the spatial
component of a four-vector under velocity transformations.
Two polarization vectors, both in the plane perpendicular to $\bm{\hat{k}}$,
are denoted by
$\displaystyle\bm{\hat{\epsilon}}_{\lambda}(\bm{\hat{k}})\,;\qquad\lambda=1,2.{}$
(346)
They have the orthonormality properties
$\displaystyle\bm{\hat{\epsilon}}_{\lambda_{2}}^{\dagger}(\bm{\hat{k}})\,\bm{\hat{\epsilon}}_{\lambda_{1}}(\bm{\hat{k}})$
$\displaystyle=$ $\displaystyle\delta_{{\lambda_{2}},{\lambda_{1}}},{}$ (347)
$\displaystyle\bm{\hat{k}}_{\rm
s}^{\dagger}\bm{\hat{\epsilon}}_{\lambda}(\bm{\hat{k}})$ $\displaystyle=$
$\displaystyle 0,{}$ (348)
and the completeness property
$\displaystyle\sum_{\lambda=1}^{2}\bm{\hat{\epsilon}}_{\lambda}(\bm{\hat{k}})\,\bm{\hat{\epsilon}}_{\lambda}^{\dagger}(\bm{\hat{k}})$
$\displaystyle=$ $\displaystyle\bm{I}-\bm{\hat{k}}_{\rm s}\,\bm{\hat{k}}_{\rm
s}^{\dagger}=(\bm{\tau}\cdot\bm{\hat{k}})^{2}=\bm{{\it\Pi}}_{\rm s}^{\rm
T}(\bm{\hat{k}}).$ (349)
From Eq. (348), we also have
$\displaystyle(\bm{\tau}\cdot\bm{\hat{k}})^{2}\,\bm{\hat{\epsilon}}_{\lambda}(\bm{\hat{k}})=\bm{\hat{\epsilon}}_{\lambda}(\bm{\hat{k}}).$
(350)
The polarization vectors can represent linear polarization, circular
polarization, or any combination by a suitable choice of
$\bm{\hat{\epsilon}}_{\lambda}(\bm{\hat{k}})$. For example, for $\bm{k}$ in
the $\bm{\hat{e}}^{3}$ direction, linear polarization vectors in the
$\bm{\hat{e}}^{1}$ and $\bm{\hat{e}}^{2}$ directions are
$\displaystyle\bm{\hat{\epsilon}}_{1}(\bm{\hat{e}}^{3})$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}-{1\over\sqrt{2}}\\\ 0\\\
{1\over\sqrt{2}}\end{array}\right);\quad\bm{\hat{\epsilon}}_{2}(\bm{\hat{e}}^{3})=\left(\begin{array}[]{c}{{\rm
i}\over\sqrt{2}}\\\ 0\\\ {{\rm i}\over\sqrt{2}}\end{array}\right),\quad{}$
(357)
according to Eq. (54). Similarly, circular polarization vectors are (see Sec.
8.2)
$\displaystyle\bm{\hat{\epsilon}}_{1}(\bm{\hat{e}}^{3})$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}1\\\ 0\\\
0\end{array}\right);\quad\bm{\hat{\epsilon}}_{2}(\bm{\hat{e}}^{3})=\left(\begin{array}[]{c}0\\\
0\\\ 1\end{array}\right).{}$ (364)
These polarization vectors can be transformed to the vectors corresponding to
any direction of $\bm{k}$ with the rotation operator in Eq. (166). (See also
Sec. 7.5.)
Positive $(+)$ and negative $(-)$ energy transverse photon wave functions are
given by
$\displaystyle\psi_{\bm{k},\lambda}^{(\pm)}(\bm{x})$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2(2\pi)^{3}}}\left(\begin{array}[]{c}\bm{\hat{\epsilon}}_{\lambda}(\bm{\hat{k}})\\\
\bm{\tau}\cdot\bm{\hat{k}}\
\bm{\hat{\epsilon}}_{\lambda}(\bm{\hat{k}})\end{array}\right){\rm e}^{\pm{\rm
i}\bm{k}\cdot\bm{x}}.\quad{}$ (367)
Although constructed geometrically to be transverse by the choice of
polarization vectors, these wave functions are also transverse in the sense
defined in Sec. 5. We have (see B for more detail)
$\displaystyle{\it\Pi}^{\rm
T}(\bm{\nabla})\,\psi_{\bm{k},\lambda}^{(\pm)}(\bm{x})$ $\displaystyle=$
$\displaystyle{\it\Pi}^{\rm
T}(\bm{\hat{k}})\,\psi_{\bm{k},\lambda}^{(\pm)}(\bm{x})=\psi_{\bm{k},\lambda}^{(\pm)}(\bm{x}),\qquad$
(368) $\displaystyle{\it\Pi}^{\rm
L}(\bm{\nabla})\,\psi_{\bm{k},\lambda}^{(\pm)}(\bm{x})$ $\displaystyle=$
$\displaystyle{\it\Pi}^{\rm
L}(\bm{\hat{k}})\,\psi_{\bm{k},\lambda}^{(\pm)}(\bm{x})=0,$ (369)
where ${\it\Pi}^{\rm T}$ and ${\it\Pi}^{\rm L}$ are defined in Eqs. (133) and
(136). The wave functions in Eq. (367) are eigenfunctions of the Hamiltonian
in Eq. (344) with eigenvalues $\pm\hbar c\,|\bm{k}|$. In particular,
$\displaystyle{\cal H}\,\psi_{\bm{k},\lambda}^{(\pm)}(\bm{x})$
$\displaystyle=$ $\displaystyle\pm\hbar
c\,\bm{\alpha}\cdot\bm{k}\,\psi_{\bm{k},\lambda}^{(\pm)}(\bm{x}),$ (370)
and
$\displaystyle\left(\begin{array}[]{cc}{\bm{0}}&\bm{\tau}\cdot\bm{k}\\\
\bm{\tau}\cdot\bm{k}&{\bm{0}}\end{array}\right)\left(\begin{array}[]{c}\bm{\hat{\epsilon}}_{\lambda}(\bm{\hat{k}})\\\
\bm{\tau}\cdot\bm{\hat{k}}\
\bm{\hat{\epsilon}}_{\lambda}(\bm{\hat{k}})\end{array}\right)$
$\displaystyle=$
$\displaystyle|\bm{k}|\left(\begin{array}[]{c}\bm{\hat{\epsilon}}_{\lambda}(\bm{\hat{k}})\\\
\bm{\tau}\cdot\bm{\hat{k}}\
\bm{\hat{\epsilon}}_{\lambda}(\bm{\hat{k}})\end{array}\right),$ (377)
so that
$\displaystyle{\cal H}\,\psi_{\bm{k},\lambda}^{(\pm)}(\bm{x})$
$\displaystyle=$ $\displaystyle\pm\hbar
c\,|\bm{k}|\,\psi_{\bm{k},\lambda}^{(\pm)}(\bm{x}).$ (378)
Also,
$\displaystyle\bm{{\cal P}}\,\psi_{\bm{k},\lambda}^{(\pm)}(\bm{x})$
$\displaystyle=$
$\displaystyle\pm\hbar\,\bm{k}\,\psi_{\bm{k},\lambda}^{(\pm)}(\bm{x}).$ (379)
The wave functions have the expected property
$\displaystyle\overline{\psi}_{\bm{k},\lambda}^{(\pm)}(\bm{x})\,\psi_{\bm{k},\lambda}^{(\pm)}(\bm{x})$
$\displaystyle=$ $\displaystyle 0,$ (380)
since the electric and magnetic field strengths are equal for a transverse
photon. Normalization and orthogonality relations are
$\displaystyle\int{\rm d}\bm{x}\
\psi_{\bm{k}_{2},\lambda_{2}}^{(\pm)\dagger}(\bm{x})\,\psi_{\bm{k}_{1},\lambda_{1}}^{(\pm)}(\bm{x})$
$\displaystyle=$
$\displaystyle\delta_{\lambda_{2}\lambda_{1}}\delta(\bm{k}_{2}-\bm{k}_{1}),\qquad{}$
(381) $\displaystyle\int{\rm d}\bm{x}\
\psi_{\bm{k}_{2},\lambda_{2}}^{(\pm)\dagger}(\bm{x})\,\psi_{\bm{k}_{1},\lambda_{1}}^{(\mp)}(\bm{x})$
$\displaystyle=$ $\displaystyle 0.{}$ (382)
The latter relation follows from a cancellation of terms between the upper-
three and lower-three components of the wave function:
$\displaystyle\int{\rm d}\bm{x}\
\psi_{\bm{k}_{2},\lambda_{2}}^{(\pm)\dagger}(\bm{x})\,\psi_{\bm{k}_{1},\lambda_{1}}^{(\mp)}(\bm{x})$
$\displaystyle=$ $\displaystyle\frac{1}{2(2\pi)^{3}}\int{\rm d}\bm{x}\
\bm{\hat{\epsilon}}_{\lambda_{2}}^{\dagger}(\bm{\hat{k}}_{2})\left[\bm{I}+\bm{\tau}\cdot\bm{\hat{k}}_{2}\,\bm{\tau}\cdot\bm{\hat{k}}_{1}\right]\bm{\hat{\epsilon}}_{\lambda_{1}}(\bm{\hat{k}}_{1})\,{\rm
e}^{\mp{\rm i}(\bm{k}_{2}+\bm{k}_{1})\cdot\bm{x}}\quad$ (383) $\displaystyle=$
$\displaystyle\frac{1}{2}\,\bm{\hat{\epsilon}}_{\lambda_{2}}^{\dagger}(\bm{\hat{k}}_{2})\left[\bm{I}+\bm{\tau}\cdot\bm{\hat{k}}_{2}\,\bm{\tau}\cdot\bm{\hat{k}}_{1}\right]\bm{\hat{\epsilon}}_{\lambda_{1}}(\bm{\hat{k}}_{1})\delta(\bm{k}_{2}+\bm{k}_{1})=0.{}$
The transverse wave functions constitute a complete set of such functions. The
completeness is established by writing
$\displaystyle\sum_{\lambda=1}^{2}\int{\rm d}\bm{k}\
\psi_{\bm{k},\lambda}^{(\pm)}(\bm{x}_{2})\psi_{\bm{k},\lambda}^{(\pm)\dagger}(\bm{x}_{1})$
$\displaystyle=\sum_{\lambda=1}^{2}\int{\rm d}\bm{k}\
\frac{1}{2(2\pi)^{3}}\,\left(\begin{array}[]{ccc}\bm{\hat{\epsilon}}_{\lambda}(\bm{\hat{k}})\,\bm{\hat{\epsilon}}_{\lambda}^{\dagger}(\bm{\hat{k}})&&\bm{\hat{\epsilon}}_{\lambda}(\bm{\hat{k}})\,\bm{\hat{\epsilon}}_{\lambda}^{\dagger}(\bm{\hat{k}})\bm{\tau}\cdot\bm{\hat{k}}\\\
\bm{\tau}\cdot\bm{\hat{k}}\,\bm{\hat{\epsilon}}_{\lambda}(\bm{\hat{k}})\,\bm{\hat{\epsilon}}_{\lambda}^{\dagger}(\bm{\hat{k}})&&\bm{\tau}\cdot\bm{\hat{k}}\,\bm{\hat{\epsilon}}_{\lambda}(\bm{\hat{k}})\,\bm{\hat{\epsilon}}_{\lambda}^{\dagger}(\bm{\hat{k}})\bm{\tau}\cdot\bm{\hat{k}}\end{array}\right){\rm
e}^{\pm{\rm i}\bm{k}\cdot(\bm{x}_{2}-\bm{x}_{1})}\qquad$ (386)
$\displaystyle=\int{\rm d}\bm{k}\
\frac{1}{2(2\pi)^{3}}\,\left(\begin{array}[]{ccc}(\bm{\tau}\cdot\bm{\hat{k}})^{2}&&\bm{\tau}\cdot\bm{\hat{k}}\\\
\bm{\tau}\cdot\bm{\hat{k}}&&(\bm{\tau}\cdot\bm{\hat{k}})^{2}\end{array}\right){\rm
e}^{\pm{\rm i}\bm{k}\cdot(\bm{x}_{2}-\bm{x}_{1})}.{}$ (389)
To evaluate the integrals for the sum over positive and negative energy
solutions, we use $\kappa$ to represent either a plus sign or a minus sign and
write
$\displaystyle\sum_{\kappa\rightarrow\pm}\int{\rm d}\bm{k}\
(\bm{\tau}\cdot\bm{\hat{k}})^{2}\,{\rm e}^{\kappa{\rm
i}\bm{k}\cdot(\bm{x}_{2}-\bm{x}_{1})}=2(2\pi)^{3}\,\bm{{\it\Pi}}_{\rm s}^{\rm
T}(\bm{\nabla}_{2})\,\delta(\bm{x}_{2}-\bm{x}_{1})$ (390)
and
$\displaystyle\sum_{\kappa\rightarrow\pm}\int{\rm d}\bm{k}\
\bm{\tau}\cdot\bm{\hat{k}}\,{\rm e}^{\kappa{\rm
i}\bm{k}\cdot(\bm{x}_{2}-\bm{x}_{1})}$ $\displaystyle=$ $\displaystyle 0,$
(391)
which yields the transverse completeness relation
$\displaystyle\sum_{\kappa\rightarrow\pm}\sum_{\lambda=1}^{2}\int{\rm
d}\bm{k}\
\psi_{\bm{k},\lambda}^{(\kappa)}(\bm{x}_{2})\psi_{\bm{k},\lambda}^{(\kappa)\dagger}(\bm{x}_{1})={\it\Pi}^{\rm
T}(\bm{\nabla}_{2})\,\delta(\bm{x}_{2}-\bm{x}_{1}).\qquad{}$ (392)
### 7.2 Longitudinal plane-wave photons
The transverse wave functions alone do not provide a complete description of
electromagnetic fields. For example, the field of a point charge $q$ at rest
at the origin, given by
$\displaystyle\bm{E}_{\rm s}(\bm{x})$ $\displaystyle=$
$\displaystyle\frac{q}{4\pi\epsilon_{0}}\,\frac{\bm{x}_{\rm
s}}{|\bm{x}|^{3}}=-\frac{q}{4\pi\epsilon_{0}}\bm{\nabla}_{\rm
s}\,\frac{1}{|\bm{x}|},$ (393)
is purely longitudinal, because
$\displaystyle\bm{{\it\Pi}}_{\rm s}^{\rm L}(\bm{\nabla})\,\bm{E}_{\rm
s}(\bm{x})$ $\displaystyle=$ $\displaystyle\bm{E}_{\rm s}(\bm{x}),$ (394)
$\displaystyle\bm{{\it\Pi}}_{\rm s}^{\rm T}(\bm{\nabla})\,\bm{E}_{\rm
s}(\bm{x})$ $\displaystyle=$ $\displaystyle 0.$ (395)
The longitudinal photons are represented by a third polarization state,
labeled $\lambda=0$, with the polarization vector taken to be
$\displaystyle\bm{\hat{\epsilon}}_{0}(\bm{\hat{k}})$ $\displaystyle=$
$\displaystyle\bm{\hat{k}}_{\rm s}.{}$ (396)
If $\bm{k}$ is in the $\bm{\hat{e}}^{3}$ direction, the longitudinal
polarization vector is (up to a phase factor)
$\displaystyle\bm{\hat{\epsilon}}_{0}(\bm{\hat{e}}^{3})$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}0\\\ 1\\\ 0\end{array}\right).{}$ (400)
Longitudinal wave functions are
$\displaystyle\psi_{\bm{k},0}^{(+)}(\bm{x})$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{(2\pi)^{3}}}\left(\begin{array}[]{c}\bm{\hat{\epsilon}}_{0}(\bm{\hat{k}})\\\
{\bm{0}}\end{array}\right){\rm e}^{{\rm i}\bm{k}\cdot\bm{x}}\quad{}$ (403)
or
$\displaystyle\psi_{\bm{k},0}^{(-)}(\bm{x})$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{(2\pi)^{3}}}\left(\begin{array}[]{c}{\bm{0}}\\\
\bm{\hat{\epsilon}}_{0}(\bm{\hat{k}})\end{array}\right){\rm e}^{-{\rm
i}\bm{k}\cdot\bm{x}}.\quad{}$ (406)
This polarization state has the property that
$\displaystyle{\it\Pi}^{\rm L}(\bm{\nabla})\,\psi_{\bm{k},0}^{(\pm)}(\bm{x})$
$\displaystyle=$ $\displaystyle{\it\Pi}^{\rm
L}(\bm{\hat{k}})\,\psi_{\bm{k},0}^{(\pm)}(\bm{x})=\psi_{\bm{k},0}^{(\pm)}(\bm{x}),\qquad$
(407) $\displaystyle{\it\Pi}^{\rm
T}(\bm{\nabla})\,\psi_{\bm{k},0}^{(\pm)}(\bm{x})$ $\displaystyle=$
$\displaystyle{\it\Pi}^{\rm
T}(\bm{\hat{k}})\,\psi_{\bm{k},0}^{(\pm)}(\bm{x})=0.$ (408)
The wave function has an energy eigenvalue of zero,
$\displaystyle{\cal H}\,\psi_{\bm{k},0}^{(\pm)}(\bm{x})$ $\displaystyle=$
$\displaystyle\pm\hbar
c\,\bm{\alpha}\cdot\bm{k}\,\psi_{\bm{k},0}^{(\pm)}(\bm{x})=0,$ (409)
because $\bm{\tau}\cdot\bm{k}\,\bm{\hat{\epsilon}}_{0}(\bm{\hat{k}})=0$.
However,
$\displaystyle\bm{{\cal P}}\,\psi_{\bm{k},0}^{(\pm)}(\bm{x})$ $\displaystyle=$
$\displaystyle\pm\hbar\,\bm{k}\,\psi_{\bm{k},0}^{(\pm)}(\bm{x}).$ (410)
Normalization and orthogonality of the $\lambda=0$ wave functions, as well as
the transverse wave functions, are given by Eqs. (381) and (382), where
$\lambda_{1}$ and $\lambda_{2}$ may take on any of the values 0, 1, or 2.
Completeness relations are given by
$\displaystyle\int{\rm d}\bm{k}\
\psi_{\bm{k},0}^{(+)}(\bm{x}_{2})\psi_{\bm{k},0}^{(+)\dagger}(\bm{x}_{1})$
$\displaystyle=$ $\displaystyle\int{\rm d}\bm{k}\
\frac{1}{(2\pi)^{3}}\,\left(\begin{array}[]{ccc}\bm{\hat{\epsilon}}_{0}(\bm{\hat{k}})\,\bm{\hat{\epsilon}}_{0}^{\dagger}(\bm{\hat{k}})&&{\bm{0}}\\\
{\bm{0}}&&{\bm{0}}\end{array}\right){\rm e}^{{\rm
i}\bm{k}\cdot(\bm{x}_{2}-\bm{x}_{1})}{}$ (413)
and
$\displaystyle\int{\rm d}\bm{k}\
\psi_{\bm{k},0}^{(-)}(\bm{x}_{2})\psi_{\bm{k},0}^{(-)\dagger}(\bm{x}_{1})$
$\displaystyle=$ $\displaystyle\int{\rm d}\bm{k}\
\frac{1}{(2\pi)^{3}}\,\left(\begin{array}[]{ccc}{\bm{0}}&&{\bm{0}}\\\
{\bm{0}}&&\bm{\hat{\epsilon}}_{0}(\bm{\hat{k}})\,\bm{\hat{\epsilon}}_{0}^{\dagger}(\bm{\hat{k}})\end{array}\right){\rm
e}^{-{\rm i}\bm{k}\cdot(\bm{x}_{2}-\bm{x}_{1})},{}$ (416)
where
$\displaystyle\int{\rm d}\bm{k}\
\bm{\hat{\epsilon}}_{0}(\bm{\hat{k}})\,\bm{\hat{\epsilon}}_{0}^{\dagger}(\bm{\hat{k}}){\rm
e}^{\pm{\rm
i}\bm{k}\cdot(\bm{x}_{2}-\bm{x}_{1})}=(2\pi)^{3}\,\bm{{\it\Pi}}_{\rm s}^{\rm
L}(\bm{\nabla}_{2})\,\delta(\bm{x}_{2}-\bm{x}_{1}),\qquad$ (417)
which yields
$\displaystyle\sum_{\kappa\rightarrow\pm}\int{\rm d}\bm{k}\
\psi_{\bm{k},0}^{(\kappa)}(\bm{x}_{2})\psi_{\bm{k},0}^{(\kappa)\dagger}(\bm{x}_{1})={\it\Pi}^{\rm
L}(\bm{\nabla}_{2})\,\delta(\bm{x}_{2}-\bm{x}_{1}).\qquad{}$ (418)
For the example of a point charge at the origin, we have
$\displaystyle{\it\Psi}_{\rm p}(\bm{x})$ $\displaystyle=$
$\displaystyle\frac{q}{4\pi\epsilon_{0}|\bm{x}|^{3}}\left(\begin{array}[]{c}\bm{x}_{\rm
s}\\\ {\bm{0}}\end{array}\right),$ (421)
which may be written as (see C for some detail)
$\displaystyle{\it\Psi}_{\rm p}(\bm{x})$ $\displaystyle=$
$\displaystyle{\it\Pi}^{\rm L}(\bm{\nabla}){\it\Psi}_{\rm p}(\bm{x})=\int{\rm
d}\bm{x}_{1}{\it\Pi}^{\rm
L}(\bm{\nabla})\,\delta(\bm{x}-\bm{x}_{1}){\it\Psi}_{\rm p}(\bm{x}_{1})$ (422)
$\displaystyle=$ $\displaystyle\sum_{\kappa\rightarrow\pm}\int{\rm d}\bm{k}\
\psi_{\bm{k},0}^{(\kappa)}(\bm{x})\int{\rm
d}\bm{x}_{1}\,\psi_{\bm{k},0}^{(\kappa)\dagger}(\bm{x}_{1}){\it\Psi}_{\rm
p}(\bm{x}_{1})$ $\displaystyle=$ $\displaystyle-\frac{{\rm
i}\,q}{\sqrt{(2\pi)^{3}}\,\epsilon_{0}}\int{\rm
d}\bm{k}\,\frac{1}{|\bm{k}|}\,\psi_{\bm{k},0}^{(+)}(\bm{x}).{}$
### 7.3 Full orthogonality and completeness of the plane wave solutions
The full orthogonality relations are
$\displaystyle\int{\rm d}\bm{x}\
\psi_{\bm{k}_{2},\lambda_{2}}^{(\kappa_{2})\dagger}(\bm{x})\,\psi_{\bm{k}_{1},\lambda_{1}}^{(\kappa_{1})}(\bm{x})$
$\displaystyle=$
$\displaystyle\delta_{\kappa_{2}\kappa_{1}}\delta_{\lambda_{2}\lambda_{1}}\delta(\bm{k}_{2}-\bm{k}_{1}),\qquad{}$
(423)
where the factor $\delta_{\kappa_{2}\kappa_{1}}$ is 1 if $\kappa_{2}$ and
$\kappa_{1}$ represent the same sign and is 0 for opposite signs, and
$\lambda_{2}$ and $\lambda_{1}$ may be any of 0,1,2. The combined result of
the transverse and longitudinal completeness relations, Eqs. (392) and (418),
is
$\displaystyle\sum_{\kappa\rightarrow\pm}\sum_{\lambda=0}^{2}\int{\rm
d}\bm{k}\
\psi_{\bm{k},\lambda}^{(\kappa)}(\bm{x}_{2})\psi_{\bm{k},\lambda}^{(\kappa)\dagger}(\bm{x}_{1})={\cal
I}\,\delta(\bm{x}_{2}-\bm{x}_{1}),\qquad$ (424)
where ${\it\Pi}^{\rm T}(\bm{\nabla})+{\it\Pi}^{\rm L}(\bm{\nabla})={\cal I}$.
### 7.4 Time dependence of the wave functions
The time dependence of the photon wave functions is given by222The notation
$f(\bm{x})=f(x)\big{|}_{t=0}$ is employed throughout this paper.
$\displaystyle\psi_{\bm{k},\lambda}^{(\kappa)}(x)$ $\displaystyle=$
$\displaystyle\psi_{\bm{k},\lambda}^{(\kappa)}(\bm{x})\,{\rm e}^{-\kappa{\rm
i}\omega t},{}$ (425)
where $\omega$ is determined by the equation
$\displaystyle\gamma^{\mu}\partial_{\mu}\,\psi_{\bm{k},\lambda}^{(\kappa)}(x)$
$\displaystyle=$ $\displaystyle\gamma^{0}\left({\cal
I}\,\frac{\partial}{\partial
ct}+\bm{\alpha}\cdot\bm{\nabla}\right)\psi_{\bm{k},\lambda}^{(\kappa)}(x)=0.{}$
(426)
For transverse photons
$\displaystyle\omega$ $\displaystyle=$ $\displaystyle c|\bm{k}|,{}$ (427)
and for longitudinal photons
$\displaystyle\omega$ $\displaystyle=$ $\displaystyle 0.{}$ (428)
The complete exponential factor is thus
$\displaystyle{\rm e}^{-\kappa{\rm i}k\cdot x},{}$ (429)
where
$\displaystyle k$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}|\bm{k}|\\\ \bm{k}_{\rm
c}\end{array}\right)\quad\mbox{or}\quad\left(\begin{array}[]{c}0\\\
\bm{k}_{\rm c}\end{array}\right),{}$ (434)
depending on whether the photon is transverse or longitudinal. This
corresponds to the eigenvalue equation
$\displaystyle{\cal H}\,\psi_{\bm{k},\lambda}^{(\kappa)}(\bm{x})$
$\displaystyle=$
$\displaystyle\kappa\hbar\omega\,\psi_{\bm{k},\lambda}^{(\kappa)}(\bm{x})$
(435)
and a time dependence given by
$\displaystyle\psi_{\bm{k},\lambda}^{(\kappa)}(x)$ $\displaystyle=$
$\displaystyle{\rm e}^{-{\rm i}{\cal
H}t/\hbar}\psi_{\bm{k},\lambda}^{(\kappa)}(\bm{x}).{}$ (436)
It is also of interest to consider the effect of a hypothetical photon mass
$m_{\gamma}$ on the longitudinal photon wave function. Such a modification,
with an infinitesimal mass, resolves an ambiguity in the construction of the
Green function as discussed in Sec. 9. Following the form of the Dirac
equation in Eq. (6), we have
$\displaystyle\left({\rm
i}\,\hbar\gamma^{\mu}\partial_{\mu}-m_{\gamma}c\right)\psi_{\bm{k},0}^{(\kappa)}(x)=0{}$
(437)
or
$\displaystyle\left(\frac{\hbar\kappa\omega}{c}\gamma^{0}-m_{\gamma}c\,{\cal
I}\right)\psi_{\bm{k},0}^{(\kappa)}(x)=0,$ (438)
which yields
$\displaystyle\hbar\omega$ $\displaystyle=$ $\displaystyle m_{\gamma}c^{2},$
(439)
since
$\displaystyle\gamma^{0}\psi_{\bm{k},0}^{(\kappa)}(x)$ $\displaystyle=$
$\displaystyle\kappa\,\psi_{\bm{k},0}^{(\kappa)}(x).$ (440)
### 7.5 Rotation of the wave functions
The rotations of the wave functions follow from the discussion of Sec. 6.4,
with an additional consideration of the vector $\bm{k}$. On physical grounds,
a rotation parameterized by the vector $\bm{u}$ of the state of a photon means
rotation of the vector $\bm{k}$ into the vector $\bm{k}^{\prime}$, according
to
$\displaystyle\bm{k}^{\prime}$ $\displaystyle=$
$\displaystyle\bm{R}(\bm{u})\bm{k},$ (441)
where $\bm{R}(\bm{u})$ is defined by Eq. (158). Similarly, the polarization
vector is transformed by the spherical rotation operator
$\displaystyle\bm{\hat{\epsilon}}_{\lambda}(\bm{\hat{k}}^{\prime})$
$\displaystyle=$ $\displaystyle\bm{R}_{\rm
s}(\bm{u})\,\bm{\hat{\epsilon}}_{\lambda}(\bm{\hat{k}}){}$ (442)
and
$\displaystyle\bm{\tau}\cdot\bm{\hat{k}}^{\prime}\,\bm{\hat{\epsilon}}_{\lambda}(\bm{\hat{k}}^{\prime})$
$\displaystyle=$ $\displaystyle\bm{R}_{\rm
s}(\bm{u})\,\bm{\tau}\cdot\bm{\hat{k}}\,\bm{R}_{\rm s}^{-1}(\bm{u})\bm{R}_{\rm
s}(\bm{u})\,\bm{\hat{\epsilon}}_{\lambda}(\bm{\hat{k}}),\qquad$ (443)
where the rotation angle $\theta$, axis direction $\bm{\hat{u}}$,
$\bm{\hat{k}}$, and $\bm{\hat{k}}^{\prime}$ are related by
$\displaystyle\bm{\hat{u}}\sin{\theta}$ $\displaystyle=$
$\displaystyle\bm{\hat{k}}\times\bm{\hat{k}}^{\prime}.{}$ (444)
For applications, the rotation operator can be expressed as a function of
$\bm{\hat{k}}$ and $\bm{\hat{k}}^{\prime}$ rather than $\bm{u}$. From
$\displaystyle\bm{R}_{\rm s}(\bm{u})$ $\displaystyle=$
$\displaystyle\bm{I}-(\bm{\tau}\cdot\bm{\hat{u}})^{2}\left(1-\cos{\theta}\right)-{\rm
i}\,\bm{\tau}\cdot\bm{\hat{u}}\,\sin{\theta},\qquad$ (445)
one has for transverse polarization, $\lambda=1,2$,333The identity
$\bm{\tau}\cdot\bm{k}\times\bm{k}^{\prime}={\rm i}(\bm{k}^{\prime}_{\rm
s}\,\bm{k}_{\rm s}^{\dagger}-\bm{k}_{\rm s}\,\bm{k}_{\rm s}^{\prime\dagger})$
is useful here.
$\displaystyle\bm{\hat{\epsilon}}_{\lambda}(\bm{\hat{k}}^{\prime})$
$\displaystyle=$
$\displaystyle\frac{(\bm{\tau}\cdot\bm{\hat{k}}^{\prime})^{2}+\bm{\tau}\cdot\bm{\hat{k}}^{\prime}\,\bm{\tau}\cdot\bm{\hat{k}}}{1+\bm{\hat{k}}^{\prime}\cdot\bm{\hat{k}}}\,\bm{\hat{\epsilon}}_{\lambda}(\bm{\hat{k}}),$
(446)
$\displaystyle\bm{\tau}\cdot\bm{\hat{k}}^{\prime}\,\bm{\hat{\epsilon}}_{\lambda}(\bm{\hat{k}}^{\prime})$
$\displaystyle=$
$\displaystyle\frac{(\bm{\tau}\cdot\bm{\hat{k}}^{\prime})^{2}+\bm{\tau}\cdot\bm{\hat{k}}^{\prime}\,\bm{\tau}\cdot\bm{\hat{k}}}{1+\bm{\hat{k}}^{\prime}\cdot\bm{\hat{k}}}\,\bm{\tau}\cdot\bm{\hat{k}}\,\bm{\hat{\epsilon}}_{\lambda}(\bm{\hat{k}}),\qquad$
(447)
and for longitudinal polarization, $\lambda=0$,
$\displaystyle\bm{\hat{\epsilon}}_{0}(\bm{\hat{k}}^{\prime})$ $\displaystyle=$
$\displaystyle\bm{\hat{k}}_{\rm s}^{\prime}\bm{\hat{k}}_{\rm
s}^{\dagger}\,\bm{\hat{\epsilon}}_{0}(\bm{\hat{k}}).$ (448)
We thus have for a rotation of the wave functions characterized by the vector
$\bm{u}$
$\displaystyle\psi_{\bm{k}^{\prime}\\!,\,\lambda}^{(\kappa)}(x)$
$\displaystyle=$ $\displaystyle{\cal
R}(\bm{u})\,\psi_{\bm{k},\lambda}^{\,(\kappa)}\big{(}R^{-1}(\bm{u})\,x\big{)},{}$
(449)
where $\bm{k}^{\prime}\cdot\bm{x}=\bm{k}\cdot\bm{R}^{-1}(\bm{u})\,\bm{x}$ in
the exponent of the wave function. This yields the result
$\displaystyle\gamma^{\mu}\partial_{\mu}\psi_{\bm{k}^{\prime}\\!,\,\lambda}^{(\kappa)}(x)$
$\displaystyle=$ $\displaystyle 0,{}$ (450)
according to the discussion in Sec. 6.4.
The expected transformation in Eq. (449) can be confirmed by an explicit
calculation based on the completeness of the wave functions. For a rotation of
a transverse wave function, we write
$\displaystyle{\cal
R}(\bm{u})\,\psi_{\bm{k},\lambda}^{(\kappa)}\big{(}R^{-1}(\bm{u})\,x\big{)}=\int{\rm
d}\bm{x}_{1}\,\delta(\bm{x}-\bm{x}_{1})\,{\cal
R}(\bm{u})\,\psi_{\bm{k},\lambda}^{(\kappa)}\big{(}\bm{R}^{-1}(\bm{u})\,\bm{x}_{1}\big{)}\,{\rm
e}^{-\kappa{\rm i}\omega t}$
$\displaystyle\qquad=\sum_{\kappa_{1}\rightarrow\pm}\sum_{\lambda_{1}=1}^{2}\int{\rm
d}\bm{k}_{1}\int{\rm d}\bm{x}_{1}\
\psi_{\bm{k}_{1},\lambda_{1}}^{(\kappa_{1})}(\bm{x})\psi_{\bm{k}_{1},\lambda_{1}}^{(\kappa_{1})\dagger}(\bm{x}_{1}){\cal
R}(\bm{u})\,\psi_{\bm{k},\lambda}^{(\kappa)}\big{(}\bm{R}^{-1}(\bm{u})\,\bm{x}_{1}\big{)}\,{\rm
e}^{-\kappa{\rm i}\omega t},\qquad{}$ (451)
where $\lambda=1$ or $2$, and from Eq. (264) and the subsequent remarks, it
follows that the rotated wave function is also transverse, so $\lambda_{1}$ is
restricted to 1 or 2. The evaluation requires the matrix element
$\displaystyle\int{\rm d}\bm{x}_{1}\
\psi_{\bm{k}_{1},\lambda_{1}}^{(\kappa_{1})\dagger}(\bm{x}_{1}){\cal
R}(\bm{u})\,\psi_{\bm{k},\lambda}^{(\kappa)}\big{(}\bm{R}^{-1}(\bm{u})\,\bm{x}_{1}\big{)}$
$\displaystyle\qquad=\frac{1}{2(2\pi)^{3}}\int{\rm
d}\bm{x}_{1}\left[\bm{\hat{\epsilon}}_{\lambda_{1}}^{\dagger}(\bm{\hat{k}}_{1})\,\bm{R}_{\rm
s}(\bm{u})\,\bm{\hat{\epsilon}}_{\lambda}(\bm{\hat{k}})+\bm{\hat{\epsilon}}_{\lambda_{1}}^{\dagger}(\bm{\hat{k}}_{1})\,\bm{\tau}\cdot\bm{\hat{k}}_{1}\,\bm{R}_{\rm
s}(\bm{u})\,\bm{\tau}\cdot\bm{\hat{k}}\,\bm{\hat{\epsilon}}_{\lambda}(\bm{\hat{k}})\right]$
$\displaystyle\qquad\qquad\qquad\qquad\times{\rm e}^{-\kappa_{1}{\rm
i}\bm{k}_{1}\cdot\bm{x}_{1}}{\rm e}^{\kappa{\rm
i}\bm{k}\cdot\bm{R}^{-1}(\bm{u})\bm{x}_{1}}$
$\displaystyle\qquad=\frac{1}{2}\,\delta_{\kappa_{1}\kappa}\left[\bm{\hat{\epsilon}}_{\lambda_{1}}^{\dagger}(\bm{\hat{k}^{\prime}})\,\bm{R}_{\rm
s}(\bm{u})\,\bm{\hat{\epsilon}}_{\lambda}(\bm{\hat{k}})+\bm{\hat{\epsilon}}_{\lambda_{1}}^{\dagger}(\bm{\hat{k}^{\prime}})\,\bm{\tau}\cdot\bm{\hat{k}^{\prime}}\,\bm{R}_{\rm
s}(\bm{u})\,\bm{\tau}\cdot\bm{\hat{k}}\,\bm{\hat{\epsilon}}_{\lambda}(\bm{\hat{k}})\right]\delta(\bm{k}_{1}-\bm{k}^{\prime})\,\qquad$
$\displaystyle\qquad=\delta_{\kappa_{1}\kappa}\,\delta_{\lambda_{1}\lambda}\,\delta(\bm{k}_{1}-\bm{k}^{\prime}).{}$
(452)
In the exponent in Eq. (452), we have
$\bm{k}\cdot\bm{R}^{-1}(\bm{u})\bm{x}_{1}=\bm{R}(\bm{u})\bm{k}\cdot\bm{x}_{1}=\bm{k}^{\prime}\cdot\bm{x}_{1}$.
The factor $\delta_{\kappa_{1}\kappa}$ results from the requirement that
$\bm{k}^{\prime}\rightarrow\bm{k}$ as the rotation angle $\theta\rightarrow
0$, i.e., that $\bm{k}$ does not change sign for an infinitesimal rotation.
The factor $\delta_{\lambda_{1}\lambda}$ follows from Eq. (442) and the
discussion that follows it, together with the orthonormality of the
polarization vectors. Substitution of Eq. (452) into (451) yields
$\displaystyle{\cal
R}(\bm{u})\,\psi_{\bm{k},\lambda}^{(\kappa)}\big{(}R^{-1}(\bm{u})\,x\big{)}$
$\displaystyle=$ $\displaystyle\psi_{\bm{k}^{\prime},\lambda}^{(\kappa)}(x),$
(453)
in accord with the general arguments leading to Eq. (449).
For a rotation of a longitudinal wave function, we have
$\displaystyle{\cal
R}(\bm{u})\,\psi_{\bm{k},0}^{(\kappa)}\big{(}R^{-1}(\bm{u})\,x\big{)}=\int{\rm
d}\bm{x}_{1}\,\delta(\bm{x}-\bm{x}_{1})\,{\cal
R}(\bm{u})\,\psi_{\bm{k},0}^{(\kappa)}\big{(}\bm{R}^{-1}(\bm{u})\,\bm{x}_{1}\big{)}$
$\displaystyle\qquad\qquad=\sum_{\kappa_{1}\rightarrow\pm}\int{\rm
d}\bm{k}_{1}\int{\rm d}\bm{x}_{1}\
\psi_{\bm{k}_{1},0}^{(\kappa_{1})}(\bm{x})\psi_{\bm{k}_{1},0}^{(\kappa_{1})\dagger}(\bm{x}_{1}){\cal
R}(\bm{u})\,\psi_{\bm{k},0}^{(\kappa)}\big{(}\bm{R}^{-1}(\bm{u})\,\bm{x}_{1}\big{)}\qquad{}$
(454)
where only longitudinal wave functions contribute, and the evaluation requires
the matrix element
$\displaystyle\int{\rm d}\bm{x}_{1}\
\psi_{\bm{k}_{1},0}^{(\kappa_{1})\dagger}(\bm{x}_{1}){\cal
R}(\bm{u})\,\psi_{\bm{k},0}^{(\kappa)}\big{(}\bm{R}^{-1}(\bm{u})\,\bm{x}_{1}\big{)}$
$\displaystyle\qquad\qquad=\frac{1}{(2\pi)^{3}}\int{\rm
d}\bm{x}_{1}\delta_{\kappa_{1}\kappa}\,\bm{\hat{\epsilon}}_{0}^{\dagger}(\bm{\hat{k}}_{1})\,\bm{R}_{\rm
s}(\bm{u})\,\bm{\hat{\epsilon}}_{0}(\bm{\hat{k}})\,{\rm e}^{-\kappa_{1}{\rm
i}\bm{k}_{1}\cdot\bm{x}_{1}}{\rm e}^{\kappa{\rm
i}\bm{k}\cdot\bm{R}^{-1}(\bm{u})\bm{x}_{1}}\qquad$
$\displaystyle\qquad\qquad=\delta_{\kappa_{1}\kappa}\,\bm{\hat{\epsilon}}_{0}^{\dagger}(\bm{\hat{k}^{\prime}})\,\bm{R}_{\rm
s}(\bm{u})\,\bm{\hat{\epsilon}}_{0}(\bm{\hat{k}})\,\delta(\bm{k}_{1}-\bm{k}^{\prime})\,$
$\displaystyle\qquad\qquad=\delta_{\kappa_{1}\kappa}\,\delta(\bm{k}_{1}-\bm{k}^{\prime}).{}$
(455)
Substitution of Eq. (455) into (454) yields
$\displaystyle{\cal
R}(\bm{u})\,\psi_{\bm{k},0}^{(\kappa)}\big{(}R^{-1}(\bm{u})\,x\big{)}$
$\displaystyle=$ $\displaystyle\psi_{\bm{k}^{\prime},0}^{(\kappa)}(x),$ (456)
in agreement with Eq. (449).
### 7.6 Velocity transformation of the wave functions
#### 7.6.1 Transverse wave functions
Under the velocity transformation of a transverse photon by a velocity
$\bm{v}$, the four-vector
$\displaystyle k$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}|\bm{k}|\\\ \bm{k}_{\rm
c}\end{array}\right)$ (459)
is transformed to
$\displaystyle k^{\prime}$ $\displaystyle=$ $\displaystyle
V(\bm{v})\,k=\left(\begin{array}[]{c}|\bm{k}|\left(\cosh{\zeta}+\bm{\hat{v}}\cdot\bm{\hat{k}}\sinh{\zeta}\right)\\\
\bm{k}_{\rm c}+|\bm{k}|\bm{\hat{v}}_{\rm
c}\\!\left[\sinh{\zeta}+\bm{\hat{v}}\cdot\bm{\hat{k}}(\cosh{\zeta}-1)\right]\end{array}\right),\qquad$
(462)
and the wave function is expected to transform according to
$\displaystyle\xi\,\psi_{\bm{k}^{\prime}\\!,\,\lambda}^{\prime\,(\kappa)}(x)$
$\displaystyle=$ $\displaystyle{\cal
V}(\bm{v})\,\psi_{\bm{k},\lambda}^{\,(\kappa)}\big{(}V^{-1}(\bm{v})\,x\big{)}.{}$
(463)
The prime on the transformed function indicates that it is a function of
modified polarization vectors, which in general are not simply rotated vectors
corresponding to $\bm{\hat{k}}\rightarrow\bm{\hat{k}}^{\prime}$. The
transformed transverse wave function is proportional to
$\displaystyle{\cal
V}(\bm{v})\left(\begin{array}[]{c}\bm{\hat{\epsilon}}_{\lambda}(\bm{\hat{k}})\\\
\bm{\tau}\cdot\bm{\hat{k}}\
\bm{\hat{\epsilon}}_{\lambda}(\bm{\hat{k}})\end{array}\right)$
$\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}\left\\{I+(\bm{\tau}\cdot\bm{\hat{v}})^{2}\,(\cosh{\zeta}-1)+\bm{\tau}\cdot\bm{\hat{v}}\,\bm{\tau}\cdot\bm{\hat{k}}\,\sinh{\zeta}\right\\}\bm{\hat{\epsilon}}_{\lambda}(\bm{\hat{k}})\\\
\left\\{\bm{\tau}\cdot\bm{\hat{v}}\sinh{\zeta}+\left[I+(\bm{\tau}\cdot\bm{\hat{v}})^{2}\,(\cosh{\zeta}-1)\right]\bm{\tau}\cdot\bm{\hat{k}}\
\right\\}\bm{\hat{\epsilon}}_{\lambda}(\bm{\hat{k}})\end{array}\right)\qquad$
(468) $\displaystyle=$
$\displaystyle\xi\left(\begin{array}[]{c}\bm{\hat{\epsilon}}_{\lambda}^{\prime}(\bm{\hat{k}}^{\prime})\\\
\bm{\tau}\cdot\bm{\hat{k}}^{\prime}\
\bm{\hat{\epsilon}}_{\lambda}^{\prime}(\bm{\hat{k}}^{\prime})\end{array}\right),{}$
(471)
where the fact that transformed wave function can be written in the form given
at the right-hand end is based on the three identities:
$\displaystyle\left|\left\\{I+(\bm{\tau}\cdot\bm{\hat{v}})^{2}\,(\cosh{\zeta}-1)+\bm{\tau}\cdot\bm{\hat{v}}\,\bm{\tau}\cdot\bm{\hat{k}}\,\sinh{\zeta}\right\\}\bm{\hat{\epsilon}}_{\lambda}(\bm{\hat{k}})\right|$
$\displaystyle=$
$\displaystyle\cosh{\zeta}+\bm{\hat{v}}\cdot\bm{\hat{k}}\sinh{\zeta},\qquad{}$
(472) $\displaystyle{\bm{k}_{\rm
s}^{\prime}}^{\dagger}\left\\{I+(\bm{\tau}\cdot\bm{\hat{v}})^{2}\,(\cosh{\zeta}-1)+\bm{\tau}\cdot\bm{\hat{v}}\,\bm{\tau}\cdot\bm{\hat{k}}\,\sinh{\zeta}\right\\}\bm{\hat{\epsilon}}_{\lambda}(\bm{\hat{k}})$
$\displaystyle=$ $\displaystyle 0,{}$ (473)
$\displaystyle\bm{\tau}\cdot\bm{\hat{v}}\sinh{\zeta}+\left[I+(\bm{\tau}\cdot\bm{\hat{v}})^{2}\,(\cosh{\zeta}-1)\right]\bm{\tau}\cdot\bm{\hat{k}}\
$
$\displaystyle\qquad\qquad=\bm{\tau}\cdot\bm{\hat{k}}^{\prime}\left\\{I+(\bm{\tau}\cdot\bm{\hat{v}})^{2}\,(\cosh{\zeta}-1)+\bm{\tau}\cdot\bm{\hat{v}}\,\bm{\tau}\cdot\bm{\hat{k}}\,\sinh{\zeta}\right\\},\quad{}$
(474)
where
$\displaystyle\bm{\hat{k}}^{\prime}$ $\displaystyle=$
$\displaystyle\frac{\bm{\hat{k}}+\bm{\hat{v}}\\!\left[\sinh{\zeta}+\bm{\hat{v}}\cdot\bm{\hat{k}}(\cosh{\zeta}-1)\right]}{\cosh{\zeta}+\bm{\hat{v}}\cdot\bm{\hat{k}}\sinh{\zeta}}.$
(475)
Equation (472) determines the scalar multiplicative factor in Eq. (463) to be
$\displaystyle\xi$ $\displaystyle=$
$\displaystyle\cosh{\zeta}+\bm{\hat{v}}\cdot\bm{\hat{k}}\sinh{\zeta}.$ (476)
According to Eq. (473), the transformed polarization vector is in the plane
perpendicular to $\bm{k}^{\prime}$, that is, the transformed transverse
polarization vector is also transverse. This is in contrast to the vector
potential, which does not maintain transversality under a velocity
transformation, so the Coulomb, or radiation, gauge condition is not
preserved. The difference is just the fact that the vector potential
transforms as a vector, so that the angle between the space component of the
vector potential and the space component of the wave vector is not necessarily
preserved under a velocity transformation, whereas the polarization vector
transforms as the electric field, i.e., as a component of a second-rank
tensor, and Eq. (473) shows that for this case the transversality is
preserved. Equation (474) shows that the lower components of the transformed
wave function, as given in Eq. (471), can be written as
$\bm{\tau}\cdot\bm{\hat{k}}^{\prime}$ times the upper components in the same
expression. This together with the relation $k^{\prime}\cdot x=k\cdot
V^{-1}(\bm{v})\,x$ in the exponent of the wave function insures that the
transformed wave function is a solution of the source-free Maxwell equation.
It also follows from Eq. (471) that if $\lambda_{2}\neq\lambda_{1}$, then
$\displaystyle\bm{\hat{\epsilon}}_{\lambda_{2}}^{\prime\,\dagger}(\bm{\hat{k}}^{\prime})\,\bm{\hat{\epsilon}}_{\lambda_{1}}^{\prime}(\bm{\hat{k}}^{\prime})=0,{}$
(477)
so the orthogonality of the polarization vectors is preserved by the velocity
transformation.
Equation (463) can be be obtained by an explicit calculation based on the
completeness of the wave functions, as for rotations. A velocity
transformation of a transverse wave function is given by
$\displaystyle{\cal
V}(\bm{v})\,\psi_{\bm{k},\lambda}^{(\kappa)}\big{(}V^{-1}(\bm{v})\,x\big{)}$
$\displaystyle\qquad\qquad=\int{\rm
d}\bm{x}_{1}\,\delta(\bm{x}-\bm{x}_{1})\,{\cal
V}(\bm{v})\,\psi_{\bm{k},\lambda}^{(\kappa)}\big{(}V^{-1}(\bm{v})\,x_{1}\big{)}$
$\displaystyle\qquad\qquad=\sum_{\kappa_{1}\rightarrow\pm}\sum_{\lambda_{1}=0}^{2}\int{\rm
d}\bm{k}_{1}\int{\rm d}\bm{x}_{1}\
\psi_{\bm{k}_{1},\lambda_{1}}^{\prime\,(\kappa_{1})}(\bm{x})\psi_{\bm{k}_{1},\lambda_{1}}^{\prime\,(\kappa_{1})\dagger}(\bm{x}_{1}){\cal
V}(\bm{v})\,\psi_{\bm{k},\lambda}^{(\kappa)}\big{(}V^{-1}(\bm{v})\,x_{1}\big{)},\qquad{}$
(478)
where $\lambda=1$ or $2$, $t_{1}=t$, and the primes on the wave functions
indicate that the velocity transformed polarization vectors provide the basis
vectors for $\lambda_{1}=1,2$. For $\lambda_{1}=0$,
$\displaystyle\int{\rm d}\bm{x}_{1}\
\psi_{\bm{k}_{1},0}^{\prime\,(+)\dagger}(\bm{x}_{1}){\cal
V}(\bm{v})\,\psi_{\bm{k},\lambda}^{(\kappa)}\big{(}V^{-1}(\bm{v})\,x_{1}\big{)}$
$\displaystyle\qquad=\frac{1}{\sqrt{2}(2\pi)^{3}}\int{\rm
d}\bm{x}_{1}\left(\begin{array}[]{cc}\bm{\hat{\epsilon}}_{0}^{\prime\dagger}(\bm{\hat{k}}_{1})&{\bm{0}}\end{array}\right){\cal
V}(\bm{v})\left(\begin{array}[]{c}\bm{\hat{\epsilon}}_{\lambda}(\bm{\hat{k}})\\\
\bm{\tau}\cdot\bm{\hat{k}}\
\bm{\hat{\epsilon}}_{\lambda}(\bm{\hat{k}})\end{array}\right){\rm e}^{-{\rm
i}\bm{k}_{1}\cdot\bm{x}_{1}}{\rm e}^{-\kappa{\rm i}k\cdot
V^{-1}(\bm{v})x_{1}}\qquad$ (482)
$\displaystyle\qquad=\frac{1}{\sqrt{2}}\,\delta_{+\kappa}\,\xi\,\bm{\hat{\epsilon}}_{0}^{\prime\dagger}(\bm{\hat{k}}^{\prime})\,\bm{\hat{\epsilon}}_{\lambda}^{\prime}(\bm{\hat{k}}^{\prime})\,\delta(\bm{k}_{1}-\bm{k}^{\prime})\,{\rm
e}^{-{\rm i}k^{\prime 0}ct}=0,{}$ (483) $\displaystyle\int{\rm d}\bm{x}_{1}\
\psi_{\bm{k}_{1},0}^{\prime\,(-)\dagger}(\bm{x}_{1}){\cal
V}(\bm{v})\,\psi_{\bm{k},\lambda}^{(\kappa)}\big{(}V^{-1}(\bm{v})\,x_{1}\big{)}$
$\displaystyle\qquad=\frac{1}{\sqrt{2}(2\pi)^{3}}\int{\rm
d}\bm{x}_{1}\left(\begin{array}[]{cc}{\bm{0}}&\bm{\hat{\epsilon}}_{0}^{\prime\dagger}(\bm{\hat{k}}_{1})\end{array}\right){\cal
V}(\bm{v})\left(\begin{array}[]{c}\bm{\hat{\epsilon}}_{\lambda}(\bm{\hat{k}})\\\
\bm{\tau}\cdot\bm{\hat{k}}\
\bm{\hat{\epsilon}}_{\lambda}(\bm{\hat{k}})\end{array}\right){\rm e}^{{\rm
i}\bm{k}_{1}\cdot\bm{x}_{1}}{\rm e}^{-\kappa{\rm i}k\cdot
V^{-1}(\bm{v})x_{1}}$ (487)
$\displaystyle\qquad=\frac{1}{\sqrt{2}}\,\delta_{-\kappa}\,\xi\,\bm{\hat{\epsilon}}_{0}^{\prime\dagger}(\bm{\hat{k}}^{\prime})\,\bm{\tau}\cdot\bm{\hat{k}}^{\prime}\bm{\hat{\epsilon}}_{\lambda}^{\prime}(\bm{\hat{k}}^{\prime})\,\delta(\bm{k}_{1}-\bm{k}^{\prime})\,{\rm
e}^{{\rm i}k^{\prime 0}ct}=0.{}$ (488)
In the exponent in Eqs. (483) and (488), $k\cdot
V^{-1}(\bm{v})x_{1}=V(\bm{v})k\cdot x_{1}=k^{\prime}\cdot x_{1}$. As for
rotations, the factors $\delta_{+\kappa}$ and $\delta_{-\kappa}$ result from
the requirement that $k^{\prime}\rightarrow k$ as the velocity
$|\bm{v}|\rightarrow 0$, and the last equalities follow from Eqs. (473) and
(474). For $\lambda_{1}=1$ or $2$, we have
$\displaystyle\int{\rm d}\bm{x}_{1}\
\psi_{\bm{k}_{1},\lambda_{1}}^{\prime(\kappa_{1})\dagger}(\bm{x}_{1}){\cal
V}(\bm{v})\,\psi_{\bm{k},\lambda}^{(\kappa)}\big{(}V^{-1}(\bm{v})\,x_{1}\big{)}$
$\displaystyle\qquad\qquad=\frac{1}{2(2\pi)^{3}}\int{\rm
d}\bm{x}_{1}\left(\begin{array}[]{cc}\bm{\hat{\epsilon}}_{\lambda_{1}}^{\prime\dagger}(\bm{\hat{k}}_{1})&\bm{\hat{\epsilon}}_{\lambda_{1}}^{\prime\dagger}(\bm{\hat{k}_{1}})\,\bm{\tau}\cdot\bm{\hat{k}}_{1}\end{array}\right){\cal
V}(\bm{v})\left(\begin{array}[]{c}\bm{\hat{\epsilon}}_{\lambda}(\bm{\hat{k}})\\\
\bm{\tau}\cdot\bm{\hat{k}}\
\bm{\hat{\epsilon}}_{\lambda}(\bm{\hat{k}})\end{array}\right)\qquad$ (492)
$\displaystyle\qquad\qquad\qquad\times{\rm e}^{-\kappa_{1}{\rm
i}\bm{k}_{1}\cdot\bm{x}_{1}}{\rm e}^{-\kappa{\rm i}k\cdot
V^{-1}(\bm{v})x_{1}}$
$\displaystyle\qquad\qquad=\delta_{\kappa_{1}\kappa}\,\delta_{\lambda_{1}\lambda}\,\xi\,\delta(\bm{k}_{1}-\bm{k}^{\prime})\,{\rm
e}^{-\kappa{\rm i}k^{\prime 0}ct}.{}$ (493)
Substitution of Eq. (493) into (478) yields
$\displaystyle{\cal
V}(\bm{v})\,\psi_{\bm{k},\lambda}^{(\kappa)}\big{(}V^{-1}(\bm{v})\,x\big{)}$
$\displaystyle=$
$\displaystyle\xi\,\psi_{\bm{k}^{\prime},\lambda}^{\,\prime\,(\kappa)}(x),$
(494)
where
$\displaystyle\xi\,\bm{\hat{\epsilon}}_{\lambda}^{\prime}(\bm{\hat{k}}^{\prime})$
$\displaystyle=$
$\displaystyle\left\\{I+(\bm{\tau}\cdot\bm{\hat{v}})^{2}\,(\cosh{\zeta}-1)+\bm{\tau}\cdot\bm{\hat{v}}\,\bm{\tau}\cdot\bm{\hat{k}}\,\sinh{\zeta}\right\\}\bm{\hat{\epsilon}}_{\lambda}(\bm{\hat{k}}),$
(495)
in accord with Eq. (463).
#### 7.6.2 Longitudinal wave functions
For a longitudinal solution, the four-vector $k$ in the invariant phase factor
$k\cdot x$, given by
$\displaystyle k$ $\displaystyle=$ $\displaystyle\left(\begin{array}[]{c}0\\\
\bm{k}_{\rm c}\end{array}\right)$ (498)
from Eqs. (428)-(434), has a zero time component. However, the transformed
phase, given by $k\cdot V^{-1}({\bm{v}})\,x=V({\bm{v}})k\cdot
x=k^{\prime}\cdot x$, where
$\displaystyle k^{\prime}$ $\displaystyle=$ $\displaystyle
V(\bm{v})\,k=\left(\begin{array}[]{c}\bm{\hat{v}}\cdot\bm{k}\sinh{\zeta}\\\
\bm{k}_{\rm c}+\bm{\hat{v}}_{\rm
c}\,\bm{\hat{v}}\cdot\bm{k}(\cosh{\zeta}-1)\end{array}\right),\qquad$ (501)
does have time dependence. Thus the wave function is expected to transform
according to
$\displaystyle\sum_{\kappa^{\prime}\rightarrow\pm}\sum_{\lambda^{\prime}=0}^{3}\xi_{\lambda^{\prime}\,0}^{\kappa^{\prime}\kappa}(t)\,\psi_{\bm{k}^{\prime}\\!,\,\lambda^{\prime}}^{\,\prime\,(\kappa^{\prime})}(x)$
$\displaystyle=$ $\displaystyle{\cal
V}(\bm{v})\,\psi_{\bm{k},0}^{(\kappa)}\big{(}V^{-1}(\bm{v})\,x\big{)},$ (502)
where the coefficients include the extra time dependence introduced by the
velocity transformation. The sum over polarization states is necessary,
because unlike the case of the transverse wave function, the transformed
longitudinal wave function is mixture of longitudinal and transverse
components. This is expected on physical grounds, since moving charges cause
radiative atomic transitions. On the other hand, the transformed space-like
wave vector does not the match the wave vector of either the longitudinal or
transverse basis functions. Since the solutions are classified according to
their three-wave-vector, there is a residual time dependence in the expansion
that is included in the transformation coefficients.
To be explicit, for $\lambda\neq 0$, we consider specific polarization
vectors. Let $\bm{\hat{\epsilon}}^{\prime}_{1}(\bm{\hat{k}}^{\prime})$ be a
linear polarization vector in the plane of $\bm{\hat{k}}^{\prime}$ and
$\bm{\hat{v}}$ and $\bm{\hat{\epsilon}}^{\prime}_{2}(\bm{\hat{k}}^{\prime})$
be a linear polarization vector perpendicular to $\bm{\hat{k}}^{\prime}$ and
$\bm{\hat{v}}$. These conditions, together with $\bm{k}_{\rm
s}^{\prime\,\dagger}\,\bm{\hat{\epsilon}}^{\prime}_{1}(\bm{\hat{k}}^{\prime})=0$,
yield (up to phase factors)
$\displaystyle\bm{\hat{\epsilon}}_{1}^{\prime}(\bm{\hat{k}}^{\prime})$
$\displaystyle=$
$\displaystyle\frac{\bm{\hat{v}}\cdot\bm{\hat{k}}^{\prime}\,\bm{\hat{k}}_{\rm
s}^{\prime}-\bm{\hat{v}}_{\rm
s}}{\sqrt{1-(\bm{\hat{v}}\cdot\bm{\hat{k}}^{\prime})^{2}}},$ (503)
$\displaystyle\bm{\hat{\epsilon}}_{2}^{\prime}(\bm{\hat{k}}^{\prime})$
$\displaystyle=$
$\displaystyle\frac{\bm{\tau}\cdot\bm{\hat{v}}\,\bm{\hat{k}}_{\rm
s}^{\prime}}{\sqrt{1-(\bm{\hat{v}}\cdot\bm{\hat{k}}^{\prime})^{2}}},$ (504)
where
$\displaystyle\bm{\hat{\epsilon}}^{\prime}_{2}(\bm{\hat{k}}^{\prime})$
$\displaystyle=$
$\displaystyle\bm{\tau}\cdot\bm{\hat{k}}^{\prime}\,\bm{\hat{\epsilon}}^{\prime}_{1}(\bm{\hat{k}}^{\prime}),$
(505) $\displaystyle\bm{\hat{\epsilon}}^{\prime}_{1}(\bm{\hat{k}}^{\prime})$
$\displaystyle=$
$\displaystyle\bm{\tau}\cdot\bm{\hat{k}}^{\prime}\,\bm{\hat{\epsilon}}^{\prime}_{2}(\bm{\hat{k}}^{\prime}).$
(506)
The transformed upper- and lower-component longitudinal wave functions follow
from the expressions (for $t=0$ and $\bm{x}=0$)
$\displaystyle{\cal
V}(\bm{v})\left(\begin{array}[]{c}\bm{\hat{\epsilon}}_{0}(\bm{\hat{k}})\\\
{\bm{0}}\end{array}\right)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}\left[\bm{I}+(\bm{\tau}\cdot\bm{\hat{v}})^{2}\,(\cosh{\zeta}-1)\right]\bm{\hat{\epsilon}}_{0}(\bm{\hat{k}})\\\
\bm{\tau}\cdot\bm{\hat{v}}\sinh{\zeta}\,\bm{\hat{\epsilon}}_{0}(\bm{\hat{k}})\end{array}\right)=\xi_{00}^{++}(0)\left(\begin{array}[]{c}\bm{\hat{\epsilon}}^{\prime}_{0}(\bm{\hat{k}}^{\prime})\\\
{\bm{0}}\end{array}\right)$ (518)
$\displaystyle+\frac{\xi_{10}^{++}(0)}{\sqrt{2}}\left(\begin{array}[]{c}\bm{\hat{\epsilon}}^{\prime}_{1}(\bm{\hat{k}}^{\prime})\\\
\bm{\tau}\cdot\bm{\hat{k}}^{\prime}\,\bm{\hat{\epsilon}}^{\prime}_{1}(\bm{\hat{k}}^{\prime})\end{array}\right)+\frac{\xi_{10}^{-+}(0)}{\sqrt{2}}\left(\begin{array}[]{c}\bm{\hat{\epsilon}}^{\prime}_{1}(\bm{\hat{k}}^{\prime})\\\
-\bm{\tau}\cdot\bm{\hat{k}}^{\prime}\,\bm{\hat{\epsilon}}^{\prime}_{1}(\bm{\hat{k}}^{\prime})\end{array}\right),\qquad{}$
and
$\displaystyle{\cal V}(\bm{v})\left(\begin{array}[]{c}{\bm{0}}\\\
\bm{\hat{\epsilon}}_{0}(\bm{\hat{k}})\end{array}\right)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}\bm{\tau}\cdot\bm{\hat{v}}\sinh{\zeta}\,\bm{\hat{\epsilon}}_{0}(\bm{\hat{k}})\\\
\left[\bm{I}+(\bm{\tau}\cdot\bm{\hat{v}})^{2}\,(\cosh{\zeta}-1)\right]\bm{\hat{\epsilon}}_{0}(\bm{\hat{k}})\end{array}\right)=\xi_{00}^{--}(0)\left(\begin{array}[]{c}{\bm{0}}\\\
\bm{\hat{\epsilon}}^{\prime}_{0}(\bm{\hat{k}}^{\prime})\end{array}\right)$
(530)
$\displaystyle+\frac{\xi_{20}^{--}(0)}{\sqrt{2}}\left(\begin{array}[]{c}\bm{\hat{\epsilon}}^{\prime}_{2}(\bm{\hat{k}}^{\prime})\\\
\bm{\tau}\cdot\bm{\hat{k}}^{\prime}\,\bm{\hat{\epsilon}}^{\prime}_{2}(\bm{\hat{k}}^{\prime})\end{array}\right)+\frac{\xi_{20}^{+-}(0)}{\sqrt{2}}\left(\begin{array}[]{c}-\bm{\hat{\epsilon}}^{\prime}_{2}(\bm{\hat{k}}^{\prime})\\\
\bm{\tau}\cdot\bm{\hat{k}}^{\prime}\,\bm{\hat{\epsilon}}^{\prime}_{2}(\bm{\hat{k}}^{\prime})\end{array}\right),\qquad{}$
based on the relations
$\displaystyle\bm{\hat{k}}$ $\displaystyle=$
$\displaystyle\frac{\bm{\hat{k}}^{\prime}-\bm{\hat{v}}\,\bm{\hat{v}}\cdot\bm{\hat{k}}^{\prime}\left(1-{\rm
sech}{\,\zeta}\right)}{\sqrt{1-(\bm{\hat{v}}\cdot\bm{\hat{k}}^{\prime})^{2}\tanh^{2}{\zeta}}},$
(531) $\displaystyle\bm{\hat{k}}_{\rm s}^{\prime}\bm{\hat{k}}_{\rm
s}^{\prime\dagger}\left[\bm{I}+(\bm{\tau}\cdot\bm{\hat{v}})^{2}\,(\cosh{\zeta}-1)\right]\bm{\hat{k}}_{\rm
s}$ $\displaystyle=$
$\displaystyle\cosh{\zeta}\,\sqrt{1-(\bm{\hat{v}}\cdot\bm{\hat{k}}^{\prime})^{2}\tanh^{2}{\zeta}}\
\bm{\hat{k}}_{\rm s}^{\prime},$ (532) $\displaystyle\bm{\hat{k}}_{\rm
s}^{\prime}\bm{\hat{k}}_{\rm
s}^{\prime\dagger}\bm{\tau}\cdot\bm{\hat{v}}\sinh{\zeta}\,\bm{\hat{k}}_{\rm
s}$ $\displaystyle=$ $\displaystyle 0,$ (533)
$\displaystyle(\bm{I}-\bm{\hat{k}}_{\rm s}^{\prime}\bm{\hat{k}}_{\rm
s}^{\prime\dagger})\left[\bm{I}+(\bm{\tau}\cdot\bm{\hat{v}})^{2}\,(\cosh{\zeta}-1)\right]\bm{\hat{k}}_{\rm
s}$ $\displaystyle=$
$\displaystyle\frac{\sinh{\zeta}\tanh{\zeta}\,\bm{\hat{v}}\cdot\bm{\hat{k}}^{\prime}(\bm{\hat{v}}\cdot\bm{\hat{k}}^{\prime}\bm{\hat{k}}^{\prime}_{\rm
s}-\bm{\hat{v}}_{\rm
s})}{\sqrt{1-(\bm{\hat{v}}\cdot\bm{\hat{k}}^{\prime})^{2}\tanh^{2}{\zeta}}},\qquad$
(534) $\displaystyle(\bm{I}-\bm{\hat{k}}_{\rm s}^{\prime}\bm{\hat{k}}_{\rm
s}^{\prime\dagger})\,\bm{\tau}\cdot\bm{\hat{v}}\sinh{\zeta}\,\bm{\hat{k}}_{\rm
s}$ $\displaystyle=$
$\displaystyle\frac{\sinh{\zeta}\,\bm{\tau}\cdot\bm{\hat{k}}^{\prime}(\bm{\hat{v}}\cdot\bm{\hat{k}}^{\prime}\bm{\hat{k}}^{\prime}_{\rm
s}-\bm{\hat{v}}_{\rm
s})}{\sqrt{1-(\bm{\hat{v}}\cdot\bm{\hat{k}}^{\prime})^{2}\tanh^{2}{\zeta}}},$
(535)
with coefficients given by
$\displaystyle\xi_{00}^{++}(0)=\xi_{00}^{--}(0)$ $\displaystyle=$
$\displaystyle\cosh{\zeta}\,\sqrt{1-(\bm{\hat{v}}\cdot\bm{\hat{k}}^{\prime})^{2}\tanh^{2}{\zeta}}\
,$ (536) $\displaystyle\xi_{10}^{++}(0)=\xi_{20}^{--}(0)$ $\displaystyle=$
$\displaystyle\frac{\sinh{\zeta}}{\sqrt{2}}\sqrt{1-(\bm{\hat{v}}\cdot\bm{\hat{k}}^{\prime})^{2}}\,\sqrt{\frac{1+\bm{\hat{v}}\cdot\bm{\hat{k}}^{\prime}\tanh{\zeta}}{1-\bm{\hat{v}}\cdot\bm{\hat{k}}^{\prime}\tanh{\zeta}}}\
,$ (537) $\displaystyle\xi_{10}^{-+}(0)=\xi_{20}^{+-}(0)$ $\displaystyle=$
$\displaystyle-\frac{\sinh{\zeta}}{\sqrt{2}}\sqrt{1-(\bm{\hat{v}}\cdot\bm{\hat{k}}^{\prime})^{2}}\,\sqrt{\frac{1-\bm{\hat{v}}\cdot\bm{\hat{k}}^{\prime}\tanh{\zeta}}{1+\bm{\hat{v}}\cdot\bm{\hat{k}}^{\prime}\tanh{\zeta}}}\
.$ (538)
Terms that do not appear in Eq. (518) or (530) make no contribution
$\displaystyle\xi_{00}^{+-}(0)=\xi_{00}^{-+}(0)=\xi_{10}^{--}(0)=\xi_{10}^{+-}(0)=\xi_{20}^{++}(0)=\xi_{20}^{-+}(0)=0.$
(539)
An explicit calculation of Eq. (502) is made by evaluating
$\displaystyle{\cal
V}(\bm{v})\,\psi_{\bm{k},0}^{(\kappa)}\big{(}V^{-1}(\bm{v})\,x\big{)}=\int{\rm
d}\bm{x}_{1}\,\delta(\bm{x}-\bm{x}_{1})\,{\cal
V}(\bm{v})\,\psi_{\bm{k},0}^{(\kappa)}\big{(}V^{-1}(\bm{v})\,x_{1}\big{)}$
$\displaystyle\qquad\qquad=\sum_{\kappa_{1}\rightarrow\pm}\sum_{\lambda_{1}=0}^{2}\int{\rm
d}\bm{k}_{1}\int{\rm d}\bm{x}_{1}\
\psi_{\bm{k}_{1},\lambda_{1}}^{\prime\,(\kappa_{1})}(\bm{x})\psi_{\bm{k}_{1},\lambda_{1}}^{\prime\,(\kappa_{1})\dagger}(\bm{x}_{1}){\cal
V}(\bm{v})\,\psi_{\bm{k},0}^{(\kappa)}\big{(}V^{-1}(\bm{v})\,x_{1}\big{)}.\qquad{}$
(540)
For $\lambda_{1}=0$,
$\displaystyle\int{\rm
d}\bm{x}_{1}\,\psi_{\bm{k}_{1},0}^{\prime\,(+)\dagger}(\bm{x}_{1}){\cal
V}(\bm{v})\,\psi_{\bm{k},0}^{(+)}\big{(}V^{-1}(\bm{v})\,x_{1}\big{)}$
$\displaystyle\qquad\qquad=\frac{1}{(2\pi)^{3}}\int{\rm
d}\bm{x}_{1}\left(\begin{array}[]{cc}\bm{\hat{\epsilon}}_{0}^{\prime\dagger}(\bm{\hat{k}}_{1})&{\bm{0}}\end{array}\right){\cal
V}(\bm{v})\left(\begin{array}[]{c}\bm{\hat{\epsilon}}_{0}(\bm{\hat{k}})\\\
{\bm{0}}\end{array}\right){\rm e}^{-{\rm i}\bm{k}_{1}\cdot\bm{x}_{1}}{\rm
e}^{-{\rm i}k\cdot V^{-1}(\bm{v})x_{1}}\qquad$ (544)
$\displaystyle\qquad\qquad=\xi_{00}^{++}(0)\,\delta(\bm{k}_{1}-\bm{k}^{\prime})\,{\rm
e}^{-{\rm i}k^{\prime 0}ct},{}$ (545) $\displaystyle\int{\rm
d}\bm{x}_{1}\,\psi_{\bm{k}_{1},0}^{\prime\,(-)\dagger}(\bm{x}_{1}){\cal
V}(\bm{v})\,\psi_{\bm{k},0}^{(-)}\big{(}V^{-1}(\bm{v})\,x_{1}\big{)}$
$\displaystyle\qquad\qquad=\frac{1}{(2\pi)^{3}}\int{\rm
d}\bm{x}_{1}\left(\begin{array}[]{cc}{\bm{0}}&\bm{\hat{\epsilon}}_{0}^{\prime\dagger}(\bm{\hat{k}}_{1})\end{array}\right){\cal
V}(\bm{v})\left(\begin{array}[]{c}{\bm{0}}\\\
\bm{\hat{\epsilon}}_{0}(\bm{\hat{k}})\end{array}\right){\rm e}^{{\rm
i}\bm{k}_{1}\cdot\bm{x}_{1}}{\rm e}^{{\rm i}k\cdot V^{-1}(\bm{v})x_{1}}\qquad$
(549)
$\displaystyle\qquad\qquad=\xi_{00}^{--}(0)\,\delta(\bm{k}_{1}-\bm{k}^{\prime})\,{\rm
e}^{{\rm i}k^{\prime 0}ct},{}$ (550)
where
$\displaystyle\bm{\hat{k}}^{\prime}$ $\displaystyle=$
$\displaystyle\frac{\bm{\hat{k}}+\bm{\hat{v}}\,\bm{\hat{v}}\cdot\bm{\hat{k}}(\cosh{\zeta}-1)}{\sqrt{1+(\bm{\hat{v}}\cdot\bm{\hat{k}})^{2}\sinh^{2}{\zeta}~{}}},$
(551)
and $k^{\prime 0}$ is the time component associated with the transformed
space-like vector $k$
$\displaystyle k^{\prime 0}$ $\displaystyle=$
$\displaystyle\bm{\hat{v}}\cdot\bm{k}\,\sinh{\zeta}=\bm{\hat{v}}\cdot\bm{k}^{\prime}\,\tanh{\zeta}.$
(552)
For $\lambda_{1}=1$ or $2$,
$\displaystyle\int{\rm
d}\bm{x}_{1}\,\psi_{\bm{k}_{1},\lambda_{1}}^{\prime\,(\kappa_{1})\dagger}(\bm{x}_{1}){\cal
V}(\bm{v})\,\psi_{\bm{k},0}^{(+)}\big{(}V^{-1}(\bm{v})\,x_{1}\big{)}$
$\displaystyle\qquad\qquad=\frac{1}{\sqrt{2}(2\pi)^{3}}\int{\rm
d}\bm{x}_{1}\left(\begin{array}[]{cc}\bm{\hat{\epsilon}}_{\lambda_{1}}^{\prime\dagger}(\bm{\hat{k}}_{1})&\quad\bm{\hat{\epsilon}}_{\lambda_{1}}^{\prime\dagger}(\bm{\hat{k}}_{1})\,\bm{\tau}\cdot\bm{\hat{k}}_{1}\end{array}\right){\cal
V}(\bm{v})\left(\begin{array}[]{c}\bm{\hat{\epsilon}}_{0}(\bm{\hat{k}})\\\
{\bm{0}}\end{array}\right)$ (556) $\displaystyle\qquad\qquad\qquad\times{\rm
e}^{-\kappa_{1}{\rm i}\bm{k}_{1}\cdot\bm{x}_{1}}{\rm e}^{-{\rm i}k\cdot
V^{-1}(\bm{v})x_{1}}$
$\displaystyle\qquad\qquad=\delta_{\kappa_{1}+}\delta_{\lambda_{1}1}\,\xi_{10}^{++}(0)\,\delta(\bm{k}_{1}-\bm{k}^{\prime})\,{\rm
e}^{-{\rm i}k^{\prime
0}ct}+\delta_{\kappa_{1}-}\delta_{\lambda_{1}1}\,\xi_{10}^{-+}(0)\,\delta(\bm{k}_{1}+\bm{k}^{\prime})\,{\rm
e}^{-{\rm i}k^{\prime 0}ct},\qquad$ (557) $\displaystyle\int{\rm
d}\bm{x}_{1}\,\psi_{\bm{k}_{1},\lambda_{1}}^{\prime\,(\kappa_{1})\dagger}(\bm{x}_{1}){\cal
V}(\bm{v})\,\psi_{\bm{k},0}^{(-)}\big{(}V^{-1}(\bm{v})\,x_{1}\big{)}$
$\displaystyle\qquad\qquad=\frac{1}{\sqrt{2}(2\pi)^{3}}\int{\rm
d}\bm{x}_{1}\left(\begin{array}[]{cc}\bm{\hat{\epsilon}}_{\lambda_{1}}^{\prime\,\dagger}(\bm{\hat{k}}_{1})&\quad\bm{\hat{\epsilon}}_{\lambda_{1}}^{\prime\,\dagger}(\bm{\hat{k}}_{1})\,\bm{\tau}\cdot\bm{\hat{k}}_{1}\end{array}\right){\cal
V}(\bm{v})\left(\begin{array}[]{c}{\bm{0}}\\\
\bm{\hat{\epsilon}}_{0}(\bm{\hat{k}})\end{array}\right)$ (561)
$\displaystyle\qquad\qquad\qquad\times{\rm e}^{-\kappa_{1}{\rm
i}\bm{k}_{1}\cdot\bm{x}_{1}}{\rm e}^{{\rm i}k\cdot V^{-1}(\bm{v})x_{1}}$
$\displaystyle\qquad\qquad=\delta_{\kappa_{1}-}\delta_{\lambda_{1}2}\,\xi_{20}^{--}(0)\,\delta(\bm{k}_{1}-\bm{k}^{\prime})\,{\rm
e}^{{\rm i}k^{\prime
0}ct}+\delta_{\kappa_{1}+}\delta_{\lambda_{1}2}\,\xi_{20}^{+-}(0)\,\delta(\bm{k}_{1}+\bm{k}^{\prime})\,{\rm
e}^{{\rm i}k^{\prime 0}ct}.\qquad$ (562)
Here both signs of $\kappa_{1}$ are included, because there is no continuity
condition on the transverse solutions, which are absent in the limit of small
velocity transformations. These results yield Eq. (502) with the non-zero
coefficients given by
$\displaystyle\xi_{00}^{++}(t)$ $\displaystyle=$
$\displaystyle\xi_{00}^{++}(0)\,{\rm e}^{-{\rm
i}\bm{\hat{v}}\cdot\bm{k}^{\prime}\tanh{\zeta}\,ct}\,,\qquad$ (563)
$\displaystyle\xi_{00}^{--}(t)$ $\displaystyle=$
$\displaystyle\xi_{00}^{--}(0)\,{\rm e}^{{\rm
i}\bm{\hat{v}}\cdot\bm{k}^{\prime}\tanh{\zeta}\,ct}\,,\qquad$ (564)
$\displaystyle\xi_{10}^{++}(t)$ $\displaystyle=$
$\displaystyle\xi_{10}^{++}(0)\,{\rm e}^{{\rm
i}(|\bm{k}^{\prime}|-\bm{\hat{v}}\cdot\bm{k}^{\prime}\tanh{\zeta})\,ct}\,,\qquad$
(565) $\displaystyle\xi_{10}^{-+}(t)$ $\displaystyle=$
$\displaystyle\xi_{10}^{-+}(0)\,{\rm e}^{-{\rm
i}(|\bm{k}^{\prime}|+\bm{\hat{v}}\cdot\bm{k}^{\prime}\tanh{\zeta})\,ct}\,,\qquad$
(566) $\displaystyle\xi_{20}^{+-}(t)$ $\displaystyle=$
$\displaystyle\xi_{20}^{+-}(0)\,{\rm e}^{{\rm
i}(|\bm{k}^{\prime}|+\bm{\hat{v}}\cdot\bm{k}^{\prime}\tanh{\zeta})\,ct}\,,\qquad$
(567) $\displaystyle\xi_{20}^{--}(t)$ $\displaystyle=$
$\displaystyle\xi_{20}^{--}(0)\,{\rm e}^{-{\rm
i}(|\bm{k}^{\prime}|-\bm{\hat{v}}\cdot\bm{k}^{\prime}\tanh{\zeta})\,ct}\,.\qquad$
(568)
These transformed longitudinal solutions can be used, for example, to describe
the fields of a moving charge by means of the expansion in Eq. (422).
### 7.7 Standing-wave parity eigenfunctions
The parity operator $\mathfrak{P}$ changes the sign of the coordinates and
includes multiplication by the matrix ${\cal P}=-\gamma^{0}$, so that the
transformed wave function is also a solution of the Maxwell equations, as
discussed in Sec. 6.6. We thus have
$\displaystyle\mathfrak{P}\psi_{\bm{k},\lambda}^{\,(\kappa)}(\bm{x})$
$\displaystyle=$
$\displaystyle-\gamma^{0}\psi_{\bm{k},\lambda}^{\,(\kappa)}(-\bm{x}).{}$ (569)
With this definition, the parity operator commutes with the Hamiltonian in Eq.
(344)
$\displaystyle\mathfrak{P}{\cal H}$ $\displaystyle=$ $\displaystyle{\cal
H}\mathfrak{P},$ (570)
so we may identify eigenstates of both parity and energy. Since
$\displaystyle\mathfrak{P}^{2}\psi_{\bm{k},\lambda}^{\,(\kappa)}(\bm{x})$
$\displaystyle=$ $\displaystyle\psi_{\bm{k},\lambda}^{\,(\kappa)}(\bm{x}),$
(571)
the parity eigenvalues are $\pm 1$.
Transverse parity and energy eigenstates are
$\displaystyle\psi_{\bm{k},\lambda}^{\,(\kappa,+)}(\bm{x})$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{(2\pi)^{3}}}\left(\begin{array}[]{c}\kappa{\rm
i}\,\bm{\hat{\epsilon}}_{\lambda}(\bm{\hat{k}})\sin{\bm{k}\cdot\bm{x}}\\\
\bm{\tau}\cdot\bm{\hat{k}}\,\bm{\hat{\epsilon}}_{\lambda}(\bm{\hat{k}})\cos{\bm{k}\cdot\bm{x}}\end{array}\right),{}$
(574) $\displaystyle\psi_{\bm{k},\lambda}^{\,(\kappa,-)}(\bm{x})$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{(2\pi)^{3}}}\left(\begin{array}[]{c}\bm{\hat{\epsilon}}_{\lambda}(\bm{\hat{k}})\cos{\bm{k}\cdot\bm{x}}\\\
\kappa{\rm
i}\,\bm{\tau}\cdot\bm{\hat{k}}\,\bm{\hat{\epsilon}}_{\lambda}(\bm{\hat{k}})\sin{\bm{k}\cdot\bm{x}}\end{array}\right),\quad{}$
(577)
where
$\displaystyle\mathfrak{P}\psi_{\bm{k},\lambda}^{\,(\kappa,\pm)}(\bm{x})$
$\displaystyle=$
$\displaystyle\pm\psi_{\bm{k},\lambda}^{\,(\kappa,\pm)}(\bm{x}),$ (578)
$\displaystyle{\cal H}\psi_{\bm{k},\lambda}^{\,(\kappa,\pm)}(\bm{x})$
$\displaystyle=$ $\displaystyle\kappa\hbar
c|\bm{k}|\psi_{\bm{k},\lambda}^{\,(\kappa,\pm)}(\bm{x}),$ (579)
and $\lambda=1,~{}2$. These states are linear combinations of plane-wave
states that form standing plane waves. Orthogonality relations are calculated
with the aid of the integrals
$\displaystyle\int{\rm
d}\bm{x}\,\cos{\bm{k}_{2}\cdot\bm{x}}\,\cos{\bm{k}_{1}\cdot\bm{x}}$
$\displaystyle=$ $\displaystyle
4\pi^{3}\left[\delta(\bm{k}_{2}-\bm{k}_{1})+\delta(\bm{k}_{2}+\bm{k}_{1})\right],$
(580) $\displaystyle\int{\rm
d}\bm{x}\,\sin{\bm{k}_{2}\cdot\bm{x}}\,\sin{\bm{k}_{1}\cdot\bm{x}}$
$\displaystyle=$ $\displaystyle
4\pi^{3}\left[\delta(\bm{k}_{2}-\bm{k}_{1})-\delta(\bm{k}_{2}+\bm{k}_{1})\right],\qquad$
(581) $\displaystyle\int{\rm
d}\bm{x}\,\cos{\bm{k}_{2}\cdot\bm{x}}\,\sin{\bm{k}_{1}\cdot\bm{x}}$
$\displaystyle=$ $\displaystyle 0,$ (582)
which show that the states that differ only by the sign of $\bm{k}$ are not
orthogonal. One of the overlapping states may be removed by including only
states with $\bm{k}$ such that $\bm{k}\cdot\bm{k}_{0}>0$, where $\bm{k}_{0}$
is a fixed direction in space. With this restriction on $\bm{k}_{2}$ and
$\bm{k}_{1}$, the orthonormality of the states is given by
$\displaystyle\int{\rm
d}\bm{x}\,\psi_{\bm{k}_{2},\lambda_{2}}^{\,(\kappa_{2},\pi_{2})\dagger}(\bm{x})\,\psi_{\bm{k}_{1},\lambda_{1}}^{\,(\kappa_{1},\pi_{1})}(\bm{x})=\delta_{\kappa_{2}\kappa_{1}}\delta_{\pi_{2}\pi_{1}}\delta_{\lambda_{2}\lambda_{1}}\delta(\bm{k}_{2}-\bm{k}_{1}).\qquad$
(583)
Completeness of these eigenfunctions, including the restriction on $\bm{k}$
provided by a factor $\theta(\bm{k}\cdot\bm{k}_{0})$, where the theta function
is defined as
$\displaystyle\theta(x)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{cc}1&\mbox{for}~{}x>0\\\
\frac{1}{2}&\mbox{for}~{}x=0\rule{0.0pt}{10.0pt}\\\
0&\mbox{for}~{}x<0\rule{0.0pt}{10.0pt}\\\ \end{array}\right.,{}$ (587)
follows from
$\displaystyle\sum_{\kappa,\pi\rightarrow\pm}\sum_{\lambda=1}^{2}\int{\rm
d}\bm{k}\,\theta(\bm{k}\cdot\bm{k}_{0})\,\psi_{\bm{k},\lambda}^{\,(\kappa,\pi)}(\bm{x}_{2})\,\psi_{\bm{k},\lambda}^{\,(\kappa,\pi)\dagger}(\bm{x}_{1})$
$\displaystyle\qquad=\frac{2}{(2\pi)^{3}}\int{\rm
d}\bm{k}\,\theta(\bm{k}\cdot\bm{k}_{0}){\it\Pi}^{\rm
T}(\bm{\hat{k}})(\cos{\bm{k}\cdot\bm{x}_{2}}\,\cos{\bm{k}\cdot\bm{x}_{1}}+\sin{\bm{k}\cdot\bm{x}_{2}}\,\sin{\bm{k}\cdot\bm{x}_{1}})\qquad$
$\displaystyle\qquad={\it\Pi}^{\rm
T}(\bm{\nabla}_{2})\,\delta(\bm{x}_{2}-\bm{x}_{1}).$ (588)
Longitudinal parity and energy eigenstates are given by
$\displaystyle\psi_{\bm{k},0}^{\,(+,+)}(\bm{x})$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{4\pi^{3}}}\left(\begin{array}[]{c}\bm{\hat{\epsilon}}_{0}(\bm{\hat{k}})\sin{\bm{k}\cdot\bm{x}}\\\
{\bm{0}}\end{array}\right),{}$ (591)
$\displaystyle\psi_{\bm{k},0}^{\,(+,-)}(\bm{x})$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{4\pi^{3}}}\left(\begin{array}[]{c}\bm{\hat{\epsilon}}_{0}(\bm{\hat{k}})\cos{\bm{k}\cdot\bm{x}}\\\
{\bm{0}}\end{array}\right),{}$ (594)
$\displaystyle\psi_{\bm{k},0}^{\,(-,+)}(\bm{x})$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{4\pi^{3}}}\left(\begin{array}[]{c}{\bm{0}}\\\
\bm{\hat{\epsilon}}_{0}(\bm{\hat{k}})\cos{\bm{k}\cdot\bm{x}}\end{array}\right),{}$
(597) $\displaystyle\psi_{\bm{k},0}^{\,(-,-)}(\bm{x})$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{4\pi^{3}}}\left(\begin{array}[]{c}{\bm{0}}\\\
\bm{\hat{\epsilon}}_{0}(\bm{\hat{k}})\sin{\bm{k}\cdot\bm{x}}\end{array}\right),{}$
(600)
where
$\displaystyle\mathfrak{P}\psi_{\bm{k},0}^{\,(\kappa,\pm)}(\bm{x})$
$\displaystyle=$ $\displaystyle\pm\psi_{\bm{k},0}^{\,(\kappa,\pm)}(\bm{x}),$
(601) $\displaystyle{\cal H}\psi_{\bm{k},0}^{\,(\kappa,\pm)}(\bm{x})$
$\displaystyle=$ $\displaystyle 0.$ (602)
As for the transverse eigenfunctions, there is overlap between states with
opposite signs of $\bm{k}$, so the same condition on $\bm{k}$ may be applied
here, that is $\theta(\bm{k}\cdot\bm{k}_{0})>0$, to eliminate the redundancy.
With this condition on $\bm{k}_{2}$ and $\bm{k}_{1}$, the orthonormality
relation is
$\displaystyle\int{\rm
d}\bm{x}\,\psi_{\bm{k}_{2},0}^{\,(\kappa_{2},\pi_{2})\dagger}(\bm{x})\,\psi_{\bm{k}_{1},0}^{\,(\kappa_{1},\pi_{1})}(\bm{x})$
$\displaystyle=$
$\displaystyle\delta_{\kappa_{2}\kappa_{1}}\delta_{\pi_{2}\pi_{1}}\delta(\bm{k}_{2}-\bm{k}_{1}).$
In addition, the longitudinal parity eigenfunctions are orthogonal to the
transverse parity eigenfunctions. Completeness of the longitudinal parity
eigenfunctions is given by
$\displaystyle\sum_{\kappa,\pi\rightarrow\pm}\int{\rm
d}\bm{k}\,\theta(\bm{k}\cdot\bm{k}_{0})\,\psi_{\bm{k},0}^{\,(\kappa,\pi)}(\bm{x}_{2})\,\psi_{\bm{k},0}^{\,(\kappa,\pi)\dagger}(\bm{x}_{1})$
$\displaystyle\qquad=\frac{2}{(2\pi)^{3}}\int{\rm
d}\bm{k}\,\theta(\bm{k}\cdot\bm{k}_{0}){\it\Pi}^{\rm
L}(\bm{\hat{k}})(\cos{\bm{k}\cdot\bm{x}_{2}}\,\cos{\bm{k}\cdot\bm{x}_{1}}+\sin{\bm{k}\cdot\bm{x}_{2}}\,\sin{\bm{k}\cdot\bm{x}_{1}})\qquad$
$\displaystyle\qquad={\it\Pi}^{\rm
L}(\bm{\nabla}_{2})\,\delta(\bm{x}_{2}-\bm{x}_{1}).$ (604)
The combined completeness relation is thus
$\displaystyle\sum_{\kappa,\pi\rightarrow\pm}\sum_{\lambda=0}^{2}\int{\rm
d}\bm{k}\,\theta(\bm{k}\cdot\bm{k}_{0})\,\psi_{\bm{k},\lambda}^{\,(\kappa,\pi)}(\bm{x}_{2})\,\psi_{\bm{k},\lambda}^{\,(\kappa,\pi)\dagger}(\bm{x}_{1})={\cal
I}\delta(\bm{x}_{2}-\bm{x}_{1}).$ (605)
### 7.8 Wave packets
The plane wave solutions considered in this section are not normalizable as
ordinary functions. Rather, integrals over products of solutions should be
interpreted in the sense of distributions or generalized functions, like the
delta function [26]. That is, they provide a well-defined value for an
integral when they are included in the integrand together with a suitable
weight, or test function. However, the plane waves can provide the basis for
an expansion of a normalizable wave packet as a sum and integral over a
complete set of solutions of the Maxwell equation. If
$f_{\lambda}^{(\kappa)}(\bm{k})$ is a suitable function, we write
$\displaystyle{\it\Psi}_{f}(x)$ $\displaystyle=$
$\displaystyle\sum_{\kappa\,\lambda}\int{\rm
d}\bm{k}\,f_{\lambda}^{(\kappa)}(\bm{k})\,\psi_{\bm{k},\lambda}^{(\kappa)}(x),{}$
(606)
and ${\it\Psi}_{f}$ is a solution of the Maxwell equation
$\displaystyle\gamma^{\mu}\partial_{\mu}{\it\Psi}_{f}(x)$ $\displaystyle=$
$\displaystyle\sum_{\kappa\,\lambda}\int{\rm
d}\bm{k}\,f_{\lambda}^{(\kappa)}(\bm{k})\,\gamma^{\mu}\partial_{\mu}\,\psi_{\bm{k},\lambda}^{(\kappa)}(x)=0.$
(607)
Further, ${\it\Psi}_{f}$ will be normalized if $f_{\lambda}^{(\kappa)}$ is,
because
$\displaystyle\int{\rm d}\bm{x}\,{\it\Psi}_{f}^{\dagger}(x){\it\Psi}_{f}(x)$
$\displaystyle=$
$\displaystyle\sum_{\kappa\,\lambda}\sum_{\kappa^{\prime}\,\lambda^{\prime}}\int{\rm
d}\bm{k}\int{\rm d}\bm{k}^{\prime}\,f_{\lambda}^{(\kappa)*}(\bm{k})\int{\rm
d}\bm{x}\,\psi_{\bm{k},\lambda}^{(\kappa)\dagger}(x)\,\psi_{\bm{k^{\prime}},\lambda^{\prime}}^{(\kappa^{\prime})}(x)f_{\lambda^{\prime}}^{(\kappa^{\prime})}(\bm{k}^{\prime})$
(608) $\displaystyle=$ $\displaystyle\sum_{\kappa\,\lambda}\int{\rm
d}\bm{k}\,\left|f_{\lambda}^{(\kappa)}(\bm{k})\right|^{2}=1.$
In view of the orthonormality (in the generalized sense) of the plane-wave
solutions, Eq. (606) may be inverted to give
$\displaystyle f_{\lambda}^{(\kappa)}(\bm{k})$ $\displaystyle=$
$\displaystyle\int{\rm
d}\bm{x}\,\psi_{\bm{k},\lambda}^{(\kappa)\dagger}(\bm{x}){\it\Psi}_{f}(\bm{x}),{}$
(609)
where we have specified ${\it\Psi}_{f}(\bm{x})={\it\Psi}_{f}(x)\big{|}_{t=0}$,
and the time dependence of the wave function is given by Eq. (606).
An example is a normalized photon wave packet which at $t=0$ has
(approximately) a wave vector $\bm{k}_{0}$, a transverse polarization vector
$\bm{\hat{\epsilon}}_{1}(\bm{\hat{k}}_{0})$, and a Gaussian envelope of length
$a$ and width $b$ centered at $\bm{x}=0$:
$\displaystyle{\it\Psi}_{f}(\bm{x})$ $\displaystyle=$
$\displaystyle\frac{1}{a^{\frac{1}{2}}b}\left({\frac{2}{\pi^{3}}}\right)^{\\!\frac{1}{4}}\left(\begin{array}[]{c}\bm{\hat{\epsilon}}_{1}(\bm{\hat{k}}_{0})\\\
\bm{\tau}\cdot\bm{\hat{k}}_{0}\
\bm{\hat{\epsilon}}_{1}(\bm{\hat{k}}_{0})\end{array}\right){\rm e}^{{\rm
i}\bm{k}_{0}\cdot\bm{x}}\,{\rm
e}^{-\left(\bm{x}_{\parallel}^{2}/a^{2}+\bm{x}_{\perp}^{2}/b^{2}\right)},\quad{}$
(612)
where $\bm{x}_{\parallel}=\bm{x}\cdot\bm{\hat{k}}_{0}\,\bm{\hat{k}}_{0}$ and
$\bm{x}_{\perp}=\bm{x}-\bm{x}_{\parallel}$. For $a$ and $b$ large compared to
$|\bm{k}_{0}|^{-1}$, the packet has a functional form that resembles a
positive-energy transverse plane wave. From Eq. (609), we have
$\displaystyle f_{\lambda}^{(\kappa)}(\bm{k})$ $\displaystyle=$
$\displaystyle\frac{a^{\frac{1}{2}}b}{2}\left({\frac{2}{\pi^{3}}}\right)^{\\!\frac{1}{4}}F_{\lambda}^{(\kappa)}(\bm{\hat{k}})\,\,{\rm
e}^{-\left[\left(\bm{k}_{0}-\kappa\bm{k}_{\parallel}\right)^{2}a^{2}/4+\bm{k}_{\perp}^{2}b^{2}/4\right]},{}$
(613)
where $\bm{k}_{\parallel}=\bm{k}\cdot\bm{\hat{k}}_{0}\,\bm{\hat{k}}_{0}$,
$\bm{k}_{\perp}=\bm{k}-\bm{k}_{\parallel}$, for $\lambda=1,2$,
$\displaystyle F_{\lambda}^{(\kappa)}(\bm{\hat{k}})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\,\bm{\hat{\epsilon}}_{\lambda}^{\dagger}(\bm{\hat{k}})\left(\bm{I}+\bm{\tau}\cdot\bm{\hat{k}}\,\bm{\tau}\cdot\bm{\hat{k}}_{0}\right)\bm{\hat{\epsilon}}_{1}(\bm{\hat{k}}_{0}),\qquad{}$
(614)
and
$\displaystyle F_{0}^{(+)}(\bm{\hat{k}})$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\,\bm{\hat{\epsilon}}_{0}^{\dagger}(\bm{\hat{k}})\,\bm{\hat{\epsilon}}_{1}(\bm{\hat{k}}_{0}),$
(615) $\displaystyle F_{0}^{(-)}(\bm{\hat{k}})$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\,\bm{\hat{\epsilon}}_{0}^{\dagger}(\bm{\hat{k}})\,\bm{\tau}\cdot\bm{\hat{k}}_{0}\bm{\hat{\epsilon}}_{1}(\bm{\hat{k}}_{0}).$
(616)
The wave packet has small longitudinal components, because
$F_{0}^{(\kappa)}(\bm{\hat{k}})$ is not necessarily zero unless
$\bm{\hat{k}}=\pm\bm{\hat{k}}_{0}$. It has negative-energy components, but
they are also suppressed, particularly as $a,b\rightarrow\infty$, because for
$\kappa\rightarrow-$, the exponential factors in Eq. (613) favor
$\bm{k}=-\bm{k}_{0}$, and for $\lambda=1,2$,
$F_{\lambda}^{(\kappa)}(\bm{-\hat{k}}_{0})=0$ as compared to
$F_{\lambda}^{(\kappa)}(\bm{\hat{k}}_{0})=\delta_{\lambda\,1}$. Thus a
cancellation between the upper-three and lower-three components of the wave
function suppresses the contribution of negative-energy eigenstates to the
wave packet.
The expectation value of the Hamiltonian ${\cal H}$, Eq. (344), is
$\displaystyle\left<{\it\Psi}_{f}\left|\,{\cal
H}\,\right|{\it\Psi}_{f}\right>$ $\displaystyle=$ $\displaystyle-{\rm
i}\,\hbar c\int{\rm
d}\bm{x}{\it\Psi}_{f}^{\dagger}(\bm{x})\,\bm{\alpha}\cdot\bm{\nabla}{\it\Psi}_{f}(\bm{x})=\hbar
c|\bm{k}_{0}|=\hbar\omega_{0},{}$ (617)
and the expectation value of the momentum $\bm{{\cal P}}$, Eq. (345), is
$\displaystyle\left<{\it\Psi}_{f}\left|\,\bm{{\cal
P}}\,\right|{\it\Psi}_{f}\right>$ $\displaystyle=$
$\displaystyle\hbar\bm{k}_{0}.{}$ (618)
The initial probability density $Q(\bm{x})$ is
$\displaystyle Q(\bm{x})$ $\displaystyle=$
$\displaystyle{\it\Psi}_{f}^{\dagger}(\bm{x}){\it\Psi}_{f}(\bm{x})=\frac{2}{ab^{2}}\left({\frac{2}{\pi^{3}}}\right)^{\\!\frac{1}{2}}\,{\rm
e}^{-2\left(\bm{x}_{\parallel}^{2}/a^{2}+\bm{x}_{\perp}^{2}/b^{2}\right)},\qquad{}$
(619)
with
$\displaystyle\int{\rm d}\bm{x}\,Q(\bm{x})$ $\displaystyle=$ $\displaystyle
1,\qquad$ (620)
and the initial energy density $E(\bm{x})$ is
$\displaystyle E(\bm{x})$ $\displaystyle=$
$\displaystyle{\it\Psi}_{f}^{\dagger}(\bm{x})\,{\cal
H}\,{\it\Psi}_{f}(\bm{x})=\hbar\omega_{0}Q(\bm{x})-\frac{{\rm i}}{2}\,\hbar
c\bm{\hat{k}}_{0}\cdot\bm{\nabla}Q(\bm{x}).\qquad$ (621)
The real part of the energy density is proportional to the probability density
for the photon, and the imaginary term, which vanishes upon integration to
arrive at the expectation value in Eq. (617), reflects the change in the
initial probability density at the point $\bm{x}$ due to the motion of the
wave packet. At a fixed point in the path of the wave packet, the probability
density increases as the packet approaches and decreases after the maximum of
the wave packet has passed by. The time-dependent probability density is
$\displaystyle Q(x)$ $\displaystyle=$
$\displaystyle{\it\Psi}_{f}^{\dagger}(x){\it\Psi}_{f}(x)=\left|{\rm e}^{-{\rm
i}\,{\cal H}\,t/\hbar}{\it\Psi}_{f}(\bm{x})\right|^{2},{}$ (622)
and the change at $t=0$ is
$\displaystyle\frac{\partial Q(x)}{\partial t}\,\bigg{|}_{t=0}$
$\displaystyle=$ $\displaystyle-\frac{{\rm
i}}{\hbar}{\it\Psi}_{f}^{\dagger}(\bm{x})\left({\cal H}-\overleftarrow{{\cal
H}}^{\dagger}\right)\\!{\it\Psi}_{f}(\bm{x})$ (623) $\displaystyle=$
$\displaystyle\frac{2}{\hbar}\,{\rm Im}{\it\Psi}_{f}^{\dagger}(\bm{x}){\cal
H}{\it\Psi}_{f}(\bm{x})=-c\bm{\hat{k}}_{0}\cdot\bm{\nabla}Q(\bm{x}).{}$
The gradient produces a vector that points toward the maximum of the wave
packet, so that on the forward side of the packet
$\bm{\hat{k}}_{0}\cdot\bm{\nabla}Q(\bm{x})$ is negative and the probability
density is increasing, as expected. Eq. (623) also shows that the wave packet
is initially moving with velocity c in the direction of $\bm{\hat{k}}_{0}$.
The time dependence of the wave packet, Eq. (612) at $t=0$, is given exactly
by Eq. (606). However, approximations may be made in order to obtain a more
transparent expression. In view of the exponential factors in Eq. (613), the
assumption that $a,b\gg|\bm{k}_{0}|^{-1}$ implies
$\bm{k}\approx\kappa\bm{k}_{0}$, and
$\displaystyle F_{\lambda}^{(\kappa)}(\bm{\hat{k}})$ $\displaystyle\approx$
$\displaystyle F_{\lambda}^{(\kappa)}(\kappa\bm{\hat{k}}_{0})=\delta_{\lambda
1}\delta_{\kappa+},{}$ (624)
so that from Eq. (606), ${\it\Psi}_{f}\rightarrow{\it\Psi}_{f}^{\prime}$,
where
$\displaystyle{\it\Psi}_{f}^{\prime}(x)$ $\displaystyle=$
$\displaystyle\frac{a^{\frac{1}{2}}b}{8\pi^{\frac{3}{2}}}\left({\frac{2}{\pi^{3}}}\right)^{\\!\frac{1}{4}}\int{\rm
d}\bm{k}\left(\begin{array}[]{c}\bm{\hat{\epsilon}}_{1}(\bm{\hat{k}})\\\
\bm{\tau}\cdot\bm{\hat{k}}\
\bm{\hat{\epsilon}}_{1}(\bm{\hat{k}})\end{array}\right){\rm e}^{-{\rm
i}\left(|\bm{k}|ct-\bm{k}\cdot\bm{x}\right)}\,{\rm
e}^{-\left[\left(\bm{k}_{0}-\bm{k}_{\parallel}\right)^{2}a^{2}/4+\bm{k}_{\perp}^{2}b^{2}/4\right]}.\qquad$
(627)
This is a normalized positive energy wave function with polarization
$\bm{\hat{\epsilon}}_{1}$ that is an exact solution of the Maxwell equation
$\gamma^{\mu}\partial_{\mu}{\it\Psi}_{f}^{\prime}(x)=0$. Further
simplifications are the replacements $\bm{\hat{k}}\rightarrow\bm{\hat{k}}_{0}$
in the polarization vector matrix and
$|\bm{k}|\rightarrow\bm{k}\cdot\bm{\hat{k}}_{0}$ in the exponent, which yield
${\it\Psi}_{f}^{\prime}\rightarrow{\it\Psi}_{f}^{\prime\prime}$, with
$\displaystyle{\it\Psi}_{f}^{\prime\prime}(x)$ $\displaystyle=$
$\displaystyle\frac{a^{\frac{1}{2}}b}{8\pi^{\frac{3}{2}}}\left({\frac{2}{\pi^{3}}}\right)^{\\!\frac{1}{4}}\left(\begin{array}[]{c}\bm{\hat{\epsilon}}_{1}(\bm{\hat{k}}_{0})\\\
\bm{\tau}\cdot\bm{\hat{k}}_{0}\
\bm{\hat{\epsilon}}_{1}(\bm{\hat{k}}_{0})\end{array}\right)\int{\rm
d}\bm{k}\,{\rm e}^{-{\rm
i}\left(\bm{k}\cdot\bm{\hat{k}}_{0}ct-\bm{k}\cdot\bm{x}\right)}\,{\rm
e}^{-\left[\left(\bm{k}_{0}-\bm{k}_{\parallel}\right)^{2}a^{2}/4+\bm{k}_{\perp}^{2}b^{2}/4\right]}\qquad$
(630) $\displaystyle=$
$\displaystyle\frac{1}{a^{\frac{1}{2}}b}\left({\frac{2}{\pi^{3}}}\right)^{\\!\frac{1}{4}}\left(\begin{array}[]{c}\bm{\hat{\epsilon}}_{1}(\bm{\hat{k}}_{0})\\\
\bm{\tau}\cdot\bm{\hat{k}}_{0}\
\bm{\hat{\epsilon}}_{1}(\bm{\hat{k}}_{0})\end{array}\right){\rm e}^{-{\rm
i}\left(\omega_{0}t-\bm{k}_{0}\cdot\bm{x}\right)}\,{\rm
e}^{-\left[\left(ct-\bm{\hat{k}}_{0}\cdot\bm{x}\right)^{2}/a^{2}+\bm{x}_{\perp}^{2}/b^{2}\right]},{}$
(633)
which is an approximate wave function with a normalized Gaussian probability
distribution
$\displaystyle Q^{\prime\prime}(x)$ $\displaystyle=$
$\displaystyle\frac{2}{ab^{2}}\left({\frac{2}{\pi^{3}}}\right)^{\\!\frac{1}{2}}\,{\rm
e}^{-2\left[\left(ct-\bm{\hat{k}}_{0}\cdot\bm{x}\right)^{2}/a^{2}+\bm{x}_{\perp}^{2}/b^{2}\right]},\qquad{}$
(634)
that moves with velocity $c$ in the $\bm{\hat{k}}_{0}$ direction.
### 7.9 Conservation of probability
The formulation of the Poynting theorem in Sec. 4 can be reinterpreted here to
demonstrate conservation of probability. We define the probability density
four-vector to be
$\displaystyle q^{\mu}(x)$ $\displaystyle=$
$\displaystyle\overline{{\it\Psi}}(x){\gamma^{\mu}}{\it\Psi}(x),{}$ (635)
where in the previous section $Q(x)=q^{0}(x)$. For the source-free case,
${\it\Xi}(x)=0$, Eq. (102) is
$\displaystyle\partial_{\mu}\overline{{\it\Psi}}(x){\gamma^{\mu}}{\it\Psi}(x)$
$\displaystyle=$ $\displaystyle 0,$ (636)
which is the statement of conservation of probability
$\displaystyle\frac{\partial}{\partial
t}\,q^{0}(x)+c\,\bm{\nabla}\cdot\bm{q}(x)$ $\displaystyle=$ $\displaystyle
0.{}$ (637)
Applying this relation to plane-wave states gives consistent, although
trivial, results. We have
$\displaystyle q^{0}(x)$ $\displaystyle=$
$\displaystyle\psi_{\bm{k},\lambda}^{(\kappa)\dagger}(x)\,\psi_{\bm{k},\lambda}^{(\kappa)}(x)=(2\pi)^{-3},$
(638)
which reflects the fact that the probability distribution for a plane wave is
uniform throughout space and not normalizable. For transverse plane waves,
$\lambda=1,2$, the probability flux vector is
$\displaystyle\bm{q}(x)$ $\displaystyle=$
$\displaystyle\psi_{\bm{k},\lambda}^{(\kappa)\dagger}(x)\,\bm{\alpha}\,\psi_{\bm{k},\lambda}^{(\kappa)}(x)=(2\pi)^{-3}\bm{\hat{k}},$
(639)
and for longitudinal plane waves $\bm{q}(x)=0$.
For the wave packet in Eq. (612), at $t=0$
$\displaystyle\bm{q}(x)$ $\displaystyle=$
$\displaystyle\bm{\hat{k}}_{0}\,q^{0}(x),$ (640)
so Eq. (623) is essentially the conservation of probability equation evaluated
at $t=0$. Conservation of probability is valid for any solution of the
homogeneous Maxwell equation from the definition of the probability density
operator. However, it also happens to be valid for the wave packet represented
by ${\it\Psi}_{f}^{\prime\prime}$, which is not an exact solution of the wave
equation. The wave equation is not satisfied by
${\it\Psi}_{f}^{\prime\prime}$, because there is a small extra term resulting
from the gradient operator acting on the perpendicular coordinate
$\bm{x}_{\perp}$. On the other hand, $\bm{q}(x)$ for
${\it\Psi}_{f}^{\prime\prime}$ is proportional to $\bm{\hat{k}}_{0}$ and
$\bm{\hat{k}}_{0}\cdot\bm{\nabla}\,\bm{x}_{\perp}=0$, so there is no
corresponding extra term in $\bm{\nabla}\cdot\bm{q}(x)$.
## 8 Angular momentum eigenfunctions
Radiation emitted in atomic transitions is characterized by its angular
momentum and parity. In this section, wave functions that are eigenfunctions
of energy, angular momentum, and parity are given; they are also classified
according to whether they are transverse or longitudinal. The three-component
angular-momentum matrices given here are to some extent parallels of the
three-vector functions of [27].
### 8.1 Angular momentum
The spatial angular-momentum operator is given by
$\displaystyle\bm{L}$ $\displaystyle=$ $\displaystyle\bm{x}\times\bm{p}=-{\rm
i}\,\hbar\,\bm{x}\times\bm{\nabla},$ (641)
and following the example of the Dirac equation, the total angular momentum is
given as a $3\times 3$ matrix by [5]
$\displaystyle\bm{J}$ $\displaystyle=$ $\displaystyle\bm{L}+\hbar\bm{\tau},{}$
(642)
where it is understood that the first term on the right side is a $3\times 3$
matrix with $\bm{L}$ for diagonal elements and zeros for the rest.444In some
works, where electric and magnetic fields or the vector potential are three-
vector valued fields, the spin operator is represented by a cross product. For
example, Corben and Schwinger [27] write $J_{z}\bm{\Phi}=L_{z}\bm{\Phi}+{\rm
i}\,\bm{e}_{z}\times\bm{\Phi}$, where $\bm{\Phi}$ is a vector potential, and
in Edmonds [28], spin is represented symbolically as ${\rm
i}\,\bm{\hat{e}}\bm{\times}$. (In order to adhere to convention, we denote
both the current three-vector and the angular-momentum matrix by $\bm{J}$. In
either case, the meaning should be clear from the context.) The extension to a
$6\times 6$ matrix is
$\displaystyle\bm{{\cal J}}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}\bm{J}&{\bm{0}}\\\
{\bm{0}}&\bm{J}\end{array}\right)=\bm{x}\times\bm{{\cal P}}+\hbar\bm{{\cal
S}}{}$ (645)
and we have
$\displaystyle\left[{\cal H},\bm{{\cal J}}\right]$ $\displaystyle=$
$\displaystyle 0,{}$ (646)
so eigenfunctions of both energy and angular momentum may be constructed. The
vanishing of the commutator follows from the relations
$\displaystyle\left[\bm{\tau}\cdot\bm{\nabla},\bm{L}\right]$ $\displaystyle=$
$\displaystyle-{\rm i}\hbar\bm{\tau}\times\bm{\nabla},$ (647)
$\displaystyle\left[\bm{\tau}\cdot\bm{\nabla},\bm{\tau}\right]$
$\displaystyle=$ $\displaystyle{\rm i}\bm{\tau}\times\bm{\nabla},$ (648)
$\displaystyle\left[\bm{\tau}\cdot\bm{\nabla},\bm{J}\right]$ $\displaystyle=$
$\displaystyle 0.{}$ (649)
It is of interest to see that $\bm{{\cal J}}$ commutes with ${\cal H}$ only
for the (relative) combination of $\bm{L}$ and $\bm{\tau}$ given in Eq. (642).
To obtain eigenstates of the square of the total angular momentum $\bm{{\cal
J}}^{2}$ and the third component of angular momentum ${\cal J}^{3}$, given by
$\displaystyle\bm{{\cal J}}^{2}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}\bm{J}^{2}&{\bm{0}}\\\
{\bm{0}}&\bm{J}^{2}\end{array}\right)$ (652)
and
$\displaystyle{\cal J}^{3}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}J^{3}&{\bm{0}}\\\
{\bm{0}}&J^{3}\end{array}\right),$ (655)
we construct matrix spherical harmonics that are analogous to conventional
vector spherical harmonics and are three-component extensions of the Dirac
two-component spin-angular-momentum eigenfunctions. Orthonormal basis matrices
are given by
$\displaystyle\bm{\hat{\epsilon}}^{(1)}=\left(\begin{array}[]{c}1\\\ 0\\\
0\end{array}\right),\qquad\bm{\hat{\epsilon}}^{(0)}=\left(\begin{array}[]{c}0\\\
1\\\
0\end{array}\right),\qquad\bm{\hat{\epsilon}}^{(-1)}=\left(\begin{array}[]{c}0\\\
0\\\ 1\end{array}\right),$ (665)
and they satisfy the eigenvalue equations
$\displaystyle\bm{\tau}^{2}\,\bm{\hat{\epsilon}}^{(\nu)}$ $\displaystyle=$
$\displaystyle 2\,\bm{\hat{\epsilon}}^{(\nu)},$ (666)
$\displaystyle\tau^{3}\,\bm{\hat{\epsilon}}^{(\nu)}$ $\displaystyle=$
$\displaystyle\nu\,\bm{\hat{\epsilon}}^{(\nu)},$ (667)
where the 2 may be regarded as $s(s+1)$, with $s=1$ as the spin eigenvalue.
The matrix spherical harmonics are
$\displaystyle\bm{Y}_{jl}^{m}(\bm{\hat{x}})$ $\displaystyle=$
$\displaystyle\sum_{\nu}\,(l\ m-\nu\ 1\ \nu|l\ 1\ j\
m)Y_{l}^{m-\nu}(\bm{\hat{x}})\,\bm{\hat{\epsilon}}^{(\nu)},$ (668)
with vector addition coefficients and spherical harmonics in the notation of
[28]. The spherical harmonics satisfy the eigenvalue equations
$\displaystyle\bm{L}^{2}Y_{l}^{m}(\bm{\hat{x}})$ $\displaystyle=$
$\displaystyle\hbar^{2}l(l+1)\,Y_{l}^{m}(\bm{\hat{x}}),$ (669) $\displaystyle
L^{3}Y_{l}^{m}(\bm{\hat{x}})$ $\displaystyle=$
$\displaystyle\hbar\,m\,Y_{l}^{m}(\bm{\hat{x}}),$ (670)
and by their construction, the matrix spherical harmonics are eigenfunctions
of the total angular momentum:
$\displaystyle\bm{J}^{2}\bm{Y}_{jl}^{m}(\bm{\hat{x}})$ $\displaystyle=$
$\displaystyle\hbar^{2}j(j+1)\,\bm{Y}_{jl}^{m}(\bm{\hat{x}}),$ (671)
$\displaystyle J^{3}\bm{Y}_{jl}^{m}(\bm{\hat{x}})$ $\displaystyle=$
$\displaystyle\hbar\,m\,\bm{Y}_{jl}^{m}(\bm{\hat{x}}).$ (672)
Explicit expressions in terms of spherical harmonics are
$\displaystyle\bm{Y}_{jj}^{m}(\bm{\hat{x}})$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}-\sqrt{\frac{(j+m)(j+1-m)}{2j(j+1)}}\,Y_{j}^{m-1}(\bm{\hat{x}})\\\\[0.0pt]
\frac{m}{\sqrt{j(j+1)}}\,Y_{j}^{m}(\bm{\hat{x}})\\\\[6.0pt]
\sqrt{\frac{(j-m)(j+1+m)}{2j(j+1)}}\,Y_{j}^{m+1}(\bm{\hat{x}})\end{array}\right),\qquad{}$
(676) $\displaystyle\bm{Y}_{jj+1}^{m}(\bm{\hat{x}})$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}\sqrt{\frac{(j+1-m)(j+2-m)}{(2j+2)(2j+3)}}\,Y_{j+1}^{m-1}(\bm{\hat{x}})\\\\[6.0pt]
-\sqrt{\frac{(j+1-m)(j+1+m)}{(j+1)(2j+3)}}\,Y_{j+1}^{m}(\bm{\hat{x}})\\\\[6.0pt]
\sqrt{\frac{(j+2+m)(j+1+m)}{(2j+2)(2j+3)}}\,Y_{j+1}^{m+1}(\bm{\hat{x}})\end{array}\right),\qquad{}$
(680) $\displaystyle\bm{Y}_{jj-1}^{m}(\bm{\hat{x}})$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}\sqrt{\frac{(j-1+m)(j+m)}{(2j-1)2j}}\,Y_{j-1}^{m-1}(\bm{\hat{x}})\\\\[6.0pt]
\sqrt{\frac{(j-m)(j+m)}{(2j-1)j}}\,Y_{j-1}^{m}(\bm{\hat{x}})\\\\[6.0pt]
\sqrt{\frac{(j-1-m)(j-m)}{(2j-1)2j}}\,Y_{j-1}^{m+1}(\bm{\hat{x}})\end{array}\right).\qquad{}$
(684)
These functions are orthonormal
$\displaystyle\int{\rm
d}\bm{{\it\Omega}}\,\bm{Y}_{j_{2}l_{2}}^{m_{2}\dagger}(\bm{\hat{x}})\,\bm{Y}_{j_{1}l_{1}}^{m_{1}}(\bm{\hat{x}})$
$\displaystyle=$
$\displaystyle\delta_{j_{2}j_{1}}\delta_{l_{2}l_{1}}\delta_{m_{2}m_{1}},\qquad$
(685)
which follows from the relations
$\displaystyle\int{\rm
d}\bm{{\it\Omega}}\,Y_{l_{2}}^{m_{2}*}(\bm{\hat{x}})\,Y_{l_{1}}^{m_{1}}(\bm{\hat{x}})=\delta_{l_{2}l_{1}}\delta_{m_{2}m_{1}},{}$
(686)
$\displaystyle\bm{\hat{\epsilon}}^{(\nu_{2})\dagger}\,\bm{\hat{\epsilon}}^{(\nu_{1})}=\delta_{\nu_{2}\nu_{1}},$
(687) $\displaystyle\sum_{\nu}(l\ 1\ j_{2}\ m|l\ m-\nu\ 1\ \nu)(l\ m-\nu\ 1\
\nu|l\ 1\ j_{1}\ m)=\delta_{j_{2}j_{1}},\qquad$ (688)
and they are complete
$\displaystyle\sum_{jlm}\bm{Y}_{jl}^{m}(\bm{\hat{x}}_{2})\bm{Y}_{jl}^{m\dagger}(\bm{\hat{x}}_{1})=\bm{I}\,\delta(\cos{\theta_{2}}-\cos{\theta_{1}})\,\delta(\phi_{2}-\phi_{1}),\qquad$
(689)
based on the relations
$\displaystyle\sum_{j}(l\ m-\nu_{2}\ 1\ \nu_{2}|l\ 1\ j\ m)(l\ 1\ j\ m|l\
m-\nu_{1}\ 1\ \nu_{1})=\delta_{\nu_{2}\nu_{1}},\qquad$ (690)
$\displaystyle\sum_{\nu}\bm{\hat{\epsilon}}^{(\nu)}\,\bm{\hat{\epsilon}}^{(\nu)\dagger}=\bm{I},$
(691)
$\displaystyle\sum_{lm}Y_{l}^{m}(\bm{\hat{x}}_{2})Y_{l}^{m*}(\bm{\hat{x}}_{1})=\delta(\cos{\theta_{2}}-\cos{\theta_{1}})\,\delta(\phi_{2}-\phi_{1}),\qquad$
(692)
where $\theta_{i},\phi_{i}$ are the spherical coordinates of
$\bm{\hat{x}}_{i}$.
An alternative set of matrix angular-momentum eigenfunctions is
$\displaystyle\bm{X}^{jm}_{1}(\bm{\hat{x}})$ $\displaystyle=$
$\displaystyle\frac{1}{\hbar\sqrt{j(j+1)}}\,\bm{L}_{\rm
s}Y_{j}^{m}(\bm{\hat{x}}),{}$ (693)
$\displaystyle\bm{X}^{jm}_{2}(\bm{\hat{x}})$ $\displaystyle=$
$\displaystyle\frac{1}{\hbar\sqrt{j(j+1)}}\,\bm{\tau}\cdot\bm{\hat{x}}\bm{L}_{\rm
s}Y_{j}^{m}(\bm{\hat{x}}),{}$ (694)
$\displaystyle\bm{X}^{jm}_{3}(\bm{\hat{x}})$ $\displaystyle=$
$\displaystyle\bm{\hat{x}}_{\rm s}Y_{j}^{m}(\bm{\hat{x}}).{}$ (695)
For $j=0$, $\bm{X}^{00}_{1}(\bm{\hat{x}})=\bm{X}^{00}_{2}(\bm{\hat{x}})=0$.
From a comparison of Eqs. (676)-(684) to Eqs. (693)-(695), one has
$\displaystyle\bm{X}^{jm}_{1}(\bm{\hat{x}})$ $\displaystyle=$
$\displaystyle\bm{Y}_{jj}^{m}(\bm{\hat{x}}),{}$ (696)
$\displaystyle\bm{X}^{jm}_{2}(\bm{\hat{x}})$ $\displaystyle=$
$\displaystyle-\sqrt{\frac{j}{2j+1}}\,\bm{Y}_{jj+1}^{m}(\bm{\hat{x}})-\sqrt{\frac{j+1}{2j+1}}\,\bm{Y}_{jj-1}^{m}(\bm{\hat{x}}),{}$
(697) $\displaystyle\bm{X}^{jm}_{3}(\bm{\hat{x}})$ $\displaystyle=$
$\displaystyle-\sqrt{\frac{j+1}{2j+1}}\,\bm{Y}_{jj+1}^{m}(\bm{\hat{x}})+\sqrt{\frac{j}{2j+1}}\,\bm{Y}_{jj-1}^{m}(\bm{\hat{x}}).{}$
(698)
In view of the relations in Eqs. (696)-(698), the functions
$\bm{X}_{i}^{jm}(\bm{\hat{x}})$ are eigenfunctions of $\bm{J}^{2}$ and $J^{3}$
with
$\displaystyle\bm{J}^{2}\bm{X}_{i}^{jm}(\bm{\hat{x}})$ $\displaystyle=$
$\displaystyle\hbar^{2}j(j+1)\,\bm{X}_{i}^{jm}(\bm{\hat{x}}),$ (699)
$\displaystyle J^{3}\bm{X}_{i}^{jm}(\bm{\hat{x}})$ $\displaystyle=$
$\displaystyle\hbar\,m\,\bm{X}_{i}^{jm}(\bm{\hat{x}}).$ (700)
This can be confirmed directly from the definitions in Eqs. (693)-(695) with
the aid of the commutation relations
$\displaystyle\left[L^{i},L^{j}\right]$ $\displaystyle=$ $\displaystyle{\rm
i}\hbar\epsilon_{ijk}L^{k},$ (701) $\displaystyle\left[L^{i},x^{j}\right]$
$\displaystyle=$ $\displaystyle{\rm i}\hbar\epsilon_{ijk}x^{k}$ (702)
and the tau matrix identities in Sec. 3, which provide the operator identities
$\displaystyle J^{i}\bm{L}_{\rm s}$ $\displaystyle=$ $\displaystyle\bm{L}_{\rm
s}L^{i},{}$ (703) $\displaystyle\left[J^{i},\bm{\tau}\cdot\bm{\hat{x}}\right]$
$\displaystyle=$ $\displaystyle 0,$ (704) $\displaystyle
J^{i}\bm{\hat{x}}_{\rm s}$ $\displaystyle=$ $\displaystyle\bm{\hat{x}}_{\rm
s}L^{i}.{}$ (705)
These functions are orthonormal
$\displaystyle\int{\rm
d}\bm{{\it\Omega}}\,\bm{X}^{j_{2}m_{2}\dagger}_{i_{2}}(\bm{\hat{x}})\,\bm{X}^{j_{1}m_{1}}_{i_{1}}(\bm{\hat{x}})$
$\displaystyle=$
$\displaystyle\delta_{i_{2}i_{1}}\delta_{j_{2}j_{1}}\delta_{m_{2}m_{1}},\qquad$
(706)
and they are complete
$\displaystyle\sum_{ijm}\bm{X}_{i}^{jm}(\bm{\hat{x}}_{2})\,\bm{X}_{i}^{jm\dagger}(\bm{\hat{x}}_{1})=\bm{I}\,\delta(\cos{\theta_{2}}-\cos{\theta_{1}})\,\delta(\phi_{2}-\phi_{1}),\qquad$
(707)
where the latter fact may be seen from the completeness of the matrix
spherical harmonics and the relation
$\displaystyle\sum_{i}\bm{X}_{i}^{jm}(\bm{\hat{x}}_{2})\,\bm{X}_{i}^{jm\dagger}(\bm{\hat{x}}_{1})$
$\displaystyle=$
$\displaystyle\sum_{l}\bm{Y}_{jl}^{m}(\bm{\hat{x}}_{2})\bm{Y}_{jl}^{m\dagger}(\bm{\hat{x}}_{1}).\qquad$
(708)
The parity of the eigenfunctions is given by
$\displaystyle\bm{X}^{jm}_{1}(-\bm{\hat{x}})$ $\displaystyle=$
$\displaystyle(-1)^{j}\bm{X}^{jm}_{1}(\bm{\hat{x}}),$ (709)
$\displaystyle\bm{X}^{jm}_{2}(-\bm{\hat{x}})$ $\displaystyle=$
$\displaystyle(-1)^{j+1}\bm{X}^{jm}_{2}(\bm{\hat{x}}),$ (710)
$\displaystyle\bm{X}^{jm}_{3}(-\bm{\hat{x}})$ $\displaystyle=$
$\displaystyle(-1)^{j+1}\bm{X}^{jm}_{3}(\bm{\hat{x}}),$ (711)
which follows from $Y_{j}^{m}(-\bm{\hat{x}})=(-1)^{j}Y_{j}^{m}(\bm{\hat{x}})$.
### 8.2 Helicity eigenstates
Special cases of the transverse plane-wave solutions in Eq. (367) are
circularly polarized states with the polarization vectors in Eq. (364). They
can be grouped with the longitudinal plane-wave states in Eqs. (403) and (406)
with the polarization vector in Eq. (400). These polarization vectors are
summarized here as
$\displaystyle\bm{\hat{\epsilon}}_{1}(\bm{\hat{e}}^{3})\\!=\\!\left(\begin{array}[]{c}1\\\
0\\\
0\end{array}\right);\quad\bm{\hat{\epsilon}}_{0}(\bm{\hat{e}}^{3})\\!=\\!\left(\begin{array}[]{c}0\\\
1\\\
0\end{array}\right);\quad\bm{\hat{\epsilon}}_{-1}(\bm{\hat{e}}^{3})\\!=\\!\left(\begin{array}[]{c}0\\\
0\\\ 1\end{array}\right),$ (721)
where we have changed the label for $\lambda$ from 2 to $-1$ for this section.
The states have a well-defined helicity; they are eigenfunctions of the
operator for the projection of angular momentum in the direction of the wave
vector $\bm{{\cal J}}\cdot\bm{\hat{k}}$ [5, 29]. In view of the relations
$\displaystyle\bm{L}\cdot\bm{\hat{k}}\,{\rm e}^{\pm{\rm i}\bm{k}\cdot\bm{x}}$
$\displaystyle=$ $\displaystyle 0,$ (722)
$\displaystyle\bm{\tau}\cdot\bm{\hat{k}}\,\bm{\hat{\epsilon}}_{\lambda}(\bm{\hat{k}})$
$\displaystyle=$
$\displaystyle\lambda\,\bm{\hat{\epsilon}}_{\lambda}(\bm{\hat{k}})$ (723)
for the polarizations considered here, we have
$\displaystyle\bm{{\cal
J}}\cdot\bm{\hat{k}}\,\psi_{\bm{k},\lambda}^{(\pm)}(\bm{x})$ $\displaystyle=$
$\displaystyle\lambda\hbar\,\psi_{\bm{k},\lambda}^{(\pm)}(\bm{x})$ (724)
for these states.
### 8.3 Transverse spherical photons
Transverse spherical wave functions are given by
$\displaystyle\psi_{\omega,jm}^{{\rm T}(\kappa,+)}(\bm{x})$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}f_{\omega,j}(r)\bm{X}_{1}^{jm}(\bm{\hat{x}})\\\\[10.0pt]
-\kappa{\rm
i}\frac{c}{\omega}\bm{\tau}\cdot\bm{\nabla}f_{\omega,j}(r)\bm{X}_{1}^{jm}(\bm{\hat{x}})\end{array}\right),\qquad{}$
(727) $\displaystyle\psi_{\omega,jm}^{{\rm T}(\kappa,-)}(\bm{x})$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}\frac{c}{\omega}\bm{\tau}\cdot\bm{\nabla}f_{\omega,j}(r)\bm{X}_{1}^{jm}(\bm{\hat{x}})\\\\[10.0pt]
\kappa{\rm
i}f_{\omega,j}(r)\bm{X}_{1}^{jm}(\bm{\hat{x}})\end{array}\right),{}$ (730)
where $r=|\bm{x}|$ and $j\geq 1$. They are transverse because
$\bm{\nabla}_{\rm s}^{\dagger}\,\bm{L}_{\rm s}=\bm{\nabla}_{\rm
s}^{\dagger}\,\bm{\tau}\cdot\bm{\nabla}=0$, so that
$\displaystyle{\it\Pi}^{\rm T}(\bm{\nabla})\,\psi_{\omega,jm}^{{\rm
T}(\kappa,\pi)}(\bm{x})$ $\displaystyle=$ $\displaystyle\psi_{\omega,jm}^{{\rm
T}(\kappa,\pi)}(\bm{x}),$ (731) $\displaystyle{\it\Pi}^{\rm
L}(\bm{\nabla})\,\psi_{\omega,jm}^{{\rm T}(\kappa,\pi)}(\bm{x})$
$\displaystyle=$ $\displaystyle 0.$ (732)
The wave functions are eigenfunctions of angular momentum with eigenvalues
given by [see Eq. (649)]
$\displaystyle\bm{{\cal J}}^{2}\,\psi_{\omega,jm}^{{\rm
T}(\kappa,\pi)}(\bm{x})$ $\displaystyle=$
$\displaystyle\hbar^{2}j(j+1)\,\psi_{\omega,jm}^{{\rm
T}(\kappa,\pi)}(\bm{x}),$ (733) $\displaystyle{\cal
J}^{3}\,\psi_{\omega,jm}^{{\rm T}(\kappa,\pi)}(\bm{x})$ $\displaystyle=$
$\displaystyle\hbar\,m\,\psi_{\omega,jm}^{{\rm T}(\kappa,\pi)}(\bm{x}),$ (734)
and they are eigenfunctions of ${\cal H}$, with eigenvalue $\kappa\hbar\omega$
$\displaystyle{\cal H}\,\psi_{\omega,jm}^{{\rm T}(\kappa,\pi)}(\bm{x})$
$\displaystyle=$ $\displaystyle-{\rm i}\,\hbar
c\,\bm{\alpha}\cdot\bm{\nabla}\psi_{\omega,jm}^{{\rm
T}(\kappa,\pi)}(\bm{x})=\kappa\hbar\omega\,\psi_{\omega,jm}^{{\rm
T}(\kappa,\pi)}(\bm{x}),$ (735)
provided
$\displaystyle\left(\bm{\nabla}^{2}+\frac{\omega^{2}}{c^{2}}\right)f_{\omega,j}(r)\bm{X}_{1}^{jm}(\bm{\hat{x}})=0,$
(736)
which is true if
$\displaystyle\left(\frac{1}{r}\,\frac{\partial^{2}}{\partial
r^{2}}\,r-\frac{j(j+1)}{r^{2}}+\frac{\omega^{2}}{c^{2}}\right)f_{\omega,j}(r)$
$\displaystyle=$ $\displaystyle 0.{}$ (737)
Solutions of Eq. (737) are spherical Bessel functions given by [30]
$\displaystyle f_{\omega,j}(r)$ $\displaystyle\propto$
$\displaystyle\left\\{\begin{array}[]{c}j_{j}(\omega r/c)\\\\[10.0pt]
h_{j}^{(1)}(\omega r/c)\end{array}\right..$ (740)
We employ the normalized solution
$\displaystyle f_{\omega,j}(r)$ $\displaystyle=$
$\displaystyle\frac{\omega}{c}\,\sqrt{\frac{2}{\pi c}}\ j_{j}(\omega r/c)$
(741)
for the wave functions; any other linear combination of spherical Bessel
functions (with $j\geq 1$) is not integrable as $r\rightarrow 0$. The parity
of the wave functions is
$\displaystyle\mathfrak{P}\,\psi_{\omega,jm}^{{\rm T}(\kappa,+)}(\bm{x})$
$\displaystyle=$ $\displaystyle(-1)^{j+1}\,\psi_{\omega,jm}^{{\rm
T}(\kappa,+)}(\bm{x}),$ (742)
$\displaystyle\mathfrak{P}\,\psi_{\omega,jm}^{{\rm T}(\kappa,-)}(\bm{x})$
$\displaystyle=$ $\displaystyle(-1)^{j}\,\psi_{\omega,jm}^{{\rm
T}(\kappa,-)}(\bm{x}).$ (743)
This provides the conventional parity and angular-momentum attributes for
electric and magnetic multipole radiation. Namely, $\psi_{\omega,jm}^{{\rm
T}(\kappa,+)}(\bm{x})$ is magnetic $2j$-pole or M$j$ radiation and
$\psi_{\omega,jm}^{{\rm T}(\kappa,-)}(\bm{x})$ is electric $2j$-pole or E$j$
radiation.
Alternative forms for the lower three components in Eq. (727) or the upper
three components in Eq. (730) are obtained by writing (see D)
$\displaystyle\bm{\tau}\cdot\bm{\nabla}$ $\displaystyle=$
$\displaystyle\frac{1}{r}\,\frac{\partial}{\partial
r}\,r\,\bm{\tau}\cdot\,\bm{\hat{x}}\,+\frac{1}{\hbar r}\left(\bm{L}_{\rm
s}\,\bm{\hat{x}}_{\rm s}^{\dagger}+\bm{\hat{x}}_{\rm s}\,\bm{L}_{\rm
s}^{\dagger}\right),\qquad$ (744)
which yields
$\displaystyle\bm{\tau}\cdot\bm{\nabla}\,f_{\omega,j}(r)\bm{X}_{1}^{jm}(\bm{\hat{x}})=\frac{1}{r}\,\frac{\partial}{\partial
r}\,r\,f_{\omega,j}(r)\bm{X}_{2}^{jm}(\bm{\hat{x}})-\frac{\sqrt{j(j+1)}}{r}\,f_{\omega,j}(r)\bm{X}_{3}^{jm}(\bm{\hat{x}}).$
(745)
Relations among spherical Bessel functions provide
$\displaystyle\frac{1}{r}\,\frac{\partial}{\partial r}\,r\,f_{\omega,j}(r)$
$\displaystyle=$
$\displaystyle\frac{\omega}{c}\,\frac{1}{2j+1}\big{[}(j+1)f_{\omega,j-1}(r)-j\,f_{\omega,j+1}(r)\big{]},\qquad$
(746) $\displaystyle\frac{1}{r}\,f_{\omega,j}(r)$ $\displaystyle=$
$\displaystyle\frac{\omega}{c}\,\frac{1}{2j+1}\big{[}f_{\omega,j-1}(r)+f_{\omega,j+1}(r)\big{]},$
(747)
which together with Eqs. (697) and (698) yield the second alternative form
$\displaystyle\frac{c}{\omega}\bm{\tau}\cdot\bm{\nabla}\,f_{\omega,j}(r)\bm{X}_{1}^{jm}(\bm{\hat{x}})=-\sqrt{\frac{j+1}{2j+1}}f_{\omega,j-1}(r)\,\bm{Y}_{jj-1}^{m}(\bm{\hat{x}})+\sqrt{\frac{j}{2j+1}}f_{\omega,j+1}(r)\,\bm{Y}_{jj+1}^{m}(\bm{\hat{x}}).\quad$
(748)
This latter form is useful in calculating the wave function orthonormality and
completeness relations.
An analogous longitudinal function is obtained by writing [Eq. (941)]
$\displaystyle\bm{\nabla}_{\rm s}$ $\displaystyle=$
$\displaystyle\bm{\hat{x}}_{\rm s}\,\frac{\partial}{\partial r}-\frac{1}{\hbar
r}\,\bm{\tau}\cdot\bm{\hat{x}}\,\bm{L}_{\rm s}{}$ (749)
and
$\displaystyle\frac{\omega}{c}\,\bm{F}_{\omega}^{jm}(\bm{x})$ $\displaystyle=$
$\displaystyle\bm{\nabla}_{\rm s}\,f_{\omega,j}(r)\,Y_{j}^{m}(\bm{\hat{x}})$
(750) $\displaystyle=$ $\displaystyle\frac{\partial}{\partial
r}\,f_{\omega,j}(r)\bm{X}^{jm}_{3}(\bm{\hat{x}})-\frac{\sqrt{j(j+1)}}{r}\,f_{\omega,j}(r)\bm{X}^{jm}_{2}(\bm{\hat{x}}),\qquad$
so that
$\displaystyle\bm{\tau}\cdot\bm{\nabla}\,\bm{F}_{\omega}^{jm}(\bm{x})$
$\displaystyle=$ $\displaystyle 0.$ (751)
From the additional relation
$\displaystyle\frac{\partial}{\partial r}\,f_{\omega,j}(r)$ $\displaystyle=$
$\displaystyle\frac{\omega}{c}\,\frac{1}{2j+1}\big{[}j\,f_{\omega,j-1}(r)-(j+1)f_{\omega,j+1}(r)\big{]},$
(752)
together with Eqs. (697) and (698), one has
$\displaystyle\bm{F}_{\omega}^{jm}(\bm{x})=\sqrt{\frac{j}{2j+1}}\
f_{\omega,j-1}(r)\,\bm{Y}_{jj-1}^{m}(\bm{\hat{x}})+\sqrt{\frac{j+1}{2j+1}}\
f_{\omega,j+1}(r)\,\bm{Y}_{jj+1}^{m}(\bm{\hat{x}}).\qquad{}$ (753)
The orthonormality of the transverse wave functions is given by
$\displaystyle\int{\rm d}\bm{x}\,\psi_{\omega_{2},j_{2}m_{2}}^{{\rm
T}(\kappa_{2},\pi_{2})\dagger}(\bm{x})\psi_{\omega_{1},j_{1}m_{1}}^{{\rm
T}(\kappa_{1},\pi_{1})}(\bm{x})$ $\displaystyle=$
$\displaystyle\delta_{\kappa_{2}\kappa_{1}}\delta_{\pi_{2}\pi_{1}}\delta_{j_{2}j_{1}}\delta_{m_{2}m_{1}}\delta(\omega_{2}-\omega_{1}),\qquad{}$
(754)
which takes into account the integral
$\displaystyle\int_{0}^{\infty}{\rm
d}r\,r^{2}\,f_{\omega_{2},j}(r)\,f_{\omega_{1},j}(r)$ $\displaystyle=$
$\displaystyle\delta(\omega_{2}-\omega_{1}).\qquad$ (755)
The completeness relation for the transverse wave functions is
$\displaystyle\int_{0}^{\infty}{\rm d}\omega\sum_{\kappa\pi
jm}\psi_{\omega,jm}^{{\rm T}(\kappa,\pi)}(\bm{x}_{2})\psi_{\omega,jm}^{{\rm
T}(\kappa,\pi)\dagger}(\bm{x}_{1})$ $\displaystyle=$
$\displaystyle{\it\Pi}^{\rm
T}(\bm{\nabla}_{2})\,\delta(\bm{x}_{2}-\bm{x}_{1}),{}$ (756)
which is shown in some detail by writing
$\displaystyle\sum_{\kappa\pi}\psi_{\omega,jm}^{{\rm
T}(\kappa,\pi)}(\bm{x}_{2})\psi_{\omega,jm}^{{\rm
T}(\kappa,\pi)\dagger}(\bm{x}_{1})$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}\bm{S}_{\omega}^{jm}(\bm{x}_{2},\bm{x}_{1})&{\bm{0}}\\\
{\bm{0}}&\bm{S}_{\omega}^{jm}(\bm{x}_{2},\bm{x}_{1})\end{array}\right),$ (759)
where
$\displaystyle\bm{S}_{\omega}^{jm}(\bm{x}_{2},\bm{x}_{1})$ $\displaystyle=$
$\displaystyle
f_{\omega,j}(r_{2})\bm{X}_{1}^{jm}(\bm{\hat{x}}_{2})f_{\omega,j}(r_{1})\bm{X}_{1}^{jm\dagger}(\bm{\hat{x}}_{1})$
(760)
$\displaystyle+\frac{c^{2}}{\omega^{2}}\left[\bm{\tau}\cdot\bm{\nabla}_{2}f_{\omega,j}(r_{2})\bm{X}_{1}^{jm}(\bm{\hat{x}}_{2})\right]\left[\bm{\tau}\cdot\bm{\nabla}_{1}f_{\omega,j}(r_{1})\bm{X}_{1}^{jm}(\bm{\hat{x}}_{1})\right]^{\dagger},\qquad$
and
$\displaystyle\bm{S}_{\omega}^{jm}(\bm{x}_{2},\bm{x}_{1})$ $\displaystyle=$
$\displaystyle\bm{{\it\Pi}}_{\rm s}^{\rm
T}(\bm{\nabla}_{2})\Big{[}\bm{S}_{\omega}^{jm}(\bm{x}_{2},\bm{x}_{1})+\bm{F}_{\omega}^{jm}(\bm{x}_{2})\bm{F}_{\omega}^{jm\dagger}(\bm{x}_{1})\Big{]}$
(761) $\displaystyle=$ $\displaystyle\bm{{\it\Pi}}_{\rm s}^{\rm
T}(\bm{\nabla}_{2})\Big{[}f_{\omega,j}(r_{2})\bm{Y}_{jj}^{m}(\bm{\hat{x}}_{2})f_{\omega,j}(r_{1})\bm{Y}_{jj}^{m\dagger}(\bm{\hat{x}}_{1})$
$\displaystyle+f_{\omega,j-1}(r_{2})\,\bm{Y}_{jj-1}^{m}(\bm{\hat{x}}_{2})f_{\omega,j-1}(r_{1})\,\bm{Y}_{jj-1}^{m\dagger}(\bm{\hat{x}}_{1})$
$\displaystyle+f_{\omega,j+1}(r_{2})\,\bm{Y}_{jj+1}^{m}(\bm{\hat{x}}_{2})f_{\omega,j+1}(r_{1})\,\bm{Y}_{jj+1}^{m\dagger}(\bm{\hat{x}}_{1})\Big{]},$
which gives
$\displaystyle\int_{0}^{\infty}{\rm
d}\omega\sum_{jm}\bm{S}_{\omega}^{jm}(\bm{x}_{2},\bm{x}_{1})=\bm{{\it\Pi}}_{\rm
s}^{\rm T}(\bm{\nabla}_{2})\frac{1}{r_{2}r_{1}}\,\delta(r_{2}-r_{1})$
$\displaystyle\qquad\qquad\times\sum_{jm}\Big{[}\bm{Y}_{jj}^{m}(\bm{\hat{x}}_{2})\bm{Y}_{jj}^{m\dagger}(\bm{\hat{x}}_{1})+\bm{Y}_{jj-1}^{m}(\bm{\hat{x}}_{2})\bm{Y}_{jj-1}^{m\dagger}(\bm{\hat{x}}_{1})+\bm{Y}_{jj+1}^{m}(\bm{\hat{x}}_{2})\bm{Y}_{jj+1}^{m\dagger}(\bm{\hat{x}}_{1})\Big{]}\qquad$
$\displaystyle\qquad=\bm{{\it\Pi}}_{\rm s}^{\rm
T}(\bm{\nabla}_{2})\,\delta(\bm{x}_{2}-\bm{x}_{1}),$ (762)
based on
$\displaystyle\int_{0}^{\infty}{\rm
d}\omega\,f_{\omega,j}(r_{2})\,f_{\omega,j}(r_{1})$ $\displaystyle=$
$\displaystyle\frac{1}{r_{2}r_{1}}\,\delta(r_{2}-r_{1}).\qquad$ (763)
### 8.4 Longitudinal spherical photons
Longitudinal spherical wave functions are
$\displaystyle\psi_{k,jm}^{{\rm L}(+)}(\bm{x})$ $\displaystyle=$
$\displaystyle\frac{1}{k}\left(\begin{array}[]{c}{\bm{0}}\\\\[10.0pt]
\bm{\nabla}_{\rm
s}\,g_{k,j}(r)Y_{j}^{m}(\bm{\hat{x}})\end{array}\right),\qquad{}$ (766)
$\displaystyle\psi_{k,jm}^{{\rm L}(-)}(\bm{x})$ $\displaystyle=$
$\displaystyle\frac{1}{k}\left(\begin{array}[]{c}\bm{\nabla}_{\rm
s}\,g_{k,j}(r)Y_{j}^{m}(\bm{\hat{x}})\\\\[10.0pt]
{\bm{0}}\end{array}\right).{}$ (769)
It follows from the identity $\bm{\tau}\cdot\bm{\nabla}\,\bm{\nabla}_{\rm
s}=0$ that they are longitudinal
$\displaystyle{\it\Pi}^{\rm L}(\bm{\nabla})\,\psi_{k,jm}^{{\rm
L}(\pi)}(\bm{x})$ $\displaystyle=$ $\displaystyle\psi_{k,jm}^{{\rm
L}(\pi)}(\bm{x}),$ (770) $\displaystyle{\it\Pi}^{\rm
T}(\bm{\nabla})\,\psi_{k,jm}^{{\rm L}(\pi)}(\bm{x})$ $\displaystyle=$
$\displaystyle 0$ (771)
and that they are eigenfunctions of ${\cal H}$, with eigenvalue $0$
$\displaystyle{\cal H}\,\psi_{k,jm}^{{\rm L}(\pi)}(\bm{x})$ $\displaystyle=$
$\displaystyle 0,$ (772)
with no condition on $g$. However, in order to have a complete set of
longitudinal wave functions, a set of functions, indexed by the parameter $k$
is specified here. The form of the plane-wave longitudinal solutions in Eqs.
(403) and (406) and of the expansion of a plane wave in spherical waves
suggest the choice
$\displaystyle g_{k,j}(r)$ $\displaystyle=$ $\displaystyle
k\,\sqrt{\frac{2}{\pi}}\ j_{j}(kr)$ (773)
for the radial wave function, where $k$ is a free parameter. This set of
functions provides an infinite orthonormal set of degenerate ($\omega=0$)
basis functions for each $j$.
From the form of $\bm{\nabla}_{\rm s}$ in Eq. (749) and the fact that
$\bm{X}_{2}^{jm}(\bm{\hat{x}})$ and $\bm{X}_{3}^{jm}(\bm{\hat{x}})$ are
eigenfunctions of angular momentum, it follows that the longitudinal spherical
wave functions are also eigenfunctions of angular momentum
$\displaystyle\bm{{\cal J}}^{2}\,\psi_{k,jm}^{{\rm L}(\pi)}(\bm{x})$
$\displaystyle=$ $\displaystyle\hbar^{2}j(j+1)\,\psi_{k,jm}^{{\rm
L}(\pi)}(\bm{x}),$ (774) $\displaystyle{\cal J}^{3}\,\psi_{k,jm}^{{\rm
L}(\pi)}(\bm{x})$ $\displaystyle=$ $\displaystyle\hbar\,m\,\psi_{k,jm}^{{\rm
L}(\pi)}(\bm{x}).$ (775)
They have parity given by
$\displaystyle\mathfrak{P}\,\psi_{k,jm}^{{\rm L}(+)}(\bm{x})$ $\displaystyle=$
$\displaystyle(-1)^{j+1}\,\psi_{k,jm}^{{\rm L}(+)}(\bm{x}),$ (776)
$\displaystyle\mathfrak{P}\,\psi_{k,jm}^{{\rm L}(-)}(\bm{x})$ $\displaystyle=$
$\displaystyle(-1)^{j}\,\psi_{k,jm}^{{\rm L}(-)}(\bm{x}).$ (777)
With spherical Bessel functions for the radial wave functions, we have,
following Eqs. (749) to (753),
$\displaystyle\frac{\bm{\nabla}_{\rm
s}}{k}\,g_{k,j}(r)Y_{j}^{m}(\bm{\hat{x}})$ $\displaystyle=$
$\displaystyle\sqrt{\frac{j}{2j+1}}\
g_{k,j-1}(r)\,\bm{Y}_{jj-1}^{m}(\bm{\hat{x}})+\sqrt{\frac{j+1}{2j+1}}\
g_{k,j+1}(r)\,\bm{Y}_{jj+1}^{m}(\bm{\hat{x}}).\qquad$ (778)
This identity facilitates calculation of the orthonormality relation for the
longitudinal wave functions, which is
$\displaystyle\int{\rm d}\bm{x}\,\psi_{k_{2},j_{2}m_{2}}^{{\rm
L}(\pi_{2})\dagger}(\bm{x})\psi_{k_{1},j_{1}m_{1}}^{{\rm L}(\pi_{1})}(\bm{x})$
$\displaystyle=$
$\displaystyle\delta_{\pi_{2}\pi_{1}}\delta_{j_{2}j_{1}}\delta_{m_{2}m_{1}}\delta(k_{2}-k_{1}),\qquad{}$
(779)
with
$\displaystyle\int_{0}^{\infty}{\rm
d}r\,r^{2}\,g_{k_{2},j}(r)\,g_{k_{1},j}(r)$ $\displaystyle=$
$\displaystyle\delta(k_{2}-k_{1}).\qquad$ (780)
The completeness relation for the longitudinal wave functions is
$\displaystyle\int_{0}^{\infty}{\rm d}k\sum_{\pi jm}\psi_{k,jm}^{{\rm
L}(\pi)}(\bm{x}_{2})\psi_{k,jm}^{{\rm L}(\pi)\dagger}(\bm{x}_{1})$
$\displaystyle=$ $\displaystyle{\it\Pi}^{\rm
L}(\bm{\nabla}_{2})\,\delta(\bm{x}_{2}-\bm{x}_{1}),\qquad{}$ (781)
which follows from
$\displaystyle\sum_{\pi}\psi_{k,jm}^{{\rm
L}(\pi)}(\bm{x}_{2})\psi_{k,jm}^{{\rm L}(\pi)\dagger}(\bm{x}_{1})$
$\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}\bm{T}_{k}^{jm}(\bm{x}_{2},\bm{x}_{1})&{\bm{0}}\\\
{\bm{0}}&\bm{T}_{k}^{jm}(\bm{x}_{2},\bm{x}_{1})\end{array}\right),$ (784)
where
$\displaystyle\bm{T}_{k}^{jm}(\bm{x}_{2},\bm{x}_{1})$ $\displaystyle=$
$\displaystyle-\frac{\bm{\nabla}_{\\!2\,\rm s}\bm{\nabla}_{\\!1\,\rm
s}^{\dagger}}{\bm{\nabla}_{2}^{2}}\,g_{k,j}(r_{2})g_{k,j}(r_{1})Y_{j}^{m}(\bm{\hat{x}}_{2})Y_{j}^{m*}(\bm{\hat{x}}_{1}),$
(785)
and
$\displaystyle\bm{\nabla}^{2}g_{k,j}(r)Y_{j}^{m}(\bm{\hat{x}})$
$\displaystyle=$ $\displaystyle-k^{2}g_{k,j}(r)Y_{j}^{m}(\bm{\hat{x}}).$ (786)
Thus from
$\displaystyle\int_{0}^{\infty}{\rm d}k\,g_{k,j}(r_{2})g_{k,j}(r_{1})$
$\displaystyle=$ $\displaystyle\frac{1}{r_{2}r_{1}}\,\delta(r_{2}-r_{1}),$
(787)
we have
$\displaystyle\int_{0}^{\infty}{\rm
d}k\sum_{jm}\bm{T}_{k}^{jm}(\bm{x}_{2},\bm{x}_{1})$ $\displaystyle=$
$\displaystyle\bm{{\it\Pi}}^{\rm L}_{\rm
s}(\bm{\nabla}_{2})\delta(\bm{x}_{2}-\bm{x}_{1}),\qquad$ (788)
which provides Eq. (781). That result, together with Eq. (756) gives the full
completeness relation.
As an illustration of a role of the spherical functions, we revisit the
example of a point charge at the origin, for which
$\displaystyle{\it\Psi}_{\rm p}(\bm{x})$ $\displaystyle=$
$\displaystyle-\frac{q}{4\pi\epsilon_{0}}\left(\begin{array}[]{c}\bm{\nabla}_{\rm
s}\,\frac{\textstyle 1}{\textstyle r}\\\\[8.0pt] {\bm{0}}\end{array}\right).$
(791)
In view of the integral
$\displaystyle\int_{0}^{\infty}{\rm d}k\,\frac{1}{k}\,g_{k,0}(r)$
$\displaystyle=$ $\displaystyle\sqrt{\frac{\pi}{2}}\,\frac{1}{r},$ (792)
one has
$\displaystyle{\it\Psi}_{\rm p}(\bm{x})$ $\displaystyle=$
$\displaystyle-\frac{q}{\sqrt{2}\,\pi\epsilon_{0}}\int_{0}^{\infty}{\rm
d}k\,\psi_{k,00}^{{\rm L}(-)}(\bm{x}),$ (793)
which is the analog, for spherical solutions, of Eq. (422) for plane wave
solutions.
## 9 Maxwell Green function
A solution of the Maxwell equation for the electric and magnetic fields
${\it\Psi}(x)$ given a specified current source ${\it\Xi}(x)$, as they are
related in Eq. (97)
$\displaystyle\gamma_{\mu}\partial^{\mu}{\it\Psi}(x)$ $\displaystyle=$
$\displaystyle{\it\Xi}(x),$
can be found with the aid of the $6\times 6$ matrix Maxwell Green function
${\cal D}_{\rm M}(x_{2}-x_{1})$, given by
$\displaystyle{\cal D}_{\rm M}(x_{2}-x_{1})$ $\displaystyle=$
$\displaystyle\sum_{\lambda=0}^{2}\int{\rm d}\bm{k}\
\psi_{\bm{k},\lambda}^{(+)}(x_{2})\overline{\psi}_{\bm{k},\lambda}^{(+)}(x_{1})\,\theta(t_{2}-t_{1})$
$\displaystyle-$ $\displaystyle\sum_{\lambda=0}^{2}\int{\rm d}\bm{k}\
\psi_{\bm{k},\lambda}^{(-)}(x_{2})\overline{\psi}_{\bm{k},\lambda}^{(-)}(x_{1})\,\theta(t_{1}-t_{2}),$
where $\psi_{\bm{k},\lambda}^{(\pm)}(x)$ is given by Eq. (367), (403) or
(406), and Eq. (425). In view of the relations
$\displaystyle\gamma_{\mu}\partial_{2}^{\mu}\,\psi_{\bm{k},\lambda}^{(\pm)}(x_{2})$
$\displaystyle=$ $\displaystyle 0,$ (795)
$\displaystyle\gamma_{\mu}\partial_{2}^{\mu}\,\theta(t_{2}-t_{1})$
$\displaystyle=$ $\displaystyle\gamma_{0}\,\delta(ct_{2}-ct_{1}),$ (796)
$\displaystyle\gamma_{\mu}\partial_{2}^{\mu}\,\theta(t_{1}-t_{2})$
$\displaystyle=$ $\displaystyle-\gamma_{0}\,\delta(ct_{2}-ct_{1})$ (797)
and the completeness of the wave functions, we have
$\displaystyle\gamma_{\mu}\partial_{2}^{\mu}\,{\cal D}_{\rm M}(x_{2}-x_{1})$
$\displaystyle=$ $\displaystyle{\cal I}\,\delta(x_{2}-x_{1}){}$ (798)
and
$\displaystyle{\cal D}_{\rm
M}(x_{2}-x_{1})\,\gamma_{\mu}\overleftarrow{\partial}_{1}^{\mu}$
$\displaystyle=$ $\displaystyle-{\cal I}\,\delta(x_{2}-x_{1}),$ (799)
where
$\displaystyle\delta(x_{2}-x_{1})$ $\displaystyle=$
$\displaystyle\delta(ct_{2}-ct_{1})\,\delta(\bm{x}_{2}-\bm{x}_{1}).$ (800)
In terms of the Green function, a solution for the electric and magnetic
fields is
$\displaystyle{\it\Psi}(x_{2})$ $\displaystyle=$ $\displaystyle\int{\rm
d}^{4}x_{1}\,{\cal D}_{\rm M}(x_{2}-x_{1})\,{\it\Xi}(x_{1}),{}$ (801)
as is confirmed by the application of $\gamma_{\mu}\partial_{2}^{\mu}$ to both
sides. In Eq. (801) ${\rm d}^{4}x_{1}=c\,{\rm d}t_{1}{\rm d}\bm{x}_{1}$. A
separation into transverse or longitudinal solutions may be made by
restricting the sum over polarizations to $\lambda=1,2$ for a transverse
solution or $\lambda=0$ for a longitudinal solution.
The Maxwell Green function also may be written as an integral over the four-
vector $k$ of the plane-wave solutions. For this it is useful to make the
separation into transverse and longitudinal components. For the transverse
part we have
$\displaystyle{\cal D}_{\rm M}^{\rm T}(x_{2}-x_{1})$ $\displaystyle=$
$\displaystyle\sum_{\lambda=1}^{2}\int{\rm d}\bm{k}\
\psi_{\bm{k},\lambda}^{(+)}(\bm{x}_{2})\overline{\psi}_{\bm{k},\lambda}^{(+)}(\bm{x}_{1})\,{\rm
e}^{-{\rm i}\omega(t_{2}-t_{1})}\,\theta(t_{2}-t_{1})$ (802)
$\displaystyle-\sum_{\lambda=1}^{2}\int{\rm d}\bm{k}\
\psi_{\bm{k},\lambda}^{(-)}(\bm{x}_{2})\overline{\psi}_{\bm{k},\lambda}^{(-)}(\bm{x}_{1})\,{\rm
e}^{{\rm i}\omega(t_{2}-t_{1})}\,\theta(t_{1}-t_{2}),$
and we employ the identities
$\displaystyle{\rm e}^{-{\rm i}\omega(t_{2}-t_{1})}\,\theta(t_{2}-t_{1})$
$\displaystyle=$ $\displaystyle\frac{{\rm i}}{2\pi}\int_{-\infty}^{\infty}{\rm
d}k_{0}\,\frac{{\rm e}^{-{\rm i}k_{0}(ct_{2}-ct_{1})}}{k_{0}(1+{\rm
i}\delta)-\omega/c},$ (803) $\displaystyle-{\rm e}^{{\rm
i}\omega(t_{2}-t_{1})}\,\theta(t_{1}-t_{2})$ $\displaystyle=$
$\displaystyle\frac{{\rm i}}{2\pi}\int_{-\infty}^{\infty}{\rm
d}k_{0}\,\frac{{\rm e}^{-{\rm i}k_{0}(ct_{2}-ct_{1})}}{k_{0}(1+{\rm
i}\delta)+\omega/c},$ (804)
where the limit $\delta\rightarrow 0^{+}$ for the integral is understood. We
also have
$\displaystyle\bm{\alpha}\cdot\bm{k}\,\psi_{\bm{k},\lambda}^{(\pm)}(\bm{x}_{2})$
$\displaystyle=$
$\displaystyle|\bm{k}|\,\psi_{\bm{k},\lambda}^{(\pm)}(\bm{x}_{2})=\frac{\omega}{c}\,\psi_{\bm{k},\lambda}^{(\pm)}(\bm{x}_{2}).\qquad$
(805)
Together these relations yield [see Eq. (389)]
$\displaystyle{\cal D}_{\rm M}^{\rm T}(x_{2}-x_{1})$ $\displaystyle=$
$\displaystyle\frac{{\rm i}}{2\pi}\sum_{\lambda=1}^{2}\int{\rm
d}^{4}k\Bigg{[}\frac{{\rm e}^{-{\rm i}k_{0}(ct_{2}-ct_{1})}}{k_{0}(1+{\rm
i}\delta)-\bm{\alpha}\cdot\bm{k}}\,\psi_{\bm{k},\lambda}^{(+)}(\bm{x}_{2})\overline{\psi}_{\bm{k},\lambda}^{(+)}(\bm{x}_{1})$
(808) $\displaystyle\qquad+\frac{{\rm e}^{-{\rm
i}k_{0}(ct_{2}-ct_{1})}}{k_{0}(1+{\rm
i}\delta)+\bm{\alpha}\cdot\bm{k}}\,\psi_{\bm{k},\lambda}^{(-)}(\bm{x}_{2})\overline{\psi}_{\bm{k},\lambda}^{(-)}(\bm{x}_{1})\Bigg{]}$
$\displaystyle=$ $\displaystyle\frac{{\rm i}}{(2\pi)^{4}}\int{\rm d}^{4}k\
{\rm e}^{-{\rm i}k_{0}(ct_{2}-ct_{1})}\,\frac{{\rm e}^{{\rm
i}\bm{k}\cdot(\bm{x}_{2}-\bm{x}_{1})}}{\gamma^{0}k_{0}(1+{\rm
i}\delta)-\bm{\gamma}\cdot\bm{k}}\left(\begin{array}[]{cc}(\bm{\tau}\cdot\bm{\hat{k}})^{2}&{\bm{0}}\\\
{\bm{0}}&(\bm{\tau}\cdot\bm{\hat{k}})^{2}\end{array}\right)$ $\displaystyle=$
$\displaystyle\frac{{\rm i}}{(2\pi)^{4}}\,{\it\Pi}^{\rm
T}(\bm{\nabla}_{2})\int_{\rm C_{F}}{\rm d}^{4}k\ \frac{{\rm e}^{-{\rm
i}k\cdot(x_{2}-x_{1})}}{\gamma^{\mu}k_{\mu}},{}$ (809)
where ${\rm d}^{4}k={\rm d}k_{0}\,{\rm d}\bm{k}$, and ${\rm C_{F}}$ indicates
that the contour of integration over $k_{0}$ is the Feynman contour, which
passes from $-\infty$ below the negative real axis, through $0$, and above the
positive real axis to $+\infty$; this is equivalent to including the factor
$(1+{\rm i}\delta)$ multiplying $k_{0}$ in the denominator and integrating
along the real axis.
For applications, it is useful to consider an alternative form for the
transverse Green function. Taking into account the relation
$\displaystyle{\it\Pi}^{\rm T}(\bm{\hat{k}})\frac{1}{\gamma^{0}k_{0}(1+{\rm
i}\delta)-\bm{\gamma}\cdot\bm{k}}$ $\displaystyle=$
$\displaystyle{\it\Pi}^{\rm
T}(\bm{\hat{k}})\,\frac{\gamma^{0}k_{0}-\bm{\gamma}\cdot\bm{k}}{k_{0}^{2}-\bm{k}^{2}+{\rm
i}\delta}$ (812) $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}k_{0}\,(\bm{\tau}\cdot\bm{\hat{k}})^{2}&-\bm{\tau}\cdot\bm{k}\\\
\bm{\tau}\cdot\bm{k}&-k_{0}\,(\bm{\tau}\cdot\bm{\hat{k}})^{2}\end{array}\right)\frac{1}{k^{2}+{\rm
i}\delta},{}$
we have, from Eq. (809),
$\displaystyle{\cal D}_{\rm M}^{\rm T}(x_{2}-x_{1})$ $\displaystyle=$
$\displaystyle\frac{{\rm i}}{(2\pi)^{4}}\int_{-\infty}^{\infty}{\rm d}k_{0}\
{\rm e}^{-{\rm
i}k_{0}(ct_{2}-ct_{1})}\,\left(\begin{array}[]{cc}k_{0}\,\bm{{\it\Pi}}_{\rm
s}^{\rm T}(\bm{\nabla})&{\rm i}\,\bm{\tau}\cdot\bm{\nabla}\\\ -{\rm
i}\,\bm{\tau}\cdot\bm{\nabla}&-k_{0}\,\bm{{\it\Pi}}_{\rm s}^{\rm
T}(\bm{\nabla})\end{array}\right)\int{\rm d}\bm{k}\,\frac{{\rm e}^{{\rm
i}\bm{k}\cdot\bm{r}}}{k^{2}+{\rm i}\delta}$ (815) $\displaystyle=$
$\displaystyle\frac{1}{8\pi^{2}{\rm i}}\int_{-\infty}^{\infty}{\rm d}k_{0}\
{\rm e}^{-{\rm
i}k_{0}(ct_{2}-ct_{1})}\,\left(\begin{array}[]{cc}k_{0}\,\bm{{\it\Pi}}_{\rm
s}^{\rm T}(\bm{\nabla})&{\rm i}\,\bm{\tau}\cdot\bm{\nabla}\\\ -{\rm
i}\,\bm{\tau}\cdot\bm{\nabla}&-k_{0}\,\bm{{\it\Pi}}_{\rm s}^{\rm
T}(\bm{\nabla})\end{array}\right)\frac{{\rm e}^{{\rm i}(k_{0}^{2}+{\rm
i}\delta)^{1/2}|\bm{r}|}}{|\bm{r}|}$ (821) $\displaystyle\rightarrow$
$\displaystyle\frac{1}{8\pi^{2}{\rm i}}\int_{-\infty}^{\infty}{\rm d}k_{0}\
{\rm e}^{-{\rm i}k_{0}(ct_{2}-ct_{1})}$
$\displaystyle\qquad\times\left(\begin{array}[]{cc}k_{0}(\bm{\bm{\tau}\cdot\hat{r}})^{2}&-(k_{0}^{2}+{\rm
i}\delta)^{1/2}\,\bm{\tau}\cdot\bm{\hat{r}}\\\ (k_{0}^{2}+{\rm
i}\delta)^{1/2}\,\bm{\tau}\cdot\bm{\hat{r}}&-k_{0}(\bm{\bm{\tau}\cdot\hat{r}})^{2}\end{array}\right)\frac{{\rm
e}^{{\rm i}(k_{0}^{2}+{\rm i}\delta)^{1/2}|\bm{r}|}}{|\bm{r}|},{}$
where $\bm{r}=\bm{x}_{2}-\bm{x}_{1}$, the gradient $\bm{\nabla}$ is with
respect to $\bm{r}$, and the branch of the square root in the exponent is
determined by the condition ${\rm Im}(k_{0}^{2}+{\rm i}\delta)^{1/2}>0$, which
specifies that $(k_{0}^{2}+{\rm i}\delta)^{1/2}\rightarrow|k_{0}|$ for real
values of $k_{0}$. In the last line of Eq. (821), higher-order terms in
$(k_{0}\,|\bm{r}|)^{-1}$ are not included, but the exact expression follows
from the formulas in E.
For the longitudinal Green function, we write
$\displaystyle{\cal D}_{\rm M}^{\rm L}(x_{2}-x_{1})$ $\displaystyle=$
$\displaystyle\int{\rm d}\bm{k}\
\psi_{\bm{k},0}^{(+)}(\bm{x}_{2})\overline{\psi}_{\bm{k},0}^{(+)}(\bm{x}_{1})\,{\rm
e}^{-\epsilon(ct_{2}-ct_{1})}\,\theta(t_{2}-t_{1})$ (822)
$\displaystyle-\int{\rm d}\bm{k}\
\psi_{\bm{k},0}^{(-)}(\bm{x}_{2})\overline{\psi}_{\bm{k},0}^{(-)}(\bm{x}_{1})\,{\rm
e}^{\epsilon(ct_{2}-ct_{1})}\,\theta(t_{1}-t_{2}),{}$
where damping factors with $\epsilon>0$ are added so that the Green function
falls off for large time differences. In addition, we assume that the
longitudinal wave functions are solutions of the Maxwell equation with an
infinitesimal mass $m_{\epsilon}$ included, as given in Eq. (437), in order to
be able to use the Feynman contour to specify the path of integration over
$k_{0}$ in relation to the poles of the integrand. We employ the identities
$\displaystyle{\rm e}^{-{\rm i}m_{\epsilon}c^{2}(t_{2}-t_{1})/\hbar}\,{\rm
e}^{-\epsilon(ct_{2}-ct_{1})}\,\theta(t_{2}-t_{1})$ $\displaystyle=$
$\displaystyle\frac{{\rm i}}{2\pi}\int_{-\infty}^{\infty}{\rm
d}k_{0}\,\frac{{\rm e}^{-{\rm i}k_{0}(ct_{2}-ct_{1})}}{k_{0}+{\rm i}\epsilon-
m_{\epsilon}c/\hbar},\qquad$ (823) $\displaystyle-{\rm e}^{{\rm
i}m_{\epsilon}c^{2}(t_{2}-t_{1})/\hbar}\,{\rm
e}^{\epsilon(ct_{2}-ct_{1})}\,\theta(t_{1}-t_{2})$ $\displaystyle=$
$\displaystyle\frac{{\rm i}}{2\pi}\int_{-\infty}^{\infty}{\rm
d}k_{0}\,\frac{{\rm e}^{-{\rm i}k_{0}(ct_{2}-ct_{1})}}{k_{0}-{\rm
i}\epsilon+m_{\epsilon}c/\hbar}$ (824)
and
$\displaystyle\gamma^{0}\,\psi_{\bm{k},0}^{(\kappa)}(\bm{x}_{2})$
$\displaystyle=$
$\displaystyle\kappa\,\psi_{\bm{k},0}^{(\kappa)}(\bm{x}_{2}),$ (825)
$\displaystyle\bm{\alpha}\cdot\bm{k}\,\psi_{\bm{k},0}^{(\kappa)}(\bm{x}_{2})$
$\displaystyle=$ $\displaystyle 0$ (826)
to obtain
$\displaystyle{\cal D}_{\rm M}^{\rm L}(x_{2}-x_{1})$ $\displaystyle=$
$\displaystyle\frac{{\rm i}}{2\pi}\int{\rm d}^{4}k\Bigg{[}\frac{{\rm e}^{-{\rm
i}k_{0}(ct_{2}-ct_{1})}}{k_{0}+{\rm i}\epsilon-
m_{\epsilon}c/\hbar}\,\psi_{\bm{k},0}^{(+)}(\bm{x}_{2})\overline{\psi}_{\bm{k},0}^{(+)}(\bm{x}_{1})$
(827) $\displaystyle\qquad\qquad+\frac{{\rm e}^{-{\rm
i}k_{0}(ct_{2}-ct_{1})}}{k_{0}-{\rm
i}\epsilon+m_{\epsilon}c/\hbar}\,\psi_{\bm{k},0}^{(-)}(\bm{x}_{2})\overline{\psi}_{\bm{k},0}^{(-)}(\bm{x}_{1})\Bigg{]}$
$\displaystyle=$ $\displaystyle\frac{{\rm i}}{2\pi}\int_{\rm C_{F}}{\rm
d}^{4}k\frac{{\rm e}^{-{\rm
i}k_{0}(ct_{2}-ct_{1})}}{k_{0}-\gamma^{0}m_{\epsilon}c/\hbar}\sum_{\kappa\rightarrow\pm}\psi_{\bm{k},0}^{(\kappa)}(\bm{x}_{2})\overline{\psi}_{\bm{k},0}^{(\kappa)}(\bm{x}_{1})$
$\displaystyle=$ $\displaystyle\frac{{\rm i}}{(2\pi)^{4}}\,{\it\Pi}^{\rm
L}(\bm{\nabla}_{2})\int_{\rm C_{F}}{\rm d}^{4}k\ \frac{{\rm e}^{-{\rm
i}k\cdot(x_{2}-x_{1})}}{\gamma^{\mu}k_{\mu}-m_{\epsilon}c/\hbar}\,.{}$
Here the limit $\epsilon\rightarrow 0$ would be undefined without the mass
term. A concise alternative expression for the longitudinal Green function is
obtained by the substitution of the partial completeness relations that follow
from Eqs. (413) and (416) into Eq. (822):
$\displaystyle{\cal D}_{\rm M}^{\rm L}(x_{2}-x_{1})$ $\displaystyle=$
$\displaystyle{\it\Pi}^{\rm
L}(\bm{\nabla}_{2})\,\delta(\bm{x}_{2}-\bm{x}_{1})\left(\begin{array}[]{cc}\bm{I}\,\theta(t_{2}-t_{1})&{\bm{0}}\\\
{\bm{0}}&\bm{I}\,\theta(t_{1}-t_{2})\end{array}\right).$ (830)
The transverse and longitudinal Green functions in Eqs. (809) and (827) differ
only by the type of projection operator and the infinitesimal mass term in Eq.
(827). However, such a mass term in the last line of Eqs. (809) would not
change the relation of the path of integration over $k_{0}$ to the location of
the poles of the integrand, so it could also be included in that expression.
In particular, the poles in Eq. (812) at $k_{0}=\pm\left(\bm{k}^{2}-{\rm
i}\delta\right)^{1/2}$ would move to
$k_{0}=\pm\left[\bm{k}^{2}+(m_{\epsilon}c/\hbar)^{2}-{\rm
i}\delta\right]^{1/2}$. These poles lie on curves in the second and fourth
quadrants of the complex $k_{0}$ plane, whereas the Feynman contour passes
through the first and third quadrants. Thus, we may write ${\cal D}_{\rm
M}^{\rm T}(x_{2}-x_{1})+{\cal D}_{\rm M}^{\rm L}(x_{2}-x_{1})={\cal D}_{\rm
M}(x_{2}-x_{1})$, with
$\displaystyle{\cal D}_{\rm M}(x_{2}-x_{1})$ $\displaystyle=$
$\displaystyle\frac{{\rm i}}{(2\pi)^{4}}\,\int_{C_{F}}{\rm d}^{4}k\ \frac{{\rm
e}^{-{\rm i}k\cdot(x_{2}-x_{1})}}{\gamma^{\mu}k_{\mu}-m_{\epsilon}c/\hbar}.{}$
(831)
This result is a covariant Green function for the Maxwell equation which is of
the same form as the well-known Green function for the Dirac equation. A
formal coordinate-representation is
$\displaystyle{\cal D}_{\rm M}(x_{2}-x_{1})$ $\displaystyle=$
$\displaystyle\frac{1}{\gamma_{\mu}\partial_{2}^{\mu}}\,\delta(x_{2}-x_{1}).$
(832)
The fields given by Eq. (801) represent a particular solution of the Maxwell
equation. Any solution of Eq. (801) for ${\it\Xi}(x_{1})=0$, such as the field
of a static charge distribution, may be added to the particular solution, and
the sum will be a solution with the same source function. In fact, even if the
three-vector current density vanishes in the distant past and future, there
could be a static charge distribution that persists, with or without a net
total charge, for which the fields would be non-zero indefinitely. To deal
with this case, we obtain an expression that takes into account the possible
fields in the past and future by writing the time derivative
$\displaystyle\frac{\partial}{\partial(ct_{1})}\int{\rm d}\bm{x}_{1}\,{\cal
D}_{\rm M}(x_{2}-x_{1})\gamma_{0}{\it\Psi}(x_{1})$ $\displaystyle=$
$\displaystyle\int{\rm d}\bm{x}_{1}\,{\cal D}_{\rm
M}(x_{2}-x_{1})\gamma_{0}\overleftarrow{\partial}_{1}^{0}{\it\Psi}(x_{1})$
(833) $\displaystyle+\int{\rm d}\bm{x}_{1}\,{\cal D}_{\rm
M}(x_{2}-x_{1})\gamma_{0}\partial_{1}^{\,0}{\it\Psi}(x_{1}),\quad$
and for fields that vanish for large space-like distances, we write
$\displaystyle\int{\rm d}\bm{x}_{1}\,\bm{\nabla}_{1}\cdot\,{\cal D}_{\rm
M}(x_{2}-x_{1})\bm{\gamma}{\it\Psi}(x_{1})$ $\displaystyle=$
$\displaystyle\int{\rm d}\bm{x}_{1}\,{\cal D}_{\rm
M}(x_{2}-x_{1})\bm{\gamma}\cdot\overleftarrow{\bm{\nabla}}_{1}{\it\Psi}(x_{1})$
(834) $\displaystyle+\int{\rm d}\bm{x}_{1}\,{\cal D}_{\rm
M}(x_{2}-x_{1})\bm{\gamma}\cdot\bm{\nabla}_{1}{\it\Psi}(x_{1})=0,\qquad$
where the result is zero, because it may be written as an integral over the
bounding surface, by the Gauss-Ostrogradsky theorem. The sum of Eqs. (833) and
(834) is
$\displaystyle\frac{\partial}{\partial(ct_{1})}\int{\rm d}\bm{x}_{1}\,{\cal
D}_{\rm M}(x_{2}-x_{1})\gamma_{0}{\it\Psi}(x_{1})$
$\displaystyle\qquad\qquad=\int{\rm d}\bm{x}_{1}\,{\cal D}_{\rm
M}(x_{2}-x_{1})\gamma_{\mu}\overleftarrow{\partial}_{1}^{\,\mu}{\it\Psi}(x_{1})+\int{\rm
d}\bm{x}_{1}\,{\cal D}_{\rm
M}(x_{2}-x_{1})\gamma_{\mu}\partial_{1}^{\,\mu}{\it\Psi}(x_{1})\qquad$
$\displaystyle\qquad\qquad=-\int{\rm
d}\bm{x}_{1}\,\delta(x_{2}-x_{1}){\it\Psi}(x_{1})+\int{\rm d}\bm{x}_{1}\,{\cal
D}_{\rm M}(x_{2}-x_{1})\,{\it\Xi}(x_{1}).$ (835)
Integration of Eq. (835) over $t_{1}$ from $t_{\rm i}$ to $t_{\rm f}$, where
$t_{\rm i}<t_{2}<t_{\rm f}$, yields
$\displaystyle{\it\Psi}(x_{2})$ $\displaystyle=$ $\displaystyle\int{\rm
d}\bm{x}_{1}\,{\cal D}_{\rm
M}(x_{2}-x_{1})\gamma_{0}{\it\Psi}(x_{1})\Big{|}_{t_{1}=t_{\rm i}}-\int{\rm
d}\bm{x}_{1}\,{\cal D}_{\rm
M}(x_{2}-x_{1})\gamma_{0}{\it\Psi}(x_{1})\Big{|}_{t_{1}=t_{\rm f}}$ (836)
$\displaystyle+c\int_{t_{\rm i}}^{t_{\rm f}}{\rm d}t_{1}\int{\rm
d}\bm{x}_{1}\,{\cal D}_{\rm M}(x_{2}-x_{1})\,{\it\Xi}(x_{1})$
or
$\displaystyle{\it\Psi}(x_{2})$ $\displaystyle=$ $\displaystyle\int{\rm
d}\bm{x}_{1}\,\sum_{\lambda=0}^{2}\int{\rm
d}\bm{k}\left[\psi_{\bm{k},\lambda}^{(+)}(x_{2})\psi_{\bm{k},\lambda}^{(+)\dagger}(x_{1}){\it\Psi}(x_{1})\Big{|}_{t_{1}=t_{\rm
i}}+\psi_{\bm{k},\lambda}^{(-)}(x_{2})\psi_{\bm{k},\lambda}^{(-)\dagger}(x_{1}){\it\Psi}(x_{1})\Big{|}_{t_{1}=t_{\rm
f}}\right]$ (837) $\displaystyle+c\int_{t_{\rm i}}^{t_{\rm f}}{\rm
d}t_{1}\int{\rm d}\bm{x}_{1}\,{\cal D}_{\rm
M}(x_{2}-x_{1})\,{\it\Xi}(x_{1}).{}$
As a consistency check of this expression, we note that it properly reduces to
the expected result for the field of a constant charge distribution. In this
case, the current source term vanishes, the initial and final fields are the
same, and they are purely longitudinal. As a result, only longitudinal
functions with no time dependence will contribute to the sum over states,
which is just the longitudinal completeness relation, and Eq. (837) reduces to
the proper identity.
## 10 Applications of the Maxwell Green function
The Maxwell Green function is used here to calculate the radiation fields of a
point dipole source as an example of an application. Only the large distance
transverse fields are considered, and they are given by
$\displaystyle{\it\Psi}_{\rm d}(x_{2})$ $\displaystyle=$
$\displaystyle\int{\rm d}^{4}x_{1}\,{\cal D}_{\rm M}^{\rm
T}(x_{2}-x_{1})\,{\it\Xi}_{\rm d}(x_{1}),$ (838)
with the source term
$\displaystyle{\it\Xi}_{\rm d}(x)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}-\mu_{0}c\,\bm{j}_{\rm s}(\bm{x})\,{\rm
e}^{-{\rm i}\omega_{\rm d}t}\\\ {\bm{0}}\end{array}\right).$ (841)
The classical source for dipole radiation is a charge $q$ with position
$\displaystyle\bm{x}_{\rm d}(t)$ $\displaystyle=$
$\displaystyle\bm{x}_{0}\cos{\omega_{\rm d}t}$ (842)
which produces a current density
$\displaystyle\bm{j}_{\rm cl}(x)$ $\displaystyle=$ $\displaystyle
q\,\delta(\bm{x}-\bm{x}_{\rm d}(t))\,\dot{\bm{x}}_{\rm d}(t)$ (843)
$\displaystyle\approx$ $\displaystyle-\omega_{\rm
d}\,\bm{d}\,\delta(\bm{x})\sin{\omega_{\rm d}t},$
where $\bm{d}=q\,\bm{x}_{0}$. This is the real part of
$\displaystyle\bm{j}(x)$ $\displaystyle=$ $\displaystyle-{\rm i}\,\omega_{\rm
d}\,\bm{d}\,\delta(\bm{x}){\rm e}^{-{\rm i}\omega_{\rm d}t},$ (844)
which is the source current for the radiation [17]. For the transverse Maxwell
Green function, we use the expression on the last line of Eq. (821).
Integration over $t_{1}$ yields a factor $2\pi\delta(k_{0}c-\omega_{\rm d})$,
and evaluation of the integration over $k_{0}$ follows. The result is
$\displaystyle c\int{\rm d}t_{1}\,{\cal D}_{\rm M}^{\rm
T}(x_{2}-x_{1})\,{\it\Xi}_{\rm d}(x_{1})$
$\displaystyle\qquad\qquad=-\frac{\mu_{0}c\,k}{4\pi{\rm
i}}\,\left(\begin{array}[]{cc}(\bm{\tau}\cdot\bm{\hat{r}})^{2}&-\bm{\tau}\cdot\bm{\hat{r}}\\\
\bm{\tau}\cdot\bm{\hat{r}}&-(\bm{\tau}\cdot\bm{\hat{r}})^{2}\end{array}\right)\left(\begin{array}[]{c}\bm{j}_{\rm
s}(\bm{x}_{1})\\\ {\bm{0}}\end{array}\right)\frac{{\rm e}^{{\rm
i}k|\bm{r}|}}{|\bm{r}|}\,{\rm e}^{-{\rm i}\omega_{\rm d}t_{2}}+\dots\ ,\qquad$
(849)
where $k=\omega_{\rm d}/c$. Since the source is point-like at the origin
$\bm{x}_{1}=0$, $\bm{r}=\bm{x}_{2}$, and
$\displaystyle{\it\Psi}_{\rm d}(x)$ $\displaystyle=$
$\displaystyle\frac{k^{2}}{4\pi\epsilon_{0}}\,\left(\begin{array}[]{c}(\bm{\tau}\cdot\bm{\hat{x}})^{2}\,\bm{d}_{\rm
s}\\\ \bm{\tau}\cdot\bm{\hat{x}}\,\bm{d}_{\rm s}\end{array}\right)\frac{{\rm
e}^{{\rm i}k|\bm{x}|}}{|\bm{x}|}\,{\rm e}^{-{\rm i}\omega_{\rm d}t}+\dots\ .$
(852)
The time-average differential radiated power, based on Eqs. (99) to (104) with
a factor $1/2$ from the time averaging [17] is
$\displaystyle\frac{{\rm d}I_{\rm d}}{{\rm d}{\it\Omega}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\,\bm{x}^{2}\,\bm{\hat{x}}\cdot\bm{S}(x)=\frac{c\epsilon_{0}}{4}\,\bm{x}^{2}\,\overline{{\it\Psi}}_{\rm
d}(x)\bm{\gamma}\cdot\bm{\hat{x}}{\it\Psi}_{\rm d}(x)$ (853) $\displaystyle=$
$\displaystyle\frac{ck^{4}}{32\,\pi^{2}\epsilon_{0}}\,\bm{d}_{\rm
s}^{\dagger}(\bm{\tau}\cdot\bm{\hat{x}})^{2}\,\bm{d}_{\rm
s}=\frac{ck^{4}}{32\,\pi^{2}\epsilon_{0}}\left[\,\bm{d}^{2}-(\bm{\hat{x}}\cdot\bm{d})^{2}\right],$
which is the well-known result.
A more realistic example is the radiation produced by a Dirac transition
current, which is given by
$\displaystyle{\it\Xi}_{\rm D}(x)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}-\mu_{0}c\,\bm{j}_{\rm s}^{if}(x)\\\
{\bm{0}}\end{array}\right)=\left(\begin{array}[]{c}\frac{\textstyle
2e}{\textstyle\epsilon_{0}}\,\phi_{f}^{\dagger}(\bm{x})\,\bm{\alpha}_{\rm
s}\,\phi_{i}(\bm{x})\,{\rm e}^{-{\rm i}\omega_{if}t}\\\
{\bm{0}}\end{array}\right),$ (858)
where $\phi_{i}$ and $\phi_{f}$ are the initial and final hydrogen atom Dirac
wave functions, here $\bm{\alpha}$ is the $4\times 4$ Dirac matrix, and
$\displaystyle\omega_{if}$ $\displaystyle=$
$\displaystyle\frac{E_{i}-E_{f}}{\hbar}$ (859)
is the frequency corresponding to the energy difference of the transition. The
factor of 2 multiplying the matrix element accounts for the difference between
a classical dipole moment and the quantum mechanical dipole moment operator in
Eq. (870). (See the footnote on p. 407 of [17].) We have
$\displaystyle c\int{\rm d}t_{1}\,{\cal D}_{\rm M}^{\rm
T}(x_{2}-x_{1})\,{\it\Xi}_{\rm D}(x_{1})$
$\displaystyle\qquad\qquad=\frac{{\rm
i}k}{4\pi\epsilon_{0}c}\,\left(\begin{array}[]{cc}(\bm{\tau}\cdot\bm{\hat{r}})^{2}&-\bm{\tau}\cdot\bm{\hat{r}}\\\
\bm{\tau}\cdot\bm{\hat{r}}&-(\bm{\tau}\cdot\bm{\hat{r}})^{2}\end{array}\right)\left(\begin{array}[]{c}\bm{j}_{\rm
s}^{if}(\bm{x}_{1})\\\ {\bm{0}}\end{array}\right)\frac{{\rm e}^{{\rm
i}k|\bm{r}|}}{|\bm{r}|}\,{\rm e}^{-{\rm i}\omega_{if}t_{2}}+\dots\ ,\qquad$
(864)
where $k=\omega_{if}/c$. For distances far from the source atom,
$|\bm{x}_{2}|\gg|\bm{x}_{1}|$,
$\bm{\hat{k}}\approx\bm{\hat{r}}\approx\bm{\hat{x}}_{2}$, and in the exponent
$k|\bm{r}|=k|\bm{x}_{2}|-\bm{k}\cdot\bm{x}_{1}+\dots\ $, which yields
$\displaystyle{\it\Psi}_{\rm D}(x_{2})$ $\displaystyle=$
$\displaystyle\int{\rm d}^{4}x_{1}\,{\cal D}_{\rm M}^{\rm
T}(x_{2}-x_{1})\,{\it\Xi}_{\rm D}(x_{1})$ (867) $\displaystyle=$
$\displaystyle\frac{{\rm i}k}{4\pi\epsilon_{0}c}\,\int{\rm
d}\bm{x}_{1}\left(\begin{array}[]{c}(\bm{\tau}\cdot\bm{\hat{k}})^{2}\,\bm{j}_{\rm
s}^{if}(\bm{x}_{1})\\\ \bm{\tau}\cdot\bm{\hat{k}}\,\bm{j}_{\rm
s}^{if}(\bm{x}_{1})\end{array}\right){\rm e}^{-{\rm
i}\bm{k}\cdot\bm{x}_{1}}\,\frac{{\rm e}^{{\rm
i}k|\bm{x}_{2}|}}{|\bm{x}_{2}|}\,{\rm e}^{-{\rm i}\omega_{if}t_{2}}+\dots\ .$
The average radiated power is
$\displaystyle\frac{{\rm d}I_{\rm D}}{{\rm d}{\it\Omega}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\,\bm{x}_{2}^{2}\,\bm{\hat{x}_{2}}\cdot\bm{S}(x_{2})=\frac{c\epsilon_{0}}{4}\,\bm{x}_{2}^{2}\,\overline{{\it\Psi}}_{\rm
D}(x_{2})\bm{\gamma}\cdot\bm{\hat{k}}{\it\Psi}_{\rm D}(x_{2})$
$\displaystyle=$
$\displaystyle\frac{k^{2}}{32\,\pi^{2}\epsilon_{0}c}\,\int{\rm
d}\bm{x}_{1}\,\bm{j}_{\rm s}^{if\dagger}(\bm{x}_{1})\,{\rm e}^{{\rm
i}\bm{k}\cdot\bm{x}_{1}}\,(\bm{\tau}\cdot\bm{\hat{k}})^{2}\,\int{\rm
d}\bm{x}_{1}^{\prime}\,\bm{j}_{\rm s}^{if}(\bm{x}_{1}^{\prime})\,{\rm
e}^{-{\rm i}\bm{k}\cdot\bm{x}_{1}^{\prime}}$ $\displaystyle=$
$\displaystyle\hbar\omega_{if}\,\frac{\alpha
kc}{2\pi}\sum_{\lambda=1}^{2}\int{\rm
d}\bm{x}\,\phi_{i}^{\dagger}(\bm{x})\,\bm{\hat{\epsilon}}_{\lambda}(\bm{\hat{k}})\cdot\bm{\alpha}\,{\rm
e}^{{\rm i}\bm{k}\cdot\bm{x}}\phi_{f}(\bm{x})\int{\rm
d}\bm{x}^{\prime}\,\phi_{f}^{\dagger}(\bm{x}^{\prime})\,\bm{\hat{\epsilon}}_{\lambda}(\bm{\hat{k}})\cdot\bm{\alpha}\,{\rm
e}^{-{\rm i}\bm{k}\cdot\bm{x}^{\prime}}\phi_{i}(\bm{x}^{\prime}),$
where $\alpha=e^{2}/4\pi\epsilon_{0}\hbar c$ is the fine-structure constant.
The radiated power integrated over directions of the vector $\bm{\hat{k}}$ may
be interpreted as $\hbar\omega_{if}A_{if}$, where $A_{if}$ is the radiative
transition rate for $i\rightarrow f$, that is, the probability that the atom
providing the source current makes a transition from state $i$ to state $f$ in
one second. This gives
$\displaystyle A_{if}$ $\displaystyle=$ $\displaystyle\frac{\alpha
kc}{2\pi}\int{\rm
d}{\it\Omega}_{\bm{k}}\sum_{\lambda=1}^{2}\left<i\left|\bm{\hat{\epsilon}}_{\lambda}(\bm{\hat{k}})\cdot\bm{\alpha}\,{\rm
e}^{{\rm
i}\bm{k}\cdot\bm{x}}\right|f\right>\left<f\left|\bm{\hat{\epsilon}}_{\lambda}(\bm{\hat{k}})\cdot\bm{\alpha}\,{\rm
e}^{-{\rm i}\bm{k}\cdot\bm{x}}\right|i\right>,{}$ (869)
which is the same as the relativistic radiative transition rate given by QED
(see F). In the dipole approximation ${\rm e}^{{\rm
i}\bm{k}\cdot\bm{x}}\rightarrow 1$,
$\left<i\left|\bm{\alpha}\right|f\right>={\rm
i}\,k\left<i\left|\bm{x}\right|f\right>$, which follows from the identity
$[H,\bm{x}]=[c\bm{\alpha}\cdot\bm{p},\bm{x}]=-{\rm i}\hbar c\bm{\alpha}$,
where $H$ is the Dirac Hamiltonian, and integration over $\bm{\hat{k}}$ yields
the familiar result
$\displaystyle A_{if}$ $\displaystyle\rightarrow$
$\displaystyle\frac{4\alpha\omega_{if}^{3}}{3c^{2}}\left|\left<f\left|\bm{x}\right|i\right>\right|^{2}.{}$
(870)
## 11 Summary
In Eq. (97), two of the Maxwell equations, Eqs. (2) and (3), are written in
the form of the Dirac equation without a mass, but with the addition of a
source term ${\it\Xi}(x)$:
$\displaystyle\gamma^{\mu}\partial_{\mu}{\it\Psi}(x)$ $\displaystyle=$
$\displaystyle{\it\Xi}(x),$
where the gamma matrices are $6\times 6$ versions of the Dirac gamma matrices
in Eq. (87), and
$\displaystyle{\it\Psi}(x)=\left(\begin{array}[]{c}\bm{E}_{\rm s}(x)\\\ {\rm
i}\,c\bm{B}_{\rm s}(x)\vbox to15.0pt{}\end{array}\right),$
$\displaystyle{\it\Xi}(x)=\left(\begin{array}[]{c}-\mu_{0}c\bm{J}_{\rm
s}(x)\\\ {\bm{0}\vbox to15.0pt{}}\end{array}\right)$
from Eqs. (90) and (96). The source-free version of this equation, with
${\it\Xi}(x)=0$, can be written as a Schrödinger-like equation from Eq. (82)
or (426)
$\displaystyle{\rm i}\hbar\,\frac{\partial}{\partial t}{\it\Psi}(x)={\cal
H}{\it\Psi}(x),{}$ (872)
where the Hamiltonian, Eq. (344),
$\displaystyle{\cal H}=-{\rm i}\,\hbar c\,\bm{\alpha}\cdot\bm{\nabla}$
is the analog of the Dirac Hamiltonian for the electron. The factors of
$\hbar$ are not essential here, but they are introduced to provide the
conventional units of frequency and energy. As with the Dirac wave functions,
where all four components are necessary to describe an electron bound in an
atom relativistically, all six of the components of the photon wave function
apparently are necessary to properly account for the space-time properties of
electromagnetic fields. There are three polarization degrees of freedom, two
for radiation and one for electrostatic interactions, and relativistic
covariance requires twice that many components. Alternatively stated, six
complex functions are necessary to describe the six components of the electric
and magnetic fields, and they are coupled by the Maxwell equation and Lorentz
transformations.
According to Eq. (872), as in Eq. (436), the time dependence of the solution
is given by
$\displaystyle{\it\Psi}(x)={\rm e}^{-{\rm i}{\cal
H}t/\hbar}{\it\Psi}(\bm{x}).$ (873)
The time-independent solutions may be expanded in eigenfunctions of the
Hamiltonian with eigenvalues $E_{n}$ given by
$\displaystyle{\cal H}{\it\Psi}_{n}(\bm{x})$ $\displaystyle=$ $\displaystyle
E_{n}{\it\Psi}_{n}(\bm{x}),{}$ (874)
where $n$ is a set of parameters that characterize the state represented by
the wave function. For each eigenfunction, one has
$\displaystyle{\it\Psi}_{n}(x)={\rm e}^{-{\rm
i}E_{n}t/\hbar}{\it\Psi}_{n}(\bm{x}).$ (875)
The eigenfunctions are orthonormal
$\displaystyle\int{\rm
d}\bm{x}\,{\it\Psi}_{n_{2}}^{\dagger}(\bm{x}){\it\Psi}_{n_{1}}(\bm{x})=\delta_{n_{2}n_{1}},{}$
(876)
and they are complete
$\displaystyle\sum_{n}{\it\Psi}_{n}(\bm{x}_{2}){\it\Psi}_{n}^{\dagger}(\bm{x}_{1})=\delta(\bm{x}_{2}-\bm{x}_{1}).{}$
(877)
The state index $n$ includes continuous variables, so Eq. (876) has delta
functions in those variables on the right-hand side, and the summation symbol
in Eq. (877) includes integration over those variables.
States considered in detail in this paper are propagating plane waves in Secs.
7.1 and 7.2, standing plane waves in Sec. 7.7, and angular-momentum
eigenstates in Secs. 8.3 and 8.4. The propagating plane-wave states are
eigenfunctions of the momentum operator in Eq. (345)
$\displaystyle\bm{{\cal P}}=-{\rm i}\,\hbar\,{\cal I}\,\bm{\nabla}.$
This operator commutes with the Hamiltonian, $\left[{\cal H},\bm{{\cal
P}}\right]=0$, and eigenstates of both energy and momentum are given in Eqs.
(367), (403), and (406). These plane-wave states are further characterized by
polarization vectors given in Eq. (346) or (396). Linear combinations of the
traveling plane waves, combined to give standing-wave parity eigenfunctions,
are in Eqs. (574), (577), and (591) to (600). The angular-momentum operator,
Eq. (645), is
$\displaystyle\bm{{\cal J}}$ $\displaystyle=$
$\displaystyle\bm{x}\times\bm{{\cal P}}+\hbar\,\bm{{\cal S}},$
where the spin matrix, Eq. (254), is
$\displaystyle\bm{{\cal S}}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}\bm{\tau}&{\bm{0}}\\\
{\bm{0}}&\bm{\tau}\end{array}\right),$
and $\bm{\tau}$ is given by Eqs. (33) to (41). One has $\left[{\cal
H},\bm{{\cal J}}\right]=0$ in Eq. (646), and simultaneous eigenfunctions of
energy, angular momentum squared $\bm{{\cal J}}^{2}$, third component of
angular momentum ${\cal J}^{3}$, and parity are given in Eq (727), (730),
(766), and (769). All three sets of eigenfunctions listed above are shown to
be orthogonal and complete, as in Eqs. (876) and (877).
The eigenfunctions considered here are not normalizable wave functions.
However, they provide basis functions for the expansion of a normalizable wave
packet, as discussed in Sec. 7.8. For the sum
$\displaystyle{\it\Psi}_{f}(\bm{x})=\sum_{n}f_{n}{\it\Psi}_{n}(\bm{x}),$ (879)
from the orthonormality of the eigenfunctions one has
$\displaystyle f_{n}=\int{\rm
d}\bm{x}{\it\Psi}_{n}^{\dagger}(\bm{x}){\it\Psi}_{f}(\bm{x})$ (880)
and
$\displaystyle\int{\rm
d}\bm{x}\,{\it\Psi}_{f}^{\dagger}(\bm{x}){\it\Psi}_{f}(\bm{x})=\sum_{n}|f_{n}|^{2}=1$
(881)
for suitably chosen $f_{n}$. For the example of a Gaussian wave packet in Eq.
(612), the expectation value of the Hamiltonian, in Eq. (617), is
$\displaystyle\left<{\it\Psi}_{f}\left|\,{\cal
H}\,\right|{\it\Psi}_{f}\right>$ $\displaystyle=$
$\displaystyle\hbar\omega_{0},$
where $\omega_{0}=c|\bm{k}_{0}|$ is the frequency corresponding to the wave
vector $\bm{k}_{0}$ of the wave packet. It is clear that Eq. (617) applies in
more generality than just to the wave packet in Eq. (612). If the Gaussian
shape function were replaced by any normalized real function, the expectation
value of the Hamiltonian would still be exactly $\hbar\omega_{0}$. For the
wave packet in Eq. (612), the expectation expectation value of the momentum
operator, Eq. (618), is
$\displaystyle\left<{\it\Psi}_{f}\left|\,\bm{{\cal
P}}\,\right|{\it\Psi}_{f}\right>$ $\displaystyle=$
$\displaystyle\hbar\bm{k}_{0},$
and the expectation value of the projection of the angular momentum in the
direction of the wave vector is
$\displaystyle\left<{\it\Psi}_{f}\left|\,\bm{{\cal
J}}\cdot\bm{\hat{k}}_{0}\,\right|{\it\Psi}_{f}\right>$ $\displaystyle=$
$\displaystyle\hbar\,\bm{\hat{\epsilon}}_{1}^{\dagger}(\bm{\hat{k}}_{0})\,\bm{\tau}\cdot\bm{\hat{k}}_{0}\,\bm{\hat{\epsilon}}_{1}(\bm{\hat{k}}_{0}),$
(882)
where the result depends on the polarization state represented by
$\bm{\hat{\epsilon}}_{1}(\bm{\hat{k}}_{0})$. For circular polarization, Eq.
(364), the expectation value is $\pm\hbar$, while for linear polarization, Eq.
(357), it is 0. The real part of the energy density is $\hbar\omega_{0}$ times
the probability density
$\displaystyle{\rm Re}\,{\it\Psi}_{f}^{\dagger}(x){\cal H}{\it\Psi}_{f}(x)$
$\displaystyle=$
$\displaystyle\hbar\omega_{0}{\it\Psi}_{f}^{\dagger}(x){\it\Psi}_{f}(x)$ (883)
for the wave packet.
For any wave packet, represented by ${\it\Psi}_{f}$, the photon probability
density four-vector defined in Eq. (635) is
$\displaystyle q_{f}^{\mu}(x)$ $\displaystyle=$
$\displaystyle\overline{{\it\Psi}}_{f}(x){\gamma^{\mu}}{\it\Psi}_{f}(x),$
(884)
and the differential conservation of probability is given by Eq. (637)
$\displaystyle\frac{\partial}{\partial
t}\,q_{f}^{0}(x)+c\,\bm{\nabla}\cdot\bm{q}_{f}(x)$ $\displaystyle=$
$\displaystyle 0.$ (885)
This is valid for any solution of the homogeneous Maxwell equation. By
integrating over a closed volume and converting the divergence to an integral
of the normal component of the vector over the surface, one obtains a
statement of conservation of probability for the volume. If the surface of the
volume is taken to infinity in all directions, where the wave function
vanishes, this expression shows the time independence of the normalization of
the wave function
$\displaystyle\frac{\partial}{\partial t}\int{\rm
d}\bm{x}\,{\it\Psi}_{f}^{\dagger}(x){\it\Psi}_{f}(x)$ $\displaystyle=$
$\displaystyle 0.$ (886)
Of course, this does not give meaningful results for a plane wave, because in
this case, the probability density is constant over space and is not
normalizable.
For electromagnetic fields and photons, Lorentz invariance is a necessary
consideration. In Secs. 6.4 and 6.5 it is shown that
$\displaystyle\gamma^{\mu}\partial_{\mu}{\it\Psi}^{\prime}(x)$
$\displaystyle=$ $\displaystyle{\it\Xi}^{\prime}(x),$
where the primes indicate that the field and source have been transformed by
either a rotation or a velocity boost. For rotations represented by the vector
$\bm{u}=\theta\bm{\hat{u}}$, the transformed field, in Eq. (238), is
$\displaystyle{\it\Psi}^{\prime}(x)$ $\displaystyle=$ $\displaystyle{\cal
R}(\bm{u}){\it\Psi}\\!\big{(}R^{-1}(\bm{u})\,x\big{)},$
where $R(\bm{u})$ is the coordinate rotation operator in Eq. (164) and
$\displaystyle{\cal R}(\bm{u})$ $\displaystyle=$ $\displaystyle{\rm e}^{-{\rm
i}\bm{{\cal S}}\cdot\bm{u}}$
in Eq. (251). The source term transforms in the same way. For velocity
transformations, corresponding to the velocity
$\bm{v}=c\tanh{\zeta}\,\bm{\hat{v}}$, Eq. (265) is
$\displaystyle{\it\Psi}^{\prime}(x)$ $\displaystyle=$ $\displaystyle{\cal
V}(\bm{v}){\it\Psi}\\!\big{(}V^{-1}(\bm{v})\,x\big{)},$
where $V(\bm{v})$ is the coordinate velocity transformation operator in Eq.
(196) and
$\displaystyle{\cal V}(\bm{v})$ $\displaystyle=$ $\displaystyle{\rm
e}^{\zeta\bm{{\cal K}}\cdot\bm{\hat{v}}}$
in Eq. (269), where
$\displaystyle\bm{{\cal K}}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}{\bm{0}}&\bm{\tau}\\\
\bm{\tau}&{\bm{0}}\end{array}\right)$
in Eq. (272). The transformation of the source term under a velocity boost is
noteworthy. In the presence of a non-zero source, the Maxwell equation is
invariant, but the left- and right-hand sides do not transform separately. As
shown in Sec. 6.5, the derivatives acting on the fields produce terms that
combine with the original source term in such a way as to produce the velocity
transformed source term, even though it is the three-vector current. In the
absence of sources, the transformation reduces to a more conventional form.
In the case of photon wave functions, sources are taken to be absent and the
wave functions are solutions of the homogeneous Maxwell equation. The Lorentz
transformations of the plane-wave functions are explicitly shown in Secs. 7.5
and 7.6. As with the Dirac equation for an electron, the eigenvalues in Eq.
(874) may be either positive or negative, and here they also may be zero, for
both plane-wave and spherical-wave eigenfunctions. The negative eigenvalues,
which are associated with relativistic invariance, are necessary in order to
have a complete set of solutions satisfying Eq. (877). It is relevant to note
that for the six-component wave packet in Eq. (612), there is an interaction
between the upper-three components and lower-three components, evident in Eq.
(614), that suppresses the role of the negative energy states.
In Sec. 5, Eqs. (133) and (136), orthogonal transverse and longitudinal
projection operators are defined:
$\displaystyle{\it\Pi}^{\rm
T}(\bm{\nabla})=\left(\begin{array}[]{ccc}\bm{{\it\Pi}}_{\rm s}^{\rm
T}(\bm{\nabla})&&{\bm{0}}\\\ {\bm{0}}&&\bm{{\it\Pi}}_{\rm s}^{\rm
T}(\bm{\nabla})\vbox to15.0pt{}\end{array}\right),$
$\displaystyle{\it\Pi}^{\rm
L}(\bm{\nabla})=\left(\begin{array}[]{ccc}\bm{{\it\Pi}}_{\rm s}^{\rm
L}(\bm{\nabla})&&{\bm{0}}\\\ {\bm{0}}&&\bm{{\it\Pi}}_{\rm s}^{\rm
L}(\bm{\nabla})\vbox to15.0pt{}\end{array}\right),$
where from Eqs. (115) and (116)
$\displaystyle\bm{{\it\Pi}}_{\rm s}^{\rm
T}(\bm{\nabla})=\frac{(\bm{\tau}\cdot\bm{\nabla})^{2}}{\bm{\nabla}^{2}}\,,$
$\displaystyle\bm{{\it\Pi}}_{\rm s}^{\rm
L}(\bm{\nabla})=\frac{\bm{\nabla}_{\rm s}\bm{\nabla}_{\rm
s}^{\dagger}}{\bm{\nabla}^{2}}\,.$
These operators commute with the Hamiltonian, the momentum operator, and the
angular-momentum operator
$\displaystyle\left[{\cal H},{\it\Pi}\right]=\left[\bm{{\cal
P}},{\it\Pi}\right]=\left[\bm{{\cal J}},{\it\Pi}\right]=0,$ (889)
where ${\it\Pi}$ represents either projection operator, ${\it\Pi}^{\rm
T}(\bm{\nabla})$ or ${\it\Pi}^{\rm L}(\bm{\nabla})$, so all the eigenstates
considered in this paper are classified as being either transverse or
longitudinal, with
$\displaystyle{\it\Pi}^{\rm T}(\bm{\nabla}){\it\Psi}^{\rm T}_{n}(x)$
$\displaystyle=$ $\displaystyle{\it\Psi}^{\rm T}_{n}(x),$ (890)
$\displaystyle{\it\Pi}^{\rm L}(\bm{\nabla}){\it\Psi}^{\rm L}_{n}(x)$
$\displaystyle=$ $\displaystyle{\it\Psi}^{\rm L}_{n}(x),$ (891)
respectively. The transverse states describe radiation and have non-zero
eigenvalues in Eq. (874), while the longitudinal states correspond to
electrostatic interactions, with an eigenvalue of zero. An exception is that
the longitudinal states may have a non-zero eigenvalue if a hypothetical mass
term is considered, as discussed in Sec. 7.4. The projection operators commute
with rotations, but not, in general, with velocity boosts. However, as shown
in Sec. 7.6.1, the velocity transformed transverse plane-wave states are also
transverse. This corresponds to the fact that radiation may be treated
relativistically independent of electrostatic interactions. On the other hand,
as shown in Sec. 7.6.2, the velocity transformed longitudinal states have both
longitudinal and transverse components, corresponding to the fact that moving
charges may excite radiative transitions. Both transverse and longitudinal
states are necessary in order to have a complete set, as in Eq. (877).
A solution of the inhomogeneous Maxwell equation may be obtained with the
Maxwell Green function, as discussed in Sec. 9. The Green function satisfies
the equation
$\displaystyle\gamma_{\mu}\partial_{2}^{\mu}\,{\cal D}_{\rm M}(x_{2}-x_{1})$
$\displaystyle=$ $\displaystyle{\cal I}\,\delta(x_{2}-x_{1})$
in Eq. (798), and a solution of the Maxwell equation is given by
$\displaystyle{\it\Psi}(x_{2})$ $\displaystyle=$ $\displaystyle\int{\rm
d}^{4}x_{1}\,{\cal D}_{\rm M}(x_{2}-x_{1})\,{\it\Xi}(x_{1})$
in Eq. (801). It is shown, by summing over the complete set of plane-wave
solutions, that the Green function is
$\displaystyle{\cal D}_{\rm M}(x_{2}-x_{1})$ $\displaystyle=$
$\displaystyle\frac{{\rm i}}{(2\pi)^{4}}\,\int_{\rm C_{F}}{\rm d}^{4}k\
\frac{{\rm e}^{-{\rm
i}k\cdot(x_{2}-x_{1})}}{\gamma^{\mu}k_{\mu}-m_{\epsilon}c/\hbar}$
in Eq. (831), which is the same form as the Dirac Green function, except that
here it is a $6\times 6$ matrix instead of a $4\times 4$ matrix. In this
equation, CF is the Feynman contour and the infinitesimal mass is included to
resolve an ambiguity in the longitudinal contribution.
In Sec. 10, applications of the Maxwell Green function are made, including a
calculation of radiation from a Dirac-electron current source. In this example
the six-component Maxwell formalism couples radiation to the Dirac current
relativistically with a result that is the same as the result of a calculation
that starts from Feynman-gauge QED.
## 12 Conclusion
We conclude that the criteria for properties of a single-photon wave function
proposed in the introduction are met by the formalism described in the
subsequent sections. In particular, the example of a photon wave packet
provides a normalizable solution of the wave equation whose properties can be
verified by explicit calculations. It yields the unanticipated result that for
virtually any probability distribution, under rather mild assumptions about
the form of the wave packet, the expectation value of the Hamiltonian is
exactly $\left<\,{\cal H}\,\right>=\hbar\omega_{0}$, where $\omega_{0}$ is the
frequency associated with the wave vector of the packet.
## Appendix A Velocity transformation of electromagnetic fields
The velocity transformation of electromagnetic fields is derived here without
invoking potentials for completeness. With the aid of the identity
$\bm{\nabla}_{\rm c}^{\top}\bm{\tilde{\tau}}\cdot c\bm{B}=(\bm{\nabla}\times
c\bm{B})_{\rm c}^{\top}$, Eqs. (1) and (2) may be written as
$\displaystyle\bm{\nabla}_{\rm c}^{\top}\bm{E}_{\rm c}$ $\displaystyle=$
$\displaystyle\mu_{0}c^{2}\rho,$ (892) $\displaystyle\frac{\partial\bm{E}_{\rm
c}^{\top}}{\partial ct}-\bm{\nabla}_{\rm c}^{\top}\bm{\tilde{\tau}}\cdot
c\bm{B}$ $\displaystyle=$ $\displaystyle-\mu_{0}c\bm{J}_{\rm c}^{\top}$ (893)
or
$\displaystyle\partial_{\rm c}^{\top}gF$ $\displaystyle=$
$\displaystyle\mu_{0}J^{\top},{}$ (894)
where
$\displaystyle F$ $\displaystyle=$
$\displaystyle\frac{1}{c}\left(\begin{array}[]{ccc}0&&-\bm{E}^{\top}_{\rm
c}\\\ \bm{E}_{\rm c}&&\bm{\tilde{\tau}}\cdot c\bm{B}\vbox
to14.0pt{}\end{array}\right)$ (897)
is the field tensor [see Eq. (71)] and
$\displaystyle J$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}c\rho\\\ \bm{J}_{\rm
c}\end{array}\right).$ (900)
Since $V(\bm{v})\,g\,V(\bm{v})=g$, Eq. (894) is equivalent to
$\displaystyle\partial_{\rm c}^{\top}V(\bm{v})\,g\,V(\bm{v})\,F(x)V(\bm{v})$
$\displaystyle=$ $\displaystyle\mu_{0}J^{\top}(x)V(\bm{v})\qquad$ (901)
or
$\displaystyle\partial_{\rm
c}^{\top}g\,V(\bm{v})\,F\left(V^{-1}(\bm{v})x\right)V(\bm{v})$
$\displaystyle=$
$\displaystyle\mu_{0}J^{\top}\\!\\!\left(V^{-1}(\bm{v})x\right)V(\bm{v}).\qquad$
(902)
Assuming the current transforms as a four-vector,
$\displaystyle J^{\prime}(x)$ $\displaystyle=$ $\displaystyle
V(\bm{v})\,J\\!\left(V^{-1}(\bm{v})x\right),$ (903)
Eq. (894) will be invariant if the field tensor transforms according to
$\displaystyle F^{\prime}(x)$ $\displaystyle=$ $\displaystyle
V(\bm{v})\,F\\!\left(V^{-1}(\bm{v})x\right)V(\bm{v}).{}$ (904)
Direct calculation yields555The identity
$\epsilon_{ijk}=\epsilon_{ljk}\hat{v}^{i}\hat{v}^{l}+\epsilon_{ilk}\hat{v}^{j}\hat{v}^{l}+\epsilon_{ijl}\hat{v}^{k}\hat{v}^{l}$
may be useful here.
$\displaystyle F^{\prime}(x^{\prime})$ $\displaystyle=$
$\displaystyle\frac{1}{c}\left(\begin{array}[]{ccc}0&&-\bm{E}^{\prime\top}_{\rm
c}(x)\\\ \bm{E}^{\prime}_{\rm c}(x)&&\bm{\tilde{\tau}}\cdot
c\bm{B}^{\prime}(x)\vbox to14.0pt{}\end{array}\right),$ (907)
where $x^{\prime}=V(\bm{v})\,x$ and
$\displaystyle\bm{E}_{\rm c}^{\,\prime}$ $\displaystyle=$
$\displaystyle\bm{E}_{\rm c}\cosh{\zeta}-\bm{\hat{v}}_{\rm
c}\,\bm{\hat{v}}\cdot\bm{E}\left(\cosh{\zeta}-1\right)-\bm{\tilde{\tau}}\cdot\bm{\hat{v}}\,c\bm{B}_{\rm
c}\sinh{\zeta},{}$ (908) $\displaystyle c\bm{B}_{\rm c}^{\,\prime}$
$\displaystyle=$ $\displaystyle c\bm{B}_{\rm c}\cosh{\zeta}-\bm{\hat{v}}_{\rm
c}\,\bm{\hat{v}}\cdot
c\bm{B}\left(\cosh{\zeta}-1\right)+\bm{\tilde{\tau}}\cdot\bm{\hat{v}}\,\bm{E}_{\rm
c}\sinh{\zeta}.{}$ (909)
These relations are equivalent (up to the velocity sign convention) to the
electric and magnetic field transformations in [17], and in the spherical
basis they are
$\displaystyle\bm{E}_{\rm s}^{\prime}$ $\displaystyle=$
$\displaystyle\bm{E}_{\rm s}+(\bm{\tau}\cdot\bm{\hat{v}})^{2}\bm{E}_{\rm
s}(\cosh{\zeta}-1)+{\rm i}\,\bm{\tau}\cdot\bm{\hat{v}}\,c\bm{B}_{\rm
s}\sinh{\zeta},$ (910) $\displaystyle c\bm{B}_{\rm s}^{\prime}$
$\displaystyle=$ $\displaystyle c\bm{B}_{\rm
s}+(\bm{\tau}\cdot\bm{\hat{v}})^{2}c\bm{B}_{\rm s}(\cosh{\zeta}-1)-{\rm
i}\,\bm{\tau}\cdot\bm{\hat{v}}\,\bm{E}_{\rm s}\sinh{\zeta}.$ (911)
The Cartesian transformation in Eqs. (908) and (909) can be written as
$\displaystyle{\it\Psi}_{\rm c}^{\prime}(x^{\prime})$ $\displaystyle=$
$\displaystyle{\rm e}^{-\zeta\bm{\tilde{}}{\bm{{\cal
K}}}\cdot\bm{\hat{v}}}{\it\Psi}_{\rm c}(x),$ (912)
where
$\displaystyle\bm{\tilde{}}{\bm{{\cal K}}}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c@{\quad}c}{\bm{0}}&\bm{\tilde{\tau}}\\\
-\bm{\tilde{\tau}}&{\bm{0}}\end{array}\right){}$ (915)
and
$\displaystyle{\it\Psi}_{\rm c}(x)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}\bm{E}_{\rm c}(x)\\\ c\bm{B}_{\rm
c}(x)\end{array}\right).$ (918)
Similarly, the Cartesian form of the rotation transformation, corresponding to
the operator in Eq. (251), is
$\displaystyle{\it\Psi}_{\rm c}^{\prime}(x^{\prime})$ $\displaystyle=$
$\displaystyle{\rm e}^{\bm{\tilde{}}{\bm{{\cal S}}}\cdot\bm{u}}{\it\Psi}_{\rm
c}(x),$ (919)
where
$\displaystyle\bm{\tilde{}}{\bm{{\cal S}}}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c@{\quad}c}\bm{\tilde{\tau}}&{\bm{0}}\\\
{\bm{0}}&\bm{\tilde{\tau}}\end{array}\right).{}$ (922)
The components of the matrices in Eqs. (915) and (922) have the commutation
relations
$\displaystyle\left[\tilde{\cal S}^{i},\tilde{\cal S}^{j}\right]$
$\displaystyle=$ $\displaystyle\epsilon_{ijk}\,\tilde{\cal S}^{k},$ (923)
$\displaystyle\left[\tilde{\cal S}^{i},\tilde{\cal K}^{j}\right]$
$\displaystyle=$ $\displaystyle\epsilon_{ijk}\,\tilde{\cal K}^{k},$ (924)
$\displaystyle\left[\tilde{\cal K}^{i},\tilde{\cal K}^{j}\right]$
$\displaystyle=$ $\displaystyle-\epsilon_{ijk}\,\tilde{\cal S}^{k},$ (925)
characteristic of the Lie algebra of Lorentz transformations.
## Appendix B Inverse Laplacian
For cases where the integral definition of the inverse Laplacian converges
poorly, we use a generalized definition that includes a damping factor to
resolve ambiguity in the intermediate steps of the calculation. From the
equation
$\displaystyle\left(\bm{\nabla}^{2}-\epsilon^{2}\right)\frac{{\rm
e}^{-\epsilon\,|\bm{x}-\bm{x}^{\prime}|}}{|\bm{x}-\bm{x}^{\prime}|}$
$\displaystyle=$ $\displaystyle-4\pi\delta(\bm{x}-\bm{x}^{\prime}),$ (926)
one has
$\displaystyle\frac{1}{\bm{\nabla}^{2}-\epsilon^{2}}\,\delta(\bm{x}-\bm{x}^{\prime})$
$\displaystyle=$ $\displaystyle-\frac{1}{4\pi}\,\frac{{\rm
e}^{-\epsilon\,|\bm{x}-\bm{x}^{\prime}|}}{|\bm{x}-\bm{x}^{\prime}|}.$ (927)
Multiplication by $f(\bm{x}^{\prime})$ and integration over $\bm{x}^{\prime}$
yields
$\displaystyle\frac{1}{\bm{\nabla}^{2}-\epsilon^{2}}\,f(\bm{x})$
$\displaystyle=$ $\displaystyle-{1\over 4\pi}\int{{\rm
d}}\bm{x}^{\prime}\,\frac{{\rm
e}^{-\epsilon\,|\bm{x}-\bm{x}^{\prime}|}}{|\bm{x}-\bm{x}^{\prime}|}\,f(\bm{x}^{\prime}).\qquad$
(928)
We thus have, for example,
$\displaystyle\frac{1}{\bm{\nabla}^{2}-\epsilon^{2}}\,{\rm e}^{{\rm
i}\bm{k}\cdot\bm{x}}$ $\displaystyle=$
$\displaystyle-\frac{1}{\bm{k}^{2}+\epsilon^{2}}\,{\rm e}^{{\rm
i}\bm{k}\cdot\bm{x}}\rightarrow-\frac{1}{\bm{k}^{2}}\,{\rm e}^{{\rm
i}\bm{k}\cdot\bm{x}},\qquad$ (929)
by direct calculation of the integral.
## Appendix C Coulomb matrix element
The calculation of the Coulomb field matrix element in Eq. (422) requires
evaluation of the integral
$\displaystyle\int{\rm
d}\bm{x}\,\frac{\bm{\hat{k}}\cdot\bm{x}}{|\bm{x}|^{3}}\,{\rm e}^{-{\rm
i}\bm{k}\cdot\bm{x}}$ $\displaystyle=$ $\displaystyle\lim_{\epsilon\rightarrow
0}\int{\rm d}\bm{x}\,{\rm
e}^{-\epsilon|\bm{x}|}\,\frac{\bm{\hat{k}}\cdot\bm{x}}{|\bm{x}|^{3}}\,{\rm
e}^{-{\rm i}\bm{k}\cdot\bm{x}}\qquad$ (930) $\displaystyle\rightarrow$
$\displaystyle-\,{\rm i}\,\frac{4\pi}{|\bm{k}|},$
which is defined with the convergence factor ${\rm e}^{-\epsilon|\bm{x}|}$.
The inverse transformation requires the integral
$\displaystyle\int{\rm d}\bm{k}\,\frac{\bm{\hat{k}}}{|\bm{k}|}\,{\rm e}^{{\rm
i}\bm{k}\cdot\bm{x}}$ $\displaystyle=$ $\displaystyle\lim_{\epsilon\rightarrow
0}\int{\rm d}\bm{k}\,{\rm
e}^{-\epsilon|\bm{k}|}\,\frac{\bm{\hat{k}}}{|\bm{k}|}\,{\rm e}^{{\rm
i}\bm{k}\cdot\bm{x}}\qquad$ (931) $\displaystyle\rightarrow$
$\displaystyle\,{\rm i}\,\frac{2\pi^{2}\bm{x}}{|\bm{x}|^{3}},$
which confirms the result that
$\displaystyle-\frac{{\rm i}\,q}{\sqrt{(2\pi)^{3}}\,\epsilon_{0}}\int{\rm
d}\bm{k}\,\frac{1}{|\bm{k}|}\,\psi_{\bm{k},0}^{(+)}(\bm{x})$ $\displaystyle=$
$\displaystyle\frac{q}{4\pi\epsilon_{0}|\bm{x}|^{3}}\left(\begin{array}[]{c}\bm{x}_{\rm
s}\\\ {\bm{0}}\end{array}\right).$ (934)
## Appendix D Separation of the transverse and longitudinal gradient
operators
The transverse gradient operator $\bm{\tau}\cdot\bm{\nabla}$ is separated into
radial and angular parts by writing
$\displaystyle\bm{\tau}\cdot\bm{\nabla}$ $\displaystyle=$
$\displaystyle\bm{\tau}\cdot\bm{\hat{x}}\,\bm{\tau}\cdot\bm{\hat{x}}\,\bm{\tau}\cdot\bm{\nabla}+\bm{\hat{x}}_{\rm
s}\,\bm{\hat{x}}_{\rm s}^{\dagger}\,\bm{\tau}\cdot\bm{\nabla}$ (936)
$\displaystyle=$ $\displaystyle\frac{\partial}{\partial
r}\,\bm{\tau}\cdot\bm{\hat{x}}-\bm{\tau}\cdot\bm{\hat{x}}\,\overline{\bm{\nabla}}_{\rm
s}\,\bm{\hat{x}}_{\rm s}^{\dagger}+\frac{1}{\hbar r}\,\bm{\hat{x}}_{\rm
s}\bm{L}_{\rm s}^{\dagger}\qquad$
where the line over the gradient operator indicates that it does not act on
the unit vector directly to the right. That term is
$\displaystyle\bm{\tau}\cdot\bm{\hat{x}}\,\overline{\bm{\nabla}}_{\rm
s}\,\bm{\hat{x}}_{\rm s}^{\dagger}$ $\displaystyle=$
$\displaystyle\bm{\tau}\cdot\bm{\hat{x}}\left(\bm{\nabla}_{\rm
s}\bm{\hat{x}}_{\rm s}^{\dagger}+\frac{1}{r}\,\bm{\hat{x}}_{\rm
s}\bm{\hat{x}}_{\rm s}^{\dagger}-\frac{1}{r}\,\bm{I}\right)$ (937)
$\displaystyle=$ $\displaystyle-\frac{1}{\hbar r}\,\bm{L}_{\rm
s}\bm{\hat{x}}_{\rm s}^{\dagger}-\frac{1}{r}\,\bm{\tau}\cdot\bm{\hat{x}}.$
Thus
$\displaystyle\bm{\tau}\cdot\bm{\nabla}$ $\displaystyle=$
$\displaystyle\frac{1}{r}\,\frac{\partial}{\partial
r}\,r\,\bm{\tau}\cdot\,\bm{\hat{x}}\,+\frac{1}{\hbar r}\left(\bm{L}_{\rm
s}\,\bm{\hat{x}}_{\rm s}^{\dagger}+\bm{\hat{x}}_{\rm s}\,\bm{L}_{\rm
s}^{\dagger}\right).\qquad$ (938)
Acting on $\bm{L}_{\rm s}$, the transverse gradient operator yields
$\displaystyle\bm{\tau}\cdot\bm{\nabla}\,\bm{L}_{\rm s}$ $\displaystyle=$
$\displaystyle\frac{1}{r}\,\frac{\partial}{\partial
r}\,r\,\bm{\tau}\cdot\,\bm{\hat{x}}\,\bm{L}_{\rm s}-\frac{1}{\hbar
r}\bm{\hat{x}}_{\rm s}\,\bm{L}^{2},\qquad$ (939)
where $\bm{L}_{\rm s}^{\dagger}\bm{L}_{\rm s}=-\bm{L}^{2}$.
For the longitudinal gradient operator $\bm{\nabla}_{\rm s}$, the identity
$\displaystyle\bm{\nabla}$ $\displaystyle=$
$\displaystyle\bm{\hat{x}}\,\frac{\partial}{\partial r}-\frac{{\rm i}}{\hbar
r}\,\bm{\hat{x}}\times\bm{L}$ (940)
provides
$\displaystyle\bm{\nabla}_{\rm s}$ $\displaystyle=$
$\displaystyle\bm{\hat{x}}_{\rm s}\,\frac{\partial}{\partial r}-\frac{1}{\hbar
r}\,\bm{\tau}\cdot\bm{\hat{x}}\,\bm{L}_{\rm s}.{}$ (941)
## Appendix E Exact transverse Green function
The exact transverse Maxwell Green function follows from Eq. (821) together
with the relations
$\displaystyle\bm{\tau}\cdot\bm{\nabla}\,\frac{{\rm
e}^{-w|\bm{r}|}}{|\bm{r}|}$ $\displaystyle=$
$\displaystyle-w\,\bm{\tau}\cdot\bm{\hat{r}}\,\frac{{\rm
e}^{-w|\bm{r}|}}{|\bm{r}|}\left(1+\frac{1}{w|\bm{r}|}\right)$ (942)
and
$\displaystyle\frac{1}{\nabla^{2}}\,\frac{{\rm e}^{-w|\bm{r}|}}{|\bm{r}|}$
$\displaystyle=$ $\displaystyle-\frac{1}{4\pi}\int{\rm
d}\bm{x}\,\frac{1}{|\bm{r}-\bm{x}|}\frac{{\rm
e}^{-w|\bm{x}|}}{|\bm{x}|}=-\frac{1}{w^{2}\,|\bm{r}|}\left(1-{\rm
e}^{-w|\bm{r}|}\right),$ (943)
which yield
$\displaystyle\frac{\nabla^{i}\nabla^{j}}{\nabla^{2}}\,\frac{{\rm
e}^{-w|\bm{r}|}}{|\bm{r}|}$ $\displaystyle=$
$\displaystyle\frac{r^{i}r^{j}}{|\bm{r}|^{2}}\frac{{\rm
e}^{-w|\bm{r}|}}{|\bm{r}|}+\left(3\,\frac{r^{i}r^{j}}{|\bm{r}|^{2}}-\delta_{ij}\right)\left(\frac{{\rm
e}^{-w|\bm{r}|}}{w|\bm{r}|^{2}}+\frac{{\rm
e}^{-w|\bm{r}|}}{w^{2}|\bm{r}|^{3}}-\frac{1}{w^{2}|\bm{r}|^{3}}\right)$ (944)
and
$\displaystyle\bm{{\it\Pi}}_{\rm s}^{\rm T}(\bm{\nabla})\,\frac{{\rm
e}^{-w|\bm{r}|}}{|\bm{r}|}$ $\displaystyle=$
$\displaystyle(\bm{\tau}\cdot\bm{\hat{r}})^{2}\,\frac{{\rm
e}^{-w|\bm{r}|}}{|\bm{r}|}+\left[(\bm{\tau}\cdot\bm{\hat{r}})^{2}-2\,\bm{\hat{r}}_{\rm
s}\,\bm{\hat{r}}_{\rm s}^{\dagger}\right]\left(\frac{{\rm
e}^{-w|\bm{r}|}}{w|\bm{r}|^{2}}+\frac{{\rm
e}^{-w|\bm{r}|}}{w^{2}|\bm{r}|^{3}}-\frac{1}{w^{2}|\bm{r}|^{3}}\right).\qquad$
(945)
## Appendix F Radiative decay in quantum electrodynamics
In QED, the radiative decay rate of an excited state may be obtained from the
imaginary part of the radiative correction to the energy level of that state
$\displaystyle\hbar\sum_{f}A_{if}$ $\displaystyle=$ $\displaystyle-2\,{\rm
Im}(\Delta E_{i}),$ (946)
where the sum is over all states with a lower unperturbed energy. This gives a
correction to the level that, roughly speaking, results in an exponentially
damped time dependence for the population of the state:
$\displaystyle\big{|}{\rm e}^{-{\rm i}\,\Delta\\!E\,t/\hbar}\big{|}^{2}={\rm
e}^{-\sum_{f}A_{if}t}.$ (947)
For one-photon decays, the rate is included in the second-order self-energy
correction to the level. An expression derived from Feynman-gauge QED that
includes some of the real part and all of the imaginary part of this level
shift in hydrogen-like atoms is given by [31]
$\displaystyle\Delta E_{i}$ $\displaystyle=$
$\displaystyle-\frac{\alpha\hbar^{2}c^{2}}{4\pi^{2}}\int_{\hbar ck<E_{i}}{\rm
d}\bm{k}\,\frac{1}{k}\left(\delta_{jl}-\frac{k^{j}k^{l}}{\bm{k}^{2}}\right)\left<\alpha^{j}{\rm
e}^{{\rm i}\bm{k}\cdot\bm{x}}\frac{1}{H-E_{i}+\hbar ck-{\rm
i}\delta}\,\alpha^{l}{\rm e}^{-{\rm i}\bm{k}\cdot\bm{x}}\right>$ (948)
$\displaystyle=$
$\displaystyle-\frac{\alpha\hbar^{2}c^{2}}{4\pi^{2}}\int_{\hbar ck<E_{i}}{\rm
d}\bm{k}\,\frac{1}{k}\sum_{\lambda=1}^{2}\sum_{f}\left<i\left|\,\bm{\hat{\epsilon}}_{\lambda}(\bm{\hat{k}})\cdot\bm{\alpha}\,{\rm
e}^{{\rm i}\bm{k}\cdot\bm{x}}\right|f\right>$
$\displaystyle\times\frac{1}{E_{f}-E_{i}+\hbar ck-{\rm
i}\delta}\,\left<f\left|\bm{\hat{\epsilon}}_{\lambda}(\bm{\hat{k}})\cdot\bm{\alpha}\,{\rm
e}^{-{\rm i}\bm{k}\cdot\bm{x}}\right|i\right>,\quad{}$
where $H$ is the Dirac Hamiltonian. The integrand is real, except for the
imaginary infinitesimal in the denominator, for which
$\displaystyle{\rm Im}\,\frac{1}{E_{f}-E_{i}+\hbar ck-{\rm i}\delta}$
$\displaystyle\rightarrow$ $\displaystyle\pi\delta(E_{f}-E_{i}+\hbar ck)$
(949)
and hence
$\displaystyle\sum_{f}A_{if}$ $\displaystyle=$
$\displaystyle\sum_{f}\frac{\alpha kc}{2\pi}\int{\rm
d}{\it\Omega}_{\bm{k}}\sum_{\lambda=1}^{2}\left<i\left|\,\bm{\hat{\epsilon}}_{\lambda}(\bm{\hat{k}})\cdot\bm{\alpha}\,{\rm
e}^{{\rm
i}\bm{k}\cdot\bm{x}}\right|f\right>\left<f\left|\bm{\hat{\epsilon}}_{\lambda}(\bm{\hat{k}})\cdot\bm{\alpha}\,{\rm
e}^{-{\rm i}\bm{k}\cdot\bm{x}}\right|i\right>,$ (950)
with the restriction $0<E_{f}<E_{i}$ on the sum over $f$. The contribution to
the decay rate from each final state $f$ coincides with Eq. (869) for the
transition rate $A_{if}$.
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|
arxiv-papers
| 2009-10-09T22:47:05 |
2024-09-04T02:49:05.755421
|
{
"license": "Public Domain",
"authors": "Peter J. Mohr",
"submitter": "Peter Mohr",
"url": "https://arxiv.org/abs/0910.1874"
}
|
0910.1891
|
# Longitudinal boost-invariance of charge balance function in hadron-hadron
and nucleus-nucleus collisions
Li Na, Li Zhiming, and Wu Yuanfang Key Laboratory of Quark and Lepton Physics
(Huazhong Normal University),Ministry of Education Institute of Particle
Physics, Huazhong Normal University, Wuhan 430079, China
###### Abstract
Using Monte Carlo generators of the PYTHIA model for hadron-hadron collisions
and a multi-phase transport (AMPT) model for nucleus-nucleus collisions, the
longitudinal boost-invariance of charge balance function and its transverse
momentum dependence are carefully studied. It shows that the charge balance
function is boost-invariant in both p+p and Au+Au collisions in these two
models, consistent with experimental data. The balance function properly
scaled by the width of the pseudorapidity window is independent of the
position or the size of the window and is corresponding to the balance
function of the whole pseudorapidity range. This longitudinal property of
balance function also holds for particles in small transverse momentum ranges
in the PYTHIA and the AMPT default models, but is violated in the AMPT with
string melting. The physical origin of the results are discussed.
###### pacs:
25.75.Gz,25.75.Ld
## I Introduction
Charge balance function (BF) has been widely used as an effective exploring
for the hadronization scheme in hadron-hadron collisions at the ISR energies
oldbf1 and $e^{+}+e^{-}$ annihilations at PETRA energies petra . Recently,
the charge BF gains special attentions in clocking hadronization at
relativistic heavy-ion collisions. A narrowing of the BF is suggested as a
signature for delayed hadronization bf1 ; bf2 .
The dependence of the BF on centrality and system size has been reported by
several relativistic heavy-ion experiments star130 ; na49 . However, most of
the current heavy-ion experiments are limited by the pseudorapidity range
star130 ; na49 ; phenix , it is impossible to quantitatively compare the
results from the experiments with the coverage at different pseudorapidity
ranges. The dependence of the BF on the pseudorapidity window is essential for
understanding the physics of the BF star130 ; na49 ; na22bf ; tom1 , and has
been carefully studied by NA22 na22bf and STAR experiments for hadron-hadron
and relativistic heavy ion collisions, respectively.
The NA22 experiment has full 4$\pi$ acceptance and excellent momentum
resolution na22bf . It is found in the experiment that the BF in
$\pi^{+}{\mathrm{p}}$ and ${\mathrm{K}}^{+}{\mathrm{p}}$ Collisions at 22 GeV
is invariant under longitudinal boost over the whole rapidity range of
produced particles, in spite of the non-boost-invariance of the single-
particle density. Moreover, the BF of the whole rapidity range can be deduced
from the BF properly scaled by the width of rapidity windows na22bf .
The STAR experiment covers a finite but relative wide pseudorapidity range.
The scaling property of the BF in Au +Au collisions at 200 GeV is further
observed in the experiment star200 . This scaling property of the balance
function is also found in different $p_{\rm T}$ ranges of final state
particles.
These results from both hadron-hadron and nuclear collisions indicate that
charge balance of produced particles in strong interactions is boost-
invariance in longitudinal phase-space, in contrary with the single particle
density.
Therefore, it is interesting to see if those properties are taken into account
in the models which are successfully described hadron-hadron and nuclear
collisions, and how they associate with the mechanisms of particle production
in the models.
## II Charge balance function and implement models
Charge balance function measures how the conserved electric charge compensate
in the phase space, i.e., how the surrounding net charge are rearranged if the
charge of a selected point changesoldbf1 . In high energy collisions, the
production of charged particles are constrained by charge balance in the phase
space. The BF therefore provides a direct access to collision dynamics.
The BF has been originally defined in terms of a combination of four kinds of
charge-related conditional densities in pseudorapidity oldbf1
$\displaystyle B(\eta_{1}|\eta_{2})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[\rho(+,\eta_{1}|-,\eta_{2})-\rho(+,\eta_{1}|+,\eta_{2})\right.$
(1)
$\displaystyle\quad\left.+\rho(-,\eta_{1}|+,\eta_{2})-\rho(-,\eta_{1}|-,\eta_{2})\right],$
where the notation $\rho(a,\eta_{a}|b,\eta_{b})$ represents the ratio
$\rho_{ab}(\eta_{a},\eta_{b})/\rho_{b}(\eta_{b})=\langle
n_{ab}(\eta_{a},\eta_{b})\rangle/\langle n_{b}(\eta_{b})\rangle$ with $a,b$
standing for $+$ or $-$ charged particles. Projecting to pseudorapidity
difference $\delta\eta=\eta_{1}-\eta_{2}$ in an pseudorapidity window
$\eta_{w}$, it becomes bf1 ; star130
$\displaystyle B(\delta\eta|\eta_{\rm w})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[\frac{\langle n_{+-}(\delta\eta)\rangle-\langle
n_{++}(\delta\eta)\rangle}{\langle n_{+}\rangle}\right.$ (2)
$\displaystyle\quad\left.+\frac{\langle n_{-+}(\delta\eta)\rangle-\langle
n_{--}(\delta\eta)\rangle}{\langle n_{-}\rangle}\right]$
where $n_{ab}(\delta\eta)$ is the total number of pairs of opposite charged
particles with pseudorapidity difference $\delta\eta$ in the pseudorapidity
window $\eta_{\rm w}$. $n_{+}$ and $n_{-}$ are the number of positively and
negatively charged particles in the window $\eta_{\rm w}$, respectively.
$\langle\cdots\rangle$ is the average over the whole event sample.
From the findings of the BF at NA22 na22bf and STAR experiments star200 , the
BF is boost-invariant in the whole rapidity range in hadron-hadron collisions
and may be in nuclear collisions as well. In the case, the properly scaled BF
is corresponding to the BF of the whole pseudorapidity range and is deduced by
$\displaystyle B_{s}(\delta\eta)=\frac{B(\delta\eta|\eta_{\rm
w})}{1-\frac{\delta\eta}{|\eta_{\rm w}|}}$ (3)
where $|\eta_{\rm w}|$ is the width of pseudorapidity window.
The PYTHIA 5.720 pythia is well set up for p+p collisions. It is a standard
Monte Carlo generator with string fragmentation as hadronization scheme. Two
versions of a multi-phase transport (AMPT) model ampt are used to study Au+Au
collisions. One is the AMPT default and the other one is the AMPT with string
melting. In both versions, the initial conditions are obtained from the HIJING
model, and then the scattering among partons are given by ZPC. In the AMPT
default model, the partons recombined with their parent strings when they stop
interacting, and the resulting strings are converted to hadrons using the Lund
string fragmentation model, while in the AMPT model with string melting, quark
coalescence is used in combining partons into hadrons. The dynamics of the
hadronic matter is described by ART model.
It is commonly believed that in relativistic heavy ion collisions, the charge
ordering during the string fragmentation in elementary collisions is no longer
valid, and it should be replaced by the quark-coalescence mechanism in
hadronization recombination . So it is interesting to see whether the boost-
invariance of the BF is sensitive to the mechanisms of hadronization.
In this paper, we firstly study the boost-invariance of the BF for p+p
collisions at $\sqrt{s}=22$ GeV and $\sqrt{s}=200$ GeV using the PYTHIA, and
for Au+Au collisions at $\sqrt{s}=200$ GeV using two versions of the AMPT. The
transverse momentum dependence of longitudinal scaling property of the BF is
then examined in the models. The obtained results are compared with
corresponding experimental data and discussed.
## III Boost-invariance and longitudinal scaling of the BF
Figure 1: Upper panel: the $B(\delta\eta|\eta_{\rm w})$ in four
pseudorapidity windows with equal size $|\eta_{\rm w}|=3$ at the different
positions for p+p collisions at (a) $\sqrt{s}=22$ GeV and (b) $\sqrt{s}=200$
GeV by PYTHIA model. Lower panel: the scaled balance function,
$B_{s}(\delta\eta)$, deduced from the directly measured BF at six different
sizes and positions of pseudorapidity windows for p+p collisions at (c)
$\sqrt{s}=22$ GeV and (d) $\sqrt{s}=200$ GeV by PYTHIA model. The solid down
triangle is the BF of the whole $\eta$ range.
In order to demonstrate directly whether the BF is invariant under a
longitudinal Lorentz transformation over the whole rapidity in hadron-hadron
collisions, we choose four equal size ($|\eta_{\rm w}|=3$) pseudorapidity
windows locating at different positions ($-3,0$), ($-2,1$), ($-1,2$) and
($0,3$). The results for p+p collisions at $\sqrt{s}=22$ GeV and
$\sqrt{s}=200$ GeV are shown in Fig. 1(a) and (b) respectively. The statistic
errors are smaller than the markers. It is clear that the BF measured in four
windows are approximately identical to each other at two incident energies.
This indicates that the charge compensation is essentially the same in any
longitudinally-Lorentz-transformed frame for p+p collisions in the PYTHIA
model, consistent with the data from NA22 experiment. These results show that
the string fragmentation mechanism implemented in PYTHIA well describes the
production mechanisms of charge particles and their charge balance in
longitudinal phase space.
Fig. 1(c) and (d) are the scaled balance function $B_{s}(\delta\eta)$ at two
incident energies. They are deduced from directly measured
$B(\delta\eta|\eta_{\rm w})$ at six different pseudorapidity windows,
($-0.8,0.8$) (open circles), ($1,3$) (open triangles), ($-3,1$) (open
squares), ($-2.4,2.4$) (open diamonds), ($0,3$) (open crosses), and ($-2,-1$)
(open stars). From the figures we can see that all the $B_{s}(\delta\eta)$
deduced from different windows are coincide with each other within errors, as
expected from boost-invariance of the BF bf2 . The solid down triangles in the
same figures are the BF of the whole pseudorapidity range,
$B(\delta\eta|\eta_{\infty})$. It is close to the scaled balance function
$B_{s}(\delta\eta)$. These results indicate that the scaled BF is in fact
corresponding to the BF of the whole pseudorapidity range
$B(\delta\eta|\infty)$ bf2 .
Figure 2: Upper panel: the $B(\delta\eta|\eta_{\rm w})$ in five
pseudorapidity windows with equal size $|\eta_{\rm w}|=2$ at the different
positions for Au+Au collisions at $\sqrt{s}=200$ GeV by (a) the AMPT default
and (b) the AMPT with string melting. Lower panel: the scaled balance
function, $B_{s}(\delta\eta)$, deduced from the directly measured BF at
various pseudorapidity windows with different sizes and positions for Au+Au
collisions at $\sqrt{s}=200$ GeV by (c) the AMPT default and (d) the AMPT with
sting melting.
It is then interesting to see whether the boost-invariance of the BF is held
in nucleus-nucleus collisions. STAR experiment only observe the boost-
invariance of BF in cental pseudorapidity range $-1<\eta<1$ star200 , where
the single particle distribution is almost flat, or boost-invariance. Now in
model investigation, we can carefully examine the property in the whole
pseudorapidity range.
The upper panel of Fig. 2 is the BF in five pseudorapidity windows with equal
size $\eta_{\rm w}=2$ at different positions ($-3,-1$), ($-2,0$), ($-1,1$),
($0,2$) and ($1,3$). Where the Fig. 2(a) and (b) are the results from the AMPT
default (v1.11) and the AMPT with string melting (v2.11), respectively. Both
figures show that the BF is boost-invariance in pseudorapidity range (-3, 3)
in two versions of the AMPT.
The lower panel of Fig. 2 is the scaled balance functions, which are obtained
from directly measured BF at six different windows as indicated at legend of
the figure, where the solid down triangles are the BF in pseudorapidity range
(-4, 4). It shows that the scaled BF does not depend on the size and position
of the windows, and corresponds to the BF of the whole pseudorapidity in two
versions of the AMPT, consistent with the results of p+p collisions in the
PYTHIA model.
## IV The transverse-momentum dependence of the boost-invariance of the BF
The longitudinal property of boost invariance of BF comes from the special
longitudinal interaction of charged particles under the constraint of global
electric charge balance. Global electric charge conservation not only applies
to all final-state charged particles, but also constrains particles which are
produced at the same proper time of evolution. It is argued that the
transverse-momentum of final-state particles may be roughly used as a scale of
the proper time of their production in the expansion of nuclear collisions
rudy ; bmuller ; hama1 ; ptscale . Examining the $p_{\rm T}$ dependence of
longitudinal property of the BF will provide direct access on whether
particles in specified $p_{\rm T}$ range are consistent to be produced
simultaneously with well balanced electric charge.
So we turn to check whether the boost-invariant of BF holds for particles in
different $p_{\rm T}$ ranges. Fig. 3 shows the BF for p+p collisions at
$\sqrt{s}=22$ GeV and $\sqrt{s}=200$ GeV from PYTHIA in three transverse
momentum bins ($0<p_{\rm T}<0.2$), ($0.2<p_{\rm T}<0.4$), and ($p_{\rm
T}>0.2$) GeV/$c$, respectively. These $p_{\rm T}$ bins are selected to make
the multiplicity in each bin comparable. The result shows that the points at a
given $\delta\eta$ in a restricted $p_{\rm T}$ interval are approximately
coincide with each other, i.e., the boost-invariance of the BF hold in small
$p_{\rm T}$ ranges. It indicates that particles produced at different $p_{\rm
T}$ ranges are also boost-invariant for hadron-hadron collisions in the PYTHIA
model.
Figure 3: For each of three $p_{\rm T}$ ranges, the $B(\delta\eta|\eta_{\rm
w})$ in four pseudorapidity windows with equal size $|\eta_{\rm w}|=3$ at the
different positions for p+p collisions at $\sqrt{s}=22$ GeV and $\sqrt{s}=200$
GeV in upper and lower panels, respectively. Figure 4: For each of four
$p_{\rm T}$ ranges, the $B(\delta\eta|\eta_{\rm w})$ in five pseudorapidity
windows with equal size $|\eta_{\rm w}|=2$ at the different positions for
Au+Au collisions at $\sqrt{s}=200$ GeV from the AMPT default (in upper panel)
and the AMPT with string melting (in lower panel).
The same study for Au+Au 200 GeV collisions from the two versions of the AMPT
are presented in the upper and lower panels of Fig. 4, respectively. Where
four $p_{\rm T}$ bins are, ($0.15,0.4$), ($0.4,0.7$), ($0.7,1$) and ($1,2$)
GeV/$c$. From the upper panel of the figure, we can see that the BF of
different pseudorapidity windows in each $p_{\rm T}$ bin are close to each
other, in consistent with the data from STAR experiment star200 . However, in
the AMPT with string melting, as shown in the lower panel of the figure, where
the BF of different pseudorapidity windows are not as close to each other as
those in the upper panel.
This is because in the AMPT with string melting, each parton in the evolution
of nuclear collision has its own freeze-out time, which last a very long
period after the interaction of two nucleus liu-yu . The particles in the same
transverse-momentum range are not freezed-out simultaneously with well
balanced charge, and therefore the longitudinal boost-invariance of the BF in
small $p_{\rm T}$ ranges is violated. In the AMPT default, the partons
recombined with their parent strings immediately after they stop interacting,
and converted to hadrons. So the charge balance of the produced particles in
the same $p_{\rm T}$ ranges is preserved and boost-invariance of the BF keeps.
## V Summary
In the paper, we systematically study the longitudinal boost-invariance of
charge balance function and its $p_{\rm T}$ dependence for p+p and Au+Au
collisions using PYTHIA the AMPT models. It shows that charge balance function
is boost-invariance in both hadron-hadron and nuclear interactions, in
contrary to the single particle density. As expected, this boost-invariance of
the BF make the BF properly scaled by window size is independent of window and
corresponds to the BF of the whole (pseudo)rapidity range. Therefore, the BF
is a good measure free from the restriction of finite longitudinal acceptance.
It is further show that the boost invariance of the BF in specified $p_{\rm
T}$ range is valid in PYTHIA for hadron-hadron collisions and the AMPT default
for Au+Au collisions. While the AMPT with string melting fails to reproduce
this property due to the different schemes at hadronization. So the $p_{\rm
T}$ dependence of the longitudinal property of the BF may be served as a
sensitive probe for charge balance in hadronization mechanism.
## VI Acknowledgments
We thank Prof. Liu Lianshou and Dr. Yu Meiling for valuable discussions and
remarks. This work is supported in part by the NSFC of China with project No.
10835005 and No. 10647124, and the MOE of China with project No. IRT0624 and
No. B08033.
## References
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|
arxiv-papers
| 2009-10-10T01:48:09 |
2024-09-04T02:49:05.776708
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Na LI Zhiming LI Yuanfang WU",
"submitter": "Yuanfang Wu",
"url": "https://arxiv.org/abs/0910.1891"
}
|
0910.1928
|
1
MEASURABLE LOWER BOUNDS ON CONCURRENCE
IMAN SARGOLZAHI***sargolzahi@stu-mail.um.ac.ir, sargolzahi@gmail.com , SAYYED
YAHYA MIRAFZALI and MOHSEN SARBISHAEI
Department of Physics, Ferdowsi University of Mashhad
Mashhad, Iran
Received (received date)
Revised (revised date)
We derive measurable lower bounds on concurrence of arbitrary mixed states,
for both bipartite and multipartite cases. First, we construct measurable
lower bonds on the purely algebraic bounds of concurrence [F. Mintert et al.
(2004), Phys. Rev. lett., 92, 167902]. Then, using the fact that the sum of
the square of the algebraic bounds is a lower bound of the squared
concurrence, we sum over our measurable bounds to achieve a measurable lower
bound on concurrence. With two typical examples, we show that our method can
detect more entangled states and also can give sharper lower bonds than the
similar ones.
Keywords: Measuring entanglement, Concurrence
Communicated by: to be filled by the Editorial
## 1 Introduction
Recently, many studies have been focused on the experimental quantification of
entanglement [1]. Bell inequalities and entanglement witnesses [1, 2] can be
used to detect entangled states experimentally, but they don’t give any
information about the amount of entanglement. In addition, quantum state
tomography [3], determination of the full density operator $\rho$ by measuring
a complete set of observables, is only practical for low dimensional systems
since the number of measurements needed for it grows rapidly as the dimension
of the system increases. Fortunately, several methods have been introduced
which let one to estimate experimentally the amount of the entanglement of an
unknown $\rho$ with no need to the full tomography [1, 4, 5, 6, 7, 8, 9, 10,
11, 12, 13, 14, 15, 16, 17, 18, 29, 19, 30, 20, 26, 27, 28, 21, 22, 23, 24,
25]. A simple and straightforward method is the one introduced in [8, 14, 18]
for finding measurable lower bounds on an entanglement measure, namely the
concurrence [31]. These lower bounds are in terms of the expectation values of
some Hermitian operators with respect to two-fold or one-fold copy of $\rho$.
It is worth noting that these bounds work well for weakly mixed states [32, 8,
14, 18, 5].
In this paper we will use a similar procedure as [8, 14] to construct
measurable lower bounds on the purely algebraic bounds of concurrence [33,
31]. In addition, using a theorem in Sec. II, we show that the sum of our
measurable bounds leads to a measurable lower bound on the concurrence itself.
Then, we show that this method gives better results than those introduced in
[8, 14] for two typical examples.
The paper is organized as follows. In Sec. II, the concurrence and its $MKB$
(Mintert-Kus-Buchleitner) lower bounds [33] are introduced. In Secs. III and
IV, we propose measurable lower bounds on the purely algebraic bounds of
concurrence [33], which are a special class of $MKB$ bounds. The
generalization to the multipartite case is given in Sec. V and we end this
paper in Sec. VI with a summary and discussion.
## 2 Concurrence and its $MKB$ Lower Bounds
For a pure bipartite state $|\Psi\rangle$,
$|\Psi\rangle\in\mathcal{H}_{A}\otimes\mathcal{H}_{B}\ $, concurrence is
defined as [31]:
$C\left(\Psi\right)=\sqrt{2[\langle\Psi|\Psi\rangle^{2}-tr\rho_{r}^{2}]}\,,$
(1)
where $\rho_{r}$ is the reduced density operator obtained by tracing over
either subsystems A or B. It is obvious that iff $|\Psi\rangle$ is a product
state, i.e. $|\Psi\rangle=|\Psi_{A}\rangle\otimes|\Psi_{B}\rangle$, then
$C(\Psi)=0$. Interestingly, $C(\Psi)$ can be written in terms of the
expectation value of an observable with respect to two identical copies of
$|\Psi\rangle$ [31, 11, 12]:
$\displaystyle
C\left(\Psi\right)=\sqrt{{}_{AB}\langle\Psi|_{AB}\langle\Psi|{\cal
A}|\Psi\rangle_{AB}|\Psi\rangle_{AB}}\,,$ (2) $\displaystyle{\cal
A}=4P_{-}^{A}\otimes P_{-}^{B}\,,\qquad\qquad$ (3)
where $P_{-}^{A}$ ($P_{-}^{B}$) is the projector onto the antisymmetric
subspace of $\mathcal{H}_{A}\otimes\mathcal{H}_{A}\ $
($\mathcal{H}_{B}\otimes\mathcal{H}_{B}\ $). A possible decomposition of
${\cal A}$ is
$\displaystyle{\cal
A}=\sum_{\alpha}|\chi_{\alpha}\rangle\langle\chi_{\alpha}|\,,\qquad\quad$ (4)
$\displaystyle|\chi_{\alpha}\rangle\
=\large(|xy\rangle-|yx\rangle)_{A}\large(|pq\rangle-|qp\rangle)_{B}\,,$ (5)
where $|x\rangle$ and $|y\rangle$ ($|p\rangle$ and $|q\rangle$) are two
different members of an orthonormal basis of the A (B) subsystem. For mixed
states the concurrence is defined as follows [31]:
$\displaystyle C(\rho)=\min\sum_{i}p_{i}C(\Psi_{i})\,,\qquad\qquad$ (6)
$\displaystyle\rho=\sum_{i}p_{i}|\Psi_{i}\rangle\langle\Psi_{i}|\,,\qquad
p_{i}\geq 0\,,\qquad\sum_{i}p_{i}=1\,,$ (7)
where the minimum is taken over all decompositions of $\rho$ into pure states
$|\Psi_{i}\rangle$. It is appropriate to write $C(\rho)$ in terms of the
subnormalized states $|\psi_{i}\rangle$ rather than the normalized ones
$|\Psi_{i}\rangle$:
$\displaystyle C(\rho)=\min\sum_{i}\sqrt{\langle\psi_{i}|\langle\psi_{i}|{\cal
A}|\psi_{i}\rangle|\psi_{i}\rangle}\,,$ (8)
$\displaystyle|\psi_{i}\rangle=\sqrt{p_{i}}|\Psi_{i}\rangle\,,\qquad\rho=\sum_{i}|\psi_{i}\rangle\langle\psi_{i}|\,;$
(9)
since all decompositions of $\rho$ into subnormalized states are related to
each other by unitary matrices [3]: consider an arbitrary decomposition of
$\rho=\sum_{j}|\varphi_{j}\rangle\langle\varphi_{j}|$ (As a special case, one
can choose $|\varphi_{j}\rangle=\sqrt{\lambda_{j}}|\Phi_{j}\rangle$, where
$|\Phi_{j}\rangle$ and $\lambda_{j}$ are eigenvectors and eigenvalues of
$\rho$ respectively:
$\rho=\sum_{j}\lambda_{j}|\Phi_{j}\rangle\langle\Phi_{j}|$.), for any other
decomposition of $\rho=\sum_{i}|\psi_{i}\rangle\langle\psi_{i}|$ we have [3]:
$|\psi_{i}\rangle=\sum_{j}U_{ij}|\varphi_{j}\rangle\,,\qquad\sum_{i}U^{\dagger}_{ki}U_{ij}=\delta_{jk}\,.$
(10)
So Eq. (9) can be written as:
$\displaystyle C(\rho)=\min_{U}\sum_{i}\sqrt{\sum_{jklm}U_{ij}U_{ik}{\cal
A}^{lm}_{jk}U^{\dagger}_{li}U^{\dagger}_{mi}}\,,$ (11) $\displaystyle{\cal
A}^{lm}_{jk}=\langle\varphi_{l}|\langle\varphi_{m}|{\cal
A}|\varphi_{j}\rangle|\varphi_{k}\rangle\,.\qquad\qquad$ (12)
From the definition of $C(\rho)$ in Eq. (7) it is obvious that $C(\rho)=0$ iff
$\rho$ can be decomposed into product states. In other words, $C(\rho)=0$ iff
$\rho$ is separable. In addition, it can be shown that the concurrence is an
entanglement monotone [34] (An entanglement monotone is a function of $\rho$
which does not increase, on average, under LOCC and vanishes for separable
states [35].). But, except for the two-qubit case [36], $C(\rho)$ can not be
computed in general; i.e., in general, one can not find the $U$ which
minimizes Eq. (12). Any numerical method for finding the $U$ which minimizes
Eq. (12) leads to an upper bound for $C(\rho)$. So, finding lower bounds on
$C(\rho)$ is desirable. So far, several lower bounds for $C(\rho)$ have been
introduced [33, 31, 37, 38, 39, 40, 41, 42, 43, 5, 8, 13, 14, 18, 19, 21, 22,
23, 24]. One of them is that introduced by F. Mintert et al. in [33, 31]. Now,
we redrive their lower bounds in a slightly different form to make them more
suitable for finding measurable lower bounds in the following sections.
Assume that the decomposition of $\rho$ which minimizes Eq. (9) is
$\rho=\sum_{j}|\xi_{j}\rangle\langle\xi_{j}|$, then from Eqs. (3) and (5), we
have:
$\displaystyle
C(\rho)=\sum_{j}\sqrt{\sum_{\alpha}|\langle\chi_{\alpha}|\xi_{j}\rangle|\xi_{j}\rangle|^{2}}\geq\sum_{j}|\langle\chi_{\beta}|\xi_{j}\rangle|\xi_{j}\rangle|\geq\min_{\left\\{|\psi_{i}\rangle\right\\}}\sum_{i}|\langle\chi_{\beta}|\psi_{i}\rangle|\psi_{i}\rangle|\,,\qquad$
(13)
where $|\chi_{\beta}\rangle\in\left\\{|\chi_{\alpha}\rangle\right\\}$, and the
minimum is taken over all decompositions of $\rho$ as
$\rho=\sum_{i}|\psi_{i}\rangle\langle\psi_{i}|$. Now, using Eq. (10), we have:
$\displaystyle\min_{\\{|\psi_{i}\rangle\\}}\sum_{i}|\langle\chi_{\beta}|\psi_{i}\rangle|\psi_{i}\rangle|=\min_{U}\sum_{i}|\sum_{jk}U_{ij}T_{jk}^{\beta}U_{ki}^{\top}|=\min_{U}\sum_{i}|\left[UT^{\beta}U^{\top}\right]_{ii}|\,,$
(14) $\displaystyle
T_{jk}^{\beta}=\langle\chi_{\beta}|\varphi_{j}\rangle|\varphi_{k}\rangle\,.\qquad\qquad\qquad\qquad\qquad\qquad$
(15)
Since $T^{\beta}$ is a symmetric matrix, the minimum in Eq. (15) can be
computed and we have [31]:
$\min_{U}\sum_{i}|\left[UT^{\beta}U^{\top}\right]_{ii}|=\max\\{0,S_{1}^{\beta}-\sum_{l>1}S_{l}^{\beta}\\}\,,$
(16)
where $S_{l}^{\beta}$ are the singular values of $T^{\beta}$, in decreasing
order. The above expression is what was named purely algebraic lower bound of
concurrence in [31, 33] and we will refer to it as $ALB(\rho)$.
Let us define
$|\tau\rangle=\sum_{\alpha}z_{\alpha}^{\ast}|\chi_{\alpha}\rangle\,,\qquad\sum_{\alpha}|z_{\alpha}|^{2}=1\,.\qquad$
(17)
Obviously, $|\tau\rangle$ is an element of another (normalized to 2) basis of
$P_{-}^{A}\otimes P_{-}^{B}$, $\\{|\chi_{\alpha}^{\prime}\rangle\\}$. Then:
$\displaystyle|\tau\rangle\equiv|\chi_{1}^{\prime}\rangle\,,\qquad\qquad\qquad$
(18) $\displaystyle{\cal
A}=\sum_{\alpha}|\chi_{\alpha}\rangle\langle\chi_{\alpha}|=|\tau\rangle\langle\tau|+\sum_{\alpha>1}|\chi_{\alpha}^{\prime}\rangle\langle\chi_{\alpha}^{\prime}|\,.$
(19)
Again, as the inequality (13), we have:
$\displaystyle
C(\rho)=\sum_{j}\sqrt{\sum_{\alpha}|\langle\chi_{\alpha}^{\prime}|\xi_{j}\rangle|\xi_{j}\rangle|^{2}}\geq\sum_{j}|\langle\tau|\xi_{j}\rangle|\xi_{j}\rangle|$
(20)
$\displaystyle\geq\min_{\left\\{|\psi_{i}\rangle\right\\}}\sum_{i}|\langle\tau|\psi_{i}\rangle|\psi_{i}\rangle|\qquad\qquad\qquad\qquad$
(21) $\displaystyle=\min_{U}\sum_{i}|\left[U{\cal
T}U^{\top}\right]_{ii}|=\max\\{0,S_{1}^{\tau}-\sum_{l>1}S_{l}^{\tau}\\}\,,$
(22) $\displaystyle{\cal
T}_{jk}=\langle\tau|\varphi_{j}\rangle|\varphi_{k}\rangle=\sum_{\alpha}z_{\alpha}T_{jk}^{\alpha}\,,\qquad\qquad$
(23)
where $S_{l}^{\tau}$ are the singular values of ${\cal T}$, in decreasing
order. The above expression is the general form of the lower bounds introduced
in [33, 31] and we call it $LB(\rho)$.
We end this section by proving a useful theorem: if
$\left\\{|\chi_{\alpha}^{\prime}\rangle\right\\}$ be an orthogonal (normalized
to 2) basis of $P_{-}^{A}\otimes P_{-}^{B}$, i.e. ${\cal
A}=\sum_{\alpha}|\chi_{\alpha}^{\prime}\rangle\langle\chi_{\alpha}^{\prime}|$,
then:
$\displaystyle
C^{2}(\rho)=\sum_{ij}\sqrt{\sum_{\alpha}|\langle\chi_{\alpha}^{\prime}|\xi_{i}\rangle|\xi_{i}\rangle|^{2}}\sqrt{\sum_{\alpha}|\langle\chi_{\alpha}^{\prime}|\xi_{j}\rangle|\xi_{j}\rangle|^{2}}$
(24)
$\displaystyle\geq\sum_{ij}\sum_{\alpha}|\langle\chi_{\alpha}^{\prime}|\xi_{i}\rangle|\xi_{i}\rangle||\langle\chi_{\alpha}^{\prime}|\xi_{j}\rangle|\xi_{j}\rangle|\qquad\qquad$
(25)
$\displaystyle=\sum_{\alpha}\left(\sum_{i}|\langle\chi_{\alpha}^{\prime}|\xi_{i}\rangle|\xi_{i}\rangle|\right)^{2}\geq\sum_{\alpha}\left[LB_{\alpha}(\rho)\right]^{2}\,,$
(26) $\displaystyle
LB_{\alpha}(\rho)=\min_{\left\\{|\psi_{i}\rangle\right\\}}\sum_{i}|\langle\chi_{\alpha}^{\prime}|\psi_{i}\rangle|\psi_{i}\rangle|\,.\qquad$
(27)
In proving the above theorem we have used the Cauchy-Schwarz inequality.
Obviously, any entangled $\rho$ which can not be detected by ${LB_{\alpha}}$,
can not be detected by $\sum_{\alpha}\left[LB_{\alpha}(\rho)\right]^{2}$
either; i.e., $\sum_{\alpha}\left[LB_{\alpha}(\rho)\right]^{2}$ is not a more
powerful criteria than ${LB_{\alpha}}$, but, quantitatively, it may lead to a
better lower bound for $C(\rho)$.
It should be mentioned that the above theorem is, in fact, the generalization
of what has been proved in [42]. There, it was shown that:
$\displaystyle\tau(\rho)=\sum C_{mn}^{2}(\rho)\leq C^{2}(\rho)\,,\qquad\qquad$
(28) $\displaystyle
C_{mn}(\rho)=\min_{\left\\{|\psi_{i}\rangle\right\\}}\sum_{i}|\langle\psi_{i}|L_{m_{A}}\otimes
L_{n_{B}}|\psi_{i}^{\ast}\rangle|\,,$ (29)
where $L_{m_{A}}$ and $L_{n_{B}}$ are generators of $SO(d_{A})$ and
$SO(d_{B})$ respectively $(d_{A/B}=dim(\mathcal{H}_{A/B}))$, and
$|\psi_{i}^{\ast}\rangle$ is the complex conjugate of $|\psi_{i}\rangle$ in
the computational basis. In this basis $L_{m_{A}}$ and $L_{n_{B}}$ are [44]:
$\displaystyle L_{m_{A}}=|x\rangle_{A}\langle y|-|y\rangle_{A}\langle
x|\,,\qquad\quad L_{m_{B}}=|p\rangle_{B}\langle q|-|q\rangle_{B}\langle p|\,.$
For an arbitrary $|\psi\rangle$, according to the definition of
$|\chi_{\alpha}\rangle$ in Eq. (5), it can be seen that:
$|\langle\psi|L_{m_{A}}\otimes
L_{n_{B}}|\psi^{\ast}\rangle|=|\langle\chi_{\alpha}|\psi\rangle|\psi\rangle|\,.$
(30)
So:
$\displaystyle C_{mn}(\rho)=ALB_{\alpha}(\rho)\,,\qquad\qquad$ (31)
$\displaystyle
ALB_{\alpha}(\rho)=\min_{\left\\{|\psi_{i}\rangle\right\\}}\sum_{i}|\langle\chi_{\alpha}|\psi_{i}\rangle|\psi_{i}\rangle|\,.$
(32)
So what was proved in [42] is, in fact, the special case of
$|\chi_{\alpha}^{\prime}\rangle=|\chi_{\alpha}\rangle$ in expression (27). In
addition, since $ALB_{\alpha}$ can detect bound entangled states [33, 31],
this claim of [42] that any state for which $\tau(\rho)>0$ is distillable, is
not correct.
## 3 Measurable Lower Bounds in terms of Two Identical Copies of $\rho$
As we have seen in Eq. (3) the concurrence of a pure state $|\Psi\rangle$ can
be written in terms of the expectation value of the observable ${\cal A}$ with
respect to two identical copies of $|\Psi\rangle$. For an arbitrary mixed
state $\rho_{AB}$, it has been shown that [8]:
$\displaystyle C^{2}(\rho_{AB})\geq
tr\left(\rho_{AB}\otimes\rho_{AB}V_{(i)}\right)\,,\qquad i=1,2\,;\qquad$ (33)
$\displaystyle V_{(1)}=4\left(P_{-}^{A}-P_{+}^{A}\right)\otimes
P_{-}^{B}\,,\qquad\qquad
V_{(2)}=4P_{-}^{A}\otimes\left(P_{-}^{B}-P_{+}^{B}\right)\,,$ (34)
where $P_{+}^{A}$ ($P_{+}^{B}$) is the projector onto the symmetric subspace
of $\mathcal{H}_{A}\otimes\mathcal{H}_{A}\ $
($\mathcal{H}_{B}\otimes\mathcal{H}_{B}\ $). The above expression means that
measuring $V_{(i)}$ on two identical copies of $\rho$, i.e. $\rho\otimes\rho$,
gives us a measurable lower bound on $C^{2}(\rho)$. It is worth noting that if
the entanglement of $\rho$ can be detected by $V_{(i)}$, then $\rho$ is
distillable [24].
As one can see from expression (23), the $LB$ of a pure state $|\Psi\rangle$
can also be written in terms of the expectation value of the observable
$|\tau\rangle\langle\tau|$ with respect to two identical copies of
$|\Psi\rangle$. Now, for an arbitrary mixed state $\rho$, can we find an
observable $V$ such that the following inequality holds?
$LB^{2}(\rho)\geq tr\left(\rho\otimes\rho V\right)\,,$ (35)
Fortunately for the special case of $|\tau\rangle=|\chi_{\alpha}\rangle$,
where $|\chi_{\alpha}\rangle$ are defined in Eq. (5), we can do so.
Assume that the decomposition of $\rho$ which gives the minimum in Eq. (15) is
$\rho=\sum_{i}|\theta_{i}^{\alpha}\rangle\langle\theta_{i}^{\alpha}|$, i.e.:
$ALB_{\alpha}(\rho)=\sum_{i}|\langle\chi_{\alpha}|\theta_{i}^{\alpha}\rangle|\theta_{i}^{\alpha}\rangle|\,.$
(36)
In addition, assume that for a Hermitian operator $V_{\alpha}$, which acts on
$\mathcal{H}_{A}\otimes\mathcal{H}_{B}\otimes\mathcal{H}_{A}\otimes\mathcal{H}_{B}\
$, and arbitrary $|\psi\rangle$ and $|\varphi\rangle$,
$|\psi\rangle\in\mathcal{H}_{A}\otimes\mathcal{H}_{B}$ and
$|\varphi\rangle\in\mathcal{H}_{A}\otimes\mathcal{H}_{B}$, we have:
$|\langle\chi_{\alpha}|\psi\rangle|\psi\rangle||\langle\chi_{\alpha}|\varphi\rangle|\varphi\rangle|\geq\langle\psi|\langle\varphi|V_{\alpha}|\psi\rangle|\varphi\rangle\,.$
(37)
Now, from the expressions (20) and (21), we have:
$\displaystyle
ALB_{\alpha}^{2}(\rho)=\sum_{ij}|\langle\chi_{\alpha}|\theta_{i}^{\alpha}\rangle|\theta_{i}^{\alpha}\rangle||\langle\chi_{\alpha}|\theta_{j}^{\alpha}\rangle|\theta_{j}^{\alpha}\rangle|\geq\sum_{ij}\langle\theta_{i}^{\alpha}|\langle\theta_{j}^{\alpha}|V_{\alpha}|\theta_{i}^{\alpha}\rangle|\theta_{j}^{\alpha}\rangle=tr\left(\rho\otimes\rho
V_{\alpha}\right)\,.$ (38)
So, for any $V_{\alpha}$ satisfying inequality (21), measuring $V_{\alpha}$ on
two identical copies of $\rho$ gives a lower bound on
$ALB_{\alpha}^{2}(\rho)$. We can prove that the inequality (21) holds for(see
the Appendix):
$\displaystyle V_{\alpha}=V_{(1)\alpha}={\cal M}V_{(1)}{\cal M}\,,\quad\qquad
V_{\alpha}=V_{(2)\alpha}={\cal M}V_{(2)}{\cal M}\,,$ (39) $\displaystyle{\cal
M}={\cal M}_{A}\otimes{\cal M}_{A}\otimes{\cal M}_{B}\otimes{\cal
M}_{B}\,,\quad\qquad\quad$ (40) $\displaystyle{\cal M}_{A}=|x\rangle\langle
x|+|y\rangle\langle y|\,,\quad\qquad\quad{\cal M}_{B}=|p\rangle\langle
p|+|q\rangle\langle q|\,,$ (41)
where $|x\rangle$, $|y\rangle$, $|p\rangle$, $|q\rangle$ are introduced in Eq.
(5) (note that $|\chi_{\alpha}\rangle\langle\chi_{\alpha}|={\cal M}{\cal
A}{\cal M}$). In addition, for any $V_{\alpha}$ such as
$\displaystyle V_{\alpha}=c_{1}V_{(1)\alpha}+c_{2}V_{(2)\alpha}\,,\qquad
c_{1}\geq 0\,,\qquad c_{2}\geq 0\,,\qquad c_{1}+c_{2}=1\,,$ (42)
inequalities (37) and, consequently, (38) also hold.
According to the definition of $V_{\alpha}$ in Eqs. (23) and (24), we have:
$\displaystyle tr\left(\rho\otimes\rho
V_{\alpha}\right)=tr\left(\varrho\otimes\varrho V_{\alpha}\right)\,,\quad$
(43) $\displaystyle\varrho={\cal M}_{A}\otimes{\cal M}_{B}\rho{\cal
M}_{A}\otimes{\cal M}_{B}\,,\quad$ (44)
which means that if $V_{\alpha}$ detects the entanglement of $\rho$, it has,
in fact, detected the entanglement of a two-qubit submatrix of $\rho$. Any
$\rho$ which has an entangled two-qubit submatrix is distillable [45]. So any
$\rho$ which is detected by $V_{\alpha}$ is distillable.
The right hand side of the inequality (34) is invariant under local unitary
transformations [8]:
$\displaystyle tr\left(\rho\otimes\rho
V_{(i)}\right)=tr\left(\rho^{\prime}\otimes\rho^{\prime}V_{(i)}\right)\,,\quad$
(45)
$\displaystyle\rho^{\prime}={U}_{A}\otimes{U}_{B}\rho{U}^{\dagger}_{A}\otimes{U}^{\dagger}_{B}\,,\quad\qquad$
(46)
where $U_{A}$ and $U_{B}$ are arbitrary unitary operators. This is so because
$U^{\dagger}_{A}\otimes
U^{\dagger}_{A}P^{A}_{\pm}U_{A}\otimes{U_{A}}=P^{A}_{\pm}$ and
$U^{\dagger}_{B}\otimes
U^{\dagger}_{B}P^{B}_{\pm}U_{B}\otimes{U_{B}}=P^{B}_{\pm}$. So, the choices of
local bases in the definition of $V_{(i)}$ in (34) are not important since all
the choices lead to the same result. But, according to the definition of
$V_{\alpha}$ in Eqs. (23) and (24), the right hand side of the inequality (38)
is not invariant under local unitary transformations. It is however expected
since the $ALB_{\alpha}(\rho)$ is not invariant under such transformations
either.
Using Eqs. (23) and (24), it can be shown simply that the right hand side of
the inequality (38) is invariant under the following transformations:
$\displaystyle tr\left(\rho\otimes\rho
V_{\alpha}\right)=tr\left(\rho^{\prime}\otimes\rho^{\prime}V_{\alpha}\right)\,,\quad\qquad\quad$
(47) $\displaystyle\rho^{\prime}={u}_{A}\otimes{u}_{B}\rho
u^{\dagger}_{A}\otimes u^{\dagger}_{B}\,,\qquad\qquad\qquad$ (48)
$\displaystyle{\cal M}_{A}u_{A}{\cal M}_{A}=u_{A},\qquad\qquad
u_{A}u_{A}^{\dagger}=u_{A}^{\dagger}u_{A}={\cal M}_{A},$ (49)
$\displaystyle{\cal M}_{B}u_{B}{\cal M}_{B}=u_{B},\qquad\qquad
u_{B}u_{B}^{\dagger}=u_{B}^{\dagger}u_{B}={\cal M}_{B},$ (50)
$\displaystyle\Rightarrow tr(\rho^{\prime})\leq 1.\qquad\qquad\qquad\qquad$
(51)
$|\chi_{\alpha}\rangle$ is also invariant, up to a phase, under the above
transformations, i.e. $u_{A}\otimes{u_{A}}\otimes
u_{B}\otimes{u_{B}}|\chi_{\alpha}\rangle=e^{i\beta}|\chi_{\alpha}\rangle$ and
$0\leq\beta\leq 2\pi$, but it is not so for the $ALB_{\alpha}(\rho)$. Consider
the decomposition of $\rho$ into pure states as
$\rho=\sum_{i}|\theta_{i}^{\alpha}\rangle\langle\theta_{i}^{\alpha}|$. From
Eq. (51) we know that there is a decomposition of $\rho^{\prime}$ into pure
states as
$\rho^{\prime}=\sum_{i}|\theta_{i}^{{}^{\prime}\alpha}\rangle\langle\theta_{i}^{{}^{\prime}\alpha}|$,
where
$|\theta_{i}^{{}^{\prime}\alpha}\rangle=u_{A}\otimes{u_{B}}|\theta_{i}^{\alpha}\rangle$.
So, using Eq. (36):
$\sum_{i}|\langle\chi_{\alpha}|\theta_{i}^{{}^{\prime}\alpha}\rangle|\theta_{i}^{{}^{\prime}\alpha}\rangle|=\sum_{i}|\langle\chi_{\alpha}|\theta_{i}^{\alpha}\rangle|\theta_{i}^{\alpha}\rangle|=ALB_{\alpha}(\rho).$
(52)
But
$\sum_{i}|\langle\chi_{\alpha}|\theta_{i}^{{}^{\prime}\alpha}\rangle|\theta_{i}^{{}^{\prime}\alpha}\rangle|\geq\min_{\left\\{|\psi^{\prime}_{j}\rangle\right\\}}\sum_{j}|\langle\chi_{\alpha}|\psi^{\prime}_{j}\rangle|\psi^{\prime}_{j}\rangle|=ALB_{\alpha}(\rho^{\prime}),$
(53)
where the minimum is taken over all decompositions of $\rho^{\prime}$ into
pure states:
$\rho^{\prime}=\sum_{j}|\psi^{\prime}_{j}\rangle\langle\psi^{\prime}_{j}|$.
So:
$ALB_{\alpha}(\rho^{\prime})\leq ALB_{\alpha}(\rho).$ (54)
Note that expressions (38), (51) and (54) show that $tr(\rho\otimes\rho
V_{\alpha})$ bounds the amount of $ALB^{2}_{\alpha}(\rho^{\prime})$, for all
possible $\rho^{\prime}$ in Eq. (51), from below.
Now, using inequalities (14) and (22):
$C^{2}(\rho)\geq\sum_{\alpha}ALB_{\alpha}^{2}(\rho)\geq\sum_{\alpha}tr\left(\rho\otimes\rho
V_{\alpha}\right)\,,$ (55)
where the summation is only over those $\alpha$ for which
$tr\left(\rho\otimes\rho V_{\alpha}\right)\geq 0$.
Example 1. In a $d\times d$ dimensional Hilbert space, isotropic states are
defined as [2]:
$\displaystyle\rho_{{}_{F}}=\frac{1-F}{d^{2}-1}\left(I-|\phi^{+}\rangle\langle\phi^{+}|\right)+F|\phi^{+}\rangle\langle\phi^{+}|\,,$
(56)
$\displaystyle|\phi^{+}\rangle=\sum_{i=1}^{d}\frac{1}{\sqrt{d}}|i_{A}i_{B}\rangle\,,\qquad\qquad$
(57) $\displaystyle 0\leq F\leq 1\,,\qquad
F=\langle\phi^{+}|\rho_{{}_{F}}|\phi^{+}\rangle\,.\quad$ (58)
The concurrence of $\rho_{{}_{F}}$ is known and we have [34]:
$C\left(\rho_{{}_{F}}\right)=max\left\\{0,\sqrt{\frac{2d}{d-1}}\left(F-\frac{1}{d}\right)\right\\}\,.$
(59)
If we rewrite $\rho_{{}_{F}}$ as
$\displaystyle\rho_{{}_{F}}=\frac{1-F}{d^{2}-1}I+\frac{Fd^{2}-1}{d^{2}-1}|\phi^{+}\rangle\langle\phi^{+}|\equiv
gI+h|\phi^{+}\rangle\langle\phi^{+}|\,,\qquad\qquad\quad$
then:
$\displaystyle
tr\left(\rho_{{}_{F}}\otimes\rho_{{}_{F}}V_{(i)}\right)=2d\left(d-1\right)\left[\frac{h^{2}}{d^{2}}-dg^{2}-\frac{2}{d}gh\right]\,.$
(60)
In Eq. (41), if we choose $\left\\{x=p,y=q\right\\}$, then:
$\displaystyle
tr\left(\rho_{{}_{F}}\otimes\rho_{{}_{F}}V_{\alpha}\right)=4\left[\frac{h^{2}}{d^{2}}-2g^{2}-\frac{2}{d}gh\right]\,,$
and the expectation values of other $V_{\alpha}$ are not positive. Since the
case $\left\\{x=p,y=q\right\\}$ occurs $n=\frac{d(d-1)}{2}$ times in a
$d\times d$ dimensional system, we have:
$\displaystyle
tr\left(\rho_{{}_{F}}\otimes\rho_{{}_{F}}\sum_{\alpha}V_{\alpha}\right)=2d(d-1)\left[\frac{h^{2}}{d^{2}}-2g^{2}-\frac{2}{d}gh\right]\,,$
(61)
where the summation is only over those $V_{\alpha}$ for which
$\left\\{x=p,y=q\right\\}$. For $d>2$, Eq. (61) gives a better result than Eq.
(60) (Fig. 1). For $d=2$ both give the same result, as we expect from Eq.
(41).
Fig. 1. Comparison of Eqs. (60), dotted line, and (61), dashed line, for
$d=4$. The solid line is the exact value of concurrence, Eq. (59). The lower
bounds given by $V_{(i)}$ and $\sum_{\alpha}V_{\alpha}$ are set to zero when
the right hand sides of Eqs. (60) and (61) are less than zero.
Fig. 1. Comparison of Eqs. (60), dotted line, and (61), dashed line, for
$d=4$. The solid line is the exact value of concurrence, Eq. (59). The lower
bounds given by $V_{(i)}$ and $\sum_{\alpha}V_{\alpha}$ are set to zero when
the right hand sides of Eqs. (60) and (61) are less than zero.
## 4 Measurable Lower Bounds in terms of One Copy of $\rho$
From the experimental point of view, any lower bound which is defined in terms
of the expectation value of an observable with respect to two identical copies
of $\rho$, encounters, at least, two problems. First, for measuring $V_{(i)}$
or $V_{\alpha}$ we need to measure in an entangled basis in both parts A and
B. Measuring in an entangled basis is more difficult than measuring in a
separable one [12]. Second, it is not clear that the state which enters the
measuring devices is really as $\rho\otimes\rho$ even if we can produce such
state at the source place [46, 10]. So, having lower bounds in terms of the
expectation value of an observable with respect to one copy of $\rho$ is more
desirable.
Using:
$\displaystyle C(\rho)C(\sigma)\geq tr\left(\rho\otimes\sigma
V_{(i)}\right)\,,\qquad i=1,2\,;$ (62) $\displaystyle\Rightarrow
C(\rho)\geq\frac{1}{C(\sigma)}tr\left(\rho\otimes\sigma
V_{(i)}\right)\,,\qquad$ (63)
for arbitrary $\rho$ and $\sigma$, F. Mintert has introduced the following
measurable lower bound on $C(\rho)$ [14]:
$\displaystyle C(\rho)\geq-tr\left(\rho W_{\sigma}\right)\,,\qquad
W_{\sigma}=\frac{-1}{C(\sigma)}tr_{2}\left(I\otimes\sigma V_{(i)}\right)\,,$
(64)
where $\sigma$ is a pre-determined entangled state and the partial trace is
taken over the second copy of $\mathcal{H}_{A}\otimes\mathcal{H}_{B}\ $. If
$C(\sigma)$ is not computable, which is the case for almost all mixed
$\sigma$, an upper bound of $C(\sigma)$ can be used in the definition of
$W_{\sigma}$. From inequality (64), it is obvious that for any separable
state: $tr\left(\rho_{s}W_{\sigma}\right)\geq 0$. If, at least, for one
entangled state $tr\left(\rho_{e}W_{\sigma}\right)<0$, then $W_{\sigma}$ is an
entanglement witness [2].
We can, also, construct measurable lower bounds in terms of one copy of $\rho$
by using inequality (37). Suppose that the decomposition of $\sigma$ which
gives the minimum in Eq. (15) is
$\sigma=\sum_{j}|\gamma_{j}^{\alpha}\rangle\langle\gamma_{j}^{\alpha}|$, i.e.:
$ALB_{\alpha}(\sigma)=\sum_{j}|\langle\chi_{\alpha}|\gamma_{j}^{\alpha}\rangle|\gamma_{j}^{\alpha}\rangle|\,.$
(65)
Using expressions (36), (21) and (65):
$\displaystyle\left[ALB_{\alpha}(\rho)\right]\left[ALB_{\alpha}(\sigma)\right]=\sum_{ij}|\langle\chi_{\alpha}|\theta_{i}^{\alpha}\rangle|\theta_{i}^{\alpha}\rangle||\langle\chi_{\alpha}|\gamma_{j}^{\alpha}\rangle|\gamma_{j}^{\alpha}\rangle|$
$\displaystyle\geq\sum_{ij}\langle\theta_{i}^{\alpha}|\langle\gamma_{j}^{\alpha}|V_{\alpha}|\theta_{i}^{\alpha}\rangle|\gamma_{j}^{\alpha}\rangle=-tr\left(\rho
W_{\sigma\alpha}^{\prime}\right)\,,\quad$ $\displaystyle
W_{\sigma\alpha}^{\prime}=-tr_{2}\left(I\otimes\sigma
V_{\alpha}\right)\,.\qquad\qquad\qquad$
So:
$\displaystyle ALB_{\alpha}(\rho)\geq-tr\left(\rho
W_{\sigma\alpha}\right)\,,\qquad\qquad
W_{\sigma\alpha}=\frac{1}{ALB_{\alpha}(\sigma)}W_{\sigma\alpha}^{\prime}\,,$
(66)
where $\sigma$ is a pre-determined entangled state for which
$ALB_{\alpha}(\sigma)>0$. Note that, in contrast to $C(\sigma)$,
$ALB_{\alpha}(\sigma)$ is always computable, so we never need to use an upper
bound of it in the definition of $W_{\sigma\alpha}$. In addition, it can be
shown simply that
$tr\left(\rho W_{\sigma\alpha}\right)=tr\left(\varrho
W_{\sigma\alpha}\right)\,,$ (67)
where $\varrho$ is defined in Eq. (44). So any $\rho$ which is detected by
$W_{\sigma\alpha}$ is distillable. Also, using inequalities (27) and (66):
$C^{2}(\rho)\geq\sum_{\alpha}\left[ALB_{\alpha}(\rho)\right]^{2}\geq\sum_{\alpha}\left[tr\left(\rho
W_{\sigma\alpha}\right)\right]^{2}\,,$ (68)
where the summation is over those $\alpha$ for which $tr\left(\rho
W_{\sigma\alpha}\right)\leq 0$.
For isotropic states,using expressions (64) or (68) (by choosing
$\sigma=|\phi^{+}\rangle\langle\phi^{+}|$) gives the exact value of
$C(\rho_{{}_{F}})$ for arbitrary $d$. In the following, we give an example for
which the expression (68) gives better results than the expression (64).
Example 2. Consider a two-qutrit system which is initially in the pure state
$|\Phi\rangle=\sqrt{\lambda_{0}}|01\rangle+\sqrt{\lambda_{1}}|12\rangle+\sqrt{\lambda_{2}}|20\rangle\,,$
(69)
and its time evolution is given by the following Master equation [14]:
$\displaystyle\dot{\rho}={\cal L}\rho\,,\qquad\qquad$ (70) $\displaystyle{\cal
L}={\cal L}_{A}\otimes 1_{B}+1_{A}\otimes{\cal L}_{B}\,,$ (71)
where ${\cal L}_{A/B}$, for a one-qutrit $\rho_{A/B}$, is
$\displaystyle{\cal
L}_{A/B}=\frac{\Gamma}{2}\left(2\gamma\rho_{A/B}\gamma^{\dagger}-\rho_{A/B}\gamma^{\dagger}\gamma-\gamma^{\dagger}\gamma\rho_{A/B}\right)\,,$
and $\gamma$ is the coupling matrix for the spontaneous decay:
$\displaystyle\gamma=\left(\begin{array}[]{ccc}0&0&0\\\ \sqrt{2}&0&0\\\
0&1&0\end{array}\right)\,.$
To construct $W_{\sigma\alpha}$ in expression (66) and $W_{\sigma}$ in
expression (64), we choose
$\displaystyle\sigma=|\Phi_{ME}\rangle\langle\Phi_{ME}|\,,\qquad\qquad$ (73)
$\displaystyle|\Phi_{ME}\rangle=\frac{1}{\sqrt{3}}\left(|01\rangle+|12\rangle+|20\rangle\right)\,.$
(74)
It can be shown simply that for three $|\chi_{\alpha}\rangle$, for which
$\left\\{p=x\oplus 1,q=y\oplus 1\right\\}$ ($\oplus$ is the sum modulo 3),
$ALB_{\alpha}(\sigma)=2/3$, and $ALB_{\alpha}(\sigma)=0$ for other
$|\chi_{\alpha}\rangle$. So, using expression (66), we can construct three
$W_{\sigma\alpha}$ as ($x=0,1,2$ and $y=x\oplus 1$):
$\displaystyle W_{\sigma\alpha}=|x,y\oplus 1\rangle\langle x,y\oplus
1|+|y,x\oplus 1\rangle\langle y,x\oplus 1|-|x,x\oplus 1\rangle\langle
y,y\oplus 1|-|y,y\oplus 1\rangle\langle x,x\oplus 1|$ (75)
$\displaystyle=|x,y\oplus 1\rangle\langle x,y\oplus 1|+|y,x\oplus
1\rangle\langle y,x\oplus
1|-\frac{1}{2}\left(\sigma_{1}^{xy}\otimes\sigma_{1}^{x\oplus 1,y\oplus
1}-\sigma_{2}^{xy}\otimes\sigma_{2}^{x\oplus 1,y\oplus 1}\right)\,,$ (76)
$\displaystyle\sigma_{1}^{ab}=|a\rangle\langle b|+|b\rangle\langle
a|\,,\qquad\quad\qquad\sigma_{2}^{ab}=-i\left(|a\rangle\langle
b|-|b\rangle\langle a|\right)\,.\qquad\qquad$ (77)
Also, using expression (64), we can show that:
$W_{\sigma}=\frac{1}{\sqrt{3}}\sum_{\alpha=1}^{3}W_{\sigma\alpha}\,.$ (78)
As we can see from Eqs. (77) and (78), the number of local observables needed
for measuring $W_{\sigma}$ or three $W_{\sigma\alpha}$ is the same and is
equal to 12, which is less than what is needed for a full tomography. Also,
note that $\left\\{|l,m\oplus 1\rangle\right\\}$ is an orthonormal basis of
$\mathcal{H}_{A}\otimes\mathcal{H}_{B}$. So, at least from the theoretical
point of view, all the observables $|l,m\oplus 1\rangle\langle l,m\oplus 1|$
can be measured using only one set up. In such cases, for measuring
$W_{\sigma}$ or three $W_{\sigma\alpha}$, we only need 7 different set up of
local measurements. The comparison of the results of inequalities (64) and
(68), for two typical $\left\\{\lambda_{i}\right\\}$, is given in Fig. 2.
Fig. 2. Comparing the lower bounds given by (64), solid line, and (68), dashed
line, for two typical $\left\\{\lambda_{i}\right\\}$: a) $\lambda_{i}=1/3$; b)
$\left\\{\lambda_{0}=1/12,\lambda_{1}=5/6,\lambda_{2}=1/12\right\\}$. When the
lower bound given by $W_{\sigma}$ is less than zero, we set it to zero.
Fig. 2. Comparing the lower bounds given by (64), solid line, and (68), dashed
line, for two typical $\left\\{\lambda_{i}\right\\}$: a) $\lambda_{i}=1/3$; b)
$\left\\{\lambda_{0}=1/12,\lambda_{1}=5/6,\lambda_{2}=1/12\right\\}$. When the
lower bound given by $W_{\sigma}$ is less than zero, we set it to zero.
## 5 Extending to Multipartite Systems
In a bipartite system, any Hermitian operator which, for arbitrary
$|\psi\rangle$ and $|\varphi\rangle$, satisfies the inequality
$C(\psi)C(\varphi)\geq\langle\psi|\langle\varphi|V|\psi\rangle|\varphi\rangle\,,$
(79)
gives a measurable lower bound on $C^{2}(\rho)$, i.e. $C^{2}(\rho)\geq
tr\left(\rho\otimes\rho V\right)$ [10]. This can be proved simply by writing
$\rho$ in terms of its extremal decomposition
$\rho=\sum_{j}|\xi_{j}\rangle\langle\xi_{j}|$. In [18] it was shown how to use
such $V$ to construct measurable lower bounds for multipartite concurrence.
Following a similar procedure, we construct measurable lower bounds on
multipartite concurrence using $V_{\alpha}$. As the previous sections, we will
use the inequality (37) instead of the inequality (79). In other words, we
will work with the algebraic lower bounds of $C(\rho)$ rather than the
concurrence itself.
The concurrence of an N-partite pure state $|\Psi\rangle$,
$|\Psi\rangle\in{\mathcal{H}}_{A_{1}}\otimes\cdots\otimes{\mathcal{H}}_{A_{N}}$,
is defined as [31]:
$C(\Psi)=2^{1-\frac{N}{2}}\sqrt{\sum_{l}C_{l}^{2}(\Psi)}\,,$ (80)
where $\sum_{l}$ is the summation over all possible subdivisions of
${\mathcal{H}}_{A_{1}}\otimes\cdots\otimes{\mathcal{H}}_{A_{N}}$ into two
subsystems, and $C_{l}$ is the related bipartite concurrence. For example, for
a 3-partite system we have three $C_{l}$, namely $C_{1,23},C_{12,3}$ and
$C_{13,2}$. As before we have:
$\displaystyle C_{l}^{2}\left(\Psi\right)=\langle\Psi|\langle\Psi|{\cal
A}_{l}|\Psi\rangle|\Psi\rangle\,,\qquad{\cal
A}_{l}=\sum_{\alpha_{l}}|\chi_{\alpha_{l}}\rangle\langle\chi_{\alpha_{l}}|\,,$
(81)
where $|\chi_{\alpha_{l}}\rangle$ are the same as $|\chi_{\alpha}\rangle$
which have been defined in Eq. (5). Obviously, they are constructed according
to the related subdivision denoted by $l$. So:
$\displaystyle
C(\Psi)=2^{1-\frac{N}{2}}\sqrt{\sum_{l,\alpha_{l}}|\langle\chi_{\alpha_{l}}|\Psi\rangle|\Psi\rangle|^{2}}=2^{1-\frac{N}{2}}\sqrt{\sum_{\gamma}|\langle\chi_{\gamma}|\Psi\rangle|\Psi\rangle|^{2}}\,,$
(82)
where instead of $l$ and $\alpha_{l}$ we have used a collective index
$\gamma$. From now on, everything is as the bipartite case, except that we
deal with the summation over $\gamma$ instead of $\alpha$. The definition of
concurrence for mixed states is as follows:
$\displaystyle
C(\rho)=\min_{\left\\{|\psi_{i}\rangle\right\\}}\sum_{i}C(\psi_{i})=\min_{\left\\{|\psi_{i}\rangle\right\\}}\sum_{i}2^{1-\frac{N}{2}}\sqrt{\langle\psi_{i}|\langle\psi_{i}|{\cal
A}^{\prime}|\psi_{i}\rangle|\psi_{i}\rangle}\,,$ (83) $\displaystyle{\cal
A}^{\prime}=\sum_{\gamma}|\chi_{\gamma}\rangle\langle\chi_{\gamma}|\,,\qquad\qquad\qquad\qquad$
(84)
where the minimization is over all decompositions of $\rho$ into subnormalized
states $|\psi_{i}\rangle$: $\rho=\sum_{i}|\psi_{i}\rangle\langle\psi_{i}|$. It
is worth noting that $C(\rho)$, as difined in Eq. (84), is an entanglement
monotone for the multipartite case too [47].
If we define
$|\chi_{\upsilon}^{\prime}\rangle=\sum_{\gamma}U^{\prime}_{\upsilon\gamma}|\chi_{\gamma}\rangle$,
where $U^{\prime}$ is a unitary matrix, then ${\cal
A}^{\prime}=\sum_{\gamma}|\chi_{\gamma}\rangle\langle\chi_{\gamma}|=\sum_{\gamma}|\chi^{\prime}_{\gamma}\rangle\langle\chi^{\prime}_{\gamma}|$.
So, by similar reasoning leading to inequality (23), we have:
$\displaystyle C(\rho)\geq
LB_{\tau}(\rho)=\min_{\left\\{|\psi_{i}\rangle\right\\}}\sum_{i}2^{1-\frac{N}{2}}|\langle\tau|\psi_{i}\rangle|\psi_{i}\rangle|\,,$
(85)
$\displaystyle|\tau\rangle\equiv|\chi^{\prime}_{1}\rangle=\sum_{\gamma}z_{\gamma}^{\ast}|\chi_{\gamma}\rangle\,,\qquad\sum_{\gamma}|z_{\gamma}|^{2}=1\,.\qquad$
(86)
As before, in contrast to $C(\rho)$, $LB_{\tau}(\rho)$ is always computable.
We also have:
$\displaystyle
C^{2}(\rho)\geq\sum_{\gamma}\left[LB_{\gamma}(\rho)\right]^{2}\,,\qquad
LB_{\gamma}(\rho)=\min_{\left\\{|\psi_{i}\rangle\right\\}}2^{1-\frac{N}{2}}\sum_{\gamma}|\langle\chi_{\gamma}^{\prime}|\psi_{i}\rangle|\psi_{i}\rangle|\,.$
(87)
The above expression is the counterpart of the inequality (27) for the
multipartite case. What was proved in [43], neglecting an unimportant constant
in the definition of $C(\rho)$, is, in fact, the inequality (87) for the
special case of ${|\chi_{\gamma}^{\prime}\rangle}={|\chi_{\gamma}\rangle}$
(see Eqs. (16) and (17)).
According to the inequality (37),for any $|\chi_{\gamma}\rangle$:
$|\langle\chi_{\gamma}|\psi\rangle|\psi\rangle||\langle\chi_{\gamma}|\varphi\rangle|\varphi\rangle|\geq\langle\psi|\langle\varphi|V_{\gamma}|\psi\rangle|\varphi\rangle\,,$
(88)
where $V_{\gamma}$ are the same as $V_{\alpha}$ introduced in Eqs. (23) and
(24), defined according to the related $|\chi_{\gamma}\rangle$. So:
$C^{2}(\rho)\geq 2^{2-N}\sum_{\gamma}tr\left(\rho\otimes\rho
V_{\gamma}\right)\,,$ (89)
where the summation is over those $\gamma$ for which $tr\left(\rho\otimes\rho
V_{\gamma}\right)\geq 0$. Also, we have:
$\displaystyle C^{2}(\rho)\geq\sum_{\gamma}\left[tr\left(\rho
W_{\sigma\gamma}\right)\right]^{2}\,,\qquad
W_{\sigma\gamma}=\frac{-2^{2-N}}{ALB_{\gamma}(\sigma)}tr_{2}\left(I\otimes\sigma
V_{\gamma}\right)\,,$ (90)
where $\sigma$ is a pre-determined density operator with
$ALB_{\gamma}(\sigma)>0$, and the summation is over those $\gamma$ for which
$tr\left(\rho W_{\sigma\gamma}\right)\leq 0$.
## 6 Summery and Discussion
Inequality (37) is the main relation of this paper. Using this expression, we
have constructed measurable lower bounds on concurrence in term of both one
copy or two identical copies of $\rho$. We have proved that the inequality
(37) holds for $V_{\alpha}$ introduced in Eq. (41). Now verifying whether it
is possible to find $V^{\prime}_{\alpha}$ for which (37) holds for arbitrary
$|\chi_{\alpha}^{\prime}\rangle$ is valuable.
Our measurable bounds are related to the $ALB_{\alpha}(\rho)$ rather than the
concurrence itself, as we have seen in expressions (38) and (66). So we can
use (27) to get the relations (55) and (68). Inequality (55)(Inequality (68))
has this advantage that we can omit the summation over those $\alpha$ for
which $tr\left(\rho\otimes\rho V_{\alpha}\right)\leq 0$ ($tr\left(\rho
W_{\sigma\alpha}\right)\geq 0$). This useful property can help us to achieve
better results in detecting the entanglement. As an example, $W_{\sigma}$ in
Eq. (78) is, up to a constant, the summation of three $W_{\sigma\alpha}$. Now,
using expression (68), we can omit each $W_{\sigma\alpha}$ for which
$tr\left(\rho W_{\sigma\alpha}\right)\geq 0$; But, using $W_{\sigma}$, we can
not omit any $W_{\sigma\alpha}$ in Eq. (78). So, as it is shown in Fig. 2, the
ability of $W_{\sigma}$ in detecting the entanglement reduces more rapidly
than the three distinct $W_{\sigma\alpha}$.
Bounds obtained from $V_{\alpha}$ or $W_{\sigma\alpha}$ are always less than
or equal to the $ALB_{\alpha}(\rho)$. In addition, we have shown that these
bounds can not detect bound entangled states. So $ALB_{\alpha}(\rho)>0$ and
$N(\rho)>0$ ($N(\rho)$ is the negativity of the system [48]) are two necessary
conditions for detection of the entanglement by $V_{\alpha}$ or
$W_{\sigma\alpha}$. However, the ability of these bounds and also comparing
them with other observable bounds, especially those introduced in [8, 14],
need further studies. For example, in the definition of $W_{\sigma\alpha}$,
mixed states $\sigma$ can be used simply instead of pure states $\sigma$ since
$ALB_{\alpha}(\sigma)$ is always computable. Studying the above case seems
interesting.
At last, in section V, we have generalized our measurable bounds to the
multipartite case. The applicability of these bounds also needs further
studies.
Acknowledgements
We would like to thank the anonymous referees for their helpful suggestions
and improvements.
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Appendix A
In this appendix, we prove inequality (21) for $V_{\alpha}$ introduced in Eq.
(41). We prove it for $V_{(2)\alpha}$; the case of $V_{(1)\alpha}$ can be done
analogously.
Any arbitrary $|\psi\rangle$ and $|\varphi\rangle$ can be decomposed in a
separable basis of $\mathcal{H}_{A}\otimes\mathcal{H}_{B}\ $, like
${|i_{A}\rangle|j_{B}\rangle}$, as:
$\displaystyle|\psi\rangle=\sum_{ij}\psi_{ij}|i_{A}j_{B}\rangle\,,$
$\displaystyle|\varphi\rangle=\sum_{ij}\varphi_{ij}|i_{A}j_{B}\rangle\,.$
Now, from Eq. (41), we have:
$\displaystyle\langle\psi|\langle\varphi|V_{(2)\alpha}|\psi\rangle|\varphi\rangle=\qquad\qquad\qquad\qquad$
(A.1) $\displaystyle
2\left[-|\psi_{xq}\varphi_{yq}-\psi_{yq}\varphi_{xq}|^{2}-|\psi_{xp}\varphi_{yp}-\psi_{yp}\varphi_{xp}|^{2}+AA\right]\,,\quad$
(A.2) $\displaystyle
AA=-2Re\left(\psi_{xp}\varphi_{yq}\psi_{xq}^{\ast}\varphi_{yp}^{\ast}\right)-2Re\left(\psi_{yp}\varphi_{xq}\psi_{yq}^{\ast}\varphi_{xp}^{\ast}\right)\qquad$
(A.3)
$\displaystyle+2Re\left(\psi_{xp}\varphi_{yq}\psi_{yq}^{\ast}\varphi_{xp}^{\ast}\right)+2Re\left(\psi_{xq}\varphi_{yp}\psi_{yp}^{\ast}\varphi_{xq}^{\ast}\right)\,.\qquad$
(A.4)
Also for
$|\chi_{\alpha}\rangle=\left(|xy\rangle-|yx\rangle\right)_{A}\left(|pq\rangle-|qp\rangle\right)_{B}$
we have:
$\displaystyle|\langle\chi_{\alpha}|\psi\rangle|\psi\rangle||\langle\chi_{\alpha}|\varphi\rangle|\varphi\rangle|\qquad\qquad$
(A.5)
$\displaystyle=4|\left(\psi_{xp}\psi_{yq}-\psi_{xq}\psi_{yp}\right)\left(\varphi_{xp}\varphi_{yq}-\varphi_{xq}\varphi_{yp}\right)|$
(A.6) $\displaystyle\equiv 4|BB|\,.\qquad\qquad\qquad\quad$ (A.7)
To get the inequality (21), we must show:
$\displaystyle AA\leq 2|BB|+|\psi_{xq}\varphi_{yq}-\psi_{yq}\varphi_{xq}|^{2}$
(A.8) $\displaystyle+|\psi_{xp}\varphi_{yp}-\psi_{yp}\varphi_{xp}|^{2}\,.$
(A.9)
If we have:
$\displaystyle AA\leq 2|BB|+2|CC|\,,\qquad\quad$ (A.10) $\displaystyle
CC=\left(\psi_{xq}\varphi_{yq}-\psi_{yq}\varphi_{xq}\right)\left(\psi_{xp}\varphi_{yp}-\psi_{yp}\varphi_{xp}\right)\,,$
(A.11)
then inequality (A.3) holds. To get the inequality (A.11), it is sufficient to
have:
$\displaystyle\frac{AA}{2}\leq|BB+CC|\quad\qquad\quad$ (A.12)
$\displaystyle=|\left(\psi_{xp}\varphi_{yq}-\psi_{yp}\varphi_{xq}\right)\left(\psi_{yq}^{\ast}\varphi_{xp}^{\ast}-\psi_{xq}^{\ast}\varphi_{yp}^{\ast}\right)|\,.$
(A.13)
But, the above expression holds since for any complex number $z$, we have
$Re(z)\leq|z|$, which completes the proof.
|
arxiv-papers
| 2009-10-10T15:42:56 |
2024-09-04T02:49:05.781943
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Iman Sargolzahi, Sayyed Yahya Mirafzali, and Mohsen Sarbishaei",
"submitter": "Iman Sargolzahi",
"url": "https://arxiv.org/abs/0910.1928"
}
|
0910.1931
|
# Search for Supersymmetry in $\textrm{p}\overline{\textrm{p}}$ Collisions at
$\sqrt{\textrm{s}}$ =1.96 TeV Using the Trilepton Signature of Chargino-
Neutralino Production
R. Forrest Department of Physics, University of California, Davis, Davis, CA
95616, USA
###### Abstract
The production of chargino-neutralino pairs and their subsequent leptonic
decays is one of the most promising supersymmetry (SUSY) signatures at the
Tevatron $p\bar{p}$ collider. We present here the most recent results on the
search for the three-lepton and missing-transverse-energy SUSY signature using
data collected with the CDF II detector. The results are interpreted within
the minimal supergravity (mSugra) scenario.
## I Introduction
In the search for new phenomena, one well-motivated extension to the Standard
Model (SM) is supersymmetry (SUSY). SUSY particles (sparticles) contribute to
the Higgs mass squared with opposite sign relative to the contributions of SM
particles, and thus protect the weak mass scale, $M_{W}$, from divergences.
SUSY is a broken symmetry since the sparticles obviously do not have the same
mass as their SM partners, but the breaking must be ‘soft’ to allow the
divergence canceling to remain effective. If $R_{p}$ parity is
conserved111$R_{p}=(-1)^{3B+L+2S}$, where $B$ is baryon number, $L$ is lepton
number, and $S$ is spin., the lightest SUSY particle (LSP) is absolutely
stable and provides a viable candidate for cosmological dark matter
susy_primer . We use as a reference the mSugra model of SUSY breaking. This
model has the virtue of containing only five free parameters to specify.
However, our search is signature-based; we do not modify our selection to
follow the details of mSugra.
One very promising mode for SUSY discovery at hadron colliders is that of
chargino-neutralino associated production with decay into three leptons.
Charginos decay into a single lepton through a slepton
$\tilde{\chi}_{1}^{\pm}\rightarrow~{}\tilde{l}^{(*)}~{}\nu_{l}\rightarrow\tilde{\chi}_{1}^{0}~{}l^{\pm}~{}\nu_{l}$
and neutralinos similarly decay into two detectable leptons
$\tilde{\chi}_{2}^{0}\rightarrow~{}\tilde{l}^{\pm(*)}~{}l^{\mp}\rightarrow\tilde{\chi}_{1}^{0}~{}l^{\pm}~{}l^{\mp}.$
The decays can also proceed via $W$ and $Z$ bosons. The detector signature is
thus three SM leptons with associated missing energy from the undetected
neutrinos and lightest neutralinos, $\tilde{\chi}_{1}^{0}$ (LSP), in the
event. Due to its electroweak production, this is one of the few ‘jet-free’
SUSY signatures.
## II Detector, Data and Analysis Overview
This analysis is preformed with the CDF II detector at the Tevatron with
$p\bar{p}$ collisions at $\sqrt{s}=1.96$ TeV. The CDF II detector is a mostly
cylindrical particle detector composed of cylindrical sub-detectors. From the
beam axis outwards there is a silicon strip vertex detector, and a gas filled
drift chamber. This tracking system is surrounded by a solenoid providing a
1.4 T magnetic field, followed by electromagnetic and hadronic calorimeters.
The outermost detectors are wire chambers used to detect muons that escape the
inner detectors.
For this analysis we use two categories of event triggers. The first is the
high $P_{t}$ inclusive lepton trigger, which consist of single lepton objects,
the second is the SUSY dilepton trigger witch allows two lower $P_{t}$
leptons. These data are combined and in the analysis overlapping trigger
effects and efficiencies are accounted for. These data were collected up until
1 Jul, 2008, totaling $3.23~{}\textrm{fb}^{-1}$ for the unprescaled triggers.
We follow the same analysis strategy and implementation used in the previous
CDF II search rut_note . From the outset, we define lepton categories and
event level trilepton channels. Each lepton and category is exclusive and
selected based on expected purity. This channel independence allows easy
statistical combination of the final results.
The general procedure is as follows. For each event, we select muons,
electrons and tracks of some quality. Each of these objects, except the tracks
(T), have tight (t) and loose (l) categories. We then define event level
exclusive trilepton channels composed of combinations of these objects and
arrange them sequentially by expected signal sensitivity. There are several
virtues of this approach. The largest advantage is that we perform several
lepton flavor, channel-specific searches simultaneously, without the need to
account for overlapping results.
We define two selection stages to test our background estimations against
data. The first stage is the dilepton selection, which consists of the first
two objects of the trilepton selection. The second stage is the final
trilepton selection, with some event cuts applied. Once we are satisfied with
the agreement in the control regions, we apply SUSY specific cuts and look at
signal region data to compare against background.
## III Object Selection, Event Categories and Cuts
We define both tight and loose lepton categories as well as a track object.
All of these objects are central to the detector, meaning that generally
$|\eta|<1.0$ and they are isolated from nearby objects. Tight muons are
objects that have tracks, deposit a minimum amount of ionizing energy in the
calorimeter system, and are detected in the outer muon systems. Loose muons
are similar, but the requirement of the muon system detection is relaxed; they
compensate for gaps in the muon detector coverage. Tight electrons are
similarly again required to leave a good track, but they are expected to
deposit a majority of their energy in the electromagnetic calorimeter. Loose
electrons have slightly fewer requirements on the matching between objects in
the sub-detectors. We also include one type of track object in the analysis as
a possible third object. This greatly increases our sensitivity by allowing
detection of leptons that failed selection cuts, as well as single pronged
hadronic tau decays. The track object is a single, isolated track in the
tracking chamber. It differs from loose muons in that, it can have an
arbitrary amount of energy deposition.
After the object selection, we categorize the trilepton events. We first look
for three tight leptons. If the event does not qualify, we look for two tight
and one loose lepton. If the event still does not qualify, we look for one
tight and two loose leptons. Events that do not make it into the trilepton
selection are tested for two tight leptons and a track and finally one tight,
one loose and one track object. The complete list is shown in Table 1 along
with the $E_{t}$ (electrons) or $P_{t}$ (muons) requirements on these objects
222Some selections differ slightly to increase sensitivity or accommodate
standard object definitions, see rut_note ..
Channel | Selection | $E_{t}$ or $P_{t}$
---|---|---
ttt | 3 tight leptons | 15, 5, 5
ttC | 2 tight and 1 loose lepton | 15, 5, 5
tll | 1 tight and 2 loose leptons | 20, 8, 5
tt$T$ | 2 tight leptons and 1 track | 15, 5, 5
tl$T$ | 1 tight lepton, 1 loose lepton and 1 track | 20, 8, 5
Table 1: Trilepton selection event categories.
At this stage we apply additional event level cleaning cuts. We require that
every analysis level object (leptons, tracks and jets) be separated from each
other by $\Delta R>0.4$. Events with a mismeasured jet can have false
$\,/\\!\\!\\!\\!E_{T}$. We remove events with $\,/\\!\\!\\!\\!E_{T}$ and any
jet separated by less than $\Delta\phi<0.35$. We also make invariant mass cuts
at this stage. The highest opposite signed object pair invariant mass is
required to be above $20\ \textrm{GeV}/c^{2}$ and the second highest
oppositely-charged object pair is required to be above $13\
\textrm{GeV}/c^{2}$. This cut helps eliminate heavy flavor backgrounds.
Additional backgrounds due to mismeasurement are removed by cutting events
that have $\,/\\!\\!\\!\\!E_{T}$ and leptons aligned, requiring
$\Delta\phi>0.17$ for each of the leading two leptons.
To further clean up events, we require the third lepton in trilepton events to
be isolated. We also require that there not be more than three leptons or
tracks in the event above 10 GeV and that the three objects’ charges sum to
$\pm 1$.
### III.1 Backgrounds
The standard model background estimation for the analysis differs slightly
between the lepton+track channels and the trilepton channels. Generally, Monte
Carlo is used to estimate the backgrounds, but isolated track, fake lepton and
gamma conversion rates are determined from data.
### III.2 Trilepton Backgrounds
Backgrounds are treated differently based on the underlying process. Those
that give three real leptons (WZ , ZZ, $t\bar{t}$) are estimated with Monte
Carlo by simply taking them through the analysis.
The remaining background processes have two real leptons (Z, WW) and require a
third object from elsewhere in the event. This can happen, for example, with
FSR photon conversion where a photon radiated off a charged particle hits
matter in the detector and converts to an $e\bar{e}$ pair. For these
processes, we estimate this 2 lepton plus conversion rate from Monte Carlo.
The final contribution to the trilepton background is from objects in the
underlying event faking a third lepton in an event that has two genuine
leptons. This fake contribution is estimated in the trilepton channels by
selecting two well identified leptons and a third fakeable object from data
events. Fake rates have been measured for jets faking electrons and for tracks
faking muons of both tight and loose quality. These jets and tracks are the
fakeable objects selected. The event is then carried through the analysis and
weighted by the appropriate fake rate.
### III.3 Dilepton + Track Backgrounds
For channels with tracks, backgrounds are handled slightly differently.
Background processes that give three real leptons (WZ , ZZ, $t\bar{t}$) are
still estimated from Monte Carlo as previously described.
As for fakes in dilepton + track channels, we account for fake leptons, and
separately estimate the rate of isolated tracks in dilepton backgrounds.
For fake leptons, we use a method similar to the trilepton method but
calculated from data by selecting lepton + track events containing a fakeable
object. As was done with trilepton fakes, we carry the event through the
analysis, and apply the appropriate fake rate to the event.
The remaining background in the dilepton + track channels is that of dilepton
events with an isolated track from the underlying event. We measure the rate
of extraneous isolated tracks from data, and apply this rate to dilepton Monte
Carlo. This procedure gives very good agreement in our dilepton + track
control regions.
## IV Control Regions
We inspect both our dilepton selection and our trilepton selection for
agreement against predictions. The control region parameter space is
$\,/\\!\\!\\!\\!E_{T}$ vs. Invariant mass, and for easy reference is coded
according to Figure 1.
Figure 1: Control regions and codes used to refer to the control regions.
We select the first two leptons in the event and check agreement against
backgrounds. See Figure 2 for a complete listing of all the dilepton control
regions. A dilepton kinematic plot is displayed and described in Figure 3.
Figure 2: Summary of dilepton control regions. (Observed - Expected) /
Expected number of events for each control region. Figure 3: Invariant mass of
the first two tight leptons in events with low $\,/\\!\\!\\!\\!E_{T}$.
After we are satisfied with the dilepton control region agreement, the
trilepton selection is applied to an event. We check trilepton plots and
tables to ensure good agreement between background and predictions. The total
trilepton background and prediction comparison is shown in Figure 4.
Figure 4: Trilepton Control Region Summary. (Observed - Expected)/Expected
number of events.
We again look at distributions comparing data and predictions in control
regions. Trilepton control region plots are shown in Figure 5.
Figure 5: Energy of leading lepton in events with low $\,/\\!\\!\\!\\!E_{T}$
in the ttt channel.
## V Results and Limits
For an mSugra reference point we use
$\textrm{M}_{0}=60,\textrm{M}_{1/2}=190,\textrm{tan}\beta=3,\textrm{A}_{0}=0$;
the results of background and expected signal are shown in Table 2. After
looking at the signal region in the data, we see a total of seven signal
events on an expected background of $10.84\pm 1.34$ events.
CDF II Preliminary, 3.2 $\textrm{fb}^{-1}$
---
Channel | Total Background $\pm$ (stat) $\pm$ (sys) | Signal Point $\pm$ (stat) $\pm$ (sys) | Observed
ttt | 0.83 $\pm$ 0.14 $\pm$ 0.11 | 3.64 $\pm$ 0.22 $\pm$ 0.49 | 1
ttC | 0.39 $\pm$ 0.07 $\pm$ 0.04 | 2.62 $\pm$ 0.18 $\pm$ 0.35 | 0
tll | 0.25 $\pm$ 0.08 $\pm$ 0.03 | 1.12 $\pm$ 0.12 $\pm$ 0.15 | 0
ttT | 5.85 $\pm$ 0.57 $\pm$ 1.11 | 7.15 $\pm$ 0.31$\pm$ 0.91 | 4
tlT | 3.53 $\pm$ 0.52 $\pm$ 0.5 | 4.06 $\pm$ 0.23 $\pm$ 0.53 | 2
mSugra Signal point:
$\textrm{M}_{0}=60,\textrm{M}_{1/2}=190,\textrm{tan}\beta=3,\textrm{A}_{0}=0$
Table 2: Expected background and signal, errors are statistical and full
systematic.
To extract a 1-D 95% confidence level limit, we set $M_{0}=60$ and vary
$M_{1/2}$ which has the direct effect of varying the chargino mass. For each
point we scan, we get the expected limit based on the acceptance of our
analysis to the signal at that point. If we plot this against the theoretical
$\sigma\times BR$ of the signal mSugra point as a function of chargino mass,
we expect to exclude regions where our analysis’s $\sigma\times BR$ is less
than the theoretical value. Our expected limit is about 156 GeV/$c^{2}$ Figure
6, while we observe a limit of 164 GeV/$c^{2}$.
Figure 6: Expected and observed limit for the mSugra model
$M_{0}=60,tan\beta=3,A_{0}=0,(\mu)>0$. In red is the theoretical $\sigma\times
BR$ and in black is our expected limit with one and two $\sigma$ errors. We
expect to set a limit of about 156 GeV/$c^{2}$, and observe a limit of 164
GeV/$c^{2}$.
To explore a broader parameter space it is useful to scan both $M_{0}$ and
$M_{1/2}$ simultaneously. We calculate NLO cross section of the process as a
function of $M_{0}$ and $M_{1/2}$. We then calculate branching ratio to three
leptons in this same range. This gives us a plot of $\sigma\times BR$. We
generate signal Monte Carlo to test the expected and observed sensitivity at
many points in $M_{0}$ and $M_{1/2}$ space.
Figure 7: Expected and observed limit contours for the mSugra model
$tan\beta=3,A_{0}=0,(\mu)>0$ in $M_{1/2}$ vs $M_{0}$ space.
We calculate (Expected - Theory $\sigma\times$ BR) /(Theory $\sigma\times$ BR)
for both the expected and observed limits. The final exclusion contains both
of these contours which can be seen in Figure 7.
Our observed 1-D limit excludes chargino masses of less than $164\
\textrm{GeV}/c^{2}$, an improvement over the expectation due to the deficit of
data events in the lepton + track channels.
## References
* (1) A Supersymmetry Primer, Stephen P. Martin, hep-ph/9709356.
* (2) Search for Supersymmetry in $p\bar{p}$ Collisions at $\sqrt{s}$ = 1.96 TeV Using the Trilepton Signature for Chargino-Neutralino Production, CDF Collaboration, Phys. Rev. Lett. 101, 251801 (2008), DOI:10.1103/PhysRevLett.101.251801
|
arxiv-papers
| 2009-10-10T16:53:34 |
2024-09-04T02:49:05.788675
|
{
"license": "Public Domain",
"authors": "R. Forrest (for the CDF Collaboration)",
"submitter": "Robert Forrest",
"url": "https://arxiv.org/abs/0910.1931"
}
|
0910.1950
|
Also at ]A.F. Ioffe Physico-Technical Institute, 194021 St. Petersburg, Russia
# Light-induced valley currents and magnetization in graphene rings
A.S. Moskalenko andrey.moskalenko@physik.uni-halle.de [ J. Berakdar
Institut für Physik, Martin-Luther-Universität Halle-Wittenberg,
Nanotechnikum-Weinberg, Heinrich-Damerow-St. 4, 06120 Halle, Germany
###### Abstract
We study the non-equilibrium dynamics in a mesoscopic graphene ring excited by
picoseconds shaped electromagnetic pulses. We predict an ultrafast buildup of
charge polarization, currents and orbital magnetization. Applying the light
pulses identified here, non-equilibrium valley currents are generated in a
graphene ring threaded by a stationary magnetic flux. We predict a finite
graphene ring magnetization even for a vanishing charge current; the
magnetization emerges due to the light-induced difference of the valley
populations.
###### pacs:
73.63.-b,73.23.-b,73.22.Gk,81.05.Uw
Introduction.- Since the recent fabrication of graphene, a monolayer of
carbon, a number of fascinating phenomena have been uncovered, mostly owing to
the quasi-relativistic behavior of the carriers and their high mobility Neto
et al. (2009); Novoselov et al. (2005); Geim and Novoselov (2007); Beenakker
(2008); Morozov et al. (2008). Two $\sigma$-bonded interpenetrating triangular
sublattices, $A$ and $B$, build the graphene honeycomb lattice. The $\pi$ and
$\pi^{*}$ bands govern the electronic properties near the neutrality point and
result in conical valleys touching at the high symmetry points $K$ and
$K^{\prime}$ of the Brillouin zone (BZ). Near $K$ and $K^{\prime}$ the energy
dispersion is linear and the electronic properties are well described by the
effective Dirac-Weyl Hamiltonian $H_{0}$ Novoselov et al. (2005); Geim and
Novoselov (2007); Beenakker (2008) 111$H_{0}=v\vec{\sigma}\cdot\vec{p}$, where
$\vec{p}$ is the momentum operator, $\vec{\sigma}=(\sigma_{x},\sigma_{y})$ and
$v$ is the Fermi velocity. $\sigma_{x}$ and $\sigma_{y}$ are Pauli matrices
built on the basis of the pseudospin wavefunctions corresponding to the
different sublattices.. The stationary states are degenerate in the spin
${\cal S}$ and the valley quantum numbers $\tau=\pm 1$. The latter correspond
to the two non-equivalent $K$-points in BZ. Due to the suppressed intervalley
scattering the control of $\tau$ eigenstates may be utilized for novel
electronic Rycerz et al. (2007); Xiao et al. (2007) and optoelectronic
applications Yao et al. (2008). New physical effects emerge due to
confinement. E.g., in a mesoscopic graphene rings pierced by a magnetic flux,
the ring confinement breaks the valley degeneracy and results in the
persistent current Recher et al. (2007). Experimentally, such graphene rings
were fabricated and the Aharonov-Bohm effect was observed Russo et al. (2008).
While a large body of work has been devoted to various equilibrium electronic
and optical properties, the non-equilibrium time-dependent phenomena in
graphene are much less explored Mikhailov and Ziegler (2008); Syzranov et al.
(2008). The present paper presents the first study on the non-equilibrium
dynamics in graphene rings driven by asymmetric monocycle electromagnetic
pulses. Charge polarization and current carrying states build up within
picoseconds and are tunable by the parameters of the driving field. The states
may become valley polarized resulting in a non-equilibrium valley currents.
The valley population together with the charge current determine the
magnetization of the ring.
Stationary states.- We consider a graphene ring Recher et al. (2007) of radius
$r_{0}$ and width $W$ (cf. Fig.2(a)) threaded by a magnetic flux of a strength
$\Phi$. As in Berry and Mondragon (1987); Recher et al. (2007); Akhmerov and
Beenakker (2008), the Dirac electrons are confined to the ring by the
potential $\tau V(r)\sigma_{z}$ at the boundaries, as resulting e.g. from a
substrate potential Zhou et al. (2007); Rotenberg et al. (2008). The polar-
coordinates ring hamiltonian is [29]
$H=-i\hbar
v\left[\sigma_{r}\partial_{r}+\sigma_{\phi}\frac{1}{r}(\partial_{\phi}+i\tilde{\Phi})\right]+\tau
V(r)\sigma_{z}\;,$ (1)
where $\sigma_{r}=\vec{\sigma}\cdot\vec{e}_{r}$ and
$\sigma_{\phi}=\vec{\sigma}\cdot\vec{e}_{\phi}$ with $\vec{e}_{r}$ and
$\vec{e}_{\phi}$ being the basis vectors of the polar coordinate system and
$\sigma_{z}$ is the Pauli matrix expressed in the pseudospin states of the two
sublattices. $v=10^{6}$ m/s is the Fermi velocity and
$\tilde{\Phi}=\Phi/\Phi_{0}$, where $\Phi_{0}$ is the flux quantum. The
eigenstates of $H$ and $J_{z}$ (the $z$ component of the total angular
momentum with eigenvalues $m$) are
$\psi_{m}(r,\phi)=R_{+}(r)e^{i(m-1/2)\phi}\chi_{+}+R_{-}(r)e^{i(m+1/2)\phi}\chi_{-}\;,$
(2)
where $m=\pm\frac{1}{2},\pm\frac{3}{2},...$ and
$\sigma_{z}\chi_{\pm}=\pm\frac{1}{2}\chi_{\pm}$. The general form for the
radial parts is
$R_{+}(r)=c_{1}H^{(1)}_{\overline{m}-\frac{1}{2}}(\tilde{r})+c_{2}H^{(2)}_{\overline{m}-\frac{1}{2}}(\tilde{r})$
and
$R_{-}(r)=is\left[c_{1}H^{(1)}_{\overline{m}+\frac{1}{2}}(\tilde{r})+c_{2}H^{(2)}_{\overline{m}+\frac{1}{2}}(\tilde{r})\right]$,
where $\tilde{r}=r|E|/v$ is a normalized radial coordinate, $H_{m}^{(1)[(2)]}$
is the Hankel function of the first [second] kind, $s=\mbox{sgn}(E)$ selects
the solution of the positive or the negative energy branch, and
$\overline{m}=m+\tilde{\Phi}$. The boundary conditions and the normalization
fix the coefficients $c_{1}$ and $c_{2}$. For $V(r)=0$ if
$r\in\left(r_{0}-\frac{W}{2},r_{0}+\frac{W}{2}\right)$, and
$V(r)\rightarrow+\infty$ outside the ring Recher et al. (2007) we find
$\psi=\pm\tau\sigma_{\phi}\psi$ at $r=r_{0}\pm\frac{W}{2}$. With this Eq. (2)
can be solved numerically or for $W/r_{0}\ll 1$ analytically Recher et al.
(2007) yielding the spectrum
$\displaystyle E_{nm}^{s\tau}$ $\displaystyle=$ $\displaystyle
s\varepsilon_{n}+s\lambda_{n}\left[m+\tilde{\Phi}^{s\tau n}_{\rm
eff}\right]^{2}-\frac{s\lambda_{n}}{4\pi^{2}(n+1/2)^{2}},$ (3)
$\displaystyle\varepsilon_{n}$ $\displaystyle=$ $\displaystyle\frac{\hbar
v}{W}(n+1/2),\lambda_{n}=\frac{\hbar
v}{W}\left(\frac{W}{r_{0}}\right)^{2}\frac{1}{\pi(2n+1)}.$ (4)
$n=0,1,2,...,$ and $\tilde{\Phi}^{s\tau n}_{\rm
eff}=\tilde{\Phi}-\frac{1}{2}\frac{s\tau}{(n+1/2)\pi}$. For fixed $s,\tau$,
and $n$ the quantity $\tilde{\Phi}^{s\tau n}_{\rm eff}$ modifies the energy
spectrum as an effective (normalized) magnetic flux. The shift of the
effective magnetic flux from $\tilde{\Phi}$ has a different sign depending on
the valley ($\tau=\pm 1$). For $W/r_{0}\ll 1$ we find
$\displaystyle
R_{+,n}^{s\tau}(r)=\frac{1}{\sqrt{Wr_{0}}}\cos\left[(n\\!+\\!1/2)\pi\tilde{r}^{\prime}\\!-\\!\frac{\tau}{4}\pi\right]\;,$
(5) $\displaystyle
R_{-,n}^{s\tau}(r)=\frac{is}{\sqrt{Wr_{0}}}\sin\left[(n\\!+\\!1/2)\pi\tilde{r}^{\prime}\\!-\\!\frac{\tau}{4}\pi\right]\;,$
(6)
where $\tilde{r}^{\prime}=\left(r-r_{0}+\frac{W}{2}\right)/W\in(0,1)$. For
applications involving tunneling from the ring it is important to inspect the
case of a finite barrier boundary, i.e. $V=V_{0}$ for $r$ outside of the ring.
To a first order of $\gamma\equiv\frac{\hbar v}{WV}\ll 1$ we find that Eq. (4)
applies with $\varepsilon_{n}$ being replaced by $(1-\gamma)\varepsilon_{n}$
and $\lambda_{n}$ by $(1+\gamma)\lambda_{n}$. For $W=0.1$ $\mu$m the condition
$\gamma\ll 1$ means $V\gg 7$ meV. Hence, our theory developed below is valid
also for a finite barrier graphene ring. In a particular example of the boron
nitride substrate we have $V=53$ meV boron_nitride and therefore
$\gamma=0.13$. Having specified the stationary single-particle states we
proceed with the non-equilibrium calculations 222For few-electron rings the
Coulomb interaction may influence the equilibrium properties Abergel et al.
(2008) but for a larger particle numbers it is less significant..
Pulse-induced polarization.- To drive the non-equilibrium states in graphene
rings we utilize asymmetric monocycle pulses, so-called half-cycle pulses
(HCPs) You et al. (1993); Jones et al. (1993). For pulse duration $\tau_{p}$
shorter than the carriers characteristic time scale 333 For rings with
$W/r_{0}\ll 1$ IA requires $\tau_{p}\ll\hbar/\lambda_{n}$ for all radial
channels influenced by the excitation. the impulsive approximation (IA)
applies, meaning that the time-dependent carrier wavefunction
$\Psi(r,\varphi;t)$ propagates stroboscopically as Matos-Abiague and Berakdar
(2004)
$\begin{split}&\Psi(r,\varphi,t^{+})=\Psi(r,\varphi,t^{-})\
e^{i\vec{\alpha}\vec{e}_{r}};\,\\\
&\vec{\alpha}=\frac{r_{0}\vec{p}}{\hbar}\;,\ \ \ \ \ \ \
\vec{p}=e\int\vec{F}(t)dt\;,\end{split}$ (7)
where $t^{-}$ and $t^{+}$ refer to times before and after the application of
the pulse and $\vec{\alpha}$ is the action delivered to a ring carrier of
charge $e$ by the HCP electric field $F(t)$. The pulse triggers a time-
dependent carrier density distribution which depends on $\tau=\pm 1$, i.e. it
is different for the two valleys. As a physical consequence, a time-dependent
charge dipole moment is created in the ring. For the post-pulse dipole moment
$\mu^{\tau_{0}}_{m_{0}}(t)$ associated with a carrier starting from the
stationary state $\tau=\tau_{0}$ and $m=m_{0}$ we find 444We limit the
considerations to the lowest radial channel of the positive energy branch,
i.e. $n=0$ and $s=1$. This is achieved experimentally by applying a gate
voltage.
$\mu^{\tau_{0}}_{m_{0}}(t)=er_{0}\alpha
h(\Omega)\sin\frac{t}{t_{0}}\cos\left[2(m_{0}+\Phi^{\tau_{0}}_{\rm
eff})\frac{t}{t_{0}}\right],$ (8)
where $t_{0}=\hbar/\lambda_{0}$, $h(\Omega)=J_{0}(\Omega)+J_{2}(\Omega)$,
$J_{l}(x)$ denotes the Bessel function of the order $l$, and
$\Omega=\alpha\sqrt{2\left[1-\cos(2t/t_{0})\right]}$. The total electric
dipole created in the ring for a fixed $\tau$, spin value ${\cal S}$, and
$t>0$ is $\mu^{\tau}(t)=\sum_{m}f^{\tau}_{m}\mu^{\tau}_{m}(t)$, where
$f^{\tau}_{m}$ is the equilibrium distribution function. For $N_{\tau}$
carriers in a given valley $\tau$ at zero temperature $T=0$ we carried out the
summation over $m$ analytically. For an arbitrary even or odd $N_{\tau}$ we
find respectively
$\displaystyle\\!\\!\\!\mu^{\tau}_{\rm even}(t)\\!=er_{0}\alpha
h(\Omega)\cos\\!\left[\frac{2t}{t_{0}}\left(\left|\tilde{\Phi}^{\tau}_{\rm
eff}\right|-\frac{1}{2}\right)\right]\sin\frac{N_{\tau}t}{t_{0}},$
$\displaystyle\\!\\!\\!\\!\mu^{\tau}_{\rm odd}(t)=er_{0}\alpha
h(\Omega)\cos\\!\left[\frac{2t}{t_{0}}\tilde{\Phi}^{\tau}_{\rm
eff}\right]\sin\frac{N_{\tau}t}{t_{0}}.$ (9)
Both expressions apply for $\tilde{\Phi}^{\tau}_{\rm
eff}=\tilde{\Phi}-\frac{\tau}{\pi}\in[-1/2,1/2]$, outside of this interval
$\mu^{s\tau}(t)$ is determined from the periodicity in
$\tilde{\Phi}^{\tau}_{\rm eff}$ with a period 1. The total dipole moment
$\mu(t)$ depends on the distribution of the carriers between the valleys that
in turn depends on the magnetic flux $\tilde{\Phi}$. The spin degeneracy is
also important. One can show that jumps in the population of particular states
take place only at the points
$\tilde{\Phi}=-\frac{1}{2},-\frac{1}{\pi},-r,0,r,\frac{1}{\pi},\frac{1}{2}$
for $\tilde{\Phi}$ in the interval $[-1/2,1/2]$, where we denote
$r=\frac{1}{2}-\frac{1}{\pi}$. The dynamics of the dipole moment for $N=8$
carriers is shown in Fig. 1 as a function of the applied stationary magnetic
flux for two different excitation strengths $\alpha=1$ and $\alpha=5$, showing
that the ring electric dipole and hence the associated light emission are
controllable by $\Phi$ and $\alpha$. For $r_{0}=1~{}\mu$m and HCPs with a
sine-square shape and a time duration of 0.5 ps, $\alpha=1$ corresponds to the
peak value of electric field $F=26$ V/cm.
Figure 1: Dependence of the dipole moment generated in the graphene ring on
the time past after the excitation and the normalized magnetic flux
$\tilde{\Phi}=\Phi/\Phi_{0}\in[-1/2,1/2]$ (outside of this range the
periodicity by $\tilde{\Phi}\rightarrow\tilde{\Phi}+1$ can be used) in the
case of $8$ carriers in the ring at $T=0$ for (a) $\alpha=1$ and (b)
$\alpha=5$.
Note, the boundary conditions break the effective time-reversal symmetry
Recher et al. (2007) making the states corresponding to different $\tau$ but
otherwise to the same quantum numbers non-degenerate. The dynamics of the
charge polarization for confined carriers is however the same in both $\tau$
valleys for $\Phi=0$. This follows from the invariance of the states under
$\tau\rightarrow-\tau$, and $m\rightarrow-m$ at $\tilde{\Phi}=0$, as evidenced
by Eqs. (Light-induced valley currents and magnetization in graphene rings).
This degeneracy is lifted by applying a stationary magnetic flux
$\tilde{\Phi}\neq 0$. The density distribution of carriers in the ring becomes
valley-polarized.
Non-equilibrium charge and valley currents.- The electric current density
$\vec{j}=ev\psi^{\dagger}\vec{\sigma}\psi$ has the $\phi$-component of the
current density
$j_{\phi}(r)=-2{\rm Im}[R_{+}(r)R_{-}^{*}(r)]\;.$ (10)
The total charge current is $I=\int j_{\phi}{\rm d}r$. To a zero order in
$W/r_{0}\ll 1$ we find $I=0$. Only higher order corrections in $W/r_{0}$ give
rise to a non-vanishing ring current. For the eigenstate specified by
$s,\tau,n,m$ the lowest order correction Recher et al. (2007) to the current
follows from $I^{s\tau}_{nm}=-\partial E_{nm}^{s\tau}/\partial\Phi$ using the
energy spectrum in the considered limit 555We checked numerically the validity
of this formula for $I^{s\tau}_{nm}$ in the limit case $W/a\ll 1$ by comparing
with the general equation (10).. For $n=0$, $s=1$ this current is equal to
$I^{\tau}_{m}=-I_{0}\left(m+\tilde{\Phi}^{\tau}_{\rm eff}\right)$, where
$I_{0}=\frac{1}{\pi^{2}}\frac{|e|vW}{r_{0}^{2}}$. For a ring with
$r_{0}=1~{}\mu\mbox{m}$ and $W=100~{}\mbox{nm}$ we obtain
$I_{0}=0.16~{}\mbox{nA}$, if $r_{0}=425~{}\mbox{nm}$ and $W=150~{}\mbox{nm}$
as in Ref. Russo et al., 2008 we find $I_{0}=1~{}\mbox{nA}$. In both cases IA
is valid if $\tau_{p}\ll 3~{}\mbox{ps}$. Such HCPs are experimentally
available You et al. (1993). The total current in the ring $I$ is the sum of
an equilibrium (persistent) current $I_{\rm eq}$ and a non-equilibrium time-
dependent current part $I_{\rm neq}(t)$ generated in the ring: $I=I_{\rm
eq}+I_{\rm neq}(t)$. The equilibrium part is given by $I_{\rm
eq}=\sum_{m\tau}f^{\tau}_{m\tau}I^{\tau}_{m}$. For $T=0$ it is given in Ref.
Recher et al., 2007 for $N=1,2,3,4$. We derive it for any $N$ in $n=0$.
A non-equilibrium ring current is generated by a sequence of two time-delayed
mutually perpendicular HCPs (see Fig. 2(a)), similarly to the pulse-current
generation in semiconductor rings Matos-Abiague and Berakdar (2005a, b);
Moskalenko et al. (2006); Moskalenko and Berakdar (2008). This scheme allows
for shorter excitation times compared to the resonant excitation schemes using
circular polarized pulses Barth et al. (2006); Nobusada and Yabana (2007);
Räsänen et al. (2007); Moskalenko and Berakdar (2008). At $t=0$ we apply
linearly polarized (along the $x$-axis) pulse that creates a time-dependent
charge polarization along the $x$-axis (cf. Fig. 1). The second pulse is
linearly polarized along the $y$-axis and is applied at $t=t_{y}$. It
generates a non-equilibrium current depending on the charge polarization
created by the first HCP. The delay time should be short enough so that
relaxation processes are negligible in between the pulses. In the IA The
generated non-equilibrium current reads
$I_{\rm neq}=\alpha_{y}\frac{\mu_{x}(t_{y})}{er_{0}}I_{0}\Theta(t-t_{y}),$
(11)
where $\alpha_{y}$ is the excitation strength of the second HCP and
$\mu_{x}(t_{y})$ is the dipole moment created by the first HCP just before the
application of the second HCP. Equation (11) delivers the total current as
well as the individual currents in each of the two valleys, in which case
$\mu_{x}(t_{y})$ should be associated with the charge carriers in the
respective valley. Defining the valley current as the difference between the
currents flowing in two opposite valleys divided by the particle charge we
find for the generated valley current
$I_{\rm neq}^{\rm
v}=\alpha_{y}\frac{\mu^{+}_{x}(t_{y})-\mu^{-}_{x}(t_{y})}{er_{0}}\frac{I_{0}}{e}\Theta(t-t_{y}).$
(12)
On a longer time scale set by the relaxation processes the non-equilibrium
current decays due to dissipation. Thereby the incoherent electron-phonon
scattering plays usually the most important role Moskalenko et al. (2006);
Moskalenko and Berakdar (2008). Specifically for a free-standing graphene,
scattering by flexural phonons is dominant at low temperatures Mariani and von
Oppen (2008).
An example of the dependence of the generated total charge current on the
delay time $t_{y}$ is depicted in the upper panel of Fig. 2(b). The
oscillating character of this dependence is determined by the dynamics of the
dipole moment generated by the first HCP. The lower panel of Fig. 2(b)
demonstrates the dependence of the generated valley current on the delay time
$t_{y}$. This current arises as a consequence of the different contributions
to the total dipole moment from the two different valleys in presence of a
static magnetic flux (here we used $\tilde{\Phi}=1/\pi$) at the time moment
$t=t_{y}$. Comparing the upper and the lower panels we conclude that tuning
the pulses delay may result in a vanishing total generated current $I_{\rm
neq}$ while the generated valley current $I_{\rm neq}^{\rm v}$ is finite. It
is also possible to create $I_{\rm neq}\neq 0$ with $I_{\rm neq}^{\rm v}=0$.
Under the conditions of Fig. 2(b) the generated currents have the same order
of magnitude as the persistent currents. The non-equilibrium contributions are
enhanced however by increasing the HCPs excitation strengths. An increase of
the excitation strength $\alpha_{x}$ of the first HCP beyond the values around
1 does not lead however to an increase of $I_{\rm neq}$ ($I_{\rm neq}^{\rm
v}$) under the conditions where $I_{\rm neq}^{\rm v}$ ($I_{\rm neq}$) vanishes
because for this, certain delay times are required. In the strong excitation
regime the nonlinear oscillations of the dipole moment collapse Moskalenko et
al. (2006) shortly after the excitation (cf. Figs. 1(a) and (b) in the range
$t/t_{0}\in[0,2]$). For a further increase of the currents under these
conditions $\alpha_{y}$ should be increased.
Figure 2: (a) Current generation in the ring by application of two HCPs
polarized along mutually perpendicular directions and delayed in respect to
each other by the time $t_{y}$. (b) Upper and lower figures show the total
current and the valley current, respectively, generated in the graphene ring
in dependence on the delay time $t_{y}$. Value of the magnetic flux is set to
$\Phi=\Phi_{0}/\pi$, number of carriers is $N=8$, excitation strengths of both
HCPs are equal to $\alpha=1$. Arrows at $t_{y}=\frac{\pi}{4r_{0}}t_{0}$
indicates the delay time for which a valley current with no total charge
current is generated whereas for a delay time $t_{y}=\frac{\pi}{2r_{0}}t_{0}$
a total charge current with equal contributions from both valleys is
generated.
The ring charge current is associated with a magnetic dipole moment via
$\vec{M}=\frac{1}{2}\int\vec{r}\times\vec{j}\;{\rm d}^{2}\vec{r}$, i.e.
$M=\pi\int j_{\phi}r^{2}{\rm d}r$. From Eqs. (10),(5) and (6) we infer for the
non-vanishing lowest order of $W/r_{0}$
$M=\pi r_{0}^{2}I+\pi r_{0}^{2}I_{0}\sum_{sn}\frac{4s}{(2n+1)^{2}}Q_{sn},$
(13)
where $Q_{sn}=N^{+}_{sn}-N^{-}_{sn}$ is the difference in the valley
population for fixed $s$ and $n$. For a vanishing total current in the ring
and $s=1$, $n=0$, Eq. (13) simplifies to $M=4Q\pi r_{0}^{2}I_{0}$. Note, the
valley polarized magnetic moment is also a generic feature of the monolayer
graphene with a broken inversion symmetry (e.g. due to the action of the
substrate potential) Xiao et al. (2007). The difference in the valley
population in Eq. (13) arises in equilibrium for certain ranges of $\Phi\neq
0$. It can be also generated e.g. by injection of external non-equilibrium
carriers to the graphene ring, opening thus a new way for an ultrafast
detection of the valley number. Finally, we note our results are valid for
weak pulses in which case a small angular population around the ground state
is created and many-body effects remain subsidiary. Strong excitations go
beyond the present model and the influence of many-body interactions may
decisively alter the above predictions.
Conclusion.- Short linearly polarized asymmetric light pulses trigger a non-
equilibrium carrier dynamics in graphene rings threaded by a magnetic flux.
The induced charge polarization is detectable by monitoring the emitted
radiation. Delayed pulses with different polarization axes drive non-
equilibrium charge currents and hence an orbital magnetization. For
appropriate pulses, equal contributions from both valleys is achievable as
well as pure valley currents. The ring magnetization depends on the difference
in the valley population. The predicted effect is operational in presence of
tunneling allowing thus for swift injection or detection (via ring
magnetization) of valley currents in coupled graphene structures, e.g. wires,
offering new realization of ultrafast valleytronics devices.
## References
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|
arxiv-papers
| 2009-10-10T22:19:01 |
2024-09-04T02:49:05.793215
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A.S. Moskalenko, J. Berakdar",
"submitter": "Andrey Moskalenko S.",
"url": "https://arxiv.org/abs/0910.1950"
}
|
0910.2005
|
# Modulation Codes for Flash Memory Based on Load-Balancing Theory
Fan Zhang and Henry D. Pfister
Department of Electrical and Computer Engineering, Texas A&M University
{fanzhang,hpfister}@tamu.edu This work was supported in part by the National
Science Foundation under Grant No. 0747470.
###### Abstract
In this paper, we consider modulation codes for practical multilevel flash
memory storage systems with $q$ cell levels. Instead of maximizing the
lifetime of the device [7, 1, 2, 4], we maximize the average amount of
information stored per cell-level, which is defined as storage efficiency.
Using this framework, we show that the worst-case criterion [7, 1, 2] and the
average-case criterion [4] are two extreme cases of our objective function. A
self-randomized modulation code is proposed which is asymptotically optimal,
as $q\rightarrow\infty$, for an arbitrary input alphabet and i.i.d. input
distribution.
In practical flash memory systems, the number of cell-levels $q$ is only
moderately large. So the asymptotic performance as $q\rightarrow\infty$ may
not tell the whole story. Using the tools from load-balancing theory, we
analyze the storage efficiency of the self-randomized modulation code. The
result shows that only a fraction of the cells are utilized when the number of
cell-levels $q$ is only moderately large. We also propose a load-balancing
modulation code, based on a phenomenon known as “the power of two random
choices” [10], to improve the storage efficiency of practical systems.
Theoretical analysis and simulation results show that our load-balancing
modulation codes can provide significant gain to practical flash memory
storage systems. Though pseudo-random, our approach achieves the same load-
balancing performance, for i.i.d. inputs, as a purely random approach based on
the power of two random choices.
## I Introduction
Information-theoretic research on capacity and coding for write-limited memory
originates in [12], [3], [5] and [6]. In [12], the authors consider a model of
write-once memory (WOM). In particular, each memory cell can be in state
either 0 or 1. The state of a cell can go from 0 to 1, but not from 1 back to
0 later. These write-once bits are called _wits_. It is shown that, the
efficiency of storing information in a WOM can be improved if one allows
multiple rewrites and designs the storage/rewrite scheme carefully.
Multilevel flash memory is a storage technology where the charge level of any
cell can be easily increased, but is difficult to decrease. Recent multilevel
cell technology allows many charge levels to be stored in a cell. Cells are
organized into blocks that contain roughly $10^{5}$ cells. The only way to
decrease the charge level of a cell is to erase the whole block (i.e., set the
charge on all cells to zero) and reprogram each cell. This takes time,
consumes energy, and reduces the lifetime of the memory. Therefore, it is
important to design efficient rewriting schemes that maximize the number of
rewrites between two erasures [7], [1], [2], [4]. The rewriting schemes
increase some cell charge levels based on the current cell state and message
to be stored. In this paper, we call a rewriting scheme a _modulation code_.
Two different objective functions for modulation codes are primarily
considered in previous work: (i) maximizing the number of rewrites for the
worst case [7, 1, 2] and (ii) maximizing for the average case [4]. As Finucane
et al. [4] mentioned, the reason for considering average performance is the
averaging effect caused by the large number of erasures during the lifetime of
a flash memory device. Our analysis shows that the worst-case objective and
the average case objective are two extreme cases of our optimization
objective. We also discuss under what conditions each optimality measure makes
sense.
In previous work (e.g., [4, 1, 8, 2]), many modulation codes are shown to be
asymptotically optimal as the number of cell-levels $q$ goes to infinity. But
the condition that $q\rightarrow\infty$ can not be satisfied in practical
systems. Therefore, we also analyze asymptotically optimal modulation codes
when $q$ is only moderately large using the results from load-balancing theory
[13, 10, 11]. This suggests an enhanced algorithm that improves the
performance of practical system significantly. Theoretical analysis and
simulation results show that this algorithm performs better than other
asymptotically optimal algorithms when $q$ is moderately large.
The structure of the paper is as follows. The system model and performance
evaluation metrics are discussed in Section II. An asymptotically optimal
modulation code, which is universal over arbitrary i.i.d. input distributions,
is proposed in Section III. The storage efficiency of this asymptotically
optimal modulation code is analyzed in Section IV. An enhanced modulation code
is also presented in Section IV. The storage efficiency of the enhanced
algorithm is also analyzed in Section IV. Simulation results and comparisons
are presented in Section V. The paper is concluded in Section VI.
## II System Model
### II-A System Description
Flash memory devices usually rely on error detecting/correcting codes to
ensure a low error rate. So far, practical systems tend to use Bose-Chaudhuri-
Hocquenghem (BCH) and Reed-Solomon (RS) codes. The error-correcting codes
(ECC’s) are used as the outer codes while the modulation codes are the inner
codes. In this paper, we focus on the modulation codes and ignore the noise
and the design of ECC for now.
Let us assume that a block contains $n\times N$ $q$-level cells and that $n$
cells (called an $n$-cell) are used together to store $k$ $l$-ary variables
(called a $k$-variable). A block contains $N$ $n$-cells and the $N$
$k$-variables are assumed to be i.i.d. random variables. We assume that all
the $k$-variables are updated together randomly at the same time and the new
values are stored in the corresponding $n$-cells. This is a reasonable
assumption in a system with an outer ECC. We use the subscript $t$ to denote
the time index and each rewrite increases $t$ by 1. When we discuss a
modulation code, we focus on a single $n$-cell. (The encoder of the modulation
code increases some of the cell-levels based on the current cell-levels and
the new value of the $k$-variable.) Remember that cell-levels can only be
increased during a rewrite. So, when any cell-level must be increased beyond
the maximum value $q-1$, the whole block is erased and all the cell levels are
reset to zero. We let the maximal allowable number of block-erasures be $M$
and assume that after $M$ block erasures, the device becomes unreliable.
Assume the $k$-variable written at time $t$ is a random variable $x_{t}$
sampled from the set $\\{0,1,\cdots,l^{k}-1\\}$ with distribution $p_{X}(x)$.
For convenience, we also represent the $k$-variable at time $t$ in the vector
form as $\bar{x}_{t}\in\mathbb{Z}_{l}^{k}$ where $\mathbb{Z}_{l}$ denotes the
set of integers modulo $l$. The cell-state vector at time $t$ is denoted as
$\bar{s}_{t}=(s_{t}(0),s_{t}(1),\ldots,s_{t}(n-1))$ and
$s_{t}(i)\in\mathbb{Z}_{q}$ denotes the charge level of the $i$-th cell at
time $t.$ When we say $\bar{s}_{i}\succeq\bar{s}_{j},$ we mean $s_{i}(m)\geq
s_{j}(m)$ for $m=0,1,,\ldots,n-1.$ Since the charge level of a cell can only
be increased, continuous use of the memory implies that an erasure of the
whole block will be required at some point. Although writes, reads and
erasures can all introduce noise into the memory, we neglect this and assume
that the writes, reads and erasures are noise-free.
Consider writing information to a flash memory when encoder knows the previous
cell state $\bar{s}_{t-1},$ the current $k$-variable $\bar{x}_{t}$, and an
encoding function
$f:\mathbb{Z}_{l}^{k}\times\mathbb{Z}_{q}^{n}\rightarrow\mathbb{Z}_{q}^{n}$
that maps $\bar{x}_{t}$ and $\bar{s}_{t-1}$ to a new cell-state vector
$\bar{s}_{t}$. The decoder only knows the current cell state $\bar{s}_{t}$ and
the decoding function $g:\mathbb{Z}_{q}^{n}\rightarrow\mathbb{Z}_{l}^{k}$ that
maps the cell state $\bar{s}_{t}$ back to the variable vector
$\bar{\hat{x}}_{t}$. Of course, the encoding and decoding functions could
change over time to improve performance, but we only consider time-invariant
encoding/decoding functions for simplicity.
### II-B Performance Metrics
#### II-B1 Lifetime v.s. Storage Efficiency
The idea of designing efficient modulation codes jointly to store multiple
variables in multiple cells was introduced by Jiang [7]. In previous work on
modulation codes design for flash memory (e.g. [7], [1], [2], [4]), the
lifetime of the memory (either worst-case or average) is maximized given fixed
amount of information per rewrite. Improving storage density and extending the
lifetime of the device are two conflicting objectives. One can either fix one
and optimize the other or optimize over these two jointly. Most previous work
(e.g., [4, 1, 8, 2]) takes the first approach by fixing the amount of
information for each rewrite and maximizing the number of rewrites between two
erasures. In this paper, we consider the latter approach and our objective is
to maximize the total amount of information stored in the device until the
device dies. This is equivalent to maximizing the average (over the
$k$-variable distribution $p_{X}(x)$) amount of information stored per cell-
level,
$\gamma\triangleq E\left(\frac{\sum_{i=1}^{R}I_{i}}{n(q-1)}\right),$
where $I_{i}$ is the amount of information stored at the $i$-th rewrite, $R$
is the number of rewrites between two erasures, and the expectation is over
the $k$-variable distribution. We also call $\gamma$ as _storage efficiency_.
#### II-B2 Worst Case v.s. Average Case
In previous work on modulation codes for flash memory, the number of rewrites
of an $n$-cell has been maximized in two different ways. The authors in [7, 1,
2] consider the worst case number of rewrites and the authors in [4] consider
the average number of rewrites. As mentioned in [4], the reason for
considering the average case is due to the large number of erasures in the
lifetime of a flash memory device. Interestingly, these two considerations can
be seen as two extreme cases of the optimization objective in (4).
Let the $k$-variables be a sequence of i.i.d. random variables over time and
all the $n$-cells. The objective of optimization is to maximize the amount of
information stored until the device dies. The total amount of information
stored in the device111There is a subtlety here. If the $n$-cell changes to
the same value, should it count as stored information? Should this count as a
rewrite? This formula assumes that it counts as a rewrite, so that $l^{k}$
values (rather than $l^{k}-1$) can be stored during each rewrite. can be
upper-bounded by
$W=\sum_{i=1}^{M}R_{i}\log_{2}(l^{k})$ (1)
where $R_{i}$ is the number of rewrites between the $(i-1)$-th and the $i$-th
erasures. Note that the upper bound in (1) is achievable by uniform input
distribution, i.e., when the input $k$-variable is uniformly distributed over
$\mathbb{Z}_{l^{k}}$, each rewrite stores $\log_{2}(l^{k})=k\log_{2}l$ bits of
information. Due to the i.i.d. property of the input variables over time,
$R_{i}$’s are i.i.d. random variables over time. Since $R_{i}$’s are i.i.d.
over time, we can drop the subscript $i$. Since $M$, which is the maximum
number of erasures allowed, is approximately on the order of $10^{7}$, by the
law of large numbers (LLN), we have
$W\approx ME\left[R\right]k\log_{2}(l).$
Let the set of all valid encoder/decoder pairs be
$\mathcal{Q}=\left\\{f,g|\bar{s}_{t}=f(\bar{s}_{t-1},\bar{x}_{t}),\bar{x}_{t}=g(\bar{s}_{t}),\bar{s}_{t}\succeq\bar{s}_{t-1}\right\\},$
where $\bar{s}_{t}\succeq\bar{s}_{t-1}$ implies the charge levels are element-
wise non-decreasing. This allows us to treat the problem
$\max_{f,g\in\mathcal{Q}}W,$
as the following equivalent problem
$\max_{f,g\in\mathcal{Q}}E\left[R\right]k\log_{2}(l).$ (2)
Denote the maximal charge level of the $i$-th $n$-cell at time $t$ as
$d_{i}(t)$. Note that time index $t$ is reset to zero when a block erasure
occurs and increased by one at each rewrite otherwise. Denote the maximal
charge level in a block at time $t$ as $d(t),$ which can be calculated as
$d(t)=\max_{i}d_{i}(t).$ Define $t_{i}$ as the time when the $i$-th $n$-cell
reaches its maximal allowed value, i.e.,
$t_{i}\triangleq\min\\{t|d_{i}(t)=q\\}$. We assume, perhaps naively, that a
block-erasure is required when any cell within a block reaches its maximum
allowed value. The time when a block erasure is required is defined as
$T\triangleq\min_{i}t_{i}.$ It is easy to see that
$E\left[R\right]=NE\left[T\right],$ where the expectations are over the
$k$-variable distribution. So maximizing $E\left[T\right]$ is equivalent to
maximizing $W$. So the optimization problem (2) can be written as the
following optimization problem
$\max_{f,g\in\mathcal{Q}}E\left[\min_{i\in\\{1,2,\cdots,N\\}}t_{i}\right].$
(3)
Under the assumption that the input is i.i.d. over all the $n$-cells and time
indices, one finds that the $t_{i}$’s are i.i.d. random variables. Let their
common probability density function (pdf) be $f_{t}(x).$ It is easy to see
that $T$ is the minimum of $N$ i.i.d. random variables with pdf $f_{t}(x).$
Therefore, we have $f_{T}(x)=Nf_{t}(x)\left(1-F_{t}(x)\right)^{N-1},$ where
$F_{t}(x)$ is the cumulative distribution function (cdf) of $t_{i}.$ So, the
optimization problem (3) becomes
$\max_{f,g\in\mathcal{Q}}E\left[T\right]=\max_{f,g\in\mathcal{Q}}\int
Nf_{t}(x)\left(1-F_{t}(x)\right)^{N-1}x\mbox{d}x.$ (4)
Note that when $N=1,$ the optimization problem in (4) simplifies to
$\max_{f,g\in\mathcal{Q}}E\left[t_{i}\right].$ (5)
This is essentially the case that the authors in [4] consider. When the whole
block is used as one $n$-cell and the number of erasures allowed is large,
optimizing the average (over all input sequences) number of rewrites of an
$n$-cell is equivalent to maximizing the total amount of information stored
$W.$ The analysis also shows that the reason we consider average performance
is not only due to the averaging effect caused by the large number of
erasures. One other important assumption is that there is only one $n$-cell
per block.
The other extreme is when $N\gg 1.$ In this case, the pdf
$Nf_{t}(x)\left(1-F_{t}(x)\right)^{N-1}$ tends to a point mass at the minimum
of $t$ and the integral $\int
Nf_{t}(x)\left(1-F_{t}(x)\right)^{N-1}t\mbox{d}t$ approaches the minimum of
$t$. This gives the worst case stopping time for the programming process of an
$n$-cell. This case is considered by [7, 1, 2]. Our analysis shows that we
should consider the worst case when $N\gg 1$ even though the device
experiences a large number of erasures. So the optimality measure is not
determined only by $M$, but also by $N.$ When $N$ and $M$ are large, it makes
more sense to consider the worst case performance. When $N=1$, it is better to
consider the average performance. When $N$ is moderately large, we should
maximize the number of rewrites using (4) which balances the worst case and
the average case.
When $N$ is moderately large, one should probably focus on optimizing the
function in (4), but it is not clear how to do this directly. So, this remains
an open problem for future research. Instead, we will consider a load-
balancing approach to improve practical systems where $q$ is moderately large.
### II-C $N=1$ v.s. $N\gg 1$
If we assume that there is only one variable changed each time, the average
amount of information per cell-level can be bounded by $\log_{2}kl$ because
there are $kl$ possible new values. Since the number of rewrites can be
bounded by $n(q-1),$ we have
$\gamma\leq\log_{2}kl.$ (6)
If we allow arbitrary change on the $k$-variables, there are totally $l^{k}$
possible new values. It can be shown that
$\gamma\leq k\log_{2}l.$ (7)
For fixed $l$ and $q$, the bound in (7) suggests using a large $k$ can improve
the storage efficiency. This is also the reason jointly coding over multiple
cells can improve the storage efficiency [7]. Since optimal rewriting schemes
only allow a single cell-level to increase by one during each rewrite,
decodability implies that $n\geq kl-1$ for the first case and $n\geq l^{k}-1$
for the second case. Therefore, the bounds in (6) and (7) also require large
$n$ to improve storage efficiency.
The upper bound in (7) grows linearly with $k$ while the upper bound in (6)
grows logarithmically with $k$. Therefore, in the remainder of this paper, we
assume an arbitrary change in the $k$-variable per rewrite and $N=1$, i.e.,
the whole block is used as an $n$-cell, to improve the storage efficiency.
This approach implicitly trades instantaneous capacity for future storage
capacity because more cells are used to store the same number of bits, but the
cells can also be reused many more times.
Note that the assumption of $N=1$ might be difficult for real implementation,
but its analysis gives an upper bound on the storage efficiency. From the
analysis above with $N=1$, we also know that maximizing $\gamma$ is equivalent
to maximize the average number of rewrites.
## III Self-randomized Modulation Codes
In [4], modulation codes are proposed that are asymptotically optimal (as $q$
goes to infinity) in the average sense when $k=2$. In this section, we
introduce a modulation code that is asymptotically optimal for arbitrary input
distributions and arbitrary $k$ and $l$. This rewriting algorithm can be seen
as an extension of the one in [4]. The goal is, to increase the cell-levels
uniformly on average for an arbitrary input distribution. Of course,
decodability must be maintained. The solution is to use common information,
known to both the encoder (to encode the input value) and the decoder (to
ensure the decodability), to randomize the cell index over time for each
particular input value.
Let us assume the $k$-variable is an i.i.d. random variable over time with
arbitrary distribution $p_{X}(x)$ and the $k$-variable at time $t$ is denoted
as $x_{t}\in\mathbb{Z}_{l^{k}}.$ The output of the decoder is denoted as
$\hat{x}_{t}\in\mathbb{Z}_{l^{k}}.$ We choose $n=l^{k}$ and let the cell state
vector at time $t$ be $\bar{s}_{t}=(s_{t}(0),s_{t}(1),\cdots,s_{t}(n-1))$,
where $s_{t}(i)\in\mathbb{Z}_{q}$ is the charge level of the $i$-th cell at
time $t.$ At $t=0$, the variables are initialized to
$\overline{s}_{0}=(0,\ldots,0)$, $x_{0}=0$ and $r_{0}=0$.
The decoding algorithm $\hat{x}_{t}=g(\bar{s}_{t})$ is described as follows.
* •
Step 1: Read cell state vector $\bar{s}_{t}$ and calculate the $\ell_{1}$ norm
$r_{t}=\|\bar{s}_{t}\|_{1}$.
* •
Step 2: Calculate $s_{t}=\sum_{i=1}^{n-1}is_{t}(i)$ and
$\hat{x}_{t}=s_{t}-\frac{r_{t}(r_{t}+1)}{2}\bmod l^{k}.$
The encoding algorithm $\overline{s}_{t}=f(\overline{s}_{t-1},x_{t})$ is
described as follows.
* •
Step 1: Read cell state $\bar{s}_{t-1}$ and calculate $r_{t-1}$ and
$\hat{x}_{t-1}$ as above. If $\hat{x}_{t-1}=x_{t,}$ then do nothing.
* •
Step 2: Calculate $\Delta x_{t}=x_{t}-\hat{x}_{t-1}\bmod l^{k}$ and
$w_{t}=\Delta x_{t}+r_{t-1}+1\bmod l^{k}$
* •
Step 3: Increase the charge level of the $w_{t}$-th cell by 1.
For convenience, in the rest of the paper, we refer the above rewriting
algorithm as “self-randomized modulation code”.
###### Theorem 1
The self-randomized modulation code achieves at least $n(q-q^{2/3})$ rewrites
with high probability, as $q\rightarrow\infty,$ for arbitrary $k,$ $l,$ and
i.i.d. input distribution $p_{X}(x)$. Therefore, it is asymptotically optimal
for random inputs as $q\rightarrow\infty$.
###### Proof:
The proof is similar to the proof in [4]. Since exactly one cell has its level
increased by 1 during each rewrite, $r_{t}$ is an integer sequence that
increases by 1 at each rewrite. The cell index to be written $w_{t}$ is
randomized by adding the value $(r_{t}+1)\bmod l^{k}$. This causes each
consecutive sequence of $l^{k}$ rewrites to have a uniform affect on all cell
levels. As $q\rightarrow\infty$, an unbounded number of rewrites is possible
and we can assume $t\rightarrow\infty$.
Consider the first $nq-nq^{2/3}$ steps, the value
$a_{t,k,l}\triangleq(r_{t}+1)\mbox{ mod }l^{k}$ is as even as possible over
$\\{0,1,\cdots,l^{k}-1\\}.$ For convenience, we say there are $(q-q^{2/3})$
$a_{t,k,l}$’s at each value, as the rounding difference by 1 is absorbed in
the $o(q)$ term. Assuming the input distribution is
$p_{X}=\\{p_{0},p_{1},\cdots,p_{l^{k}-1}\\}$. For the case that $a_{t,k,l}=i$,
the probability that $w_{t}=j$ is $p_{(j-i)\mbox{ mod }l^{k}}$ for
$j\in\\{0,1,\cdots,l^{k}-1\\}$. Therefore, $w_{j}$ has a uniform distribution
over $\\{0,1,\cdots,l^{k}-1\\}$. Since inputs are independent over time, by
applying the same Chernoff bound argument as [4], it follows that the number
of times $w_{t}=j$ is at most $q-3$ with high probability (larger than
$1-1/\mbox{poly}(q)$) for all $j$. Summing over $j$, we finish the proof. ∎
###### Remark 1
Notice that the randomizing term $r_{t}$ a deterministic term which makes
$w_{t}$ look _random_ over time in the sense that there are equally many terms
for each value. Moreover, $r_{t}$ is known to both the encoder and the decoder
such that the encoder can generate “uniform” cell indices over time and the
decoder knows the accumulated value of $r_{t}$, it can subtract it out and
recover the data correctly. Although this algorithm is asymptotically optimal
as $q\rightarrow\infty$, the maximum number of rewrites $n(q-o(q))$ cannot be
achieved for moderate $q$. This motivates the analysis and the design of an
enhanced version of this algorithm for practical systems in next section.
###### Remark 2
A self-randomized modulation code uses $n=l^{k}$ cells to store a
$k$-variable. This is much larger than the $n=kl$ used by previous
asymptotically optimal algorithms because we allow the $k$-variable to change
arbitrarily. Although this seems to be a waste of cells, the average amount of
information stored per cell-level is actually maximized (see (6) and (7)). In
fact, the definition of asymptotic optimality requires $n\geq l^{k}-1$ if we
allow arbitrary changes to the $k$-variable.
###### Remark 3
We note that the optimality of the self-randomized modulation codes is similar
to the weak robust codes presented in [9].
###### Remark 4
We use $n=l^{k}$ cells to store one of $l^{k}-1$ possible messages. This is
slightly worse than the simple method of using $n=l^{k}-1$. Is it possible to
have self-randomization using only $n=l^{k}-1$ cells? A preliminary analysis
of this question based on group theory indicates that it is not. Thus, the
extra cell provides the possibility to randomize the mappings between message
values and the cell indices over time.
## IV Load-balancing Modulation Codes
While asymptotically optimal modulation codes (e.g., codes in [7], [1], [2],
[4] and the self-randomized modulation codes described in Section III) require
$q\rightarrow\infty$, practical systems use $q$ values between $2$ and $256$.
Compared to the number of cells $n$, the size of $q$ is not quite large enough
for asymptotic optimality to suffice. In other words, codes that are
asymptotically optimal may have significantly suboptimal performance when the
system parameters are not large enough. Moreover, different asymptotically
optimal codes may perform differently when $q$ is not large enough. Therefore,
asymptotic optimality can be misleading in this case. In this section, we
first analyze the storage efficiency of self-randomized modulation codes when
$q$ is not large enough and then propose an enhanced algorithm which improves
the storage efficiency significantly.
### IV-A Analysis for Moderately Large $q$
Before we analyze the storage efficiency of asymptotically optimal modulation
codes for moderately large $q$, we first show the connection between rewriting
process and the load-balancing problem (aka the balls-into-bins or balls-and-
bins problem) which is well studied in mathematics and computer science [13,
10, 11]. Basically, the load-balancing problem considers how to distribute
objects among a set of locations as evenly as possible. Specifically, the
balls-and-bins model considers the following problem. If $m$ balls are thrown
into $n$ bins, with each ball being placed into a bin chosen independently and
uniformly at random, define the _load_ as the number of balls in a bin, what
is the maximal load over all the bins? Based on the results in Theorem 1 in
[11], we take a simpler and less accurate approach to the balls-into-bins
problem and arrive at the following theorem.
###### Theorem 2
Suppose that $m$ balls are sequentially placed into $n$ bins. Each time a bin
is chosen independently and uniformly at random. The maximal load over all the
bins is $L$ and:
($i$) If $m=d_{1}n,$ the maximally loaded bin has $L\leq\frac{c_{1}\ln
n}{\ln\ln n}$ balls, $c_{1}>2$ and $d_{1}\geq 1$, with high probability
($1-1/\mbox{poly}(n)$) as $n\rightarrow\infty.$
($ii$) If $m=n\ln n$, the maximally loaded bin has $L\leq\frac{c_{4}(\ln
n)^{2}}{\ln\ln n}$ balls, $c_{4}>1$, with high probability
($1-1/\mbox{poly}(n)$) as $n\rightarrow\infty.$
($iii$) If $m=c_{3}n^{d_{2}},$ the maximally loaded bin has $L\leq
ec_{3}n^{d_{2}-1}+c_{2}\ln n$, $c_{2}>1$, $c_{3}\geq 1$ and $d_{2}>1$, with
high probability ($1-1/\mbox{poly}(n)$) as $n\rightarrow\infty.$
###### Proof:
Denote the event that there are at least $k$ balls in a particular bin as
$E_{k}$. Using the union bound over all subsets of size $k,$ it is easy to
show that the probability that $E_{k}$ occurs is upper bounded by
$\Pr\\{E_{k}\\}\leq\binom{m}{k}\left(\frac{1}{n}\right)^{k}.$
Using Stirling’s formula, we have
$\binom{m}{k}\leq\left(\frac{me}{k}\right)^{k}$. Then $\Pr\\{E_{k}\\}$ can be
further bounded by
$\Pr\\{E_{k}\\}\leq\left(\frac{me}{nk}\right)^{k}.$ (8)
If $m=d_{1}n$, substitute $k=\frac{c_{1}\ln n}{\ln\ln n}$ to the RHS of (8),
we have
$\displaystyle\Pr\\{E_{k}\\}$ $\displaystyle\leq\left(\frac{d_{1}e\ln\ln
n}{c_{1}\ln n}\right)^{\frac{c_{1}\ln n}{\ln\ln n}}$
$\displaystyle=e^{\left(\frac{c_{1}\ln n}{\ln\ln n}\left(\ln(d_{1}e\ln\ln
n)-\ln(c_{1}\ln n)\right)\right)}$ $\displaystyle<e^{\left(\frac{c_{1}\ln
n}{\ln\ln n}\left(\ln(d_{1}e\ln\ln n)-\ln\ln n\right)\right)}$
$\displaystyle\leq e^{(-(c_{1}-1)\ln n)}=\frac{1}{n^{c_{1}-1}}.$
Denote the event that all bins have at most $k$ balls as $\tilde{E}_{k}$. By
applying the union bound, it is shown that
$\Pr\\{\tilde{E}_{k}\\}\geq 1-\frac{n}{n^{c_{1}-1}}=1-\frac{1}{n^{c_{1}-2}}.$
Since $c_{1}>2,$ we finish the proof for the case of $m=d_{1}n.$
If $m=n\ln n$, substitute $k=\frac{c_{4}(\ln n)^{2}}{\ln\ln n}$ to the RHS of
(8), we have
$\displaystyle\Pr\\{E_{k}\\}$ $\displaystyle\leq\left(\frac{e\ln\ln
n}{c_{4}\ln n}\right)^{\frac{c_{4}(\ln n)^{2}}{\ln\ln n}}$
$\displaystyle=e^{\left(\frac{c_{4}(\ln n)^{2}}{\ln\ln n}\left(\ln e\ln\ln
n-\ln c_{4}\ln n\right)\right)}$ $\displaystyle\leq e^{\left(\frac{c_{4}(\ln
n)^{2}}{\ln\ln n}\left(\ln e\ln\ln n-\ln\ln n\right)\right)}$
$\displaystyle\leq e^{\left(-(c_{4}-1)(\ln
n)^{2}\right)}=o\left(\frac{1}{n^{2}}\right).$
By applying the union bound, we finish the proof for the case of $m=n\ln n.$
If $m=c_{3}n^{d_{2}},$ substitute $k=ec_{3}n^{d_{2}-1}+c_{2}\ln
n=ec_{3}n^{d_{2}-1}+c_{2}\ln n$ to the RHS of (8), we have
$\displaystyle\Pr\\{E_{k}\\}$
$\displaystyle\leq\left(\frac{ec_{3}n^{d_{2}-1}}{ec_{3}n^{d_{2}-1}+c_{2}\ln
n}\right)^{c_{3}en^{d_{2}-1}+c_{2}\ln n}$
$\displaystyle=e^{\left((c_{5}n^{d_{2}-1}\\!+c_{2}\ln n)\left(\ln
c_{5}n^{d_{2}-1}\\!\\!\\!\\!-\ln(c_{5}n^{d_{2}-1}\\!+c_{2}\ln
n)\right)\right)}$ $\displaystyle\leq e^{\left((c_{5}n^{d_{2}-1}+c_{2}\ln
n)\left(-\frac{c_{2}\ln n}{c_{5}n^{d_{2}-1}}\right)\right)}$
$\displaystyle\leq e^{\left(-c_{2}\ln n\right)}=\frac{1}{n^{c_{2}}}$
where $c_{5}=ec_{3}.$ By applying the union bound, it is shown that
$\Pr\\{\tilde{E}_{k}\\}\leq 1-\frac{n}{n^{c_{2}}}=1-\frac{1}{n^{c_{2}-1}}.$
Since $c_{2}>1,$ we finish the proof for the case of $m=c_{3}n^{d_{2}}.$ ∎
###### Remark 5
Note that Theorem 2 only shows an upper bound on the maximum load $L$ with a
simple proof. More precise results can be found in Theorem 1 of [11], where
the exact order of $L$ is given for different cases. It is worth mentioning
that the results in Theorem 1 of [11] are different from Theorem 2 because
Theorem 1 of [11] holds with probability $1-o(1)$ while Theorem 2 holds with
probability ($1-1/\mbox{poly}(n)$).
###### Remark 6
The asymptotic optimality in the rewriting process implies that each rewrite
only increases the cell-level of a cell by 1 and all the cell-levels are fully
used when an erasure occurs. This actually implies
$\lim_{m\rightarrow\infty}\frac{L}{m/n}=1$. Since $n$ is usually a large
number and $q$ is not large enough in practice, the theorem shows that, when
$q$ is not large enough, asymptotic optimality is not achievable. For example,
in practical systems, the number of cell-levels $q$ does not depend on the
number of cells in a block. Therefore, rather than $n(q-1),$ only roughly
$n(q-1)\frac{\ln\ln n}{\ln n}$ charge levels can be used as
$n\rightarrow\infty$ if $q$ is a small constant which is independent of $n$.
In practice, this loss could be mitigated by using writes that increase the
charge level in multiple cells simultaneously (instead of erasing the block).
###### Theorem 3
The self-randomized modulation code has storage efficiency
$\gamma=c\ln\frac{k\ln l}{c}$ when $q-1=c$ and $\gamma=\frac{c}{\theta}k\ln l$
when $q-1=c\ln n$ as $n$ goes to infinity with high probability (i.e.,
$1-o(1)$).
###### Proof:
Consider the problem of throwing $m$ balls into $n$ bins and let the r.v. $M$
be the number of balls thrown into $n$ bins until some bin has more than $q-1$
balls in it. While we would like to calculate $E[M]$ exactly, we still settle
for an approximation based on the following result. If $m=cn\ln n$, then there
is a constant $d(c)$ such that maximum number of balls $L$ in any bin
satisfies
$\left(d(c)-1\right)\ln n\leq L\leq d(c)\ln n$
with probability $1-o(1)$ as $n\rightarrow\infty$ [11] . The constant $d(c)$
is given by the largest $x$-root of
$x(\ln c-\ln x+1)+1-c=0,$
and solving this equation for $c$ gives the implicit expression
$c=-d(c)W\left(-e^{-1-\frac{1}{d(c)}}\right)$. Since the lower bound matches
the expected maximum value better, we define $\theta\triangleq d(c)-1$ and
apply it to our problem using the equation $\theta\ln n=q-1$ or
$\theta=\frac{q-1}{\ln n}$. Therefore, the storage efficiency is
$\gamma=\frac{m\ln n}{n(q-1)}=\frac{c}{\theta}k\ln l$
If $m=cn$, the maximum load is approximately $\frac{\ln n}{\ln\frac{n\ln
n}{m}}$ with probability $1-o(1)$ for large $n$ [11]. By definition,
$q-1=\frac{\ln n}{\ln\frac{n\ln n}{m}}$. Therefore, the storage efficiency is
$\gamma=\frac{m\ln n}{n(q-1)}=c\ln\frac{\ln n}{c}=c\ln\frac{k\ln l}{c}.$ ∎
###### Remark 7
The results in Theorem 3 show that when $q$ is on the order of $O(\ln n)$, the
storage efficiency is on the order of $\Theta(k\ln l)$. Taking the limit as
$q,n\rightarrow\infty$ with $q=O(\ln n)$, we have $\lim\frac{\gamma}{k\ln
l}=\frac{\theta}{c}>0.$ When $q$ is a constant independent of $n$, the storage
efficiency is on the order of $\Theta(\ln k\ln l).$ Taking the limit as
$n\rightarrow\infty$ with $q-1=c$, we have $\lim\frac{\gamma}{k\ln l}=0$. In
this regime, the self-randomized modulation codes actually perform very poorly
even though they are asymptotically optimal as $q\rightarrow\infty$.
### IV-B Load-balancing Modulation Codes
Considering the bins-and-balls problem, can we distribute balls more evenly
when $m/n$ is on the order of $o(n)?$ Fortunately, when $m=n$, the maximal
load can be reduced by a factor of roughly $\frac{\ln n}{(\ln\ln n)^{2}}$ by
using _the power of two random choices_ [10]. In detail, the strategy is,
every time we pick two bins independently and uniformly at random and throw a
ball into the less loaded bin. By doing this, the maximally loaded bin has
roughly $\frac{\ln\ln n}{\ln 2}+O(1)$ balls with high probability. Theorem 1
in [13] gives the answer in a general form when we consider $d$ random
choices. The theorem shows there is a large gain when the number of random
choice is increased from 1 to 2. Beyond that, the gain is on the same order
and only the constant can be improved.
Based on the idea of 2 random choices, we define the following load-balanced
modulation code.
Again, we let the cell state vector at time $t$ be
$\bar{s}_{t}=(s_{t}(0),s_{t}(1),\cdots,s_{t}(n-1))$, where
$s_{t}(i)\in\mathbb{Z}_{q}$ is the charge level of the $i$-th cell at time
$t.$ This time, we use $n=l^{k+1}$ cells to store a $k$-variable
$x_{t}\in\mathbb{Z}_{l^{k}}$ (i.e., we write $(k+1)\log_{2}l$ bits to store
$k\log_{2}l$ bits of information). The information loss provides $l$ ways to
write the same value. This flexibility allows us to avoid sequences of writes
that increase one cell level too much. We are primarily interested in binary
variables with 2 random choices or $l=2$. For the power of $l$ choices to be
effective, we must try to randomize (over time), the $l$ possible choices over
the set of all $\binom{n}{l}$ possibilities. The value
$r_{t}=\|\bar{s}_{t}\|_{1}$ is used to do this. Let $H$ be the Galois field
with $l^{k+1}$ elements and $h:\mathbb{Z}_{l^{k+1}}\rightarrow H$ be a
bijection that satisfies $h(0)=0$ (i.e., the Galois field element 0 is
associated with the integer 0).
The decoding algorithm calculates $\hat{x}_{t}$ from $\bar{s}_{t}$ and
operates as follows:
* •
Step 1: Read cell state vector $\bar{s}_{t}$ and calculate the $\ell_{1}$ norm
$r_{t}=\|\bar{s}_{t}\|_{1}$.
* •
Step 2: Calculate $s_{t}=\sum_{i=1}^{n}is_{t}(i)$ and
$\hat{x}_{t}^{\prime}=s_{t}\mbox{ mod }l^{k+1}.$
* •
Step 3: Calculate $a_{t}=h\left(\left(r_{t}\bmod l^{k}-1\right)+1\right)$ and
$b_{t}=h\left(r_{t}\bmod l^{k}\right)$
* •
Step 4: Calculate
$\hat{x}_{t}=h^{-1}\left(a_{t}^{-1}\left(h(\hat{x}_{t}^{\prime})-b_{t}\right)\right)\bmod
l^{k}$.
The encoding algorithm stores $x_{t}$ and operates as follows.
* •
Step 1: Read cell state $\bar{s}_{t-1}$ and decode to $\hat{x}_{t-1}^{\prime}$
and $\hat{x}_{t-1}$. If $\hat{x}_{t-1}=x_{t},$ then do nothing.
* •
Step 2: Calculate $r_{t}=\|\bar{s}_{t-1}\|_{1}+1$,
$a_{t}=h\left(\left(r_{t}\bmod l^{k}-1\right)+1\right)$, and
$b_{t}=h\left(r_{t}\bmod l^{k}\right)$
* •
Step 3: Calculate $x_{t}^{(i)}=h^{-1}\left(a_{t}h(x_{t}+il^{k})+b_{t}\right)$
and $\Delta x_{t}^{(i)}=x_{t}^{(i)}-\hat{x}_{t-1}^{\prime}\mbox{ mod }l^{k+1}$
for $i=0,1,\ldots l-1$.
* •
Step 4: Calculate222Ties can be broken arbitrarily.
$w_{t}=\arg\min_{j\in\mathbb{Z}_{l}}\\{s_{t-1}(\Delta x_{t}^{(j)})\\}$.
Increase the charge level by 1 of cell $\Delta x_{t}^{(w_{t})}$.
Note that the state vector at $t=0$ is initialized to $s_{0}=(0,\ldots,0)$ and
therefore $x_{0}=0$. The first arbitrary value that can be stored is $x_{1}$.
The following conjecture suggests that the ball-loading performance of the
above algorithm is identical to the random loading algorithm with $l=2$ random
choices.
###### Conjecture 1
If $l=2$ and $q-1=c\ln n$, then the load-balancing modulation code has storage
efficiency $\gamma=k$ with probability 1-$o(1)$ as $n\rightarrow\infty$. If
$q-1=c,$ the storage efficiency $\gamma=\frac{c\ln 2}{\ln\ln n}k$ with
probability 1-$o(1)$.
###### Proof:
Consider the affine permutation $\pi_{x}^{(a,b)}=h^{-1}(ah(x)+b)$ for $a\in
H\backslash 0$ and $b\in H$. As $a,b$ vary, this permutation maps the two
elements $x_{t}$ and $x_{t}+l^{k}$ uniformly over all pairs of cell indices.
After $m=n(n-1)$ steps, we see that all pairs of $a,b$ occur equally often.
Therefore, by picking the less charged cell, the modulation code is almost
identical to the random loading algorithm with two random choices.
Unfortunately, we are interested in the case where $m\ll n^{2}$ so the
analysis is somewhat more delicate. If $m=cn\ln n$, the highest charge level
is $c\ln n-1+\frac{\ln\ln n}{\ln 2}\approx c\ln n$ with probability $1-o(1)$
[13]. Since $q-1=c\ln n$ in this case, the storage efficiency is
$\gamma=\frac{cn\ln n\log_{2}2^{k}}{nc\ln n}=k$. If $m=cn$, then $q-1=c$ and
the maximum load is $c-1+\ln\ln n/\ln 2\approx\frac{\ln\ln n}{\ln 2}$. By
definition, we have $\frac{\ln\ln n}{\ln 2}=q-1.$ Therefore, we have
$\gamma=\frac{cn\log_{2}2^{k}}{n(q-1)}=\frac{c\ln 2}{\ln\ln n}k.$ ∎
###### Remark 8
If $l=2$ and $q$ is on the order of $O(\ln n),$ Conjecture 1 shows that the
bound (7) is achievable by load-balancing modulation codes as $n$ goes to
infinity. In this regime, the load-balancing modulation codes provide a better
constant than self-randomized modulation codes by using twice many cells.
###### Remark 9
If $l=2$ and $q$ is a constant independent of $n$, the storage efficiency is
$\gamma_{1}=c\ln\frac{k}{c}$ for the self-randomized modulation code and
$\gamma_{2}=\frac{c\ln 2}{\ln\ln n}\log_{2}\frac{n}{2}$ for the load-balancing
modulation code. But, the self-randomized modulation code uses $n=2^{k}$ cells
and the load-balancing modulation code uses $n=2^{k+1}$ cells. To make fair
comparison on the storage efficiency between them, we let $n=2^{k+1}$ for both
codes. Then we have $\gamma_{1}=c\ln\frac{\log_{2}n}{c}$ and
$\gamma_{2}=\frac{c\ln 2}{\ln\ln n}\log_{2}\frac{n}{2}$. So, as
$n\rightarrow\infty$, we see that $\frac{\gamma_{1}}{\gamma_{2}}\rightarrow
0$. Therefore, the load-balancing modulation code outperforms the self-
randomized code when $n$ is sufficiently large.
## V Simulation Results
In this section, we present the simulation results for the modulation codes
described in Sections III and IV-B. In the figures, the first modulation code
is called the “self-randomized modulation code” while the second is called the
“load-balancing modulation code”. Let the “loss factor” $\eta$ be the fraction
of cell-levels which are not used when a block erasure is required:
$\eta\triangleq 1-\frac{E[R]}{n(q-1)}.$ We show the loss factor for random
loading with 1 and 2 random choices as comparison. Note that $\eta$ does not
take the amount of information per cell-level into account. Results in Fig. 1
show that the self-randomized modulation code has the same $\eta$ with random
loading with 1 random choice and the load-balancing modulation code has the
same $\eta$ with random loading with 2 random choices. This shows the
optimality of these two modulation codes in terms of ball loading.
Figure 1: Simulation results for random loading and algorithms we proposed
with $k=3$, $l=2$ and 1000 erasures. Figure 2: Simulation results for random
loading and codes in [4] with $k=2$, $l=2,$ $n=2$ and 1000 erasures. Figure 3:
Storage efficiency of self-randomized modulation code and load-balancing
modulation code with $n=16$. Figure 4: Storage efficiency of self-randomized
modulation code and load-balancing modulation code with $n=2^{10}$.
We also provide the simulation results for random loading with 1 random choice
and the codes designed in [4], which we denote as FLM-($k=2,l=2,n=2$)
algorithm, in Fig. 2. From results shown in Fig. 2, we see that the
FLM-($k=2,l=2,n=2$) algorithm has the same loss factor as random loading with
1 random choice. This can be actually seen from the proof of asymptotic
optimality in [4] as the algorithm transforms an arbitrary input distribution
into an uniform distribution on the cell-level increment. Note that FLM
algorithm is only proved to be optimal when 1 bit of information is stored. So
we just compare the FLM algorithm with random loading algorithm in this case.
Fig. 3 and Fig. 4 show the storage efficiency $\gamma$ for these two
modulation codes. Fig. 3 and Fig. 4 show that the load-balancing modulation
code performs better than self-randomized modulation code when $n$ is large.
This is also shown by the theoretical analysis in Remark 9.
## VI Conclusion
In this paper, we consider modulation code design problem for practical flash
memory storage systems. The storage efficiency, or average (over the
distribution of input variables) amount of information per cell-level is
maximized. Under this framework, we show the maximization of the number of
rewrites for the the worst-case criterion [7, 1, 2] and the average-case
criterion [4] are two extreme cases of our optimization objective. The self-
randomized modulation code is proposed which is asymptotically optimal for
arbitrary input distribution and arbitrary $k$ and $l$, as the number of cell-
levels $q\rightarrow\infty$. We further consider performance of practical
systems where $q$ is not large enough for asymptotic results to dominate. Then
we analyze the storage efficiency of the self-randomized modulation code when
$q$ is only moderately large. Then the load-balancing modulation codes are
proposed based on the power of two random choices [13] [10]. Analysis and
numerical simulations show that the load-balancing scheme outperforms
previously proposed algorithms.
## References
* [1] V. Bohossian A. Jiang and J. Bruck. Floating codes for joint information storage in write asymmetric memories. In Proc. IEEE Int. Symp. Information Theory, pages 1166–1170, June 2007.
* [2] P. H. Siegel E. Yaakobi, A. Vardy and J. K. Wolf. Multidimensional flash codes. In Proc. 46th Annual Allerton Conf. on Commun., Control, and Comp., Monticello, IL, September 2008.
* [3] A. Fiat and A. Shamir. Generalized write-once memories. IEEE Trans. Inform. Theory, 30(3):470–480, May 1984.
* [4] Z. Liu H. Finucane and M. Mitzenmacher. Designing floating codes for expected performance. In Proc. 46th Annual Allerton Conf. on Commun., Control, and Comp., Monticello, IL, September 2008.
* [5] C. Heegard and A. El Gamal. On the capacity of computer memory with defects. IEEE Trans. Inform. Theory, 29(5):731–739, September 1983.
* [6] J. Ziv J. K. Wolf, A.D. Wyner and J. Korner. Coding for a write-once memory. AT&T Bell Laboratories Technical Journal, 63(6):1089–1112, 1984\.
* [7] A. Jiang. On the generalization of error-correcting WOM codes. In Proc. IEEE Int. Symp. Information Theory, pages 1391–1395, June 2007.
* [8] A. Jiang and J Bruck. Joint coding for flash memory storage. In Proc. IEEE Int. Symp. Information Theory, 2008.
* [9] A. Jiang, M. Langberg, M. Schwartz, and J. Bruck. Universal rewriting in constrained memories. In Proc. IEEE Int. Symp. Information Theory, June 2009.
* [10] M. D. Mitzenmacher. The power of two choices in randomized load balancing. IEEE Transactions on Parallel and Distributed Systems, 1996.
* [11] M. Raab and A. Steger. ”Balls into Bins” - A simple and tight analysis. Proceedings of the Second International Workshop on Randomization and Approximation Techniques in Computer Science, 1518:159–170, 1998.
* [12] R.L. Rivest and A. Shamir. How to reuse a write-once memory. Information and Control, 55:227–231, 1984.
* [13] A. R. Karlinz E. Upfal Y. Azar, A. Z. Brodery. Balanced allocations. Proceedings of the twenty-sixth annual ACM symposium on Theory of computing, pages 593–602, 1994.
|
arxiv-papers
| 2009-10-12T18:41:32 |
2024-09-04T02:49:05.799281
|
{
"license": "Public Domain",
"authors": "Fan Zhang and Henry D. Pfister",
"submitter": "Fan Zhang",
"url": "https://arxiv.org/abs/0910.2005"
}
|
0910.2038
|
# Automorphisms of the disk complex
Mustafa Korkmaz Department of Mathematics
Middle East Technical University
06531 Ankara, Turkey korkmaz@arf.math.metu.edu.tr and Saul Schleimer
Department of Mathematics
University of Warwick
Coventry, CV4 7AL, UK s.schleimer@warwick.ac.uk
###### Abstract.
We show that the automorphism group of the disk complex is isomorphic to the
handlebody group. Using this, we prove that the outer automorphism group of
the handlebody group is trivial.
This work is in the public domain
## 1\. Introduction
We show that the automorphism group of the disk complex is isomorphic to the
handlebody group. Using this, we prove that the outer automorphism group of
the handlebody group is trivial. These results and many of the details of the
proof are inspired by Ivanov’s work [5] on the mapping class group and the
curve complex.
Let $V=V_{g,n}$ be the genus $g$ handlebody with $n$ spots: a regular
neighborhood of a finite, polygonal, connected graph in $\mathbb{R}^{3}$ with
$n$ disjoint disks chosen on the boundary. See Figure 1 for a picture of
$V_{2,2}$. We write $V=V_{g}$ when $n=0$. Let $\partial_{0}V$ denote the union
of the spots. Let $\partial_{+}V$ be the closure of $\partial
V{\smallsetminus}\partial_{0}V$. So $\partial_{+}V\mathrel{\cong}S=S_{g,n}$ is
a compact connected orientable surface of genus $g$ with $n$ boundary
components. We write $S=S_{g}$ when $n=0$. Define
$e(V)=-\chi(\partial_{+}V)=2g-2+n$.
$\begin{array}[]{c}\psfig{height=99.58464pt}\end{array}$ Figure 1. A genus two
handlebody with two spots.
A simple closed curve $\alpha$ in $S=S_{g,n}$ is inessential if it cuts a disk
off of $S$; otherwise $\alpha$ is essential. The curve $\alpha$ is peripheral
if it cuts an annulus off of $S$; otherwise $\alpha$ is non-peripheral. A
properly embedded disk $D$ in $V=V_{g,n}$, with $\partial
D\subset\partial_{+}V$, is essential or non-peripheral exactly as its boundary
is in $\partial_{+}V$. We require any proper isotopy of $D\subset V$ to have
track disjoint from the spots of $V$. This yields a proper isotopy of
$\partial D$ in $\partial_{+}V$.
###### Definition 1.1 (Harvey [3]).
The curve complex $\mathcal{C}(S)$ is the simplicial complex with vertex set
being isotopy classes of essential, non-peripheral curves in $S$. The
$k$–simplices are given by collections of $k+1$ vertices having pairwise
disjoint representatives.
###### Definition 1.2 (McCullough [10]).
The disk complex $\mathcal{D}(V)$ is the simplicial complex with vertex set
being proper isotopy classes of essential, non-peripherial disks in $V$. The
$k$–simplices are given by collections of $k+1$ vertices having pairwise
disjoint representatives.
Note that there is a natural inclusion
$\mathcal{D}(V)\to\mathcal{C}(\partial_{+}V)$ taking a disk to its boundary.
This map is simplicial and injective.
If $\mathcal{K}$ is a simplicial complex then
$\operatorname{Aut}(\mathcal{K})$ denotes the group of simplicial
automorphisms of $\mathcal{K}$. The elements of
$\operatorname{Aut}(\mathcal{C}(S))$ and $\operatorname{Aut}(\mathcal{D}(V))$
are to be contrasted with mapping classes on the underlying spaces.
###### Definition 1.3.
The mapping class group $\mathcal{MCG}(S)$ is the group of homeomorphisms of
$S$, up to isotopy. The handlebody group $\mathcal{H}(V)$ is the group of
homeomorphisms of $V$, fixing the spots setwise, up to spot preserving
isotopy.
Some authors refer to our $\mathcal{MCG}(S)$ as the extended mapping class
group, as orientation reversing homeomorphisms are allowed. Note there is a
natural map $\mathcal{H}(V)\to\mathcal{MCG}(\partial_{+}V)$ which takes
$f\in\mathcal{H}(V)$ to $f|\partial_{+}V$. Again, this map is an injective
homomorphism.
Note finally that there is a natural homomorphism
$\mathcal{MCG}(S)\to\operatorname{Aut}(\mathcal{C}(S))$ (and similarly for
$V$). We will call any element of the image of this map a geometric
automorphism. Our main theorem is:
If a handlebody $V=V_{g,n}$ satisfies $e(V)\geq 3$ then the natural map
$\mathcal{H}(V)\to\operatorname{Aut}(\mathcal{D}(V))$ is a surjection.
In the language above: every element of $\mathcal{H}(V)$ is geometric. The
plan of the proof of Theorem 9.3 is given in Section 3 and completed in
Section 9. Section 4 shows that Theorem 9.3 is sharp; all handlebodies $V$
with $e(V)\leq 2$ exhibit some kind of exceptional behaviour. Theorem 9.3 has
a corollary:
If a handlebody $V=V_{g,n}$ satisfies $e(V)\geq 3$ then the natural map
$\mathcal{H}(V)\to\operatorname{Aut}(\mathcal{D}(V))$ is an isomorphism.
In Section 10 we use Theorem 9.3 to prove:
If $e(V)\geq 3$ then the outer automorphism group of the handlebody group is
trivial.
These results are inspired by work of Ivanov, Korkmaz and Luo [5, 8, 9]:
###### Theorem 1.4.
If $3g-3+n\geq 3$, or if $(g,n)=(0,5)$, then all elements of
$\operatorname{Aut}(\mathcal{C}(S_{g,n}))$ are geometric. Also, the outer
automorphism group of $\mathcal{MCG}(S)$ is trivial. ∎
## 2\. Background
The genus zero case of Theorem 1.4 is contained in the thesis of the first
author [8, Theorem 1].
###### Theorem 2.1.
If $g=0$ and $n\geq 5$ then all elements of
$\operatorname{Aut}(\mathcal{C}(S_{0,n}))$ are geometric. ∎
Spotted balls are the simplest handlebodies. Accordingly:
###### Lemma 2.2.
The natural maps $\mathcal{D}(V_{0,n})\to\mathcal{C}(S_{0,n})$ and
$\mathcal{H}(V_{0,n})\to\mathcal{MCG}(S_{0,n})$ are isomorphisms.
###### Proof.
The three-manifold $V_{0,n}$ is an $n$–spotted ball. Every simple closed curve
in $\partial_{+}V$ bounds a disk in $V$. This proves that
$\mathcal{D}(V_{0,n})\to\mathcal{C}(S_{0,n})$ is a surjection and thus, by the
remark immediately after Definition 1.2, an isomorphism.
It follows from the Alexander trick that the inclusion of mapping class groups
is an isomorphism. ∎
The genus zero case of Theorem 9.3 is an immediate corollary. We now give
basic definitions.
Suppose that $V$ is a handlebody. Two disks $D,E\in\mathcal{D}(V)$ are
topologically equivalent if there is a mapping class $f\in\mathcal{H}(V)$ so
that $f(D)=E$. The topological type of $D$ is its equivalence class in
$\mathcal{D}(V)$.
For any simplicial complex, $\mathcal{K}$, if $\sigma\in\mathcal{K}$ is a
simplex then recall that
$\operatorname{link}(\sigma)=\\{\tau\in\mathcal{K}\mathbin{\mid}\sigma\cap\tau=\emptyset,~{}\sigma\cup\tau\in\mathcal{K}\\}.$
So if $\mathbb{D}$ is a simplex of $\mathcal{D}(V)$ then
$\operatorname{link}(\mathbb{D})$ is the subcomplex of $\mathcal{D}(V)$
spanned by disks $E$ disjoint from some $D\in\mathbb{D}$ and distinct from all
$D\in\mathbb{D}$.
If $X\subset Y$ is a properly embedded submanifold then we write
$\operatorname{neigh}(X)$ and ${\overline{\operatorname{neigh}}}(X)$ to denote
open and closed regular neighborhoods of $X$ in $Y$. If $X$ is codimension
zero then the frontier of $X$ in $Y$ is the closure of $\partial
X{\smallsetminus}\partial Y$.
A simplex $\mathbb{D}\in\mathcal{D}(V)$ is a cut system if
$V{\smallsetminus}\operatorname{neigh}(\mathbb{D})$ is a spotted ball. Note
that every disk of $\mathbb{D}$ yields two spots of
$V{\smallsetminus}\operatorname{neigh}(\mathbb{D})$.
Recall that for simple curves $\alpha,\beta$ properly embedded in $S$ the
geometric intersection number $i(\alpha,\beta)$ is the minimum possible
intersection number between proper isotopy representatives.
Two disks $D,E\in\mathcal{D}(V)$ are dual if $i(\partial D,\partial E)=2$;
equivalently, after a suitable proper isotopy $D$ and $E$ intersect along a
single arc; equivalently, after a suitable proper isotopy a regular
neighborhood of $D\cup E$ is a four-spotted ball with all spots essential in
$V$. See Figure 2.
$\begin{array}[]{c}\psfig{height=99.58464pt}\end{array}$ Figure 2. Every spot
of the $V_{0,4}$ containing a pair of dual disks is essential in $V$.
If $\mathbb{D}=\\{D_{i}\\}$ is a cut system we define
$\operatorname{dual}_{i}(\mathbb{D})$ to be the subcomplex spanned by the
disks $E\in\mathcal{D}(V)$ which are dual to $D_{i}$ and disjoint from $D_{j}$
for all $j\neq i$. We take $\operatorname{dual}(\mathbb{D})$ to be the complex
spanned by $\cup_{i}\operatorname{dual}_{i}(\mathbb{D})$.
## 3\. The proof of Theorem 9.3
Let $V=V_{g,n}$ be a genus $g$ handlebody with $n$ spots. We suppose that
$g\geq 1$ and $e(V)\geq 3$. Let $\phi$ be any automorphism of
$\mathcal{D}(V)$. Lemma 5.1 proves that $\phi$ preserves the topological types
of disks. In addition, $\phi$ sends cut systems to cut systems (Claim 5.6).
Next Lemma 7.2 shows that $\phi$ preserves duality. Also, for any cut system
$\mathbb{D}=\\{D_{i}\\}$, the complex $\operatorname{dual}_{i}(\mathbb{D})$ is
connected (Lemma 7.3).
Pick any geometric automorphism $f_{\operatorname{cut}}$ so that
$f_{\operatorname{cut}}(\mathbb{D})=\phi(\mathbb{D})$, vertex-wise;
$f_{\operatorname{cut}}$ exists by Claim 5.6. Define
$\phi_{\operatorname{cut}}=f^{-1}_{\operatorname{cut}}\circ\phi$. Thus
$\phi_{\operatorname{cut}}|\mathbb{D}=\operatorname{Id}.$
Let $V^{\prime}\cong V_{0,2g+n}$ be the spotted ball obtained by cutting $V$
along a regular neighborhood of $\mathbb{D}$. Now, since
$\phi_{\operatorname{cut}}$ preserves
$\operatorname{link}(\mathbb{D})\cong\mathcal{D}(V^{\prime})$, by Theorem 2.1
and Lemma 2.2 there is a homeomorphism $f\colon V^{\prime}\to V^{\prime}$ so
that the induced automorphism
$f\in\operatorname{Aut}(\mathcal{D}(V^{\prime}))$ satisfies
$f=\phi_{\operatorname{cut}}|\operatorname{link}(\mathbb{D})$. Section 6
proves that $f$ preserves the $g$ pairs of spots of $V^{\prime}$ coming from
$\mathbb{D}$. Thus $f$ can be glued to give a homeomorphism
$f_{\operatorname{link}}\colon V\to V$ as well as an induced geometric
automorphism $f_{\operatorname{link}}\in\operatorname{Aut}(\mathcal{D}(V))$.
Define
$\phi_{\operatorname{link}}=f^{-1}_{\operatorname{link}}\circ\phi_{\operatorname{cut}}$.
Thus
$\phi_{\operatorname{link}}|\mathbb{D}\cup\operatorname{link}(\mathbb{D})=\operatorname{Id}.$
Recall that $\phi_{\operatorname{link}}$ preserves duals by Lemma 7.2. For
every $D_{i}\in\mathbb{D}$ pick some dual
$E_{i}\in\operatorname{dual}_{i}(\mathbb{D})$. By Lemma 8.1 there is an
integer $m_{i}\in\mathbb{Z}$ so that
$T_{i}^{m_{i}}(E_{i})=\phi_{\operatorname{link}}(E_{i})$, where $T_{i}$ is the
Dehn twist about $D_{i}$. Define $f_{\operatorname{dual}}=\prod T_{i}^{m_{i}}$
and define
$\phi_{\operatorname{dual}}=f_{\operatorname{dual}}^{-1}\circ\phi_{\operatorname{link}}$.
Letting $\mathbb{E}=\\{E_{i}\\}$ we have
$\phi_{\operatorname{dual}}|\mathbb{D}\cup\operatorname{link}(\mathbb{D})\cup\mathbb{E}=\operatorname{Id}.$
Recall that Lemma 7.3 proves that $\operatorname{dual}_{i}(\mathbb{D})$ is
connected. Therefore, a crawling argument, given in Lemma 8.2, proves that
$\phi_{\operatorname{dual}}|\mathbb{D}\cup\operatorname{link}(\mathbb{D})\cup\operatorname{dual}(\mathbb{D})=\operatorname{Id}.$
Wajnryb [13] proves that the cut system complex is connected. Thus we may
likewise crawl through $\mathcal{D}(V)$ and prove (Section 9) that
$\phi_{\operatorname{dual}}=\operatorname{Id}$
and so prove that
$\phi=f_{\operatorname{cut}}\circ f_{\operatorname{link}}\circ
f_{\operatorname{dual}}.$
Thus $\phi$ is geometric.
## 4\. Small handlebodies
In this section we deal with the small cases, where $e(V)=2g-2+n\leq 2$. We
start with genus zero. If $n\leq 3$ then $\mathcal{D}(V_{0,n})$ is empty. By
Lemma 2.2 the mapping class groups of $V$ and $\partial_{+}V$ are equal. Thus
$\mathcal{H}(V_{0}),~{}\mathcal{H}(V_{0,1})\cong\mathbb{Z}/2\mathbb{Z}$
while
$\mathcal{H}(V_{0,2})\cong
K_{4}\quad\rm{and}\quad\mathcal{H}(V_{0,3})\cong\mathbb{Z}/2\mathbb{Z}\times\Sigma_{3}.$
Here $K_{4}$ is the Klein four-group and $\Sigma_{3}$ is the symmetric group
on three objects [12, Appendix A].
If $n=4$ then $\mathcal{D}$ is a countable collection of vertices with no
higher dimensional simplices. Thus
$\operatorname{Aut}(\mathcal{D})=\Sigma_{\infty}$ is uncountable. However,
there are only countably many geometric automorphisms. In fact, by Lemma 2.2,
the mapping class group $\mathcal{H}(V_{0,4})$ is isomorphic to
$K_{4}\rtimes\operatorname{PGL}(2,\mathbb{Z})$ [12, Appendix A].
For genus one, if $n=0$ or $1$ then $\mathcal{D}$ is a single point and
$\operatorname{Aut}(\mathcal{D})$ is trivial. On the other hand
$\mathcal{H}(V_{1}),~{}\mathcal{H}(V_{1,1})\cong\mathbb{Z}\rtimes K_{4}.$
For $V=V_{1,2}$ matters are more subtle. The subcomplex
$\operatorname{NonSep}(V)\subset\mathcal{D}(V)$, spanned by non-separating
disks, is a copy of the Bass-Serre tree for the meridian curve in
$S_{1,1}=\partial_{+}V_{1,1}$ [7]. Thus $\operatorname{NonSep}(V)$ is a copy
of $T_{\infty}$: the regular tree with countably infinite valance. Now, if
$E\in\mathcal{D}(V)$ is separating then there is a unique disk $D$ disjoint
from $E$; also, $D$ is necessarily non-separating. It follows that
$\mathcal{D}(V)$ is a copy of $\operatorname{NonSep}(V)$ with countably many
leaves attached to every vertex. Thus $\operatorname{Aut}(\mathcal{D})$
contains a copy of $\operatorname{Aut}(T_{\infty})$ as well as countably many
copies of $\Sigma_{\infty}$ and is therefore uncountable. As usual
$\mathcal{H}(V)$ is countable and so $\operatorname{Aut}(\mathcal{D})$
contains non-geometric elements. However, following Luo’s treatment of
$\mathcal{C}(S)$ [9] suggests the following problem:
###### Problem 4.1.
Suppose that $V=V_{1,2}$. Let $\mathcal{G}$ be the subgroup of
$\operatorname{Aut}(\mathcal{D}(V))$ consisting of automorphisms preserving
duality: if $\phi\in\mathcal{H}$ and $D,E$ are dual then so are
$\phi(D),\phi(E)$. Is every element of $\mathcal{G}$ geometric?
Note that this approach of recording duality is precisely correct for the
four-spotted ball; the complex where simplices record duality in $V_{0,4}$ is
the Farey tessellation, $\mathcal{F}$, and every element of
$\operatorname{Aut}(\mathcal{F})$ is geometric. See [9, Section 3.2].
The last exceptional case is $V=V_{2}$. Let $\operatorname{NonSep}(V)$ be the
subcomplex of $\mathcal{D}(V)$ spanned by non-separating disks. Then
$\operatorname{NonSep}(V)$ is an increasing union, as follows:
$\mathcal{N}_{0}$ is a single triangle, $\mathcal{N}_{i+1}$ is obtained by
attaching (to every free edge of $\mathcal{N}_{i}$) a countable collection of
triangles, and $\operatorname{NonSep}(V)$ is the increasing union of the
$\mathcal{N}_{i}$. A careful discussion of $\operatorname{NonSep}(V)$ is given
by Cho and McCullough [2, Section 4]
We obtain $\mathcal{D}(V)$ by attaching a countable collection of triangles to
every edge of $\operatorname{NonSep}(V)$. To see this note that every
separating disk $E$ divides $V$ into two copies of $V_{1,1}$. These copies of
$V_{1,1}$ have meridian disks, say $D$ and $D^{\prime}$. Thus
$\operatorname{link}(E)$ is an edge and the triangle $\\{E,D,D^{\prime}\\}$
has two free edges in $\mathcal{D}(V)$, as indicated. Finally, there is a
countable collection of separating disks lying in $V{\smallsetminus}(D\cup
D^{\prime})$, again as indicated.
It follows that $\operatorname{Aut}(\mathcal{D}(V_{2}))$ is uncountable.
Again, as in Problem 4.1, we may ask: are all “duality-respecting” elements
$f\in\operatorname{Aut}(\mathcal{D}(V_{2}))$ geometric? We end with another
open problem:
###### Problem 4.2.
Suppose that $V$ is a handlebody with $e(V)$ and genus both sufficiently
large. Show that
$\operatorname{Aut}(\operatorname{NonSep}(V))=\mathcal{H}(V)$.
A solution to Problem 4.2 may lead to a simplified proof of Theorem 10.1.
## 5\. Topological types
The goal of this section is:
###### Lemma 5.1.
Suppose that $\phi\in\operatorname{Aut}(\mathcal{D}(V))$. Then $\phi$
preserves topological types of disks.
The complexity of $V_{g,n}$ is $\xi(V)=3g-3+n$. If $\xi(V)\geq 1$ then
$\xi(V)$ is the number of vertices of a maximal simplex of $\mathcal{D}(V)$.
Note that $V_{1}$, $V_{1,1}$ and $V_{0,4}$ are the only handlebodies where
$\mathcal{D}(V)$ has dimension zero. (When $\mathcal{D}(V)$ is empty its
dimension is $-1$.) Further $V_{1}$ and $V_{1,1}$ are the only handlebodies
where $\mathcal{D}(V)$ is a single point.
We will call $V_{0,3}$, the three-spotted ball, a solid pair of pants. Thus
$\xi(V)$ is the number of disks in a pants decomposition of $V$ while
$e(V)=2g-2+n$ is the number of solid pants in the decomposition. We will call
$V_{1,1}$ a solid handle. Suppose now that $E$ is separating with
$V{\smallsetminus}\operatorname{neigh}(D)=X\cup Y$. If $X$ or $Y$ is a solid
pants then we call $E$ a pants disk. If $X$ or $Y$ is a solid handle then we
call $E$ a handle disk.
Recall that if $\mathcal{K}$ and $\mathcal{L}$ are non-empty simplicial
complexes with disjoint vertex sets then
$\mathcal{K}\mathbin{\vee}\mathcal{L}$, their join, is the complex
$\mathcal{K}\cup\\{\sigma\cup\tau\mathbin{\mid}\sigma\in\mathcal{K},~{}\tau\in\mathcal{L}\\}\cup\mathcal{L}.$
###### Claim 5.2.
For any handlebody $V$ the complex $\mathcal{D}(V)$ is not a join.
###### Proof.
When $e(V)\leq 2$ this can be checked case-by-case, following Section 4. The
remaining handlebodies all admit disks $D,E$ that fill: every disk $F$ meets
at least one of $D$ or $E$. It follows that any edge-path in
$\mathcal{D}^{(1)}(V)$ connecting $D$ to $E$ has length at least three.
However, the diameter of the one-skeleton of a join is either one or two. ∎
The complex $\mathcal{D}(V)$ is flag: minimal non-faces have dimension one.
Observe that $\phi$ preserves the combinatorics of $\mathcal{D}(V)$. Thus any
topological property of $V$ that has a combinatorial characterization will be
preserved by $\phi$. We proceed with a sequence of claims.
###### Claim 5.3.
The disk $E$ is a separating disk yet not a pants disk if and only if
$\operatorname{link}(E)$ is a join. Furthermore, in this case
$\operatorname{link}(E)$ is realized as a join in exactly one way, up to
permuting the factors.
###### Proof.
Suppose that $V{\smallsetminus}\operatorname{neigh}(E)=X\cup Y$, where neither
$X$ nor $Y$ is a solid pants. Since $E$ is essential and non-peripheral both
$\mathcal{D}(X)$ and $\mathcal{D}(Y)$ are non-empty. It follows that
$\operatorname{link}(E)=\mathcal{D}(X)\mathbin{\vee}\mathcal{D}(Y)$, and
neither factor is empty. Furthermore, this join is realized uniquely, because
$\mathcal{D}(X)$ is never itself a join (by Claim 5.2), $\mathcal{D}(X)$ is
flag and join is associative.
On the other hand, if $E$ is non-separating then $\operatorname{link}(E)$ is
isomorphic to $\mathcal{D}(V_{g-1,n+2})$. If $E$ is a pants disk then
$\operatorname{link}(E)\cong\mathcal{D}(V_{g,n-1})$. Neither of these is a
join by Claim 5.2. ∎
A cone is the join of a point with some non-empty simplicial complex.
###### Claim 5.4.
Suppose that $V\neq V_{1,2}$. Then $E\in\mathcal{D}(V)$ is a handle disk if
and only if $\operatorname{link}(E)$ is a cone.
###### Proof.
Suppose that $E$ cuts off a solid handle $X$ with meridian $D$. Let $Y$ be the
other component of $V{\smallsetminus}\operatorname{neigh}(E)$. Since $V\neq
V_{1,2}$ we have that $\mathcal{D}(Y)$ is non-empty; in particular $E$ is not
a pants disk. By Claim 5.3 we have
$\operatorname{link}(E)=\mathcal{D}(X)\mathbin{\vee}\mathcal{D}(Y)$. As
$\mathcal{D}(X)=\\{D\\}$ we are done with the forward direction.
Now suppose that $\operatorname{link}(E)$ is a cone from $D$. Since a cone is
the join of the apex with the base, by Claim 5.3 the disk $E$ is separating.
Let $V{\smallsetminus}\operatorname{neigh}(E)=X\cup Y$. Thus
$\operatorname{link}(E)=\mathcal{D}(X)\mathbin{\vee}\mathcal{D}(Y)$. However,
by Claim 5.3 the decomposition of $\operatorname{link}(E)$ is unique; breaking
symmetry we may assume that $\mathcal{D}(X)=\\{D\\}$. Thus $X$ is a solid
handle and we are done. ∎
It immediately follows that:
###### Claim 5.5.
Suppose that $V\neq V_{1,2}$. Then $D\in\mathcal{D}(V)$ is non-separating if
and only if there is an $E\in\mathcal{D}(V)$ so that $\operatorname{link}(E)$
is a cone with apex $D$. ∎
###### Claim 5.6.
Suppose that $e(V)\geq 3$. A simplex $\mathbb{D}\in\mathcal{D}(V)$ is a cut
system if and only if the following properties hold:
* •
for every pair of disks $D,E\in\operatorname{link}(\mathbb{D})$ the complex
$\operatorname{link}(E)\cap\operatorname{link}(\mathbb{D})$ is not a cone with
apex $D$ and
* •
for every proper subset $\sigma\subsetneq\mathbb{D}$ there is a pair of disks
$D,E\in\operatorname{link}(\sigma)$ so that the complex
$\operatorname{link}(E)\cap\operatorname{link}(\sigma)$ is a cone with apex
$D$.
###### Proof.
The forward direction follows from Claim 5.5 and the definition of a cut
system. (When $V$ is a spotted ball the only cut system is the empty set; the
empty set has no proper subsets.)
Now for the backwards direction: From the first property and by Claim 5.5
deduce that $V^{\prime}=V{\smallsetminus}\operatorname{neigh}(\mathbb{D})$ is
a collection of spotted balls. If $V^{\prime}$ has at least two components
then there is a proper subset $\sigma\subset\mathbb{D}$ which is a cut system
for $V$. Thus $V{\smallsetminus}\operatorname{neigh}(\sigma)$ is a spotted
ball and this contradicts the second property. ∎
###### Lemma 5.7.
Suppose that $V,W$ are handlebodies with $\mathcal{D}(V)\cong\mathcal{D}(W)$.
Then either:
* •
$V\mathrel{\cong}W$ or
* •
$V,W\in\\{V_{1},V_{1,1}\\}$ or
* •
$V,W\in\\{V_{0},V_{0,1},V_{0,2},V_{0,3}\\}$.
This is the handlebody version of [8, Lemma 4.5] and [9, Lemma 2.1].
###### Proof of Lemma 5.7.
When $e(V)\leq 2$ this can be checked case-by-case, following Section 4. When
$V$ has $e(V)\geq 3$ then $\xi(V)=\xi(W)$. By Claim 5.6 the handlebodies $V$
and $W$ have cut systems of the same size. It follows that $V,W$ have the same
genus and thus the same number of spots. ∎
We now have:
###### Proof of Lemma 5.1.
Let $V=V_{g,n}$ and fix $\phi\in\operatorname{Aut}(\mathcal{D}(V))$. When
$e(V)\leq 2$, Lemma 5.1 can be checked case-by-case, following Section 4. So
suppose that $e(V)\geq 3$.
The automorphism $\phi$ must preserve the set of non-separating disks by Claim
5.5.
Suppose that $E\in\mathcal{D}(V)$ is a separating disk yet not a pants disk.
Writing $V{\smallsetminus}\operatorname{neigh}(E)=X\cup Y$ we have
$\operatorname{link}(E)=\mathcal{D}(X)\mathbin{\vee}\mathcal{D}(Y)$. By Claim
5.3 this join is realized uniquely and so we can recover $\mathcal{D}(X)$ and
$\mathcal{D}(Y)$. By Lemma 5.7 we may deduce, combinatorially, the genus and
number of spots of $X$ and $Y$. Thus $\phi$ preserves the topological type of
$E$.
The only topological type remaining is the set of pants disks. Since all other
types are preserved, so are the pants disks. We are done. ∎
## 6\. Regluing
Suppose that $\phi_{\mathbb{D}}\in\operatorname{Aut}(\mathcal{D}(V))$ fixes
$\mathbb{D}$. By Lemma 2.2 there is a homeomorphism $f$ of
$V^{\prime}=V{\smallsetminus}\operatorname{neigh}(\mathbb{D})$ so that the
induced geometric automorphism equals
$\phi_{\mathbb{D}}|\operatorname{link}(\mathbb{D})$. We must show that $f$
gives a homeomorphism of $V$: that is, for every $i$ the spots $D_{i}^{\pm}$
are preserved by $f$.
Let
$\operatorname{handle}_{i}(\mathbb{D})\subset\operatorname{link}(\mathbb{D})$
be the collection of handle disks $E\in\mathcal{D}(V)$ such that
* •
one component of $V{\smallsetminus}\operatorname{neigh}(E)$ is a solid handle
containing $D_{i}$ and
* •
$E$ is disjoint from all of the $D_{j}$.
Let $\operatorname{pants}_{i}(\mathbb{D})\subset\mathcal{D}(V^{\prime})$ be
the collection of pants disks $E$ such that one component of
$V^{\prime}{\smallsetminus}\operatorname{neigh}(E)$ is a solid pants meeting
the spots $D_{i}^{\pm}$.
By the claims in the previous section the set
$\operatorname{handle}_{i}(\mathbb{D})$ is, for all $i$, combinatorially
characterized and so preserved by $\phi_{\mathbb{D}}$. It follows that the
homeomorphism $f\in\operatorname{Homeo}(V^{\prime})$ preserves the set
$\operatorname{pants}_{i}(\mathbb{D})$, for all $i$. Now, suppose that
$f(D_{1}^{+}),f(D_{1}^{-})=A,B$ where $A,B$ are spots of $V^{\prime}$. Let
$E\in\operatorname{pants}_{1}(\mathbb{D})$ be any pants disk. Then $f(E)$ is a
pants disk cutting off $A$ and $B$. It follows that the spots $A,B$ (in some
order) equal the spots $D_{1}^{\pm}$ as desired.
## 7\. Duality
Recall that two disks $D,E\in\mathcal{D}(V)$ are dual if $i(\partial
D,\partial E)=2$ (see Figure 2). A pentagon $P\subset\mathcal{D}(V_{0,5})$ is
a collection of five disks $P=\\{E_{i}\\}_{i=0}^{4}$ so that $E_{i}$ and
$E_{i+1}$ are disjoint, for all $i$ (modulo five). We say that the disks
$E_{i},E_{i+2}$ are non-adjacent in $P$, for all $i$ (modulo five).
###### Lemma 7.1 (Pentagon Lemma).
Suppose that $V=V_{0,5}$. Two disks $D,E\in\mathcal{D}(V)$ are dual if and
only if there is a pentagon $P$ so that $D,E\in P$ and $D,E$ are non-adjacent
in $P$.
###### Proof.
Recall that $\mathcal{D}(V_{0,5})\cong\mathcal{C}(S_{0,5})$, by Lemma 2.2. The
pentagon lemma for $S_{0,5}$ (see [8, Theorem 3.2] or [9, Lemma 4.2]) implies
that there is only one pentagon in $\mathcal{D}(V_{0,5})$, up to the action of
the handlebody group. ∎
###### Lemma 7.2.
Suppose that $V=V_{g,n}$ has $e(V)\geq 3$. Two disks $D,E\in\mathcal{D}(V)$
are dual if and only if there is a simplex $\sigma\in\mathcal{D}(V)$ with
* •
$\operatorname{link}(\sigma)\cong\mathcal{D}(V_{0,5})$,
* •
$D,E$ are non-adjacent in some pentagon of $\operatorname{link}(\sigma)$.
It follows that every $\phi\in\operatorname{Aut}(\mathcal{D}(V))$ preserves
duality. We will say that a handlebody $W\subset V$ is cleanly embedded if:
* •
all spots of $W$ are essential in $V$ and
* •
if a spot of $W$ is peripheral in $V$ then it is also a spot of $V$.
###### Proof of Lemma 7.2.
Suppose that $D,E$ are dual. Let $X$ be the four-spotted ball containing them.
Isotope $X$ to be cleanly embedded. Let $\mathbb{E}$ be a pants decomposition
of $V^{\prime}=V{\smallsetminus}\operatorname{neigh}(X)$. Now, there is at
least one solid pants $P$ in
$V^{\prime}{\smallsetminus}\operatorname{neigh}(\mathbb{E})$ which has a spot,
say $F$, which is parallel to a spot of $X$. If not then $e(V)\leq 2$, a
contradiction.
Let $Y=X\cup\operatorname{neigh}(F)\cup P$ and notice that this is a five-
spotted ball containing $D$ and $E$, our original disks. Isotope $Y$ to be
cleanly embedded. Let $\mathbb{E}^{\prime}$ be any pants decomposition of
$V{\smallsetminus}\operatorname{neigh}(Y)$. Add to $\mathbb{E}^{\prime}$ any
spots of $Y$ which are non-peripheral in $V$. This then is the desired simplex
$\sigma\in\mathcal{D}(V)$. Since $D$ and $E$ are dual the pentagon lemma
implies that there is a pentagon in $\mathcal{D}(Y)$ making $D,E$ non-
adjacent.
The backwards direction follows from Lemma 5.7, the combinatorial
characterization of genus and number of spots, and from the pentagon lemma. ∎
We now discuss the dual complex. Fix a cut system $\mathbb{D}=\\{D_{i}\\}$.
Recall that $\operatorname{dual}_{i}(\mathbb{D})$ is the subcomplex of
$\mathcal{D}(V)$ spanned by the disks $E\in\mathcal{D}(V)$ which are dual to
$D_{i}$ and disjoint from $D_{j}$ for all $j\neq i$.
Define $V_{i}$ to be the spotted solid torus obtained by cutting $V$ along all
disks of $\mathbb{D}$ except $D_{i}$. Note that $V_{i}$ has exactly
$e(V)$–many spots, and this is at least three. Also, $D_{i}$ is a meridian
disk for $V_{i}$. Note that
$\operatorname{dual}_{i}(\mathbb{D})\subset\mathcal{D}(V_{i})$. A disk
$E\in\operatorname{dual}_{i}(\mathbb{D})$ is a simple dual if $E$ is a pants
disk in $V_{i}$.
Let $\mathcal{A}_{i}(\mathbb{D})$ be the complex where vertices are isotopy
classes of arcs $\alpha\subset\partial_{+}V_{i}$ so that
* •
$\alpha$ meets $\partial D_{i}$ exactly once, transversely, and
* •
$\partial\alpha$ meets distinct spots of $V_{i}$.
A collection of vertices spans a simplex if they can be realized disjointly.
If an arc $\alpha\in\mathcal{A}_{i}(\mathbb{D})$ meets spots
$A,B\in\partial_{0}V_{i}$ then the frontier of
${\overline{\operatorname{neigh}}}(A\cup\alpha\cup B)$ is a simple dual,
$E_{\alpha}$.
###### Lemma 7.3.
If $e(V)\geq 3$ then the complex $\operatorname{dual}_{i}(\mathbb{D})$ is
connected.
It suffices to check this for $i=1$. To simplify notation we write $D=D_{1}$,
$U=V_{1}$, $\operatorname{dual}(D)=\operatorname{dual}_{i}(\mathbb{D})$ and
$\mathcal{A}(D)=\mathcal{A}_{1}(\mathbb{D})$. We will prove Lemma 7.3 via a
sequence of claims.
###### Claim.
For any pair of arcs $\alpha,\gamma\in\mathcal{A}(D)$ there is a sequence
$\\{\alpha_{k}\\}_{k=0}^{N}\subset\mathcal{A}(D)$ so that:
* •
the arcs $\alpha_{k},\alpha_{k+1}$ are disjoint, for all $k<N$,
* •
$\alpha_{0}=\alpha$ and $\alpha_{N}=\gamma$, and
* •
there is at most one spot in common between the endpoints of $\alpha_{k}$ and
$\alpha_{k+1}$, for all $k<N$.
###### Proof.
Fix, for the remainder of the proof, an arc $\beta\in\mathcal{A}(D)$ so that
$\alpha$ and $\beta$ are disjoint and so that the endpoints of $\alpha$ and
$\beta$ share at most one spot. This is possible as $U$ has at least three
spots. Define the complexity of $\gamma$ to be
$c(\gamma)=i(\alpha,\gamma)+i(\beta,\gamma)$. Notice if $c(\gamma)=0$ then we
are done: one of the sequences
$\\{\alpha,\gamma\\}\quad\mbox{or}\quad\\{\alpha,\beta,\gamma\\}$
has the desired properties.
Now induct on $c(\gamma)$. Suppose, breaking symmetry, that $\alpha$ meets a
spot, say $A\in\partial_{0}U$, so that $\gamma\cap A=\emptyset$. If
$i(\alpha,\gamma)=0$ then the sequence $\\{\alpha,\gamma\\}$ has the desired
properties. If not, then let $x$ be the point of $\alpha\cap\gamma$ that is
closest, along $\alpha$, to the endpoint $\alpha\cap A$. Let
$\alpha^{\prime}\subset\alpha$ be the subarc connecting $x$ and $\alpha\cap
A$. Let $N$ be a regular neighborhood, taken in $\partial_{+}U$, of
$\gamma\cup\alpha^{\prime}$. The frontier of $N$, in $\partial_{+}U$, is a
union of three arcs: one arc properly isotopic to $\gamma$ and two more arcs
$\gamma^{\prime},\gamma^{\prime\prime}$.
The arcs $\gamma^{\prime}$ and $\gamma^{\prime\prime}$ are disjoint from
$\gamma$ and satisfy $c(\gamma^{\prime})+c(\gamma^{\prime\prime})\leq
c(\gamma)-1$. Also, since $\gamma^{\prime}$ and $\gamma^{\prime\prime}$ each
have one endpoint on the spot $A$ the arcs $\gamma^{\prime}$ and
$\gamma^{\prime\prime}$ have exactly one spot in common with $\gamma$. Now, if
$\alpha^{\prime}\cap\partial D=\emptyset$ then one of
$\gamma^{\prime},\gamma^{\prime\prime}$ meets $\partial D$ once and the other
is disjoint. On the other hand, if $\alpha^{\prime}\cap\partial
D\neq\emptyset$ then $\alpha^{\prime}$ meets $\partial D$ once. Thus one of
$\gamma^{\prime},\gamma^{\prime\prime}$ meets $\partial D$ once and the other
meets $\partial D$ twice. In either case we are done. ∎
Recall that if $\alpha\in\mathcal{A}(D)$ is an arc then $E_{\alpha}$ is the
associated simple dual.
###### Claim.
If $\alpha,\beta\in\mathcal{A}(D)$ are disjoint arcs, with at most one spot in
common between their endpoints, then there is an edge-path in
$\operatorname{dual}(D)$ of length at most four between $E_{\alpha}$ and
$E_{\beta}$.
###### Proof.
If $\alpha$ and $\beta$ share no spots then $\\{E_{\alpha},E_{\beta}\\}$ is a
path of length one. Suppose that $\alpha$ and $\beta$ share a single spot. Let
$A,B,C$ be the three spots that $\alpha$ and $\beta$ meet, with both meeting
$C$. Let $\alpha^{\prime},\beta^{\prime}$ be the subarcs of $\alpha,\beta$
connecting $C$ to $\partial D$. There are two cases: either $\alpha^{\prime}$
and $\beta^{\prime}$ are incident on the same side of $\partial D$ or are
incident on opposite sides.
Suppose that $\alpha^{\prime}$ and $\beta^{\prime}$ are incident on the same
side of $\partial D$. Then $\alpha^{\prime}$ and $\beta^{\prime}$, together
with subarcs of $\partial C$ and $\partial D$ bound a disk
$\Delta\subset\partial U$. Note that $\Delta$ may contain spots, but it meets
$A\cup B\cup C$ only along the subarc in $\partial C$. It follows that the
disk $F$, defined to be the frontier of
${\overline{\operatorname{neigh}}}\left((A\cup B\cup
C)\cup(\alpha\cup\beta)\cup\Delta\right),$
is dual to $D$. The disk $F$ is also essential as it separates at least three
spots from a solid handle. So $\\{E_{\alpha},F,E_{\beta}\\}$ is the desired
path.
Suppose that $\alpha^{\prime}$ and $\beta^{\prime}$ are incident on opposite
sides of $\partial D$. Let $d\subset\partial D$ be either component of
$\partial D{\smallsetminus}(\alpha\cup\beta)$. Let
$\alpha^{\prime\prime}={\overline{\alpha{\smallsetminus}\alpha^{\prime}}}$ and
define $\beta^{\prime\prime}$ similarly. Define $\gamma\in\mathcal{A}(D)$ by
forming the arc $\alpha^{\prime\prime}\cup d\cup\beta^{\prime\prime}$ and
using an proper isotopy of $\partial_{+}U$ to make $\gamma$ transverse to
$\partial D$. Now apply the previous paragraph to the pairs
$\\{\alpha,\gamma\\}$ and $\\{\gamma,\beta\\}$ to obtain the desired path of
length four. ∎
###### Claim.
For every dual $E\in\operatorname{dual}(D)$ there is a simple dual connected
to $E$ by an edge-path of length at most two.
###### Proof.
The graph $\partial E\cup\partial D$ cuts $\partial U$ into a pair of disks
$B,C$ and an annulus $A$. Each of $B,C$ contain at least one spot.
Suppose $E$ is separating. Then the disks $B,C$ are adjacent along an subarc
of $\partial D$. Connect a spot in $B$ to a spot in $C$ by an arc $\alpha$
that meets $\partial D$ once and that is disjoint from $\partial E$. Thus
$E_{\alpha}$ is disjoint from $E$.
Suppose $E$ is non-separating. Then the two disks $B,C$ meet only at the
points of $\partial D\cap\partial E$. Now, if the annulus $A$ contains a spot
then we may connect a spot in $B$ to a spot in $A$ by an arc $\alpha$ meeting
$\partial D$ once and $\partial E$ not at all. In this case we are done as in
the previous paragraph.
If $A$ contains no spots then, breaking symmetry, we may assume that $B$
contains at least two spots while $C$ contains at least one. Let $\delta$ be
an arc connecting some spot, say $B^{\prime}\subset B$, to $E$. Let $N$ be a
regular neighorhood of $E\cup\delta\cup B^{\prime}$. Then the frontier of $N$
contains two disks. One of these is isotopic to $E$ while the other, say
$E^{\prime}$, is non-separating, dual to $D$, and divides the spots as
described in the previous paragraph. ∎
Equipped with these claims we have:
###### Proof of Lemma 7.3.
The first two claims imply that the set of simple duals in
$\operatorname{dual}(D)$ is contained in a connected set. The third claim
shows that every vertex in $\operatorname{dual}(D)$ is distance at most two
from the set of simple duals. This completes the proof. ∎
## 8\. Crawling through the complex of duals
###### Lemma 8.1.
Suppose that $\phi_{\operatorname{link}}$ fixes $\mathbb{D}$ and
$\operatorname{link}(\mathbb{D})$. For any
$E\in\operatorname{dual}_{i}(\mathbb{D})$ the disks $E$ and $\phi(E)$ differ
by some power of $T_{i}$, the Dehn twist about $D_{i}$.
###### Proof.
As usual, it suffices to prove this for $D=D_{1}$. Let $U=V_{1}$.
Let $X\subset U$ be the four-spotted ball filled by $D$ and the dual disk $E$.
Isotope $X$ to be cleanly embedded. Let $\mathbb{F}$ be the components of
$\partial_{0}X$ which are not spots of $U$. Note that
$\phi_{\operatorname{link}}$ fixes $D$ as well as every disk of $\mathbb{F}$.
This, together with Lemma 7.2, implies that $\phi_{\operatorname{link}}$
preserves the set of disks that are contained in $X$ and dual to $D$.
Since $\mathcal{D}(X)$ equipped with the duality relation is a copy of
$\mathcal{F}$, the Farey graph, it follows that $E$ and
$F=\phi_{\operatorname{link}}(E)$ differ by some number of half-twists about
$D$. If $E$ and $F$ differ by an odd number of half-twists then $E$ and $F$
have differing topological types, contradicting Lemma 5.1 applied to
$\phi_{\operatorname{link}}|\mathcal{D}(U)$. Thus $E$ and $F$ differ by an
even number of half-twists, as desired. ∎
###### Lemma 8.2.
Suppose that $\phi_{\operatorname{dual}}$ fixes $\mathbb{D}$,
$\operatorname{link}(\mathbb{D})$, and $\mathbb{E}$, a collection of duals
(that is, $E_{i}\in\operatorname{dual}_{i}(\mathbb{D})$). Then
$\phi_{\operatorname{dual}}$ fixes every vertex of
$\operatorname{dual}_{i}(\mathbb{D})$, for all $i$.
###### Proof.
As usual, it suffices to prove this for $D=D_{1}$. Let $E=E_{1}$ and let
$U=V_{1}$. We crawl through
$\operatorname{dual}(D)=\operatorname{dual}_{1}(\mathbb{D})$, as follows.
Suppose that $F,G\in\operatorname{dual}(D)$ are adjacent vertices and suppose
that $\phi_{\operatorname{dual}}(F)=F$. By Lemma 8.1, the disks $G$ and
$G^{\prime}=\phi_{\operatorname{dual}}(G)$ differ by some number of Dehn
twists about $D$. Also, as $\phi_{\operatorname{dual}}$ is a simplical
automorphism the disks $F$ and $G^{\prime}$ are disjoint. Let $X$ be the four-
spotted ball filled by $D$ and $F$. If $G$ and $G^{\prime}$ are not equal then
$G\cap X$ and $G^{\prime}\cap X$ are also not equal and in fact differ by some
non-zero number of twists; thus one of $G\cap X$ or $G^{\prime}\cap X$ must
cross $F$, a contradiction.
Recall that $\phi_{\operatorname{dual}}(E)=E$. Suppose that $G$ is any vertex
of $\operatorname{dual}(D)$. Since $\operatorname{dual}(D)$ is connected
(Lemma 7.3) there is a path $\mathcal{P}\subset\operatorname{dual}(D)$
connecting $E$ to $G$. Induction along $\mathcal{P}$ completes the proof. ∎
## 9\. Crawling through the disk complex
Before continuing we will need the following complex:
###### Definition 9.1 (Wajnryb [13]).
The cut system graph $\mathcal{CG}(V)$ is the graph with vertex set being
isotopy classes of unordered cut systems in $V$. Edges are given by pairs of
cut systems with $g-1$ disks in common and the remaining pair of disks
disjoint.
Wajnryb also gives a two-skeleton, but we will only require:
###### Theorem 9.2 (Wajnryb [13]).
The cut system graph $\mathcal{CG}(V)$ is connected.
For the remainder of this section suppose that
$\Phi=\phi_{\operatorname{dual}}$ is an automorphism of $\mathcal{D}(V)$ and
$\mathbb{D}$ is a cut system so that $\Phi$ fixes $\mathbb{D}$,
$\operatorname{link}(\mathbb{D})$ and $\operatorname{dual}(\mathbb{D})$.
For the crawling step, suppose that $\mathbb{E},\mathbb{F}$ are adjacent in
$\mathcal{CG}(V)$ and that $\Phi$ fixes $\mathbb{E}$,
$\operatorname{link}(\mathbb{E})$ and $\operatorname{dual}(\mathbb{E})$. Let
$\mathbb{G}$ be a pants decomposition obtained by adding the new disk of
$\mathbb{F}$ to $\mathbb{E}$ and then adding non-separating disks until we
have $3g-3+n$ disks. Let $\\{P_{k}\\}$ enumerate the solid pants of
$\mathbb{G}$. Let $X_{i}=P_{k}\cup P_{\ell}$ be the four-spotted ball
containing $G_{i}$ in its interior.
Let $\mathbb{H},\mathbb{I}=\\{H_{i}\\},\\{I_{i}\\}$ be collections of disks so
that $H_{i},I_{i}$ are contained in $X_{i}$ and $G_{i},H_{i},I_{i}$ are
pairwise dual in $X_{i}$. Now, all of these disks
$\mathbb{G}\cup\mathbb{H}\cup\mathbb{I}$ lie in
$\mathbb{E}\cup\operatorname{link}(\mathbb{E})\cup\operatorname{dual}(\mathbb{E})$.
Thus $\Phi$ fixes all of them. Thus $\Phi$ fixes $\mathbb{F}$. Consider
$\Phi|\operatorname{link}(\mathbb{F})$. By Theorem 2.1 the automorphism
$f=\Phi|\operatorname{link}(\mathbb{F})$ is geometric. Let $f$ also denote the
given homeomorphism of
$V^{\prime}=V{\smallsetminus}\operatorname{neigh}(\mathbb{F})$. Let
$\mathbb{G}^{\prime}=\mathbb{G}{\smallsetminus}\mathbb{F}$ and
$\mathbb{H}^{\prime},\mathbb{I}^{\prime}$ be the disks of
$\mathbb{H},\mathbb{I}$ contained in $V^{\prime}$. Thus $f$ fixes all disks of
$\mathbb{G}^{\prime},\mathbb{H}^{\prime},\mathbb{I}^{\prime}$. It follows that
$f$ permutes the solid pants $\\{P_{k}\\}$.
If $f$ nontrivially permutes $\\{P_{k}\\}$ then, since each $G_{i}$ is fixed,
we find that adjacent solid pants are interchanged. This implies that
$V^{\prime}=P_{1}\cup P_{2}$, a contradiction.
So $f$ fixes every $P_{k}$. Since all disks in $\mathbb{G}^{\prime}$ are
fixed, $f$ is either orientation reversing, isotopic to the identity, or
isotopic to a half-twist on each of the $P_{k}$. Let
$G_{i}\in\mathbb{G}^{\prime}$ be any disk meeting $P_{k}$. Then $f|P_{k}$
cannot be orientation reversing because the triple $G_{i},H_{i},I_{i}$
determines an orientation on $X_{i}$ and hence on $P_{k}$. If $f|P_{k}$ is a
half-twist then $P_{k}$ meets two spots of $V^{\prime}$. Thus $G_{i}$ meets
two solid pants $P_{k},P_{\ell}$ so that $X_{i}=P_{k}\cup P_{\ell}$. Now, as
$e(V^{\prime})\geq 3$, the solid pants $P_{\ell}$ meets at most one spot of
$V^{\prime}$. Thus $f|P_{\ell}$ is isotopic to the identity. So if $f|P_{k}$
is a half-twist then $f(H_{i})\neq H_{i}$, a contradiction. Deduce that $f$,
when restricted to any solid pants, is isotopic to the identity. Now, since
$f$ fixes all of the $H_{i}$, $f$ is isotopic to the identity on $V^{\prime}$,
as desired.
Deduce that $\Phi|\operatorname{link}(\mathbb{F})$ is the identity. As $\Phi$
fixes duals to $\mathbb{F}$ by Lemma 8.2 the automorphism $\Phi$ fixes all of
$\operatorname{dual}(\mathbb{F})$. This completes the crawling step and so
completes the proof of:
###### Theorem 9.3.
If a handlebody $V=V_{g,n}$ satisfies $e(V)\geq 3$ then the natural map
$\mathcal{H}(V)\to\operatorname{Aut}(\mathcal{D}(V))$ is a surjection. ∎
As a corollary:
###### Theorem 9.4.
If a handlebody $V=V_{g,n}$ satisfies $e(V)\geq 3$ then the natural map
$\mathcal{H}(V)\to\operatorname{Aut}(\mathcal{D}(V))$ is an isomorphism.
Note that Theorems 9.3 and 9.4 are sharp: when $e(V)\leq 2$ the conclusions
are false. See Section 4.
###### Proof of Theorem 9.4.
Theorem 9.3 shows that the natural map is surjective. Suppose that the mapping
class $f$ lies in the kernel. As in the discussion of crawling through
$\mathcal{CG}(V)$ given above, let $\mathbb{G}=\\{G_{i}\\}$ be a pants
decomposition of $V$ so that all of the $G_{i}$ are non-separating. Let
$\\{P_{k}\\}$ enumerate the solid pants of this decomposition. Let
$X_{i}=P_{j}\cup P_{k}$ be the four-spotted ball containing $G_{i}$ in its
interior. Let $\mathbb{H},\mathbb{I}=\\{H_{i}\\},\\{I_{i}\\}$ be collections
of disks so that $H_{i},I_{i}$ are contained in $X_{i}$ and
$G_{i},H_{i},I_{i}$ are pairwise dual in $X_{i}$. All of these disks are fixed
by $f$. It follows that $f$ is isotopic to the identity. ∎
## 10\. An application
###### Theorem 10.1.
If $e(V)\geq 3$ then the outer automorphism group of the handlebody group is
trivial.
This may be restated as: $\operatorname{Aut}(\mathcal{H})\cong\mathcal{H}$.
When $g=0$ then Theorem 10.1 follows from Lemma 2.2 and the first author’s
thesis [8, Theorem 3]. For the rest of this section we restrict to the case
$g\geq 1$.
The idea of the proof is to turn an element
$\phi\in\operatorname{Aut}(\mathcal{H})$ into an automorphism of the disk
complex $\mathcal{D}(V)$. We do this, following [4], by giving an algebraic
characterization of first Dehn twists about non-separating disks and then Dehn
twists generally. We then apply Theorem 9.3 to $\phi$ to find the correspoding
geometric automorphism. An algebraic trick then gives the desired result.
A finite index subgroup $\Gamma<\mathcal{H}$ is pure if every reducible class
in $\Gamma$ fixes every component of every reducing set. For example, the
kernel of
$\mathcal{H}\to\operatorname{Aut}(H_{1}(\partial_{+}V,\mathbb{Z}/3\mathbb{Z}))$
is pure.
###### Lemma 10.2.
Suppose $\Gamma<\mathcal{H}$ is pure and finite index. Then
$\\{f_{i}\\}\subset\mathcal{H}$ is a collection of Dehn twists along a pants
decomposition of non-separating disks in $V$ if and only if
* •
the subgroup $A=\langle f_{i}\rangle$ is free Abelian of rank $\xi(V)$,
* •
$f_{i}$ and $f_{j}$ are conjugate in $\mathcal{H}$, for all $i,j$,
* •
$f_{i}$ is primitive in $C_{\mathcal{H}}(A)$: $f_{i}$ is not a proper power of
any $h\in C_{\mathcal{H}}(A)$, and
* •
the center of the centralizer of the class $f_{i}^{n}$ in $\Gamma$ is infinite
cyclic (for all $i$ and for all $n$ so that $f_{i}^{n}\in\Gamma$):
$C(C_{\Gamma}(f_{i}^{n}))\cong\mathbb{Z}.$
###### Proof.
The forwards direction is identical to the forwards direction of [4, Theorem
2.1]. The backwards direction is similar in spirit to the backwards direction
of [4, Theorem 2.1] but some details differ. Accordingly we sketch the
backwards direction.
The mapping class $f_{i}$ can not be periodic or pseudo-Anosov as that would
contradict the first property. Let $\Theta\subset S=\partial_{+}V$ be the
canonical reduction system for the Abelian group $A$ [1]. Let $\\{X_{j}\\}$ be
the components of $S{\smallsetminus}\operatorname{neigh}(\Theta)$ and let
$\\{Y_{k}\\}$ be the collection of annuli
${\overline{\operatorname{neigh}}}(\Theta)$. By [1, Lemma 3.1(2)] the number
of annuli in $\\{Y_{k}\\}$ plus the number of non-pants in $\\{X_{j}\\}$
equals $\xi(V)$. It follows that every non-pants $X_{j}$ has complexity one
(so is homeomorphic to $S_{0,4}$ or $S_{1,1}$).
Fix a power $n$ (independent of $i$) to ensure that $f_{i}^{n}\in\Gamma$. For
each $X_{j}$ of complexity one there is some $f_{i}^{n}$ so that
$f_{i}^{n}|X_{j}$ is pseudo-Anosov. Suppose that $f=f_{1}^{n}$, $X=X_{1}$ has
complexity one, and $f|X$ is pseudo-Anosov. Let $\lambda^{\pm}$ be the stable
and unstable laminations of $f|X$. For every $i$, the mapping $f_{i}^{n}|X$ is
either the identity or pseudo-Anosov. Note that in the latter case the stable
and unstable laminations of $f_{i}^{n}|X$ agree with $\lambda^{\pm}$:
otherwise a ping-pong argument gives a rank two free group in $A$, a
contradiction. Thus, perhaps taking a larger power $n$, we may assume that for
each $i$ either $f_{i}^{n}|X$ is the identity or identical to $f|X$. For each
$i$ where $f_{i}^{n}|X=f|X$ we temporarily replace $f_{i}$ by
$f_{i}^{n}f^{-1}$.
Continuing in this manner we find a free Abelian group $B<A\cap\Gamma$ of rank
at least $|\Theta|$ where all elements are supported inside of the union of
annuli $\\{Y_{k}\\}$. Since $B$ is pure, it follows that all elements of $B$
are compositions of powers of Dehn twists along disjoint curves. A theorem of
McCullough [11] implies that every curve in $\Theta$ either bounds a disk or
cobounds an annulus with some other curve of $\Theta$. However, each annulus
reduces the possible rank of $B$ by one; it follows that every curve in
$\Theta$ bounds a disk.
Let $\gamma$ be any essential non-peripherial component of $\partial X$. It
follows that $f$ commutes with $T_{\gamma}$, that $T_{\gamma}$ lies in
$\mathcal{H}$ by the above paragraph, and that $T_{\gamma}$ to some power lies
in $C(C_{\Gamma}(f))$. But this contradicts the fourth property. It follows
that every component $X_{j}$ is a pants and that $|\Theta|=\xi(V)$. Thus every
$f_{i}$ is a compositions of powers of disjoint twists. Again, by the fourth
property each $f_{i}$ is some power of a single twist. By the third property
(following [4]) $f_{i}$ is in fact a twist. Finally, by the second property,
each twist is supported on a disk of the same topological type. As every pants
decomposition of $V$ must contain a non-separating disk all of the twists
$f_{i}$ are supported by non-separating disks. ∎
We now give the general characterization:
###### Lemma 10.3.
Suppose $\Gamma<\mathcal{H}$ is pure and finite index. Then
$\\{f_{i}\\}\subset\mathcal{H}$ is a collection of Dehn twists along a pants
decomposition of $V$ if and only if
* •
the subgroup $A=\langle f_{i}\rangle$ is free Abelian of rank $\xi(V)$,
* •
$f_{i}$ is primitive in $C_{\mathcal{H}}(A)$,
* •
for all $i$ and for all $n$ so that $f_{i}^{n}\in\Gamma$ either
$C(C_{\Gamma}(f_{i}^{n}))\cong\mathbb{Z}$ or there is a $j$ so that
$C(C_{\Gamma}(f_{i}^{n}))\cong\mathbb{Z}^{2}$ with the latter given by
$\langle f_{i},f_{j}\rangle$ and $f_{j}$ is a twist on a non-separating disk.
###### Proof.
Suppose that $\\{D_{i}\\}$ is a pants decomposition and $f_{i}$ is the
positive twist on $D_{i}$. Then $A=\langle f_{i}\rangle$ is free Abelian of
the correct rank. The second property follows as $A=C_{\mathcal{H}}(A)$. The
third property follows from Ivanov’s discussion [4] except if $D_{i}$ is a
handle disk. In this case the meridian of the handle, say $D_{j}$, gives a
twist $f_{j}$ which lies in the center of the centralizer.
The backwards direction is similar to that of the proof of Lemma 10.2. The
only change occurs when $f|X$ is pseudo-Anosov: when the center of the
centralizer has rank two then the additional element is a twist about a
separating disk and this contradicts the third property. ∎
The following lemmas follow from the idential statements for the mapping class
group of $S$ [6]:
###### Lemma 10.4.
Suppose $D$ and $E$ are essential disks. The twists $T_{D},T_{E}$ commute if
and only if $D$ and $E$ can be made disjoint via proper isotopy. ∎
###### Lemma 10.5.
For any twist $T_{D}$ and for any homeomorphism $h$ we have
$hT_{D}h^{-1}=T_{h(D)}$. ∎
###### Lemma 10.6.
For any pair of disks $D$ and $E$ and any pair of integers $n$ and $m$, if
$T_{D}^{n}=T_{E}^{m}$ then $D=E$ and $n=m$. ∎
The proof of Theorem 10.1 now follows, essentially line-by-line, the proof of
either [5, Theorem 2] or [8, Theorem 3]. ∎
To extend our algebraic characterization of twists in $\mathcal{H}(V)$ (Lemmas
10.2 and 10.3) to a characterization of powers of twists inside of finite
index pure subgroups $\Gamma<\mathcal{H}(V)$ appears to be a delicate matter.
Solving this problem would, following Ivanov [5], solve:
###### Problem 10.7.
Show that the abstract commensurator of $\mathcal{H}(V)$ is $\mathcal{H}(V)$
itself. Show that $\mathcal{H}(V)$ is not arithmetic.
## References
* [1] Joan S. Birman, Alex Lubotzky, and John McCarthy. Abelian and solvable subgroups of the mapping class groups. Duke Math. J., 50(4):1107–1120, 1983. http://www.math.columbia.edu/$\sim$jb/papers.html.
* [2] Sangbum Cho and Darryl McCullough. The tree of knot tunnels. arXiv:math/0611921.
* [3] Willam J. Harvey. Boundary structure of the modular group. In Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978), pages 245–251, Princeton, N.J., 1981. Princeton Univ. Press.
* [4] N. V. Ivanov. Automorphisms of Teichmüller modular groups. In Topology and geometry—Rohlin Seminar, volume 1346 of Lecture Notes in Math., pages 199–270. Springer, Berlin, 1988.
* [5] Nikolai V. Ivanov. Automorphism of complexes of curves and of Teichmüller spaces. Internat. Math. Res. Notices, (14):651–666, 1997.
* [6] Nikolai V. Ivanov and John D. McCarthy. On injective homomorphisms between Teichmüller modular groups. I. Invent. Math., 135(2):425–486, 1999.
* [7] Richard Kent, Chris Leininger, and Saul Schleimer. Trees and mapping class groups. J. Reine Angew. Math. To appear. arXiv:math/0611241.
* [8] Mustafa Korkmaz. Automorphisms of complexes of curves on punctured spheres and on punctured tori. Topology Appl., 95(2):85–111, 1999.
* [9] Feng Luo. Automorphisms of the complex of curves. Topology, 39(2):283–298, 2000. arXiv:math/9904020.
* [10] Darryl McCullough. Virtually geometrically finite mapping class groups of $3$-manifolds. J. Differential Geom., 33(1):1–65, 1991.
* [11] Darryl McCullough. Homeomorphisms which are Dehn twists on the boundary. 2002\. http://www.math.ou.edu/$\sim$dmccullough/research/pdffiles/dehn.pdf.
* [12] Kasra Rafi and Saul Schleimer. Curve complexes are rigid, 2008. arXiv:math/0710.3794.
* [13] Bronisław Wajnryb. Mapping class group of a handlebody. Fund. Math., 158(3):195–228, 1998.
|
arxiv-papers
| 2009-10-11T20:28:31 |
2024-09-04T02:49:05.806653
|
{
"license": "Public Domain",
"authors": "Mustafa Korkmaz, Saul Schleimer",
"submitter": "Saul Schleimer",
"url": "https://arxiv.org/abs/0910.2038"
}
|
0910.2307
|
# Horizon Thermodynamics and Gravitational Field Equations in Hořava-Lifshitz
Gravity
Rong-Gen Cai cairg@itp.ac.cn Key Laboratory of Frontiers in Theoretical
Physics, Institute of Theoretical Physics,
Chinese Academy of Sciences, P.O. Box 2735, Beijing 100190, China Department
of Physics, Kinki University, Higashi-Osaka, Osaka 577-8502, Japan Nobuyoshi
Ohta ohtan@phys.kindai.ac.jp Department of Physics, Kinki University,
Higashi-Osaka, Osaka 577-8502, Japan
###### Abstract
We explore the relationship between the first law of thermodynamics and
gravitational field equation at a static, spherically symmetric black hole
horizon in Hořava-Lifshtiz theory with/without detailed balance. It turns out
that as in the cases of Einstein gravity and Lovelock gravity, the
gravitational field equation can be cast to a form of the first law of
thermodynamics at the black hole horizon. This way we obtain the expressions
for entropy and mass in terms of black hole horizon, consistent with those
from other approaches. We also define a generalized Misner-Sharp energy for
static, spherically symmetric spacetimes in Hořava-Lifshtiz theory. The
generalized Misner-Sharp energy is conserved in the case without matter field,
and its variation gives the first law of black hole thermodynamics at the
black hole horizon.
††preprint: KU-TP 035
## I Introduction
The holographic principle might be one of the principles of nature, which
states that a theory with gravity could be equivalent to a theory without
gravity in one less dimension. The well-known AdS/CFT correspondence Mald is
a realization of the holographic principle, while the latter is motivated by
black hole thermodynamics. The black hole thermodynamics says that a black
hole behaves as an ordinary thermodynamic system with temperature and entropy.
The temperature of a black hole is proportional to surface gravity at its
horizon, while the entropy of the black hole is measured by its horizon area.
Black hole mass, temperature and entropy satisfy the first law of
thermodynamics. These results come from a combination of quantum mechanics,
black hole geometry and general relativity. This implies that there might
exist a deep connection between thermodynamics and gravity theory.
Indeed some pieces of evidence have been accumulated for the connection
between thermodynamics and gravity theory in the literature. Assuming there is
a proportionality between entropy and horizon area, Jacobson Jac derived the
Einstein field equation by using the fundamental Clausius relation, $\delta
Q=TdS$, connecting heat, temperature and entropy. The key idea is to demand
that this relation holds for all the local Rindler causal horizon through each
spacetime point, with $\delta Q$ and $T$ interpreted as the energy flux and
Unruh temperature seen by an accelerated observer just inside the horizon. In
this way, the Einstein field equation is nothing but an equation of state of
spacetime. More recently, Jacobson’s argument has been generalized to all
diffeomorphism-invariant theories of gravity Brus (however, see also Parikh ).
For $f(R)$ theory and scalar-tensor theory, see also ES ; WuYZ . In fact,
investigating the thermodynamics of spacetime for $f(R)$ theory Jac1 ; AC1 ;
AC and scalar-tensor theory AC1 ; CC1 , it is found that a nonequilibrium
thermodynamic setup has to be employed. Further, it is argued that if shear of
spacetime is not assumed to vanish, the nonequilibrium thermodynamic setting
is required even for the Einstein general relativity Eling ; CL . There an
internal entropy production term has to be introduced to balance energy
conservation. The internal entropy production term $dS_{i}$ is proportional to
the squared shear of the horizon and the ratio of the proportionality constant
to the area entropy density is $1/4\pi$. The latter is a universal value for
many kinds of conformal field theories with AdS duals KSS .
There exists another route in exploring the relationship between
thermodynamics and gravity theory. Padmanabhan Pad1 first noticed that the
gravitational field equation in a static, spherically symmetric spacetime can
be rewritten as a form of the ordinary first law of thermodynamics at a black
hole horizon. This indicates that Einstein’s equation is nothing but a
thermodynamic identity. For a recent review on this, see Pad2 . This
observation was then extended to the cases of stationary axisymmetric horizons
and evolving spherically symmetric horizons in the Einstein gravity KSP ,
static spherically symmetric horizons KP and dynamical apparent horizons CCHK
in Lovelock gravity, and three-dimensional Banados-Teitelboim-Zanelli black
hole horizons Akbar1 . On the other hand, the relationship between the first
law of thermodynamics and dynamical equation of spacetime has been intensively
investigated in a Friedmann-Robertson-Walker (FRW) cosmological setup in
various gravity theories CK ; others ; AC07 ; CC2 ; AC1 ; AC ; CC1 ; GW ;
Others1 ; it is shown that (modified) Friedmann equations can be cast to a
form of the first law of thermodynamics, and there exists a Hawking radiation
associated with apparent horizon in a FRW universe CCH .
Recently a field theory model for a UV complete theory of gravity was proposed
by Hořava Hor , which is a nonrelativistic renormalizable theory of gravity
and is expected to recover Einstein’s general relativity at large scales. This
theory is named Hořava-Lifshitz theory in the literature since at the UV fixed
point of the theory space and time have different scalings. Since then a lot
of work has been done in exploring various aspects of the theory; for a more
or less complete list of references, see, for example, WDR .
In this paper we discuss the relationship between the first law of
thermodynamics and gravitational field equation in Hořava-Lifshitz theory. In
static spherically symmetric black hole spacetimes, we show that the
gravitational field equation can be rewritten as $dE-TdS=PdV$ at the black
hole horizon. Note that in Hořava-Lifshitz theory the full diffeomorphism
invariance is broken to the “foliation-preserving” diffeomorphism. Therefore
our result is a nontrivial generalization of Padmanabhan’s observation. In
addition, we discuss the question of whether one can define a generalized
Misner-Sharp quasilocal energy in Hořava-Lifshitz theory. The answer is
positive. We define a generalized Misner-Sharp energy. It is a conserved
charge when the matter field is absent, while its variation at a black hole
horizon gives the first law of black hole thermodynamics.
This paper is organized as follows. In the next section we review the
Padmanabhan’s observation by extending his discussion to a more general
spherically symmetric spacetime. In Sec. III we consider black hole spacetimes
in Hořava-Lifshitz theory. In Sec. IV the case of IR modified Hořava-Lifshitz
theory is discussed. In Sec. V we define a generalized Misner-Sharp quasilocal
energy for static, spherically symmetric spacetimes in Hořava-Lifshitz theory
and discuss its properties. The conclusion is given in Sec. VI.
## II Black Holes in Einstein Gravity
As a warm-up exercise, in this section, we will briefly review the observation
made by Padmanabhan Pad1 by generalizing his discussion to a more general
spherically symmetric case. In Einstein’s general relativity, the
gravitational field equations are
$G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=8\pi GT_{\mu\nu},$ (1)
where $G_{\mu\nu}$ is Einstein tensor and $T_{\mu\nu}$ is the energy-momentum
tensor of matter field. On the other hand, for a general static, spherically
symmetric spacetime, its metric can be written down as
$ds^{2}=-f(r)dt^{2}+f^{-1}(r)dr^{2}+b^{2}(r)(d\theta^{2}+\sin^{2}\theta
d\varphi^{2}),$ (2)
where $f(r)$ and $b(r)$ are two functions of the radius coordinate $r$. (Note
that in the Padmanabhan’s discussion Pad1 , the metric is assumed in a form
(2) with $b(r)=r$; when matter is present, however, such a metric form is not
always satisfied. See also KSP .) Suppose the metric (2) describes a
nonextremal black hole with horizon at $r_{+}$, then the function $f(r)$ has a
simple zero at $r=r_{+}$. Namely, $f^{\prime}(r)|_{r=r_{+}}=0$, but
$f^{\prime\prime}(r)|_{r=r_{+}}\neq 0$. It is easy to show that the Hawking
temperature of the black hole associated with the horizon $r_{+}$ is
$T=\frac{1}{4\pi}f^{\prime}(r)|_{r=r_{+}}\equiv\frac{1}{4\pi}f^{\prime}(r_{+}),$
(3)
where a prime stands for the derivative with respective to $r$. Einstein’s
equations in the metric (2) have the components
$\displaystyle G^{t}_{t}$ $\displaystyle=$
$\displaystyle\frac{1}{b^{2}}(-1+fb^{\prime
2}+b(f^{\prime}b^{\prime}+2fb^{\prime\prime})),$ $\displaystyle G^{r}_{r}$
$\displaystyle=$
$\displaystyle\frac{1}{b^{2}}(-1+bf^{\prime}b^{\prime}+fb^{\prime 2}).$ (4)
Note that at the horizon, one has $f(r)=0$, and then
$G^{t}_{t}|_{r=r_{+}}=G^{r}_{r}|_{r=r_{+}}=\frac{1}{b^{2}}(-1+bf^{\prime}b^{\prime})|_{r=r_{+}}.$
(5)
Therefore at the horizon, the $t-t$ component of Einstein’s equations can be
expressed as
$-1+bf^{\prime}b^{\prime}=8\pi Gb^{2}P,$ (6)
where $P=T^{r}_{r}|_{r=r_{+}}$ is the radial pressure of matter at the
horizon. Note that here (5) guarantees $T^{t}_{t}=T^{r}_{r}$ at the horizon.
Now we multiply $dr_{+}$ on both sides of (6) and rewrite this equation as
$\frac{1}{2G}bf^{\prime}b^{\prime}dr_{+}-\frac{1}{2G}dr_{+}=4\pi
b^{2}Pdr_{+}.$ (7)
Note that $b$ is a function of $r$ only and $f^{\prime}$ has a relation to the
Hawking temperature as (3). One then can rewrite the above equation as
$Td\left(\frac{4\pi b^{2}}{4G}\right)-d\left(\frac{r_{+}}{2G}\right)=PdV,$ (8)
where $dV=4\pi b^{2}dr_{+}$. Therefore $V$ is just the volume of the black
hole with horizon radius $r_{+}$ in the coordinate (2). The equation (8) can
be further rewritten as
$TdS-dE=PdV,$ (9)
with identifications
$S=\frac{4\pi b^{2}}{4G}=\frac{A}{4G},\ \ \ E=\frac{r_{+}}{2G}.$ (10)
Clearly here $S$ is precisely the entropy of the black hole, while $E$ is the
Misner-Sharp energy MS at the horizon. Thus we have shown in general that at
black hole horizon, Einstein’s equations can be cast into the form of the
first law of thermodynamics.
## III Black Holes in Hořava-Lifshitz Gravity
In the $(3+1)$-dimensional Arnowitt-Deser-Misner formalism, where the metric
can be written as
$ds^{2}=-N^{2}dt^{2}+g_{ij}(dx^{i}+N^{i}dt)(dx^{j}+N^{j}dt),$ (11)
and for a spacelike hypersurface with a fixed time, its extrinsic curvature
$K_{ij}$ is
$K_{ij}=\frac{1}{2N}(\dot{g}_{ij}-\nabla_{i}N_{j}-\nabla_{j}N_{i}),$ (12)
where a dot denotes a derivative with respect to $t$ and covariant derivatives
defined with respect to the spatial metric $g_{ij}$.
The action of Hořava-Lifshitz theory is Hor ; LMP
$\displaystyle I$ $\displaystyle=$ $\displaystyle\int dtd^{3}x({\cal
L}_{0}+{\cal L}_{1}+{\cal L}_{m}),$ (13) $\displaystyle{\cal L}_{0}$
$\displaystyle=$
$\displaystyle\sqrt{g}N\left\\{\frac{2}{\kappa^{2}}(K_{ij}K^{ij}-\lambda
K^{2})+\frac{\kappa^{2}\mu^{2}(\Lambda
R-3\Lambda^{2})}{8(1-3\lambda)}\right\\},$ $\displaystyle{\cal L}_{1}$
$\displaystyle=$
$\displaystyle\sqrt{g}N\left\\{\frac{\kappa^{2}\mu^{2}(1-4\lambda)}{32(1-3\lambda)}R^{2}-\frac{\kappa^{2}}{2\omega^{4}}\left(C_{ij}-\frac{\mu\omega^{2}}{2}R_{ij}\right)\left(C^{ij}-\frac{\mu\omega^{2}}{2}R^{ij}\right)\right\\}.$
where $\kappa^{2}$, $\lambda$, $\mu$, $\omega$ and $\Lambda$ are constant
parameters and the Cotten tensor, $C_{ij}$, is defined by
$C^{ij}=\epsilon^{ikl}\nabla_{k}\left(R^{j}_{\
l}-\frac{1}{4}R\delta^{j}_{l}\right)=\epsilon^{ikl}\nabla_{k}R^{j}_{\
l}-\frac{1}{4}\epsilon^{ikj}\partial_{k}R.$ (14)
The first two terms in ${\cal L}_{0}$ are the kinetic terms, others in $({\cal
L}_{0}+{\cal L}_{1})$ give the potential of the theory in the so-called
“detailed-balance” form, and ${\cal L}_{m}$ stands for the Lagrangian of other
matter field.
Comparing the action to that of general relativity, one can see that the speed
of light, Newton’s constant and the cosmological constant are
$c=\frac{\kappa^{2}\mu}{4}\sqrt{\frac{\Lambda}{1-3\lambda}},\ \
G=\frac{\kappa^{2}c}{32\pi},\ \ \tilde{\Lambda}=\frac{3}{2}\Lambda,$ (15)
respectively. Let us notice that when $\lambda=1$, ${\cal L}_{0}$ could be
reduced to the usual Lagrangian of Einstein’s general relativity. Therefore it
is expected that general relativity could be approximately recovered at large
distances when $\lambda=1$. Here we will mainly consider the case of
$\lambda=1$, but will also discuss the $\lambda\neq 1$ case briefly at the end
of this paper.
Now we consider black hole spacetime with metric ansatz LMP ; CCO
$ds^{2}=-\tilde{N}^{2}(r)f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\Omega_{k}^{2},$
(16)
where $d\Omega_{k}^{2}$ denotes the line element for a two-dimensional
Einstein space with constant scalar curvature $2k$. Without loss of
generality, one may take $k=0$, $\pm 1$, respectively. Substituting the metric
(16) into (13), we have
$\displaystyle I$ $\displaystyle=$
$\displaystyle\frac{\kappa^{2}\mu^{2}\Lambda\Omega_{k}}{8(1-3\lambda)}\int
dtdr\tilde{N}\left\\{-3\Lambda r^{2}-2(f-k)-2r(f-k)^{\prime}\right.$ (17)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\left.+\frac{(\lambda-1)f^{\prime
2}}{2\Lambda}+\frac{(2\lambda-1)(f-k)^{2}}{\Lambda
r^{2}}-\frac{2\lambda(f-k)}{\Lambda r}f^{\prime}+\alpha r^{2}{\cal
L}_{m}\right\\},$
where a prime denotes the derivative with respect to $r$, $\Omega_{k}$ is the
volume of the two-dimensional Einstein space and the constant
$\alpha=8(1-3\lambda)/\kappa^{2}\mu^{2}\Lambda$. In the case of $\lambda=1$ we
can rewrite the action as
$I=\frac{\kappa^{2}\mu^{2}\sqrt{-\Lambda}\Omega_{k}}{16}\int
dtdx\tilde{N}\left\\{\left(x^{3}-2x(f-k)+\frac{(f-k)^{2}}{x}\right)^{\prime}+x^{2}(\frac{\alpha}{-\Lambda}){\cal
L}_{m}\right\\}.$ (18)
Note that here $x=\sqrt{-\Lambda}r$, a prime becomes the derivative with
respect to $x$. Varying the action with $\tilde{N}$, we obtain the equations
of motion
$-\frac{\kappa^{2}\mu^{2}\sqrt{-\Lambda}\Omega_{k}}{16}\left(3x^{2}-2(f-k)-\frac{(f-k)^{2}}{x^{2}}-2xf^{\prime}+\frac{2(f-k)f^{\prime}}{x}\right)=x^{2}\frac{\Omega_{k}}{(-\Lambda)^{3/2}}\frac{\delta(\tilde{N}{\cal
L}_{m})}{\delta\tilde{N}}.$ (19)
Suppose the nonextremal black hole (16) has a horizon radius $r_{+}$, namely
$x_{+}=\sqrt{-\Lambda}r_{+}$. Then the Hawking temperature of the black hole
is
$T=\frac{1}{4\pi}\tilde{N}(r)\left.\frac{df}{dr}\right|_{r=r_{+}}=\frac{\sqrt{-\Lambda}}{4\pi}\tilde{N}(x)f^{\prime}|_{x=x_{+}}.$
(20)
Now we consider a class of solutions with $\tilde{N}(r)=$ const. For example,
the charged black hole solution discussed in CCO belongs to this class of
solutions. In this case one can set $\tilde{N}=1$ by rescaling the time
coordinate $t$. Note that here not all solutions with matter field have the
form $\tilde{N}=1$. At the horizon $x_{+}$, Eq. (19) is reduced to
$-\frac{\kappa^{2}\mu^{2}\sqrt{-\Lambda}\Omega_{k}}{16}\left(3x^{2}_{+}+2k-\frac{k^{2}}{x^{2}_{+}}-2x_{+}f^{\prime}-\frac{2kf^{\prime}}{x_{+}}\right)=x^{2}_{+}\frac{\Omega_{k}}{(-\Lambda)^{3/2}}\left.\frac{\delta(\tilde{N}{\cal
L}_{m})}{\delta\tilde{N}}\right|_{x=x_{+}}.$ (21)
Multiplying both sides with $dx_{+}$, a variation of the horizon radius, we
have
$\frac{\kappa^{2}\mu^{2}\sqrt{-\Lambda}\Omega_{k}}{16}\left(2(x_{+}+\frac{k}{x_{+}})f^{\prime}dx_{+}-(3x^{2}_{+}+2k-\frac{k^{2}}{x^{2}_{+}})dx_{+}\right)=x^{2}_{+}\frac{\Omega_{k}}{(-\Lambda)^{3/2}}Pdx_{+},$
(22)
where $P=\left.\frac{\delta(\tilde{N}{\cal
L}_{m})}{\delta\tilde{N}}\right|_{x=x_{+}}$. Note that the Hawking temperature
turns to be $T=f^{\prime}\sqrt{-\Lambda}/4\pi$ when $\tilde{N}=1$. The above
equation then can be rewritten as
$TdS-dE=PdV,$ (23)
where
$\displaystyle S$ $\displaystyle=$
$\displaystyle\frac{\pi\kappa^{2}\mu^{2}\Omega_{k}}{4}\left(x_{+}^{2}+2k\ln
x_{+}\right)+S_{0},$ $\displaystyle E$ $\displaystyle=$
$\displaystyle\frac{\kappa^{2}\mu^{2}\sqrt{-\Lambda}\Omega_{k}}{16x_{+}}\left(x_{+}^{2}+k\right)^{2},$
(24)
$V=\frac{\Omega_{k}}{3}r_{+}^{3}$, and $S_{0}$ is an undetermined constant.
Clearly $V$ is the volume of black hole with radius $r_{+}$. Comparing (III)
with black hole entropy and mass defined through a Hamiltonian approach in our
previous papers CCO , we see that $S$ and $E$ are just black hole entropy and
mass in terms of horizon radius $x_{+}$, and the gravitational field equation
at the black hole horizon can be cast to the form of the first law of
thermodynamics. Note that here we have obtained expressions for black hole
entropy and mass, but have not used any explicit black hole solutions. In
other words, the above way provides a universal method to derive black hole
entropy and mass.
Now we turn to the case without the detailed-balance condition by considering
the action as LMP ; CCO
$I=\int dtd^{3}x({\cal L}_{0}+(1-\epsilon^{2}){\cal L}_{1}+{\cal L}_{m})$ (25)
where the parameter $\epsilon^{2}\neq 0$. In this case, instead of (18), we
have
$I=\frac{\kappa^{2}\mu^{2}\sqrt{-\Lambda}\Omega_{k}}{16}\int
dtdx\tilde{N}\left\\{\left(x^{3}-2x(f-k)+(1-\epsilon^{2})\frac{(f-k)^{2}}{x}\right)^{\prime}+x^{2}(\frac{\alpha}{-\Lambda}){\cal
L}_{m}\right\\}.$ (26)
Varying the action with respect to $\tilde{N}$ yields
$\displaystyle-\frac{\kappa^{2}\mu^{2}\sqrt{-\Lambda}\Omega_{k}}{16}\left(3x^{2}-2(f-k)-(1-\epsilon^{2})\frac{(f-k)^{2}}{x^{2}}\right.$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\left.2xf^{\prime}+(1-\epsilon^{2})\frac{2(f-k)f^{\prime}}{x}\right)=x^{2}\frac{\Omega_{k}}{(-\Lambda)^{3/2}}\frac{\delta(\tilde{N}{\cal
L}_{m})}{\delta\tilde{N}}.$ (27)
At the black hole horizon where $f=0$, the equation reduces to
$-\frac{\kappa^{2}\mu^{2}\sqrt{-\Lambda}\Omega_{k}}{16}\left(3x^{2}_{+}+2k-(1-\epsilon^{2})\frac{k^{2}}{x^{2}_{+}}-2x_{+}f^{\prime}-(1-\epsilon^{2})\frac{2kf^{\prime}}{x_{+}}\right)=x^{2}_{+}\frac{\Omega_{k}}{(-\Lambda)^{3/2}}\left.\frac{\delta(\tilde{N}{\cal
L}_{m})}{\delta\tilde{N}}\right|_{x=x_{+}}.$ (28)
Multiplying $dx_{+}$ on both sides, one can express this equation as the form
(23) with the condition $\tilde{N}=1$, again. But this time we have
$\displaystyle S$ $\displaystyle=$
$\displaystyle\frac{\pi\kappa^{2}\mu^{2}\Omega_{k}}{4}\left(x_{+}^{2}+2k(1-\epsilon^{2})\ln
x_{+}\right)+S_{0},$ $\displaystyle E$ $\displaystyle=$
$\displaystyle\frac{\kappa^{2}\mu^{2}\sqrt{-\Lambda}\Omega_{k}}{16x_{+}}\left(x_{+}^{4}+2kx_{+}+(1-\epsilon^{2})k^{2}\right).$
(29)
These are nothing but the entropy and mass, expressed in terms of horizon
radius $x_{+}$, of the black hole solutions found in CCO .
Now we turn to the case with $z=4$ terms, where $z$ is the dynamical critical
exponent. Such terms are super-renormalizable ones. The vacuum black hole
solution for this case has been discussed in CLS . Including $z=4$ terms
changes ${\cal L}_{1}$ in (13) to
$\displaystyle{\cal{L}}_{1}$ $\displaystyle=$
$\displaystyle-\sqrt{g}N\frac{\kappa^{2}}{8}\Big{\\{}\frac{4}{\omega^{4}}C^{ij}C_{ij}-\frac{4\mu}{\omega^{2}}C^{ij}R_{ij}-\frac{4}{\omega^{2}M}C^{ij}L_{ij}+\mu^{2}G_{ij}G^{ij}+\frac{2\mu}{M}G^{ij}L_{ij}$
(30) $\displaystyle~{}+\frac{2\mu}{M}\Lambda
L+\frac{1}{M^{2}}L^{ij}L_{ij}-\tilde{\lambda}\big{(}\frac{L^{2}}{M^{2}}-\frac{\mu
L}{M}(R-6\Lambda)+\frac{\mu^{2}}{4}R^{2}\big{)}\Big{\\}},$
where
$\displaystyle G^{ij}$ $\displaystyle=$ $\displaystyle
R^{ij}-\frac{1}{2}g^{ij}R$ $\displaystyle L^{ij}$ $\displaystyle=$
$\displaystyle(1+2\beta)(g^{ij}\nabla^{2}-\nabla^{i}\nabla^{j})R+\nabla^{2}G^{ij}$
$\displaystyle~{}~{}+2\beta
R(R^{ij}-\frac{1}{4}g^{ij}R)+2(R^{imjn}-\frac{1}{4}g^{ij}R^{mn})R_{mn},$
$\displaystyle L$ $\displaystyle\equiv$ $\displaystyle
g^{ij}L_{ij}=\bigg{(}\frac{3}{2}+4\beta\bigg{)}\nabla^{2}R+\frac{\beta}{2}R^{2}+\frac{1}{2}R_{ij}R^{ij},$
(31)
and $\tilde{\lambda}=\lambda/(3\lambda-1)$, $\beta$ and $M$ are two new
parameters. When $\beta=-3/8$ and $\lambda=1$, the action in the metric (16)
reduces to
$\displaystyle I$ $\displaystyle=$
$\displaystyle\frac{{\kappa}^{2}\Omega_{k}}{16\sqrt{-\Lambda^{3}}}\int
dtdx\tilde{N}\left\\{\Big{[}\tilde{\mu}^{2}\Big{(}x^{3}-2x(f-k)+\frac{(f-k)^{2}}{x}\Big{)}\right.$
(32)
$\displaystyle\left.~{}~{}-2\tilde{\beta}\tilde{\mu}\Big{(}\frac{(f-k)^{3}}{x^{3}}-\frac{(f-k)^{2}}{x}\Big{)}+\tilde{\beta}^{2}\frac{(f-k)^{4}}{x^{5}}\Big{]}^{{}^{\prime}}+x^{2}\tilde{\alpha}{\cal
L}_{m}\right\\},$
where we define
$\tilde{\mu}=-\mu\Lambda,\tilde{\beta}=\frac{\Lambda^{2}}{4M}$,
$\tilde{\alpha}=16/\kappa^{2}$, and the prime is still the derivative with
respect to $x$. Varying the action with respect to $\tilde{N}$ yields
$\displaystyle-\frac{{\kappa}^{2}\mu^{2}\sqrt{-\Lambda}\Omega_{k}}{16}\Big{(}x^{3}-2x(f-k)+\frac{(f-k)^{2}}{x}$
$\displaystyle~{}~{}~{}~{}~{}~{}-2\frac{\tilde{\beta}}{\tilde{\mu}}(\frac{(f-k)^{3}}{x^{3}}-\frac{(f-k)^{2}}{x})+\frac{\tilde{\beta}^{2}}{\tilde{\mu}^{2}}\frac{(f-k)^{4}}{x^{5}}\Big{)}^{{}^{\prime}}=x^{2}\frac{\Omega_{k}}{(-\Lambda)^{3/2}}\frac{\delta(\tilde{N}{\cal
L}_{m})}{\delta\tilde{N}}$ (33)
Again, we take the values of all quantities at the black hole horizon and then
multiply $dx_{+}$ on both sides, the above equation turns to be
$TdS-dE=PdV,$ (34)
where $P=\left.\frac{\delta(\tilde{N}{\cal
L}_{m})}{\delta\tilde{N}}\right|_{x=x_{+}}$, $V=\Omega_{k}r_{+}^{3}/3$ is the
volume of the black hole, and
$\displaystyle E$ $\displaystyle=$
$\displaystyle\frac{\kappa^{2}\mu^{2}\sqrt{-\Lambda}\Omega_{k}}{16}\left(x^{3}_{+}+2kx_{+}+\frac{k^{2}}{x_{+}}+2\frac{\tilde{\beta}}{\tilde{\mu}}(\frac{k^{3}}{x_{+}^{3}}+\frac{k^{2}}{x})+\frac{\tilde{\beta}^{2}}{\tilde{\mu}^{2}}\frac{k^{4}}{x_{+}^{5}}\right)$
$\displaystyle S$ $\displaystyle=$
$\displaystyle\frac{\pi\kappa^{2}\mu^{2}\Omega_{k}}{4}\left(x_{+}^{2}+2k\ln
x_{+}-3\frac{\tilde{\beta}k^{2}}{\tilde{\mu}x_{+}^{2}}-\frac{\tilde{\beta}^{2}k^{3}}{\tilde{\mu}^{2}x_{+}^{4}}+4\frac{\tilde{\beta}k}{\tilde{\mu}}\ln
x_{+}\right)+S_{0}$ (35)
This way we have obtained entropy and mass of black hole solutions CLS ,
again, and shown that at the black hole horizon, gravitational field equation
can be cast into the form of the first law of thermodynamics.
## IV Black Holes in IR Modified Hořava-Lifshitz Gravity
In this section we consider the case with broken detailed-balance by
introducing a term $\mu^{4}R$ to the action KS . Such theory is called IR
modified Hořava-Lifshitz theory. In this way, it is found that one can get
asymptotically flat solutions. In fact, introducing the parameter
$\epsilon^{2}$ to the original action of Hořava-Lifshitz theory with detailed-
balance can also lead to asymptotically flat solutions LMP ; CCO . Here we
show that for the IR modified Hořava-Lifshitz theory, gravitational field
equations at the black hole horizon can also be cast into a form of the first
law of thermodynamics.
Now we add a new term
${\cal L}_{3}=\sqrt{g}N\frac{\kappa^{2}\mu^{2}\nu}{8(3\lambda-1)}R,$ (36)
to the action (13). Here $\nu$ is a new parameter. The term (36) “softly”
violates the so-called detailed-balance. The action in the metric (16) changes
to Park
$\displaystyle I$ $\displaystyle=$
$\displaystyle\frac{\kappa^{2}\mu^{2}\Omega}{8(1-3\lambda)}\int
dtdr\tilde{N}\left((2\lambda-1)\frac{(f-1)^{2}}{r^{2}}-2\lambda\frac{f-1}{r}f^{\prime}+\frac{\lambda-1}{2}f^{\prime
2}\right.$ (37)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-2(\nu-\Lambda)(1-f-rf^{\prime})-3\Lambda^{2}r^{2}+\alpha\Lambda
r^{2}{\cal L}_{m}),$
where we have restricted to the case with $k=1$ and a prime stands for the
derivative with respect to $r$. Considering the case with $\lambda=1$, and
varying the action with respect to $\tilde{N}$, one has the equation of motion
$\frac{\kappa^{2}\mu^{2}\Omega}{16}\left(\frac{(f-1)^{2}}{r^{2}}-2\frac{f-1}{r}f^{\prime}-2(\nu-\Lambda)(1-f-rf^{\prime})-3\Lambda^{2}r^{2}\right)=r^{2}\Omega\frac{\delta(\tilde{N}{\cal
L}_{m})}{\delta\tilde{N}},$ (38)
At a black hole horizon where $f=0$ and $f^{\prime}|_{r=r_{+}}=4\pi T$, by the
same approach, we can rewrite the equation as
$TdS-dE=PdV,$ (39)
where
$\displaystyle S$ $\displaystyle=$
$\displaystyle\frac{\pi\kappa^{2}\mu^{2}\Omega}{4}\left((\nu-\Lambda)r_{+}^{2}+2\ln
r_{+}\right)+S_{0},$ $\displaystyle E$ $\displaystyle=$
$\displaystyle\frac{\kappa^{2}\mu^{2}\Omega}{16}\left(\Lambda^{2}r_{+}^{3}+2(\nu-\Lambda)r_{+}+\frac{1}{r_{+}}\right).$
(40)
This energy is the same as that given in Park , up to a factor, which is not
figured out there. The entropy is given for the first time, although related
results on thermodynamics of black holes in the modified Hořava-Lifshitz
theory have been discussed in Myung .
## V Generalized Misner-Sharp energy and the first law of black hole
thermodynamics
Quasilocal energy is an important concept in general relativity. In
particular, the so-called Misner-Sharp energy is intensively discussed in the
literature. In a spherically symmetric spacetime with metric
$ds^{2}=h_{ab}dx^{a}dx^{b}+r^{2}(x)(d\theta^{2}+\sin^{2}\theta d\varphi^{2}),$
(41)
where $a=0$ and $1$, the Misner-Sharp energy is defined as MS
$M(r)=\frac{r}{2G}\left(1-h^{ab}\partial_{a}r\partial_{b}r\right),$ (42)
which is valid for general relativity in four dimensions. For Schwarzschild
solution, (42) just gives the Schwarzschild mass, while it gives the effective
Schwarzschild mass $m(r)$ at $r$ for a static spherically symmetric spacetime
(16) with $f(r)=1-\frac{2Gm(r)}{r}$. Therefore at a black hole horizon
$r_{+}$, the Misner-Sharp energy (42) gives us the energy of gravitational
field at the horizon $r_{+}$: $M=r_{+}/2G$.
In the previous sections, we have shown that the gravitational field equations
at a black hole horizon can be cast to a form of the first law of
thermodynamics in Hořava-Lifshtiz theory. In this section, we show that the
form of action for the Hořava-Lifshtiz theory allows us to give a generalized
Misner-Sharp quasilocal energy in the case of static, spherically symmetric
spacetime (16).
Let us start with the action (13) with the detailed-balance. In this case, the
gravitational part of the action can be rewritten in a derivative form (18),
which enables us to define a generalized Misner-Sharp energy as
$M(r)=\frac{\kappa^{2}\mu^{2}\Omega_{k}}{16\ r}\left(\Lambda^{2}r^{4}-2\Lambda
r^{2}(k-f)+(k-f)^{2}\right).$ (43)
It is easy to see that at a black hole horizon $r_{+}$, this quasilocal energy
$E(r)$ gives the mass (III) of the black hole solution. The variation of the
generalized Misner-Sharp energy with respect to $r$ gives
$dM(r)=-r^{2}\Omega_{k}\frac{\delta(\tilde{N}{\cal
L}_{m})}{\delta\tilde{N}}dr,$ (44)
from which one can see clearly that $-\frac{\delta(\tilde{N}{\cal
L}_{m})}{\delta\tilde{N}}$ is the energy density of matter field. In the case
without matter field, the generalized Misner-Sharp energy is conserved,
$dM(r)=0$. At the horizon we have
$dM(r)|_{r=r_{+}}=dE-TdS,$ (45)
where $E$ and $S$ are just mass and entropy of the black hole, given in (III).
When the matter field is absent, it gives us $dE=TdS$, which is the first law
of black hole thermodynamics. In CCO , we have used the first law to derive
the black entropy.
In the case including the $z=4$ term, the generalized Misner-Sharp energy can
be read down from the action (32)
$\displaystyle M(r)$ $\displaystyle=$
$\displaystyle\frac{\kappa^{2}\mu^{2}\Omega_{k}}{16\
r}\left(\Lambda^{2}r^{4}-2\Lambda r^{2}(k-f)+(k-f)^{2}\right.$ (46)
$\displaystyle\left.-2\frac{\tilde{\beta}}{\Lambda\mu}\left((k-f)^{2}-\frac{(k-f)^{3}}{\Lambda
r^{2}}\right)+\frac{\tilde{\beta}^{2}}{\Lambda^{2}\mu^{2}}\frac{(k-f)^{4}}{\Lambda^{2}r^{4}}\right),$
while for the IR modified Hořava-Lifshtiz theory, we have, from (37),
$M(r)=\frac{\kappa^{2}\mu^{2}\Omega}{16\
r}\left(\Lambda^{2}r^{4}+2(\nu-\Lambda)r^{2}(1-f)+(1-f)^{2}\right).$ (47)
It is easy to show that, when the matter field is absent, these two
generalized Misner-Sharp energies are conserved, while when the matter field
appears, their variation satisfies (44). At the black hole horizon, the
variation of the generalized Misner-Sharp energy obeys (45), which is closely
related to the first law of black hole thermodynamics.
## VI Conclusions and Discussions
The black hole thermodynamics implies that there might exist a deep connection
between thermodynamics and gravity theory, although they are quite different
subjects. Such a connection must be closely related to the holographic
properties of gravity. The holography is an essential feature of gravity. In
this paper we investigated the relationship between the first law of
thermodynamics and gravitational field equation at a static, spherically
symmetric black hole horizon in Hořava-Lifshtiz theory with/without detailed-
balance. It turns out that, as in the cases of Einstein gravity and Lovelock
gravity, the gravitational field equation can be cast to a form of the first
law of thermodynamics at the black hole horizon. This way we obtained entropy
and mass expressions in terms of black hole horizon, and they are exactly the
same as those resulting from the integration method for black hole entropy and
the Hamiltonian approach for black hole mass CCO .
Note that Hořava-Lifshtiz theory, different from general relativity, is not
fully diffeomorphism invariant and only keeps the “foliation-preserving”
diffeomorphism. Our results on the relation between the first law of
thermodynamics and gravity field equation in the Hořava-Lifshtiz theory
indicate that this relation is a robust one, and is of some universality. In
addition, unlike the case in general relativity, the first law of black hole
mechanics has not yet been established so far in Hořava-Lifshtiz theory. Our
result is a first step towards that goal. Furthermore, let us stress that in
the process to derive the entropy and mass of black holes in Hořava-Lifshtiz
theory, we have not employed an explicit solution of the theory. This is quite
different from the previous works in the literature. This manifests that the
relation between the first law and gravity field equation has a deep
implication.
We also defined generalized Misner-Sharp energy for static, spherically
symmetric spacetimes in Hořava-Lifshtiz theory. The generalized Misner-Sharp
energy is conserved in the case without matter field, and its variation gives
the first law of black hole thermodynamics at the black hole horizon.
Note that we have restricted ourselves to the case with $\lambda=1$ in Sec.
III. Here let us make a simple discussion of the case with $\lambda\neq 1$. In
this case, the reduced action can be expressed as CCO
$I=\frac{\kappa^{2}\mu^{2}\Omega_{k}}{8(1-3\lambda)}\int
dtdr\tilde{N}\left\\{\frac{(\lambda-1)}{2}F^{\prime
2}-\frac{2\lambda}{r}FF^{\prime}+\frac{(2\lambda-1)}{r^{2}}F^{2}\right\\},$
(48)
where $F(r)=k-\Lambda r^{2}-f(r)$. Varying the action with respect to $F$ and
$\tilde{N}$ yields the equations of motion
$\displaystyle 0$ $\displaystyle=$
$\displaystyle\left(\frac{2\lambda}{r}F-(\lambda-1)F^{\prime}\right)\tilde{N}^{\prime}+(\lambda-1)\left(\frac{2}{r^{2}}F-F^{\prime\prime}\right)\tilde{N},$
(49) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\frac{(\lambda-1)}{2}F^{\prime
2}-\frac{2\lambda}{r}FF^{\prime}+\frac{(2\lambda-1)}{r^{2}}F^{2}.$ (50)
These equations have the solution with LMP ; CCO
$F(r)=\alpha r^{s},\ \ \ \tilde{N}(r)=\gamma r^{1-2s},$ (51)
where $\alpha$ and $\gamma$ are both integration constants and
$s=\frac{2\lambda\pm\sqrt{2(3\lambda-1)}}{\lambda-1}.$
As discussed in the second reference of CCO , to have a well-defined physical
quantities and well-behaved asymptotical behavior for the solution, we have to
take the negative branch in $s$ and $s$ is in the range $s\in[-1,2)$. The
temperature of the black hole in this case is
$T=\frac{1}{4\pi}\tilde{N}(r)f^{\prime}(r)|_{r=r_{+}}=\frac{\gamma}{4\pi
r_{+}^{2s}}\left(-\Lambda r_{+}^{2}(2-s)-sk\right).$ (52)
We can rewrite Eq. (50) as
$\frac{2\lambda}{r}(k-\Lambda r^{2}-f)f^{\prime}+4\lambda\Lambda(k-\Lambda
r^{2}-f)+\frac{(\lambda-1)}{2}F^{\prime 2}+\frac{(2\lambda-1)}{r^{2}}F^{2}=0.$
(53)
On the black hole horizon where $f(r_{+})=0$, the above equation reduces to
$\frac{2\lambda}{r_{+}}(k-\Lambda
r^{2}_{+})f^{\prime}+4\lambda\Lambda(k-\Lambda
r^{2}_{+})+\frac{(\lambda-1)}{2}F^{\prime
2}(r_{+})+\frac{(2\lambda-1)}{r^{2}}F^{2}(r_{+})=0.$ (54)
Multiplying Eq. (54) by
$\frac{\sqrt{2}\kappa^{2}\mu^{2}\Omega_{k}\tilde{N}(r_{+})}{16\lambda\sqrt{3\lambda-1}}dr_{+},$
and considering the expression of the temperature (52), we find that the first
term in (54) can be expressed as $TdS$, where
$S=\frac{\pi\kappa^{2}\mu^{2}\Omega_{k}}{\sqrt{2(3\lambda-1)}}\left(k\ln(\sqrt{-\Lambda}r_{+})+\frac{1}{2}(\sqrt{-\Lambda}r_{+})^{2}\right)+S_{0},$
(55)
where $S_{0}$ is an integration constant. On the other hand, with the solution
(51), the other three terms in (54) can be expressed as $-dM$, where $M$ is
$M=\frac{\sqrt{2}\kappa^{2}\mu^{2}\gamma\Omega_{k}}{16\sqrt{3\lambda-1}}\frac{(k-\Lambda
r_{+}^{2})^{2}}{r_{+}^{2s}}.$ (56)
Thus we have shown that on the black hole horizon, the equation of motion (50)
can be expressed as $TdS-dM=0$, where $S$ and $M$ are just the black hole
entropy and mass, as found in the second reference of CCO . Here we would like
to mention that unlike the case of $\lambda=1$, due to the presence of $F^{2}$
and $F^{\prime 2}$ in the equation of motion, we have to use the black hole
solution (51) in order to express the equation of motion in the form of the
first law of thermodynamics. In addition, we here have discussed the case with
the vacuum solution.
## Acknowledgments
We thank T. Padmanabhan for useful correspondences. RGC is supported partially
by grants from NSFC, China (No. 10525060, No. 108215504 and No. 10975168) and
a grant from MSTC, China (No. 2010CB833004). NO was supported in part by the
Grant-in-Aid for Scientific Research Fund of the JSPS No. 20540283, and also
by the Japan-U.K. Research Cooperative Program. This work is completed during
RGC’s visit to Kinki University, Japan with the support of JSPS invitation
fund, the warm hospitality extended to him is greatly appreciated.
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|
arxiv-papers
| 2009-10-13T05:56:14 |
2024-09-04T02:49:05.818939
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Rong-Gen Cai, Nobuyoshi Ohta",
"submitter": "Rong-Gen Cai",
"url": "https://arxiv.org/abs/0910.2307"
}
|
0910.2387
|
# Generalized Misner-Sharp Energy in $f(R)$ Gravity
Rong-Gen Cai cairg@itp.ac.cn Key Laboratory of Frontiers in Theoretical
Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, P.O.
Box 2735, Beijing 100190, China Department of Physics, Kinki University,
Higashi-Osaka, Osaka 577-8502, Japan Li-Ming Cao caolm@itp.ac.cn Department
of Physics, Kinki University, Higashi-Osaka, Osaka 577-8502, Japan Ya-Peng Hu
yapenghu@itp.ac.cn Key Laboratory of Frontiers in Theoretical Physics,
Institute of Theoretical Physics, Chinese Academy of Sciences, P.O. Box 2735,
Beijing 100190, China Graduate School of the Chinese Academy of Sciences,
Beijing 100039, China Nobuyoshi Ohta ohtan@phys.kindai.ac.jp Department of
Physics, Kinki University, Higashi-Osaka, Osaka 577-8502, Japan
###### Abstract
We study generalized Misner-Sharp energy in $f(R)$ gravity in a spherically
symmetric spacetime. We find that unlike the cases of Einstein gravity and
Gauss-Bonnet gravity, the existence of the generalized Misner-Sharp energy
depends on a constraint condition in the $f(R)$ gravity. When the constraint
condition is satisfied, one can define a generalized Misner-Sharp energy, but
it cannot always be written in an explicit quasi-local form. However, such a
form can be obtained in a FRW universe and for static spherically symmetric
solutions with constant scalar curvature. In the FRW universe, the generalized
Misner-Sharp energy is nothing but the total matter energy inside a sphere
with radius $r$, which acts as the boundary of a finite region under
consideration. The case of scalar-tensor gravity is also briefly discussed.
PACS numbers: 04.20.Cv, 04.50.+h, 04.70.Dy
††preprint: KU-TP 036
## I Introduction
A gravitational field has certainly an associated energy. However, it is a
rather difficult task to define energy for a gravitational field in general
relativity. A local energy density of gravitational field does not make any
sense because the energy-momentum pseudo-tensor of gravitational field, which
explicitly depends on metric and its first derivative, will vanish due to the
strong equivalence principle at any point of spacetime in a locally flat
coordinate NonLocal1 ; NonLocal2 ; Szabados . In general relativity, however,
there exist two well-known definitions of total energy; one is the Arnowitt-
Deser-Misner (ADM) energy $E_{ADM}$ at spatial infinity ADM , and the other is
the Bondi-Sachs (BS) energy $E_{BS}$ at null infinity Bondi describing an
isolated system in an asymptotically flat spacetime.
Due to the absence of the local energy density of gravitational field, it is
tempting to define some meaningful quasi-local energy, which is defined on a
boundary of a given region in spacetime. Indeed, it is possible to properly
define such quasi-local energies. Some useful definitions for quasi-local
energy exist in the literature, for instance, Brown-York energy York , Misner-
Sharp energy Misner , Hawking-Hayward energy hawking ; hayward and Chen-
Nester energy Chen , etc. A nice review on this issue can be found in Szabados
. In this article, we focus on the Misner-Sharp energy.
The Misner-Sharp energy $E$ is defined in a spherically symmetric spacetime.
Various properties of the Misner-Sharp energy are discussed in some detail by
Hayward in Hayward ; Hayward1 . For example, the following properties are
established. In the Newtonian limit of a perfect fluid, the Misner-Sharp
energy $E$ yields the Newtonian mass to leading order and the Newtonian
kinetic and potential energy in the next order. For test particles, the
corresponding Hajicek energy is conserved and has the behavior appropriate to
energy in the Newtonian and special-relativistic limits. In the small-sphere
limit, the leading term in $E$ is the product of volume and the energy density
of the matter. In vacuo, the Misner-Sharp energy $E$ reduces to the
Schwarzschild energy. At null and spatial infinity, $E$ reduces to the BS and
ADM energies, respectively. In particular, it is shown that the conserved
Kodama current produces the conserved charge $E$.
In a four-dimensional, spherically-symmetric spacetime with metric
$ds^{2}=h_{ab}dx^{a}dx^{b}+r^{2}(x)d\Omega_{2}^{2},$ (1)
where $a=0$, $1$, $x^{a}$ is the coordinate on a two-dimensional spacetime
$(M^{2},h_{ab})$ and $d\Omega_{2}^{2}$ denotes the line element for a two-
dimensional sphere with unit radius, the Misner-Sharp energy $E$ can be
defined as
$E(r)=\frac{r}{2G}\left(1-h^{ab}\partial_{a}r\partial_{b}r\right).$ (2)
With this energy, the Einstein equations can be rewritten as
$dE=A\Psi_{a}dx^{a}+WdV,$ (3)
where $A=4\pi r^{2}$ is the area of the sphere with radius $r$ and $V=4\pi
r^{3}/3$ is its volume, $W$ is called work density defined as
$W=-h^{ab}T_{ab}/2$ and $\Psi$ energy supply vector, $\Psi_{a}=T_{a}^{\
b}\partial_{b}r+W\partial r_{a}$, with $T_{ab}$ being the projection of the
four-dimensional energy-momentum tenor $T_{\mu\nu}$ of matter in the normal
direction of the 2-dimensional sphere. The form (3) is called “unified first
law” Hayward2 ; Hayward3 . Projecting this form along a trapping horizon, one
is able to arrive at the first law of thermodynamics for dynamical black hole
$\langle dE,\xi\rangle=\frac{\kappa}{8\pi G}\langle dA,\xi\rangle+W\langle
dV,\xi\rangle,$ (4)
where $\xi$ is a projecting vector and
$\kappa=\frac{1}{2\sqrt{-h}}\partial_{a}(\sqrt{-h}h^{ab}\partial_{b}r)$ is
surface gravity on the trapping horizon. Defining $\delta Q=\langle
A\Psi,\xi\rangle=TdS$, we can derive entropy formula associated with apparent
horizon in various gravity theories Cai-Cao1 ; Cai-Cao2 ; CCHK . Indeed, the
Minser-Sharp energy plays an important role in connection between the Einstein
equations and first law of thermodynamics in FRW cosmological setup Cai-Cao1 ;
Cai-Cao2 ; Cai-Kim ; CCHK ; AC07 and black hole setup Pad .
Note that the original form (2) for the Misner-Sharp energy is applicable for
Einstein gravity without cosmological constant in four dimensions, thus it is
tempting to give corresponding forms for the case with a cosmological constant
and/or in other gravity theories. Indeed, a generalized form is given for
Gauss-Bonnet gravity and more general Lovelock gravity in Maeda ; Maeda1 . In
particular, we would like to mention here that Gong and Wang in GW introduce
a modified Misner-Sharp energy and discuss its relation to horizon
thermodynamics. With the generalized Misner-Sharp energy, it is shown that the
Clausius relation $\delta Q=TdS$ indeed gives correct entropy formula for
Lovelock gravity Cai-Cao1 ; CCHK .
Recently, a kind of modified gravity theories, $f(R)$, whose Lagrangian is a
function of curvature scalar $R$, has attracted a lot of attention. A main
motivation is to explain the observed accelerated expansion of the universe
without introducing the exotic dark energy with a large negative pressure. For
a review on $f(R)$ gravity, see Sotiriou . Of course, $f(R)$ gravity is a
simple generalization of Einstein gravity; when $f(R)=R$, it goes back to
Einstein theory. However, $f(R)$ is quite different from another
generalization of Einstein gravity, Lovelock gravity. The equations of motion
of the latter do not contain more than second-order derivatives, while the
equations of motion for the former do. In addition, let us notice that in some
sense, the $f(R)$ gravity is quite similar to scalar-tensor gravity, a
generalization of Einstein gravity again.
In this paper we are mainly concerned with the question whether there exists a
similar Misner-Sharp energy for $f(R)$ gravity in a spherically symmetric
spacetime. For this goal, we will take two methods, which are basically
equivalent, in fact. One is called integration method, and the other is
conserved charge method associated with the Kodama current. The integration
method is introduced in a previous paper of ours CCHK for the case of
radiation matter in Lovelock gravity. We find that existence of a generalized
Misner-Sharp energy is not trivial for $f(R)$ gravity. Its existence depends
on a constraint. Once the constraint is satisfied, we could have a generalized
Misner-Sharp energy. Otherwise, the answer is negative. The same situation
happens for the scalar-tensor gravity theory.
The organization of the paper is as follows. In Sec. II, as a warm-up
exercise, we derive the generalized Misner-Sharp energy in Gauss-Bonnet
gravity by using the integration method and by generalizing the discussion in
CCHK to more general matter content. In Sec. III, we discuss the generalized
Misner-Sharp energy in $f(R)$ gravity by the integration method and conserved
charge method, respectively. Sec. IV is devoted to investigating some special
cases, homogeneous and isotropic FRW cosmology and static spherically
symmetric case. In these cases the generalized Misner-Sharp energy has a
simple form. The conclusion and some discussions are given in Sec. V. In the
appendix, we briefly discuss the generalized Misner-Sharp energy for scalar-
tensor gravity in a FRW universe.
## II Generalized Misner-Sharp energy in Gauss-Bonnet gravity: integration
method
The equations of motion of Gauss-Bonnet gravity can be written down as
$G_{\mu\nu}+\alpha H_{\mu\nu}+\Lambda g_{\mu\nu}=8\pi GT_{\mu\nu},$ (5)
where
$\displaystyle G_{\mu\nu}$ $\displaystyle=$ $\displaystyle
R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu},$ $\displaystyle H_{\mu\nu}$
$\displaystyle=$ $\displaystyle 2(RR_{\mu\nu}-2R_{\mu\alpha}R_{\nu}^{\
\alpha}-2R^{\alpha\beta}R_{\mu\alpha\nu\beta}+R_{\mu}^{\
\alpha\beta\gamma}R_{\nu\alpha\beta\gamma})-\frac{1}{2}g_{\mu\nu}L_{GB},$ (6)
and $\alpha$ is a coupling constant with dimension of length squared. The
Gauss-Bonnet term is
$L_{GB}=R^{2}-4R_{\mu\nu}R^{\mu\nu}+R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$.
Consider an $n$-dimensional spherically symmetric spacetime of metric in the
double-null form
$ds^{2}=-2e^{-\varphi(u,v)}dudv+r^{2}(u,v)\gamma_{ij}dz^{i}dz^{j},$ (7)
where $\gamma_{ij}$ is the metric on an $(n-2)$-dimensional constant curvature
space $K^{n-2}$ with its sectional curvature $k=\pm 1,0$, and the two-
dimensional spacetime spanned by two null coordinates $(u,v)$ and its metric
are denoted as $(M^{2},h_{ab})$. Thus, the equations of gravitational field
(5) can be written explicitly as Maeda1
$\displaystyle-\frac{8\pi G}{n-2}rT_{uu}$ $\displaystyle=$
$\displaystyle(r_{,uu}+\varphi_{,u}r_{,u})\Big{[}1+\frac{2{\tilde{\alpha}}}{r^{2}}(k+2e^{\varphi}r,_{u}r,_{v})\Big{]},$
$\displaystyle-\frac{8\pi G}{n-2}rT_{vv}$ $\displaystyle=$
$\displaystyle(r,_{vv}+\varphi,_{v}r,_{v})\Big{[}1+\frac{2{\tilde{\alpha}}}{r^{2}}(k+2e^{\varphi}r,_{u}r,_{v})\Big{]},$
$\displaystyle\frac{8\pi G}{n-2}r^{2}T_{uv}$ $\displaystyle=$ $\displaystyle
rr,_{uv}+(n-3)r,_{u}r,_{v}+\frac{n-3}{2}ke^{-\varphi}+\frac{{\tilde{\alpha}}}{2r^{2}}[(n-5)k^{2}e^{-\varphi}+4rr,_{uv}(k+2e^{\varphi}r,_{u}r,_{v})$
(8)
$\displaystyle+4(n-5)r,_{u}r,_{v}(k+e^{\varphi}r,_{u}r,_{v})]-\frac{n-1}{2}{\tilde{\Lambda}}r^{2}e^{-\varphi},$
where ${\tilde{\alpha}}=(n-3)(n-4)\alpha,$
${\tilde{\Lambda}}=2\Lambda/[(n-1)(n-2)].$
The essential point of the integration method is that, similar to the case of
Einstein gravity (3), one assumes the equations (8) of gravitational field can
be cast into the form
$dE_{eff}=A\Psi_{a}dx^{a}+WdV,$ (9)
where $A=V_{n-2}^{k}r^{n-2}$ and $V=V_{n-2}^{k}r^{n-1}/(n-1)$ are area and
volume of the $(n-2)$-dimensional space with radius $r$, and energy supply
vector $\Psi$ and energy density $W$ are defined on $(M^{2},h_{ab})$ as in the
case of Einstein gravity. The right hand side in (9) can be explicitly
expressed as
$A\Psi_{a}dx^{a}+WdV=A(u,v)du+B(u,v)dv,$ (10)
where
$\displaystyle A(u,v)$ $\displaystyle=$ $\displaystyle
V_{n-2}^{k}r^{n-2}e^{\varphi}(r,_{u}T_{uv}-r,_{v}T_{uu}),$ (11) $\displaystyle
B(u,v)$ $\displaystyle=$ $\displaystyle
V_{n-2}^{k}r^{n-2}e^{\varphi}(r,_{v}T_{uv}-r,_{u}T_{vv}).$ (12)
With the equations in (8), we can express $A$ and $B$ in terms of geometric
quantities as
$\displaystyle A(u,v)$ $\displaystyle=$ $\displaystyle\frac{V_{n-2}^{k}}{8\pi
G}e^{\varphi}(n-2)r^{n-4}\Big{\\{}\frac{e^{-\varphi}}{2r^{2}}r,_{u}[-(n-1){\tilde{\Lambda}}r^{4}+(n-3)r^{2}(k+2e^{\varphi}r,_{u}r,_{v})$
$\displaystyle+(n-5){\tilde{\alpha}}(k+2e^{\varphi}r,_{u}r,_{v})^{2}+2e^{\varphi}r^{3}r,_{uv}+4e^{\varphi}{\tilde{\alpha}}r(k+2e^{\varphi}r,_{u}r,_{v})r,_{uv}]$
$\displaystyle+rr,_{v}\Big{[}1+\frac{2{\tilde{\alpha}}}{r^{2}}(k+2e^{\varphi}r,_{u}r,_{v})\Big{]}(\varphi,_{u}r,_{u}+r,_{uu})\Big{\\}},$
$\displaystyle B(u,v)$ $\displaystyle=$ $\displaystyle\frac{V_{n-2}^{k}}{8\pi
G}e^{\varphi}(n-2)r^{n-4}\Big{\\{}\frac{e^{-\varphi}}{2r^{2}}r,_{v}[-(n-1){\tilde{\Lambda}}r^{4}+(n-3)r^{2}(k+2e^{\varphi}r,_{u}r,_{v})$
(13)
$\displaystyle+(n-5){\tilde{\alpha}}(k+2e^{\varphi}r,_{u}r,_{v})^{2}+2e^{\varphi}r^{3}r,_{uv}+4{\tilde{\alpha}}re^{\varphi}(k+2e^{\varphi}r,_{u}r,_{v})r,_{uv}]$
$\displaystyle+rr,_{u}\Big{[}1+\frac{2{\tilde{\alpha}}}{r^{2}}(k+2e^{\varphi}r,_{u}r,_{v})\Big{]}(\varphi,_{v}r,_{v}+r,_{vv})\Big{\\}}.$
Now we try to derive the generalized Misner-Sharp energy by integrating the
equation (9). Clearly, if it is integrable, the following integrable condition
has to be satisfied
$\frac{\partial A(u,v)}{\partial v}=\frac{\partial B(u,v)}{\partial u}.$ (14)
It is easy to check that $A$ and $B$ given in (13) indeed satisfy the
integrable condition (14). Thus directly integrating (9) gives the
generalizing Misner-Sharp energy
$\displaystyle E_{eff}$ $\displaystyle=$ $\displaystyle\int
A(u,v)du+\int\Big{[}B(u,v)-\frac{\partial}{\partial v}\int A(u,v)du\Big{]}dv$
(15) $\displaystyle=$ $\displaystyle\frac{(n-2)V_{n-2}^{k}r^{n-3}}{16\pi
G}[-{\tilde{\Lambda}}r^{2}+(k+2e^{\varphi}r,_{u}r,_{v})+{\tilde{\alpha}}r^{-2}(k+2e^{\varphi}r,_{u}r,_{v})^{2}].$
Note that here the second term in the first line of (15) in fact vanishes and
we have fixed an integration constant so that $E_{eff}$ reduces to the Misner-
Sharp energy in Einstein gravity when ${\tilde{\alpha}}=0$. In addition, the
generalized Misner-Sharp energy can be rewritten in a covariant form
$\displaystyle E_{eff}$ $\displaystyle=$
$\displaystyle\frac{(n-2)V_{n-2}^{k}r^{n-3}}{16\pi
G}[-{\tilde{\Lambda}}r^{2}+(k-h^{ab}D_{a}rD_{b}r)+{\tilde{\alpha}}r^{-2}(k-h^{ab}D_{a}rD_{b}r)^{2}].$
(16)
This is the generalized Misner-Sharp energy given by Maeda and Nozawa in
Maeda1 through Kodama conserved charge method.
## III Generalized Misner-Sharp energy in $f(R)$ gravity: the general case
In this section, we first try to derive the generalized Misner-Sharp energy in
$f(R)$ gravity by using the integration method. Then we consider the conserved
charge method. Here we consider the four-dimensional case with spherical
symmetry, and the line element is
$ds^{2}=-2e^{-\varphi(u,v)}dudv+r^{2}(u,v)(d\theta^{2}+\sin^{2}\theta
d\phi^{2}).$ (17)
The action of the $f(R)$ gravity in the metric formalism is
$S=\frac{1}{16\pi G}\int d^{4}x\sqrt{-g}f(R)+S_{matter},$ (18)
Varying the action with respect to metric yields equations of gravitational
field
$f_{R}R_{\mu\nu}-\frac{1}{2}fg_{\mu\nu}-\nabla_{\mu}\nabla_{\nu}f_{R}+g_{\mu\nu}\square
f_{R}=8\pi GT_{\mu\nu},$ (19)
where $f_{R}=df(R)/dR,$ and $T_{\mu\nu}$ is the energy-momentum tensor for
matter field from $S_{matter}$. Note that the field equations also can be
rewritten in the form
$G_{\mu\nu}\equiv
R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=\frac{1}{f_{R}}\Big{[}\frac{1}{2}g_{\mu\nu}(f-Rf_{R})+\nabla_{\mu}\nabla_{\nu}f_{R}-g_{\mu\nu}\square
f_{R}\Big{]}+\frac{8\pi G}{f_{R}}T_{\mu\nu}.$ (20)
In this case, the right hand side can be regarded as an effective energy-
momentum tensor.
### III.1 Integration method
In the metric (17), the field equations (19) can be explicitly expressed as
$\displaystyle 8\pi GT_{uu}$ $\displaystyle=$
$\displaystyle-2f_{R}\frac{\varphi,_{u}r,_{u}+r,_{uu}}{r}-f_{R},_{uu}-\varphi,_{u}f_{R},_{u},$
$\displaystyle 8\pi GT_{vv}$ $\displaystyle=$
$\displaystyle-2f_{R}\frac{\varphi,_{v}r,_{v}+r,_{vv}}{r}-f_{R},_{vv}-\varphi,_{v}f_{R},_{v},$
$\displaystyle 8\pi GT_{uv}$ $\displaystyle=$ $\displaystyle
f_{R}\varphi_{,uv}-2f_{R}\frac{r,_{uv}}{r}+\frac{1}{2}fe^{-\varphi}+f_{R},_{uv}+\frac{2r,_{u}f_{R},_{v}+2r,_{v}f_{R},_{u}}{r}.$
(21)
In this case, following the method discussed in the previous section, we
obtain
$\displaystyle A(u,v)$ $\displaystyle=$ $\displaystyle 4\pi
r^{2}e^{\varphi}(r,_{u}T_{uv}-r,_{v}T_{uu})$ $\displaystyle=$
$\displaystyle\frac{r^{2}e^{\varphi}}{2G}\Big{[}r,_{u}\Big{(}f_{R}\varphi_{,uv}-2f_{R}\frac{r,_{uv}}{r}+\frac{1}{2}fe^{-\varphi}+f_{R},_{uv}+\frac{2r,_{u}f_{R},_{v}+2r,_{v}f_{R},_{u}}{r}\Big{)}$
$\displaystyle+r,_{v}\Big{(}2f_{R}\frac{\varphi,_{u}r,_{u}+r,_{uu}}{r}+f_{R},_{uu}+\varphi,_{u}f_{R},_{u}\Big{)}\Big{]},$
$\displaystyle B(u,v)$ $\displaystyle=$ $\displaystyle 4\pi
r^{2}e^{\varphi}(r,_{v}T_{uv}-r,_{u}T_{vv})$ (22) $\displaystyle=$
$\displaystyle\frac{r^{2}e^{\varphi}}{2G}\Big{[}r,_{v}\Big{(}f_{R}\varphi_{,uv}-f_{R}\frac{2r,_{uv}}{r}+\frac{1}{2}fe^{-\varphi}+f_{R},_{uv}+\frac{2r,_{u}f_{R},_{v}+2r,_{v}f_{R},_{u}}{r}\Big{)}$
$\displaystyle+r,_{u}\Big{(}2f_{R}\frac{\varphi,_{v}r,_{v}+r,_{vv}}{r}+f_{R},_{vv}+\varphi,_{v}f_{R},_{v}\Big{)}\Big{]}.$
Checking the integrable condition, however, unlike the case of Gauss-Bonnet
gravity, we find that it is not always satisfied for the $f(R)$ gravity:
$\displaystyle\frac{\partial A(u,v)}{\partial v}-\frac{\partial
B(u,v)}{\partial u}$ $\displaystyle=$
$\displaystyle-r^{2}e^{\varphi}[(\varphi,_{u}r,_{u}+r,_{uu})(f_{R},_{vv}+\varphi,_{v}f_{R},_{v})-(\varphi,_{v}r,_{v}+r,_{vv})$
(23) $\displaystyle(f_{R},_{uu}+\varphi,_{u}f_{R},_{u})]/2G.$
If the right hand side of the above equation vanishes, in principle, one is
able to obtain a generalized Misner-Sharp energy by integrating (9). On the
other hand, if the integrable condition is not satisfied, one is not able to
rewrite the form $Adu+Bdv$ as a total differential form, which implies that
generalized Misner-Sharp energy does not exist in this case. Now we assume
that the integrable condition is satisfied, that is to say, the right hand
side of the equation (23) vanishes. Thus, we can obtain the generalized
Misner-Sharp energy for the $f(R)$ gravity as
$\displaystyle E_{eff}\ $ $\displaystyle=$ $\displaystyle\int
A(u,v)du+\int\Big{[}B(u,v)-\frac{\partial}{\partial v}\int A(u,v)du\Big{]}dv$
(24) $\displaystyle=$
$\displaystyle\frac{r}{2G}\Big{[}(1+2e^{\varphi}r,_{u}r,_{v})f_{R}+\frac{1}{6}r^{2}(f-f_{R}R)+re^{\varphi}(f_{R},_{u}r,_{v}+f_{R},_{v}r,_{u})\Big{]}$
$\displaystyle-\frac{1}{2G}\int\Big{[}f_{R},_{u}e^{\varphi}(r^{2}r,_{v}),_{u}+f_{R},_{u}(r-\frac{1}{6}r^{3}R)+f_{R},_{v}r^{2}(r,_{u}e^{\varphi}),_{u}\Big{]}du$
$\displaystyle=$
$\displaystyle\frac{r}{2G}\Big{[}(1-h^{ab}\partial_{a}r\partial_{b}r)f_{R}+\frac{1}{6}r^{2}(f-f_{R}R)-rh^{ab}\partial_{a}f_{R}\partial_{b}r\Big{]}$
$\displaystyle-\frac{1}{2G}\int\Big{[}f_{R},_{u}e^{\varphi}(r^{2}r,_{v}),_{u}+f_{R},_{u}\Big{(}r-\frac{1}{6}r^{3}R\Big{)}+f_{R},_{v}r^{2}(r,_{u}e^{\varphi}),_{u}\Big{]}du,$
where we have used
$R=2[\frac{1}{r}+e^{\varphi}(2\frac{r_{,u}r_{,v}}{r^{2}}-\varphi_{,uv}+4\frac{r_{,uv}}{r})]$
and $f_{,u}=f_{R}R_{,u}$. We see that $E_{eff}$ reduces to the Misner-Sharp
energy in the Einstein gravity when $f_{R}=1$. Unfortunately, we see from (24)
that due to the existence of the integration in (24), we cannot arrive at an
explicit quasi-local energy for the general case. In the next section,
however, we will show that the integration can be carried out in some special
cases. Before that, we will first obtain the same result using the conserved
charge method in the next subsection.
### III.2 Conserved charge method
In a spherically symmetric spacetime, one can define a Kodama vector. The
energy-momentum tensor together with the Kodama vector can lead to a conserved
current, whose corresponding conserved charge is just the Misner-Sharp energy
in Einstein gravity. In Gauss-Bonnet gravity, Maeda and Nozawa Maeda1 obtain
the generalized Misner-Sharp energy with help of this method. In this section
we would like to see whether the conserved current method leads to a
generalized Misner-Sharp energy for the $f(R)$ gravity.
For a spherically symmetric spacetime, one can define the Kodama vector as
Kodama ; Sasaki
$K^{\mu}=-\epsilon^{\mu\nu}\nabla_{\nu}r,$ (25)
where $\epsilon_{\mu\nu}=\epsilon_{ab}(dx^{a})_{\mu}(dx^{b})_{\nu}$, and
$\epsilon_{ab}$ is the volume element of $(M^{2},h_{ab})$. For the spherically
symmetric spacetime (17), we have
$K^{\mu}=e^{\varphi}r,_{v}\Big{(}\frac{\partial}{\partial
u}\Big{)}^{\mu}-e^{\varphi}r,_{u}\Big{(}\frac{\partial}{\partial
v}\Big{)}^{\mu}.$ (26)
Conservation of the energy-momentum tensor for matter fields $T_{\mu\nu}$ in
(19) guarantees that the left hand side of the equation (19) is also
divergence-free, which can be easily checked by using the identity
$(\square\nabla_{\nu}-\nabla_{\nu}\square)F=R_{\mu\nu}\nabla^{\mu}F,$ (27)
where $F$ is an arbitrary scalar function. With the Kodama vector, define an
energy current as
$J^{\mu}=-T_{\nu}^{\mu}K^{\nu}.$ (28)
However, we find that unlike in the cases of Einstein gravity and Gauss-Bonnet
gravity Maeda1 , the energy current defined in (28) is not always divergence-
free for the $f(R)$ gravity except the case with condition
$\nabla_{\mu}\nabla_{\nu}f_{R}\nabla^{\mu}K^{\nu}=0.$ (29)
Namely, if the constraint equation (29) is satisfied, the energy current is
divergence-free
$\nabla_{\mu}J^{\mu}=0.$ (30)
In this case, we can define an associated conserved charge
$Q_{J}=\int_{\Sigma}J^{\mu}d\Sigma_{\mu},$ (31)
where $\Sigma$ is some hypersurface and
$d\Sigma_{\mu}=\sqrt{-g}dx^{v}dx^{\lambda}dx^{\rho}\delta_{\mu v\lambda\rho}$
is a directed surface line element on $\Sigma$. By using the line element in
(17) and equations in (21), we obtain
$\displaystyle Q_{J}$ $\displaystyle=$
$\displaystyle\int_{\Sigma}J^{\mu}d\Sigma_{\mu}$ (32) $\displaystyle=$
$\displaystyle\frac{r}{2G}\Big{[}(1-h^{ab}\partial_{a}r\partial_{b}r)f_{R}+\frac{1}{6}r^{2}(f-f_{R}R)-rh^{ab}\partial_{a}f_{R}\partial_{b}r\Big{]}$
$\displaystyle-\frac{1}{2G}\int\Big{[}f_{R},_{u}e^{\varphi}(r^{2}r,_{v}),_{u}+f_{R},_{u}\Big{(}r-\frac{1}{6}r^{3}R\Big{)}+f_{R},_{v}r^{2}(r,_{u}e^{\varphi}),_{u}\Big{]}du,$
where we have chosen the hypersurface $\Sigma$ with a given $v$. One can
immediately see that the charge $Q_{J}$ is precisely the generalized Misner-
Sharp energy $E_{eff}$ given by the integration method in (24). Again, this is
not a satisfying situation since we cannot express the generalized Misner-
Sharp energy in a true quasi-local form.
## IV Generalized Misner-Sharp energy in f(R) gravity: special cases
The existence of the integration in (24) is painful. An interesting question
is whether it will be absent in some special cases. The answer is positive. We
will here discuss two special cases. One is the homogeneous and isotropic FRW
universe and the other is the static spherically symmetric spacetime with
constant scalar curvature.
### IV.1 FRW Universe
Consider the metric
$ds^{2}=-dt^{2}+e^{2\psi(t,\rho)}d\rho^{2}+r^{2}(t,\rho)(d\theta^{2}+\sin^{2}\theta
d\phi^{2}).$ (33)
In this metric, the Kodama vector is
$K^{a}=e^{-\psi}r,_{\rho}\Big{(}\frac{\partial}{\partial
t}\Big{)}^{a}-e^{-\psi}r,_{t}\Big{(}\frac{\partial}{\partial\rho}\Big{)}^{a}.$
(34)
Following the same procedure, we can rewrite the equation in (10) as
$A\Psi_{a}dx^{a}+WdV=A(t,\rho)dt+B(t,\rho)d\rho.$ (35)
where
$\displaystyle A(t,\rho)$ $\displaystyle=$ $\displaystyle 4\pi
r^{2}e^{-2\psi}(T_{t\rho}r,_{\rho}-T_{\rho\rho}r,_{t}),$ $\displaystyle
B(t,\rho)$ $\displaystyle=$ $\displaystyle 4\pi
r^{2}(T_{tt}r,_{\rho}-T_{t\rho}r,_{t}).$ (36)
With the equations of gravitational field of the $f(R)$ gravity, $A$ and $B$
can be expressed in terms of geometric quantities. One can then arrive at the
generalized Misner-Sharp energy
$\displaystyle E_{eff}$ $\displaystyle=$ $\displaystyle\int
B(t,\rho)d\rho+\int\Big{[}A(t,\rho)-\frac{\partial}{\partial t}\int
B(t,\rho)d\rho\Big{]}dt$ (37) $\displaystyle=$
$\displaystyle\frac{r}{2G}\Big{[}(1-h^{ab}\partial_{a}r\partial_{b}r)f_{R}+\frac{r^{2}}{6}(f-f_{R}R)-rh^{ab}\partial_{a}f_{R}\partial_{b}r\Big{]}$
$\displaystyle+\frac{1}{2G}\int\Big{\\{}f_{R,_{\rho}}\Big{[}(-e^{-2\psi}r^{2}r,_{\rho}\psi,_{\rho}+e^{-2\psi}r^{2}r,_{\rho\rho}-r^{2}r,_{t}\psi,_{t})-r(1+r,_{t}^{2}-e^{-2\psi}r,_{\rho}^{2})+\frac{1}{6}r^{3}R\Big{]}$
$\displaystyle\qquad+r^{2}f_{R,t}(\psi,_{t}r,_{\rho}-r,_{t\rho})\Big{\\}}d\rho,$
where the integrable condition is assumed to be satisfied
$\frac{\partial A(t,\rho)}{\partial\rho}-\frac{\partial B(t,\rho)}{\partial
t}=0\text{ }.$ (38)
Now we express the generalized Misner-Sharp energy in a FRW metric
$ds^{2}=-dt^{2}+\frac{a^{2}(t)d\rho^{2}}{1-k\rho^{2}}+r^{2}(t,\rho)(d\theta^{2}+\sin^{2}\theta
d\phi^{2}),$ (39)
where $r(t,\rho)\equiv a(t)\rho$, and
$e^{\psi(t,\rho)}=\frac{a(t)}{\sqrt{1-k\rho^{2}}}$ corresponding to (33).
Because in the FRW universe, the Ricci scalar
$R=6(\frac{k}{a^{2}}+\frac{\overset{.}{a}^{2}}{a^{2}}+\frac{\overset{..}{a}}{a})$
just depends on time, we can check that the integrand in the final step in
(37) exactly vanishes. Thus, the generalized Misner-Sharp energy $E_{eff}$ in
this case can be explicitly expressed as
$\displaystyle E_{eff}$ $\displaystyle=$
$\displaystyle\frac{r}{2G}\Big{[}(1-h^{ab}\partial_{a}r\partial_{b}r)f_{R}+\frac{1}{6}r^{2}(f-f_{R}R)-rh^{ab}\partial_{a}f_{R}\partial_{b}r\Big{]}$
(40) $\displaystyle=$
$\displaystyle\frac{r^{3}}{2G}\left(\frac{1}{r_{A}^{2}}f_{R}+\frac{1}{6}(f-f_{R}R)+H\partial_{t}f_{R}\right),$
where $r_{A}=1/\sqrt{H^{2}+\frac{k}{a^{2}}}$, which in fact, is the location
of apparent horizon of the FRW universe.
### IV.2 Static spherically symmetric case
The general line element of a static spherically symmetric spacetime can be
written down as
$ds^{2}=-\lambda(r)dt^{2}+g(r)dr^{2}+r^{2}d\Omega_{2}^{2},$ (41)
where $\lambda$ and $g$ are two functions of the radial coordinate. In this
case, the Kodama vector is
$K^{\mu}=\frac{1}{\sqrt{g\lambda}}\left(\frac{\partial}{\partial
t}\right)^{\mu}.\\\ $ (42)
Using the static spherically symmetric metric, we can easily check that the
constraint (29) is naturally satisfied
$\nabla_{\mu}\nabla_{\nu}f_{R}\nabla^{\mu}K^{\nu}=0.\\\ $ (43)
Following the same procedure, we can rewrite the equation in (10) as
$dE_{eff}=A(t,r)dt+B(t,r)dr,$ (44)
where
$\displaystyle A(t,r)$ $\displaystyle=$ $\displaystyle\frac{4\pi
r^{2}}{g}(T_{tr}r,_{r}-T_{rr}r,_{t})=0,$ (45) $\displaystyle B(t,r)$
$\displaystyle=$ $\displaystyle\frac{4\pi
r^{2}}{\lambda}(T_{tt}r,_{r}-T_{tr}r,_{t})$ (46) $\displaystyle=$
$\displaystyle\frac{r^{2}}{2G}\left(\frac{1}{2}(f-f_{R}R)+\frac{1}{r^{2}}(1+\frac{rg^{{}^{\prime}}}{g^{2}}-\frac{1}{g})f_{R}+f_{R,r}(\frac{g^{{}^{\prime}}}{2g^{2}}-\frac{2}{rg})-\frac{1}{g}f_{R,rr}\right),$
where a prime denotes the derivative with respect to $r$. Integrating (44)
gives the generalized Misner-Sharp energy
$\displaystyle E_{eff}$ $\displaystyle=$ $\displaystyle\int
B(t,r)dr=\frac{r}{2G}\left((1-h^{ab}\partial_{a}r\partial_{b}r)f_{R}+\frac{r^{2}}{6}(f-f_{R}R)-rh^{ab}\partial_{a}f_{R}\partial_{b}r\right)$
(47) $\displaystyle-\frac{1}{2G}\int
dr(\frac{r^{2}g^{{}^{\prime}}}{2g^{2}}+r-\frac{r}{g}-\frac{1}{6}r^{3}R)f_{R,r}.$
Clearly the integral in (47) will be absent in two cases, one is
$\frac{r^{2}g^{{}^{\prime}}}{2g^{2}}+r-\frac{r}{g}-\frac{1}{6}r^{3}R=0$, the
other is $f_{R,r}=0$. We here consider the latter case. The trivial case with
$f(R)=R$ naturally satisfies the condition. In this case, $f_{R}=1$, and (47)
gives the Misner-Sharp energy. A little nontrivial case is that the solution
is a constant curvature one with scalar curvature $R=R_{0}=const.$ In that
case, $f_{R,r}=0$, and (47) reduces to
$E_{eff}=\frac{r}{2G}\left((1-h^{ab}\partial_{a}r\partial_{b}r)f_{R}+\frac{r^{2}}{6}(f-f_{R}R)\right).$
(48)
Note that here $R$, $f_{R}$ and $f$ are all constants. Compare to the Misner-
Sharp energy (2) and the (16), we can see clearly that this expression is
nothing but the generalized Misner-Sharp energy with a cosmological constant.
Here the effective Newtonian constant is $G/f_{R}$ and the effective
cosmological constant $\Lambda=-(f-Rf_{R})/(2f_{R})$.
## V Conclusion and discussion
The Misner-Sharp quasi-local energy plays a key role in understanding the
“unified first law”, the relation between the first law of thermodynamics and
dynamical equations of gravitational field and thermodynamics of apparent
horizon in FRW universe, etc. In this paper we studied the generalized Misner-
Sharp energy in $f(R)$ gravity by two approaches. One is the integration
method and the other is the conserved charge method. It turns out that in
general we cannot arrive at an explicit expression for the generalized Misner-
Sharp energy in a quasi-local form [see (24) and (37)], even assuming the
integrable condition (14) is satisfied. This situation is quite different from
the cases of Einstein gravity and Gauss-Bonnet gravity. This is certainly
related to the fact that for the $f(R)$ gravity, the energy current (28) is
not always divergence-free, while it does in Einstein and Gauss-Bonnet
gravities. The existence of the conserved current requires (29) is satisfied.
Some remarks on our results are in order.
(1) The relation between the two methods to derive the generalized Misner-
Sharp energy. We obtained the same generalized Misner-Sharp energy by
employing two methods: integration and conserved charge methods. At first
glance, these two methods looks different, but in fact, they are equivalent.
First let us notice that the constraint equation (23) has a relation to the
one (29):
$\frac{\partial A(u,v)}{\partial v}-\frac{\partial B(u,v)}{\partial
u}=-e^{-\varphi}r^{2}\nabla_{\mu}\nabla_{v}f_{R}\nabla^{\mu}K^{v}/2.$ (49)
Namely these two integrable conditions are equivalent. Second, substituting
the conserved current in (28) into (31), we can write the associated charge
$\displaystyle Q_{J}$ $\displaystyle=$
$\displaystyle\int_{\Sigma}J^{\mu}d\Sigma_{\mu}$ (50) $\displaystyle=$
$\displaystyle\int 4\pi r^{2}e^{\varphi}(r,_{u}T_{uv}-r,_{v}T_{uu})du,$
where the integrand is precisely $A(u,v)$ in (22). Thus, we have finished our
proof of the equivalence of the two methods.
In addition, the useful components of
$K_{\mu\nu}\equiv\nabla_{\mu}K_{\upsilon}$ and
$F^{\mu\nu}\equiv\nabla^{\mu}\nabla^{\nu}F$ in coordinates
$(t,\rho,\theta,\phi)$ are
$\displaystyle K_{tt}$ $\displaystyle=$ $\displaystyle
e^{-\psi}(\psi,_{t}r,_{\rho}-r,_{t\rho}),~{}~{}K_{t\rho}=-e^{\psi}r,_{tt},~{}~{}K_{\rho
t}=-\partial_{\rho}(e^{-\psi}r,_{\rho})+\psi,_{t}e^{\psi}r,_{t},$
$\displaystyle K_{\rho\rho}$ $\displaystyle=$ $\displaystyle
e^{\psi}(\psi_{,t}r_{,\rho}-r_{,t\rho}),~{}~{}F^{tt}=F_{,tt},~{}~{}F^{t\rho}=F^{\rho
t}=-e^{-2\psi}F_{,\rho t}+e^{-2\psi}\psi_{,t}F_{,\rho},$ $\displaystyle
F^{\rho\rho}$ $\displaystyle=$ $\displaystyle
e^{-2\psi}[\partial_{\rho}(e^{-2\psi}F_{,\rho})+\psi_{,\rho}e^{-2\psi}F_{,\rho}-\psi_{,t}F_{,t}].$
(51)
With the help of those quantities, we can easily check that the constraint
equation (29) is satisfied for the FRW universe.
(2) The meaning of the generalized Misner-Sharp energy in FRW universe. To see
clearly this, let us write down the Friedmann equations of the $f(R)$ gravity
$\displaystyle H^{2}+\frac{k}{a^{2}}$ $\displaystyle=$
$\displaystyle\frac{1}{6f_{R}}[(f_{R}R-f)-6H\partial_{t}f_{R}+16\pi
G\tilde{\rho}],$ $\displaystyle\overset{.}{H}-\frac{k}{a^{2}}$
$\displaystyle=$
$\displaystyle\frac{1}{2f_{R}}[H\partial_{t}f_{R}-\partial_{t}\partial_{t}f_{R}-8\pi
G(\tilde{\rho}+\tilde{p})],$ (52)
where $\tilde{\rho}$ and $\tilde{p}$ are energy density and pressure of the
ideal fluid in the universe. With the first line in (52), we can easily see
that the generalized Misner-Sharp energy in (40) can be rewritten as
$E_{eff}=\tilde{\rho}V,$ (53)
where $V=4\pi r^{3}/3$ is the volume of a sphere with radius $r$. Therefore,
in fact, the generalized Misner-Sharp energy in the FRW universe is nothing
but the total matter energy within a sphere with radius $r$.
(3) Thermodynamics of apparent horizon in the $f(R)$ gravity. On the apparent
horizon of a FRW universe, the energy crossing the apparent horizon within
time interval $dt$ is Cai ; Cai-Kim
$\delta Q=dE_{eff}|_{r_{A}}=A(\tilde{\rho}+\tilde{p})Hr_{A}dt.$ (54)
Note that the horizon entropy of the $f(R)$ gravity is
$S=\frac{A}{4G}f_{R}=\pi r_{A}^{2}f_{R}/G$, while the temperature of the
apparent horizon is Cai-Kim ; Cai2 ; Li : $T=\frac{1}{2\pi r_{A}}$. Obviously,
the usual Clausius relation $\delta Q=TdS$ does not hold. On the other hand,
an internal entropy production is needed to balance the energy conservation,
$\delta Q=TdS+Td_{i}S$ with
$d_{i}S=\pi
r_{A}[Hr_{A}^{3}(H\partial_{t}f_{R}-\partial_{t}\partial_{t}f_{R})-\partial_{t}f_{R}r_{A}]/G.$
(55)
This is an effect of the non-equilibrium thermodynamics of spacetime Jacobson1
; Cai ; Elizalde ; Eling .
(4) The case of scalar-tensor gravity. Indeed the $f(R)$ gravity is quite
similar to scalar-tensor gravity theory in some sense Sotiriou . Our
conclusion on the $f(R)$ gravity therefore also holds for scalar-tensor
gravity. In particular, the existence of a generalized Misner-Sharp energy has
to obey a constraint condition as well for the scalar-tensor theory. However,
in the FRW universe, a simple expression for the generalized Misner-Sharp
energy can be given, which can be seen in appendix A.
## Acknowledgments
YPH thanks D. Orlov for useful discussions. RGC and YPH are supported
partially by grants from NSFC, China (No. 10525060, No. 108215504 and No.
10975168) and a grant from MSTC, China (No. 2010CB833004). NO was supported in
part by the Grant-in-Aid for Scientific Research Fund of the JSPS No.
20540283, and also by the Japan-U.K. Research Cooperative Program. This work
is completed during RGC’s visit to Kinki University, Japan with the support of
JSPS invitation fellowship.
## Appendix A Generalized Misner-Sharp energy of scalar-tensor theory in FRW
universe
The Lagrangian of a generic scalar-tensor gravity in 4-dimensional space-time
can be written as
$L=\frac{1}{16\pi}F(\phi)R-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-V(\phi)+L_{m}.$
(56)
where we set Newtonian constant $G=1$, $F(\phi)$ is an arbitrary positive
continuous function of the scalar field $\phi$, $V(\phi)$ is its potential,
and $L_{m}$ denotes the Lagrangian of other matter fields. Varying the
associated action with respect to spacetime metric and the scalar field yields
equations of motion
$\displaystyle FG_{\mu\nu}+g_{\mu\nu}\square F-\nabla_{\mu}\nabla_{\nu}F$
$\displaystyle=$ $\displaystyle 8\pi(T_{\mu\nu}^{\phi}+T_{\mu\nu}^{m}),$ (57)
$\displaystyle\square\phi-V^{\prime}(\phi)+\frac{1}{16\pi}F^{\prime}(\phi)R$
$\displaystyle=$ $\displaystyle 0.$ (58)
where $T_{\mu\nu}^{m}$ is the energy-momentum tensor of matter fields, and
$T_{\mu\nu}^{\phi}$ is defined as
$T_{\mu\nu}^{\phi}=\partial_{\mu}\phi\partial_{\nu}\phi-
g_{\mu\nu}\Big{(}\frac{1}{2}g^{\rho\sigma}\partial_{\rho}\phi\partial_{\sigma}\phi+V(\phi)\Big{)}.$
(59)
Note that here $T_{\mu\nu}^{\phi}$ is not the energy-momentum tensor of the
scalar field. Similar to the case of f(R) gravity, we find that the current
$J^{\mu}=-T_{\nu}^{\mu(m)}K^{\nu}$ is not always divergence-free unless the
condition is satisfied
$(\nabla_{\mu}\nabla_{\nu}F+8\pi\partial_{\mu}\phi\partial_{\nu}\phi)\nabla^{\mu}K^{\nu}=0.$
(60)
However, we can easily check that the condition (60) can be always satisfied
for the FRW universe (39) by using (12). Some useful components of equations
of gravitational field (57) are given by
$\displaystyle 8\pi T_{tt}^{m}$ $\displaystyle=$ $\displaystyle
3F\Big{(}\frac{k}{a^{2}}+H^{2}\Big{)}+3H\overset{.}{F}-8\pi\Big{(}\frac{1}{2}\overset{.}{\phi}^{2}+V\Big{)},~{}~{}8\pi
T_{t\rho}^{m}=0,$ $\displaystyle 8\pi T_{\rho\rho}^{m}$ $\displaystyle=$
$\displaystyle\frac{a^{2}}{1-k\rho^{2}}\Big{[}-F\Big{(}\frac{k}{a^{2}}+H^{2}+\frac{2\overset{..}{a}}{a}\Big{)}-\overset{..}{F}-2H\overset{.}{F}+8\pi\Big{(}-\frac{1}{2}\overset{.}{\phi}^{2}+V\Big{)}\Big{]}.$
(61)
In this case, corresponding $A$ and $B$ in (35) for the scalar-tensor theory,
respectively, are
$\displaystyle A(t,\rho)$ $\displaystyle=$
$\displaystyle\frac{1}{2}Hr^{3}\Big{[}F\Big{(}\frac{k}{a^{2}}+H^{2}+\frac{2\overset{..}{a}}{a}\Big{)}+\overset{..}{F}+2H\overset{.}{F}-8\pi\Big{(}-\frac{1}{2}\overset{.}{\phi}^{2}+V\Big{)}\Big{]},$
$\displaystyle B(t,\rho)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\rho^{2}a^{3}\Big{[}3F\Big{(}\frac{k}{a^{2}}+H^{2}\Big{)}+3H\overset{.}{F}-8\pi\Big{(}\frac{1}{2}\overset{.}{\phi}^{2}+V\Big{)}\Big{]}.$
(62)
They obey the integrable condition (38). With these quantities we can obtain
the generalized Misner-Sharp energy of the scalar-tensor theory in the FRW
universe
$E_{eff}\ =\int
B(t,\rho)d\rho=\frac{r^{3}}{2}[F(\frac{k}{a^{2}}+H^{2})+H\overset{.}{F}-\frac{8\pi}{3}(\frac{1}{2}\overset{.}{\phi}^{2}+V)].$
(63)
Comparing this with the first equation in (61), one can immediately see that
the generalized Misner-Sharp energy is just the total matter energy in a
sphere with radius $r$ in the FRW universe.
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|
arxiv-papers
| 2009-10-13T13:07:57 |
2024-09-04T02:49:05.826984
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Rong-Gen Cai, Li-Ming Cao, Ya-Peng Hu, Nobuyoshi Ohta",
"submitter": "Rong-Gen Cai",
"url": "https://arxiv.org/abs/0910.2387"
}
|
0910.3013
|
# Measurement of $B^{+}_{c}$ properties at CDF
T.S. Nigmanova111Speaker, on behalf of the CDF Collaboration, K.R. Gibsonb,
M.P. Hartzc, P.F. Shepardb aUniversity of Michigan, Ann Arbor, MI 48109, USA
bUniversity of Pittsburgh, Pittsburgh, PA 15260, USA cUniversity of Toronto,
Toronto M5S, Canada
###### Abstract
The $B^{+}_{c}$ meson is composed of two heavy quarks of distinct flavor.
Measurements of its lifetime and production properties have been made based on
semileptonic $B^{+}_{c}\to J/\psi+l^{+}+X$ decays using data collected with
the CDF II detector corresponding to an integrated luminosity of 1 fb-1. The
$B^{+}_{c}$ average lifetime $c\tau$ is measured to be
142.5${}^{+15.8}_{-14.8}$(stat)$\pm 5.5(syst)$ $\mu m$. The measurements of
the ratio of the production cross section times branching ratio of
$B^{+}_{c}\to J/\psi\mu^{+}\nu$ relative to $B^{+}\to J/\psi K^{+}$ were done
for two $p_{T}(B)$ thresholds: for $p_{T}(B)>$ 4 GeV/$c$ as $0.295\pm
0.040~{}\mbox{(stat.)}^{+0.033}_{-0.026}~{}\mbox{(syst.)}\pm
0.036~{}(p_{T}~{}\mbox{spectrum})$ and for $p_{T}(B)>$ 6 GeV/$c$ as $0.227\pm
0.033~{}\mbox{(stat.)}^{+0.024}_{-0.017}~{}\mbox{(syst.)}\pm
0.014~{}(p_{T}~{}\mbox{spectrum})$.
## I Introduction
The $B^{+}_{c}$ meson charge_conjugate is composed of an anti-bottom quark
$\bar{b}$ and a charm quark $c$. The presence of two relatively heavy quarks
with different flavors is unique to the $B^{+}_{c}$ system and affects the
decay and production properties. The theoretically predicted lifetime kiselev
is about a factor of three times smaller than that of other B mesons. The
expected $B^{+}_{c}$ production cross section chang is about 3 orders of
magnitude lower than the production cross section of the $B^{+}$ CDF-bplus-
Xsect . The first observation of the $B^{+}_{c}$ was made using data taken
with the CDF detector at the Fermilab Tevatron during run I bc-observation .
Precise mass measurements have been made by the CDF Collaboration using fully
reconstructed $B^{+}_{c}\rightarrow J/\psi\pi^{+}$ decays, where $J/\psi$
decays through $J/\psi\rightarrow\mu^{+}\mu^{-}$ bc-mass .
In this work we report preliminary measurements of the $B^{+}_{c}$ lifetime in
the semileptonic decay modes $J/\psi\mu^{+}X$ and $J/\psi eX$, and the
production cross section times branching ratio of the decay mode
$B^{+}_{c}\rightarrow J/\psi\mu^{+}\nu$ relative to the $B^{+}\rightarrow
J/\psi K^{+}$ decay. The results presented here are based on a data sample
with an integrated luminosity of 1 fb-1 at $\sqrt{s}$=1.96 TeV collected by
the CDF II detector.
## II The $B^{+}_{c}$ lifetime measurement concept
To measure the lifetime of the $B^{+}_{c}$, we construct a per event lifetime
that is defined using variables measured in the transverse plane. If all of
the decay products of the $B^{+}_{c}$ decay are identified, the lifetime $ct$
is the lifetime of the $B^{+}_{c}$ meson in its rest frame measured in units
of microns of light travel time. It is expressed as
$ct=\frac{mL_{xy}}{p_{T}}$ (1)
where $m$ is the mass of the $B^{+}_{c}$, $p_{T}$ is the momentum of the
$B^{+}_{c}$ in the plane transverse to the direction of the proton beam, and
$L_{xy}$ is the decay length of the $B^{+}_{c}$ projected along the transverse
momentum. The mass of the $B^{+}_{c}$ used in this measurement is $m=6.286$
GeV/c2 bc-mass . However, we do not measure all of the particles in the
semileptonic $B^{+}_{c}$ final state. Instead we must define a pseudo lifetime
${ct^{*}=\frac{mL_{xy}(J/\psi l^{+})}{p_{T}(J/\psi l^{+})}}$ (2)
where $L_{xy}$ and $p_{T}$ are evaluated using the $J/\psi+l^{+}$ system. We
can obtain the true $B^{+}_{c}$ lifetime by defining a factor $K$, where
$ct=Kct^{*}$. We evaluate the $K$ factor distribution, $H(K)$, for $B^{+}_{c}$
events using Monte Carlo simulation. We are then able to express the
distribution of $ct^{*}$ for $B^{+}_{c}\rightarrow J/\psi l^{+}X$ as
${F_{B_{c}}(ct^{*},\sigma)=\int
dKH(K)\frac{K}{c\tau}\theta(ct^{*})exp(-\frac{Kct^{*}}{c\tau})\otimes
G(\sigma)}$ (3)
where $c\tau$ is the average $B^{+}_{c}$ lifetime, $\sigma$ represents the
estimated error on the measurement of $ct^{*}$ for each event, and $G(\sigma)$
is defined as
${G(\sigma)=\frac{1}{\sqrt{2\pi}s\sigma}e^{-\frac{1}{2}(\frac{ct^{*}}{s\sigma})^{2}}}$
(4)
The measurement of the $B^{+}_{c}$ average lifetime is be carried out by
minimizing $-2Log(L)$, which is evaluated for the candidate $J/\psi+l$ events.
$L$ is the likelihood function for the $ct^{*}$ and $\sigma$ measured in
candidate events and includes $c\tau$ as a free parameter.
## III The $B^{+}_{c}$ cross section measurement concept
We measure
${\frac{\sigma(B^{+}_{c})BF(B^{+}_{c}\to
J/\psi\mu^{+}\nu)}{\sigma(B^{+})BF(B^{+}\to J/\psi
K^{+})}=\frac{N(B^{+}_{c})}{N(B^{+})}\times\epsilon_{rel}}$ (5)
The basic strategy of the $B^{+}_{c}$ cross section measurement is to
reconstruct the number of $B^{+}_{c}\to J/\psi\mu^{+}\nu$ relative to the
number of $B^{+}\to J/\psi K^{+}$ candidates, $N(B^{+}_{c})/N(B^{+})$,
determine the relative detector and reconstruction efficiency,
$\epsilon_{rel}$ = $\epsilon(B^{+})/\epsilon(B^{+}_{c})$, and use these to
determine the ratio of the final production cross section times branching
ratio.
## IV The event selection
The analysis presented here is based on the events recorded with a di-muon
trigger that is dedicated to $J/\psi\to\mu^{+}\mu^{-}$ decays. Both analysis
use the same $J/\psi\to\mu^{+}\mu^{-}$ sample. The muon pair is reconstructed
within a pseudo-rapidity range $|\eta|<$ 1.0. We select about 6.9$\times$106
$J/\psi$ candidates, measured with a mass resolution of approximately 12
MeV/$c^{2}$. The di-muon invariant mass distribution is shown in Fig. 1.
Figure 1: The $J/\psi\rightarrow\mu^{+}\mu^{-}$ invariant mass distribution.
Events within a mass range of $\pm$50 MeV/$c^{2}$ around the central $J/\psi$
mass value were used for both the lifetime and cross section analysis.
In addition to the di-muons from the $J/\psi$ decay, we require a third track
that is matched to the same vertex as the $J/\psi$. This third track can be
from any of three samples of interest: $B^{+}_{c}\to J/\psi l^{+}$X decays,
$B^{+}\to J/\psi K^{+}$ decays, or just $J/\psi+track$ decays. The last sample
represents sources of backgrounds to the $B^{+}_{c}$ semileptonic decay.
## V The $B^{+}_{c}$ backgrounds overview
### V.1 Common for both analysis
Both the lifetime and cross section analysis have some common sources of
background: misidentified $J/\psi$, misidentified third muons, and $b\bar{b}$
backgrounds. The misidentified $J/\psi$ background occurs when one of the
muons is actually a mis-reconstructed hadron or muon from other sources that
produce a mass consistent with that of the $J/\psi$. The misidentified third
muon background can arise from the following sources. The $J/\psi$ in the
$J/\psi+track$ system is highly populated by non-$B^{+}_{c}$ sources. The
third track associated with the $J/\psi$ could be a $\pi^{+}$ or $K^{+}$ that
can either decay-in-flight or punch-through the calorimeter and the steel
absorber and produce the muon signature. The $b\bar{b}$ events represent cases
when a $J/\psi$ is produced from one $b$ jet and the third muon originates
from the other $b$ in same event.
Figure 2 shows the misidentified muon rates for $\pi^{\pm}$, $K^{\pm}$, and
p($\bar{p}$) and the pseudo-proper decay length ct∗ for misidentified third
muons.
Figure 2: The misidentified muon rates from $\pi^{\pm}$, $K^{\pm}$, and
p($\bar{p}$) as a function of hadron $p_{T}$ (left), and the pseudo-proper
decay length ct∗ for misidentified third muons (right).
### V.2 Specific for each analysis
There is an additional background for the cross section measurement. The
selected $B^{+}_{c}\to J/\psi\mu^{+}X$ sample contains contributions from
other $B^{+}_{c}$ decays with a tri-muon in the final state. For example, a
$B^{+}_{c}$ can decay into $\psi(2S)\mu^{+}\nu$ followed by
$\psi(2S)\rightarrow J/\psi X$.
The following backgrounds are specific to the lifetime analysis: misidentified
e±, residual conversions, and prompt $J/\psi$. The misidentified e± can arise
from cases when a $\pi^{\pm}$, K±, or $\bar{p}$ from the $J/\psi+track$ system
satisfies the e± likelihood function based on the calorimeter responses. The
residual conversions are e± from $\gamma$-conversion or $\pi^{o}$ Dalitz
decays. The prompt $J/\psi$ are additional $J/\psi l^{\pm}$ candidates where
the $J/\psi$ originates from prompt non-$B^{+}_{c}$ sources. Figure 3
illustrates the $e^{+}e^{-}$ veto efficiencies and the pseudo-proper decay
length ct∗ distribution for the $J/\psi$+Conversion e background sample.
Figure 3: The $e^{+}e^{-}$ veto efficiencies as a function of electron $p_{T}$
(left), and the pseudo-proper decay length ct∗ distribution for the
$J/\psi$+Conversion e background sample (right).
## VI The $B^{+}_{c}$ lifetime results
### VI.1 Lifetime systematic uncertainties
The systematic uncertainties in the $B^{+}_{c}$ lifetime measurement originate
from uncertainties in our models for background and signal events. Some of the
largest systematic uncertainties are summarized below:
* •
Resolution function - choice of model for detector resolution: $\pm$3.8 $\mu
m$
* •
Pythia model for $b\bar{b}$ background - relative contribution of QCD
processes: $\pm$2.4 $\mu m$
* •
Vertex detector alignment - uncertainties in the positions of silicon
detectors: $\pm$2.0 $\mu m$
* •
$e^{+}e^{-}$ veto efficiency - uncertainties related to modeling $e^{+}e^{-}$
veto efficiencies: $\pm$1.5 $\mu m$
* •
$B_{c}$ spectrum - variations of the K factor distribution due to variations
in the $B_{c}$ production spectrum: $\pm$1.3 $\mu m$
We add the individual uncertainties in quadrature to obtain a total
uncertainty of $\pm$5.5 $\mu m$.
### VI.2 Lifetime results
We fit the $ct^{*}$ distributions for signal candidates in the $J/\psi\mu^{+}$
and $J/\psi e^{+}$ channels separately using likelihood functions based on our
models for signal and background events. The fitted data is shown in Fig. 4
for $B^{+}_{c}\to J/\psi\mu^{+}X$ and for $B^{+}_{c}\to J/\psi e^{+}X$ decays.
Figure 4: The pseudo-proper decay length ct∗ distributions for $B^{+}_{c}\to
J/\psi\mu^{+}X$ decays (left) and for $B^{+}_{c}\to J/\psi e^{+}X$ decays
(right) with their the background models superimposed.
The $B^{+}_{c}$ lifetime is found to be 179.1${}^{+32.6}_{-27.2}$(stat) $\mu
m$ for the $J/\psi\mu^{+}$ final state and 121.7${}^{+18.0}_{-16.3}$ (stat)
$\mu m$ for the $J/\psi e^{+}$ decay mode, respectively. We performed the
simultaneous fit of both samples and found an average $B^{+}_{c}$ lifetime of
142.5${}^{+15.8}_{-14.8}(stat)\pm 5.5(syst)$ $\mu m$. Figure 5 shows our
$B^{+}_{c}$ average lifetime comparison with other measurements.
Figure 5: World average of $B^{+}_{c}$ lifetime, which includes the CDF Run I
$B_{c}$ lifetime, the most recent D0 Run II result, and the result presented
in this paper. The lifetimes are weighted by the total variance of the
individual measurements in the average.
## VII The $B^{+}_{c}$ relative cross section
In order to measure the $B^{+}_{c}$ relative cross section we need to find the
numbers of $B_{c}^{+}$ and $B^{+}$, $N(B^{+}_{c}$) and $N(B^{+}$), and
determine the relative efficiency, $\epsilon_{rel}$ =
$\epsilon(B^{+})/\epsilon(B^{+}_{c})$. We select 229 (214) $B^{+}_{c}$
candidates with the requirement $p_{T}(B^{+}_{c})>$ 4 (6) GeV/$c$,
respectively. The number of $B^{+}_{c}$ signal events after backgrounds
subtraction is presented in the following subsection. The number of $B^{+}\to
J/\psi K^{+}$ signal events is found to be 2333 $\pm$ 55 (2299 $\pm$ 53) for
$p_{T}(B^{+})>$ 4 (6) GeV/$c$, respectively. The combinatoric and $B^{+}\to
J/\psi\pi^{+}$ contributions are subtracted.
### VII.1 The $B^{+}_{c}$ backgrounds and excess
The backgrounds and the resulting number of signal events for the
$B^{+}_{c}\to J/\psi\mu^{+}\nu$ decays are summarized in Table 1.
Table 1: Observed $N(B^{+}_{c}\to J/\psi\mu^{+}\nu$) for the $p_{T}(J/\psi\mu)>$ 4 GeV/c (6 GeV/c) threshold. | $p_{T}(B)>$ 4 GeV/$c$ | $p_{T}(B)>$ 6 GeV/$c$
---|---|---
$N(B^{+}_{c}$) observed | 229$\pm$15.1(stat) | 214$\pm$14.6(stat)
Misidentified $J/\psi$ | 21.5$\pm$3.3(stat) | 20.5$\pm$3.2(stat)
Misid. third muon | 55.8$\pm$2.0(stat) | 53.6$\pm$1.9(stat)
Doubly misid. | -8.8$\pm$0.4(stat) | -7.5$\pm$0.3(stat)
$b\bar{b}$ background | 37.7$\pm$7.3(st+sys) | 35.4$\pm$7.0(st+sys)
Other $B^{+}_{c}$ modes | 5.2$\pm$0.5(stat) | 4.8$\pm$0.4(stat)
Total background | 111.4$\pm$8.3(stat) | 106.9$\pm$8.0(stat)
$B^{+}_{c}$ signal | 117.6$\pm$17.2(stat) | 107.1$\pm$16.7(stat)
The background identified as “Doubly misidentified” in Table 1 represents the
subsample of misidentified $J/\psi$ and misidentified third muons that needs
to be subtracted only once to avoid double counting.
Figure 6 shows the invariant mass distribution of the $B^{+}_{c}\to
J/\psi\mu^{+}X$ data events with the experimental backgrounds and a Monte
Carlo simulation of the signal sample superimposed (left), and with the
background subtracted (right).
Figure 6: The invariant mass distribution of the $B^{+}_{c}\to J/\psi\mu^{+}X$
data events with the experimental backgrounds and a Monte Carlo simulation of
the signal sample superimposed (left), and with the background subtracted
applied (right).
### VII.2 The relative efficiency $\epsilon_{rel}$
In order to determine the relative efficiency, we simulate $B^{+}\to J/\psi
K^{+}$, $B^{+}_{c}\to J/\psi\mu^{+}\nu$, and $B^{*+}_{c}\to B^{+}_{c}\gamma$
decays. As a description of the $\eta-p_{T}$ spectrum for $B^{+}_{c}$ we use
the most recent theoretical work by Chang et al. chang . For the $B^{+}$ we
used the spectrum from Ref. FONLL , which is found to be in good agreement
with CDF measurements. All Monte Carlo simulation events are passed through
the full detector and trigger simulation. The Monte Carlo simulation samples
were processed in the same way as for the data. The efficiencies
$\epsilon_{B^{+}_{c}}$ and $\epsilon_{B^{+}}$ for $p_{T}(B)>$ 4 (6) GeV/$c$,
along with the relative efficiency, are presented in Table 2.
Table 2: Efficiencies for $B^{+}_{c}$ and $B^{+}$ for $p_{T}(B)>$ 4 (6) GeV/$c$. Efficiency | $p_{T}(B)>$ 4 GeV/$c$ | $p_{T}(B)>$ 6 GeV/$c$
---|---|---
$\epsilon_{B^{+}_{c}}$ (%) | $0.0551\pm 0.0010$ | $0.1232\pm 0.0024$
$\epsilon_{B^{+}}$ (%) | $0.3231\pm 0.0022$ | $0.6005\pm 0.0042$
$\epsilon_{rel}$ | $5.867\pm 0.068$ (stat) | $4.873\pm 0.060$ (stat)
The $p_{T}$ spectra for data and Monte Carlo simulation are shown in Fig. 7.
Figure 7: The comparison of the $p_{T}$ spectra of data versus Monte Carlo
simulation for $B^{+}_{c}\to J/\psi\mu^{+}\nu$ (left), and for $B^{+}\to
J/\psi K^{+}$ decays (right).
### VII.3 Cross section systematic uncertainties
We divide the systematic uncertainties into two categories: uncertainties on
the number of $B^{+}_{c}$ signal events and uncertainties in the determination
of the relative efficiency.
#### VII.3.1 $B^{+}_{c}$ background uncertainty
The systematic uncertainty considered in the determination of $N(B^{+}_{c})$
arise from events which do not originate from $B^{+}_{c}$ decays. All
backgrounds are assigned a systematic uncertainty except the misidentified
$J/\psi$ background. Since the estimation of this background is determined
directly from the sidebands of the $J/\psi$, we do not know of any source of
systematic error that should be included.
The largest source of uncertainty in the misidentified third muon calculation
is due to the poor knowledge of the proton fraction in the $J/\psi+track$
sample. The particle identification method, dE/dx, does not allow us to
separate protons from kaons in our kinematic region. Consequently, we measure
the proton fraction in other momentum ranges using the time-of-flight (TOF)
particle identification combined with dE/dx information and then extrapolate
the fraction to our momentum range according to measured trends in Monte Carlo
simulation.
The other $N(B^{+}_{c})$ systematic uncertainty arises from a poor knowledge
of non-exclusive $B^{+}_{c}\to J/\psi\mu^{+}X$ branching ratios and is
estimated by varying the branching ratios of eleven $B^{+}_{c}$ decay modes
that may contribute to the sample of tri-muon events. In order to assign a
systematic uncertainty we double and halve the branching ratios of the non-
exclusive decays with respect to the rate of $B^{+}_{c}\to J/\psi\mu^{+}\nu$.
We choose the larger variation in either direction for the systematic
uncertainty.
Table 3 summarizes all of the $B^{+}_{c}$ background systematic uncertainties
assigned.
Table 3: Systematic uncertainties on the number of $B^{+}_{c}\to J/\psi\mu^{+}\nu$ events for different $p_{T}(B)$ thresholds. The total uncertainty is calculated by adding all the individual uncertainties in quadrature. $N(B^{+}_{c})$ uncertainties | $p_{T}(B)>$ 4 GeV/$c$ | $p_{T}(B)>$ 6 GeV/$c$
---|---|---
Misid. third muon | $\pm 5.7$ (sys) | $\pm 5.5$ (sys)
Doubly misid. | $\pm 0.9$ (sys) | $\pm 0.8$ (sys)
Other $B^{+}_{c}$ decays | ${}^{+6.0}_{-2.8}$ (sys) | ${}^{+5.6}_{-2.5}$ (sys)
Total | ${}^{+8.3}_{-6.4}$ (sys) | ${}^{+7.9}_{-6.1}$ (sys)
#### VII.3.2 Relative efficiency systematic uncertainty
We consider the systematic uncertainty in the prediction of the relative
efficiency due to the measured statistical uncertainty of the $B^{+}_{c}$
lifetime, knowledge of the production spectra for $B^{+}_{c}$ and $B^{+}$, and
differences between $K$ and $\mu$ triggering rates at the first level of the
CDF trigger system, the extremely fast trigger, XFT.
The relative efficiency systematic uncertainty due to $B^{+}_{c}$ lifetime
uncertainty is estimated by varying the lifetime by $\pm$14 $\mu$m of its
default value. This variation represents approximately one standard deviation
in the average lifetime result.
The relative efficiency systematic uncertainty due to the knowledge of
$B^{+}_{c}$ $p_{T}$ spectrum is fully based on the theoretically predicted
$p_{T}$ spectra from work in Ref. chang . We consider the variations between:
* •
doubling the $q\bar{q}$ contribution relative to the nominal approach;
* •
the pure gluon fusion model, called the fixed flavor number ( FFN) model, and
the more complete $gg+g\bar{b}+gc$ model, known as the general-mass variable-
flavor-number (GMVFN) model;
* •
the combined $B^{+}_{c}+B^{*+}_{c}$ spectrum and a pure $B^{+}_{c}$ spectrum.
The systematic uncertainty due to knowledge of the $B^{+}$ $p_{T}$ spectrum is
estimated by varying the Monte Carlo simulated spectrum below 10 GeV/$c$ to
bring it into agreement with the data (see the right plot in Fig. 7. The
difference between the nominal and recalculated relative efficiency is
assigned as the uncertainty due to $B^{+}$ $p_{T}$ spectrum.
Another source of systematic uncertainty that we consider is the different XFT
efficiencies of kaons and muons that exist in the data and are not modeled in
the simulation.
The total $\epsilon_{rel}$ systematic uncertainty is summarized in Table 4.
Table 4: Systematic uncertainty assigned to $\epsilon_{rel}$ for different $p_{T}(B)$ thresholds. The numbers 0.720 and 0.298 in the “Total” represent the uncertainties due to of $B^{+}_{c}$ spectrum. $\epsilon_{rel}$ uncertainties | $p_{T}(B)>$ 4 GeV/$c$ | $p_{T}(B)>$ 6 GeV/$c$
---|---|---
$B^{+}_{c}$ lifetime | ${}^{+0.393}_{-0.223}$ | ${}^{+0.354}_{-0.160}$
$B^{+}_{c}$ spectrum | $\pm$ 0.720 | $\pm$ 0.298
$B^{+}$ spectrum | $\pm$ 0.340 | $\pm$ 0.161
XFT systematics | $\pm$ 0.192 | $\pm$ 0.160
Total | ${}^{+0.554}_{-0.450}\pm$0.720 | ${}^{+0.420}_{-0.278}\pm$0.298
## VIII The $B^{+}_{c}$ relative cross section results
We have performed a measurement of the relative production cross section of
$B^{+}_{c}\to J/\psi\mu^{+}\nu$ in inclusive $J/\psi$ data with an integrated
luminosity of 1 fb-1. We have identified a sample of 229 (214) events with an
estimated background from all sources of 111$\pm$8 (107$\pm$8) events for
$p_{T}(J/\psi\mu^{+})>$ 4 (6) GeV/$c$, respectively. The final numbers used in
the cross section measurement, including systematic uncertainties, are given
in Table 5.
Table 5: Final numbers used in the calculation of the relative $B^{+}_{c}\to J/\psi\mu^{+}\nu$ production cross section times the branching ratio for two different $p_{T}(B)$ thresholds. Final values | $p_{T}(B)>$ 4 GeV/$c$
---|---
$N(B^{+}_{c}$) | 117.6$\pm$17.2 (stat) ${}^{+8.3}_{-6.4}$(sys)
$N(B^{+}$) | 2333$\pm$ 55 (stat)
$\epsilon_{rel}$ | 5.867$\pm$0.068 (stat) ${}^{+0.554}_{-0.450}$(sys)
| $\pm$0.720 ($B^{+}_{c}$ spectrum)
Final values | $p_{T}(B)>$ 6 GeV/$c$
$N(B^{+}_{c}$) | 107.2$\pm$16.7 (stat) ${}^{+7.9}_{-6.1}$ (sys)
$N(B^{+}$) | 2299$\pm$53 (stat)
$\epsilon_{rel}$ | 4.872$\pm$0.060 (stat) ${}^{+0.420}_{-0.278}$ (sys)
| $\pm$0.298 ($B^{+}_{c}$ spectrum)
We give the result for the ratio $\frac{\sigma(B^{+}_{c})BF(B^{+}_{c}\to
J/\psi\mu^{+}\nu)}{\sigma(B^{+})BF(B^{+}\to J/\psi K^{+})}$ with $p_{T}(B)>$ 4
GeV/$c$ thresholds as
$0.295\pm 0.040~{}\mbox{(stat.)}^{+0.033}_{-0.026}~{}\mbox{(syst.)}\pm
0.036~{}(p_{T}~{}\mbox{spec})$
and for $p_{T}(B)>$ 6 GeV/$c$ as
$0.227\pm 0.033~{}\mbox{(stat.)}^{+0.024}_{-0.017}~{}\mbox{(syst.)}\pm
0.014~{}(p_{T}~{}\mbox{spec})$.
Of the two results, the measurement with the $p_{T}(B)>$ 6 GeV/$c$ threshold
has the lower systematic error. Below 6 GeV/$c$ there is uncertainty in the
$B^{+}$ efficiency that appears to introduce a significant systematic
discrepancy between the simulated spectrum and the spectrum as determined from
the data.
Using theoretical assumptions and independent measurements, we are then able
to calculate the total $B^{+}_{c}$ cross section. Using the measured
quantities $BR(B^{+}\to J/\psi K^{+})$ = $(1.007\pm 0.035)\times 10^{-3}$
Ref:PDG and $\sigma(B^{+})=2.78\pm 0.24~{}\mu$b for $p_{T}(B^{+})>$ 6 GeV/$c$
CDF-bplus-Xsect , we calculate
$\displaystyle\sigma(B^{+}_{c})*BR(B^{+}_{c}\to J/\psi\mu^{+}\nu)=0.64\pm
0.20~{}\mbox{nb}$
for $p_{T}(B^{+}_{c})>$ 6 GeV/$c$. Assuming that the branching ratio
$BR(B^{+}_{c}\to J/\psi\mu^{+}\nu)=2.07\times 10^{-2}$ Ref:BcIvanov , we find
the total $B^{+}_{c}$ cross section to be
$\displaystyle\sigma(B^{+}_{c})=31\pm 10~{}\mbox{nb}$
## IX Conclusions
We have performed measurements of the $B^{+}_{c}$ lifetime and production
properties based on semileptonic $B^{+}_{c}\to J/\psi+l^{+}+X$ decays using
data from $p\bar{p}$ collisions collected with the CDF II detector
corresponding to an integrated luminosity of 1 fb-1 at $\sqrt{s}$=1.96 TeV.
## References
* (1) Reference to a particular charge state also implies the charge conjugate state.
* (2) S. Godfrey, Phys. Rev. D 70, 054017 (2004); V.V.Kiselev, arXiv:hep-ph/0308214.
* (3) Chao-Hsi Chang, Phys. Rev. D 72, 114009 (2005).
* (4) A.Abulencia et al. (CDF Collaboration), Phys. Rev. D 75, 012010 (2007).
* (5) A.Abulencia et al. (CDF Collaboration), Phys. Rev. Lett. 97, 012002 (2006).
* (6) T.Aaltonen et al. (CDF Collaboration), Phys. Rev. Lett. 100, 182002 (2008).
* (7) M. Cacciari et al., J. High Energy Phys. 07, 033 (2004).
* (8) C. Amster et al. (Particle Data Group), Phys. Lett. B667, 1 (2008).
* (9) M.A. Ivanov et al., Phys. Rev. D 73, 054024 (2006).
|
arxiv-papers
| 2009-10-16T13:04:10 |
2024-09-04T02:49:05.842497
|
{
"license": "Public Domain",
"authors": "T.S. Nigmanov, K.R. Gibson, M.P. Hartz, P.F. Shepard (for the CDF\n Collaboration)",
"submitter": "Turgun Nigmanov",
"url": "https://arxiv.org/abs/0910.3013"
}
|
0910.3021
|
# Chandra Observations of the Radio Galaxy 3C 445 and the Hotspot X-ray
Emission Mechanism
Eric S. Perlman11affiliation: Department of Physics and Space Sciences,
Florida Institute of Technology, 150 W. University Blvd., Melbourne, FL 32901
, Markos Georganopoulos22affiliation: Department of Physics, University of
Maryland, Baltimore County, 1000 Hilltop Circle, Baltimore, MD 21250
33affiliation: Laboratory for High Energy Astrophysics, NASA Goddard Space
Flight Center, Codel 661, Greenbelt, MD 20771 , Emily M. May 44affiliation:
Department of Physics and Astronomy, University of Wyoming, WY 82071
55affiliation: Southeastern Association for Research in Astronomy (SARA) NSF-
REU Summer Intern at Florida Institute of Technology , Demosthenes
Kazanas33affiliation: Laboratory for High Energy Astrophysics, NASA Goddard
Space Flight Center, Codel 661, Greenbelt, MD 20771
###### Abstract
We present new Chandra observations of the radio galaxy 3C 445, centered on
its southern radio hotspot. Our observations detect X-ray emission displaced
upstream and to the west of the radio-optical hotspot. Attempting to reproduce
both the observed spectral energy distribution (SED) and the displacement,
excludes all one zone models. Modeling of the radio-optical hotspot spectrum
suggests that the electron distribution has a low energy cutoff or break
approximately at the proton rest mass energy. The X-rays could be due to
external Compton scattering of the cosmic microwave background (EC/CMB) coming
from the fast (Lorentz factor $\Gamma\approx 4$) part of a decelerating flow,
but this requires a small angle between the jet velocity and the observer’s
line of sight ($\theta\approx 14^{\circ}$). Alternatively, the X-ray emission
can be synchrotron from a separate population of electrons. This last
interpretation does not require the X-ray emission to be beamed.
###### Subject headings:
active galaxies, radiation mechanisims, X-ray, infrared, 3C 445, inverse
compton, synchrotron, hotspots
## 1\. Introduction
The hotspots of powerful Fanaroff-Rilley type II [FRII; Fanaroff & Riley 1974]
radio galaxies and quasars are the locations where the jets of these sources,
after propagating for distances up to $\sim 1$ Mpc, terminate in a collision
with the intergalactic medium. The optical emission observed in several
hotspots suggests that at least a fraction of the electrons that goes through
the shock(s) formed at the termination of the jets undergoes efficient
particle acceleration (Heavens & Meisenheimer, 1987; Meisenheimer et al.,
1989; Prieto et al., 2002). Chandra results show that while in some sources
(e.g. Cygnus A; Wilson et al. 2000) the hotspot X-ray emission is consistent
with synchrotron-self Compton radiation from relativistic electrons in energy
equipartition with the magnetic field (SSCE), in other sources (e.g. the
hotspot on the jet side of Pictor A; Wilson et al. 2001, Hardcastle & Croston
2005, Migliori et al. 2007, Tingay et al. 2008) the X-ray emission is at a
much higher level (by up to a factor of $\sim 1000$). The nature of this
anomalously bright (significantly brighter than SSCE) X-ray hotspot emission
remains a matter of active discussion in the literature, being connected to
the issue of particle acceleration efficiency and jet power [for two recent
reviews see Harris & Krawczynski (2006) and Worall (2009)] and extending to a
similar and possibly related issue for the knots of powerful jets (e.g.
Kataoka & Stawarz 2005).
Two different considerations appear to be relevant to the collective
properties of hotspot X-ray emission, relativistic beaming and hotspot
luminosity; however, it is not as yet clear how exactly these may be combined
to reproduce the observed phenomenology. Based on early Chandra results,
suggesting that in most cases the anomalously bright X-ray hotspots were seen
at the approaching jet of sources with jets forming relatively small angles to
the observer’s line of sight, Georganopoulos & Kazanas (2003) proposed that
the X-ray emission is beamed, and that the plasma in the hotspot is
relativistic and decelerating from a bulk Lorentz factor $\Gamma\sim 2-3$ to
velocities that match the subrelativistic advance speed of lobes ($u/c\approx
0.1$; Arshakian & Longair 2001). In this scenario the X-ray emission is mostly
due to upstream Compton (UC) scattering of electrons in the upstream fast
($\Gamma\sim 2-3$) part of the flow, inverse-Compton scattering the
synchrotron photons produced downstream. The relevance of beaming is also
supported by a study of quasar hotspots by Tavecchio et al. (2005) that
concluded that the anomalously bright X-ray emission is indeed found mostly on
the hotspot corresponding to the approaching jet side, although these authors
favor the cosmic microwave background as a source of seed photons. On the
other hand, Hardcastle et al. (2004) find only weak evidence that the
anomalously X-ray bright hotspots are more frequently found at the termination
of the approaching jet.
The second consideration, namely the hotspot luminosity, was introduced by
Brunetti et al. (2003), who found that the synchrotron emission seen at radio
energies extends to the optical for lower power sources like 3C 445 (Prieto et
al., 2002) but cuts off before the optical regime for powerful sources like
Cygnus A (Wilson, Young & Shopebell, 2000). They explained this as a
consequence of radiative losses increasing with hotspot luminosity. The
luminosity range over which this decrease of the synchrotron peak frequency
with increasing luminosity is observed, was extended to powerful hotspots by
Cheung, Wardle, & Chen (2005). Based on an extensive sample of hotspot
multiwavelength data, Hardcastle et al. (2004) argued for the relevance of the
hotspot luminosity to the X-ray emission, by showing that hotspots with X-ray
luminosity much higher than that predicted by SSCE were usually of low
luminosity, in contrast to more powerful hotspots. However, instead of UC,
Hardcastle et al. (2004) favored synchrotron emission from an altogether
separate electron population as the source of the anomalously bright X-ray
emission. Other workers have also come to the conclusion that synchrotron
radiation from a second electron population is the most likely emission
mechanism for this component (e.g., in the case of Pic A, see Fan et al. 2008,
Tingay et al. 2008), partly as a result of observing compact radio hotspots
with the VLBA.
Synchrotron radiation, but from a single electron population, is indeed the
X-ray emission mechanism for the jets of low power FR I radio galaxies, as
seen from their single component radio-optical-X-ray spectra (e.g. Perlman &
Wilson 2005 for the jet of M 87). This turns out to be the case also for some
of the weakest hostspots of FR II radio galaxies: the northern hotspot of 3C
390.3 (Hardcastle, Croston, & Kraft, 2007), the northern hotspots of 3C 33
(Kraft et al., 2007), the eastern hotspots of 3C 403 (Kraft et al., 2007) and
both hotspots of $0836+299$ (Tavecchio et al., 2005). It comes as no surprise
that in these sources their X-ray emission is much higher than the anticipated
SSC flux, since the X-ray emission is indeed the continuation of the radio-
optical synchrotron spectrum to higher energies. We therefore do not consider
sources that exhibit a single spectral component from radio to X-ray emission
to be part of the family of hotspots exhibiting anomalously high X-ray
emission.
An additional handle in understanding the X-ray emission process of the
anomalously X-ray bright hotspots (those for which the X-rays (i) cannot be a
continuation of the synchrotron spectrum and (ii) are significantly brighter
than predicted by SSCE) can come from spatial displacements between the
emission at different frequencies. In a handful of these hotspots,
displacements between the radio and the X-ray hotspot emission have been
observed, with the X-rays being upstream of the radio: 3C 351 (Hardcastle et
al., 2002), 4C 74.26 (Erlund et al., 2007), 3C 390.3 and 3C 227 (Hardcastle,
Croston, & Kraft, 2007), 3C 321 (Evans et al., 2008), 3C 353 (Kataoka et al.,
2008). So far no displacements have been observed in the hotspots of sources
for which their X-ray emission is consistent with SSCE. An additional
important characteristic is that when optical-IR emission is detected from
these hotspots (as in 3C227, 3C 390.3, 3C 351) it coincides with or is shifted
somewhat upstream of the radio, to a location downstream of the X-rays.
Beaming in the hotspots that exhibit displacements can be a relevant but not a
dominant influence, because, while in three of them the hotspots are in the
approaching jet that points toward us (4C 74.26, 3C 227, 3C 351), in two
others (3C 321 and 3C 353), the jets are believed to be close to the plane of
the sky, while in superluminal 3C 390.3 the hotspot exhibiting the X-ray radio
displacement is at the termination of the counter jet. Similarly, a low
hotspot power does not seem to be strictly required, since the hotspot of 3C
351 is rather powerful and still exhibits the displacements mentioned above.
In this paper we present Chandra X-ray observations and discuss the X-ray
emission mechanism of the hotspots of 3C 445, a broad line FR II radio galaxy
at redshift $z=0.0562$ (Eracleous & Halpern, 1994), which for the standard
cosmology ($H_{0}=71$ km s-1 Mpc-1, $\Omega_{\Lambda}=0.73$ and
$\Omega_{M}=0.27$) corresponds to a luminosity distance of $247.7$ Mpc. This
is a promising source for constraining the hotspot X-ray emission mechanism
with high resolution Chandra observations. Both its southern and northern
hotspots have been detected in the radio and in the near-IR/optical, with the
southern hotspot being $\sim 3$ times brighter than the northern and having
multiple sites of synchrotron optical emission, a manifestation of ongoing
particle acceleration (Prieto et al., 2002; Mack et al., 2009). The hotspot -
counter hotspot luminosity difference can be intrinsic, or due to mild beaming
with the southern hotspot being on the approaching jet side.
In the first case, adopting the suggestion of Hardcastle et al. (2004), we
expect X-ray emission brighter than SSCE from both hotspots, because this is a
source with a modest extended power at $178$ MHz, $P_{178}=3.0\times 10^{25}$
W Hz-1 sr-1 (Hardcastle et al., 1998), as well as a low $5$ GHz luminosity
from both hotspots ($L_{5GHz}=3.4\times 10^{22}$ W Hz-1 sr-1 for the northern
hotspot and $L_{5GHz}=7.9\times 10^{22}$ W Hz-1 sr-1 for the southern hotspot
(Mack et al., 2009)). In this scenario, the X-ray emission from the weaker
northern hotspot is expected to be higher than predicted by the X-ray to radio
ratio from the brighter southern hotspot. This model makes no prediction
regarding offsets between emission in different bands.
In the second case, we expect that due to beaming, the presumed approaching-
jet side, southern hotspot will have more pronounced X-ray emission. In the
context of a relativistic decelerating hot spot flow (Georganopoulos &
Kazanas, 2003, 2004), we also expect to observe offsets, with the X-rays
peaking upstream of the radio. If, in addition, distributed particle
acceleration takes place (as suggested by Prieto et al. 2002), the optical and
radio emission will peak at the same location (see Figure 3 of Georganopoulos
& Kazanas 2004).
3C 445 has been the subject of several X-ray observations. GINGA observations
were reported by Pounds (1990), while ASCA observations were reported by
Sambruna et al. (1998). Both observations reported an absorbed Seyfert nucleus
with a hard X-ray continuum. XMM-Newton observations of 3C 445 were previously
published by Sambruna et al. (2007) and Grandi et al. (2007), but in those
data a partial window configuration was chosen for the MOS and pn that caused
both the northern and southern hotspots to be at locations where the CCDs were
not read out. This work thus represents the first study of extended X-ray
emission from this object, and we report the discovery of X-ray emission from
the southern hotspot, as well as a meaningful X-ray flux upper limit from the
northern hotspot. In §2 we discuss the Chandra observations, as well as the
data reduction and analysis procedures. In §3 we present and contrast the
morphology seen in each band for the southern hotspot. In §4 we discuss the
X-ray emission mechanism of the southern hotspot, present our conclusions and
suggest directions for future work.
## 2\. Observations and Data Reduction
3C 445 was observed with the Advanced CCD Imaging Spectrometer (ACIS) on the
Chandra X-ray Observatory on 18 October 2007 for 50ks. The ACIS-S
configuration was used due to its linear alignment which allowed all emission
regions (in particular both hotspots) to be included in the observation. The
northern and southern hot spots are at projected distances of $\sim$ 300 kpc
(more than 5 arcminutes in angular distance) from the core of the galaxy.
However, because of the greater surface brightness of the southern hotspot
(seen in the radio and near-IR) we decided to center it in the observations to
maximize our sensitivity and angular resolution in this region.
The Chandra observations were reduced in the Chandra Interactive Analysis of
Observations (CIAO) software package. Standard recipes (i.e., the science
threads) were followed for imaging spectroscopy of extended sources as well as
data preparation and filtering. No significant flare events were seen during
the observation, so all data could be used in the analysis. We extracted
spectra for the southern hotspot and nucleus, and created an exposure map to
allow a search for emission from other source regions. All X-ray spectra were
fitted in XSPEC. This process is discussed in §3.
Deep optical and near-infrared observations of the northern and southern hot
spots were obtained by Prieto et al. (2002) with the Very Large Telescope
(VLT) of the European Southern Observatory (ESO) and the Infrared Spectrometer
and Array Camera (ISAAC) in the KS (2.2 $\mu$m), H (1.7 $\mu$m), JS (1.2
$\mu$m), and I (0.9 $\mu$m) bands; deeper images in these same bands plus the
R (0.7 $\mu$m) and B (0.45 $\mu$m) bands were obtained by Mack et al. (2009).
We refer the reader to those papers for a discussion of their data reduction
procedures. We obtained near-IR and optical fluxes from their paper, as well
as from Mack et al. (2009). We obtained their VLT $J_{S}$-band image for use
in this paper.
We extracted radio data for 3C 445 from the NRAO data archives. 3C 445 was
observed with the NRAO Very Large Array on 09 September 2002, at both 8.5 and
5.0 GHz. The VLA was in the B configuration, yielding a clean beam (resolution
element) of 0.82 $\times$ 0.72 arcseconds in PA 77.44∘. Those data were
originally analyzed by Brunetti et al. (2003); we reanalyze them in this
paper. Data reduction was done in AIPS, using standard procedures. We also
obtained radio fluxes in other bands from Brunetti et al. (2003).
In registering the three datasets to a common frame of reference, we assumed
the VLA map to be the fiducial, adhering to the usual IAU standard. The VLT
image was registered to this frame by comparing to Palomar Sky Survey and
USNO-A2.0 data. The radio galaxy itself could not be used for this procedure,
as it was out of the field of view of both the VLT and VLA images. It was
therefore necessary to use stars in the field to do the registration. We then
checked the alignment of the radio and optical images by overplotting the two,
in the process reproducing the Prieto et al. (2002) overlay. Following this,
the 1$\sigma$ error in the positions from the VLT image are $\pm
0.2^{\prime\prime}$ in RA and Dec (see e.g., Deutsch 1999) relative to either
the radio or X-ray images, while those in the X-ray image are $\pm
0.4^{\prime\prime}$ 111see the Chandra Science Thread on Astrometry,
http://cxc.harvard.edu/cal/ASPECT/celmon, relative to either the radio or
optical.
## 3\. Results
Both the northern and southern hotspots of 3C 445 are seen in multiple
optical/near-IR bands (Prieto et al. 2002, Mack et al. 2009). However, X-ray
emission was seen from the southern hotspot only. In Table 1, we list all
fluxes for both hotspots, using all available data. In Figure 1, we show the
X-ray, radio and near-IR images of the southern radio hotspot. The radio and
near-IR images are shown both at the pixel scale of the Chandra data (i.e.,
$0.492^{\prime\prime}$/pix, middle and bottom left panels) as well as at the
finer pixel scale of the VLA image (i.e., $0.220^{\prime\prime}$/pix, middle
and bottom right panels). Our Chandra observations detect about 200 counts
from the southern hotspot. The X-ray emission extends along a
$6^{\prime\prime}$ region, extending very nearly east-west and peaking near
the middle or perhaps slightly to the west of its center point.
Table 1Hotspot fluxes
Feature | Telescope | Flux ($\mu$Jy) | Frequency (Hz) | Source
---|---|---|---|---
3C 445 N | VLA | 4000000 | 74$\times 10^{8}$ | 4
3C 445 N | VLA | 1400000 | 330$\times 10^{8}$ | 4
3C 445 N | VLA | 160000 | 1.4$\times 10^{9}$ | 2
3C 445 N | VLA | 58000 | 4.8$\times 10^{9}$ | 2
3C 445 N | VLA | 35000 | 8.4$\times 10^{9}$ | 2
3C 445 N | VLT | $7.2\pm 2.2$ | 1.38$\times 10^{14}$ | 2
3C 445 N | VLT | $<2.9$ | 1.81$\times 10^{14}$ | 2
3C 445 N | VLT | $2.4\pm 0.7$ | 2.47$\times 10^{14}$ | 2
3C 445 N | Chandra | $<2.60\times 10^{-4}$ | 1.21$\times 10^{18}$ | 3
3C 445 S | VLA | 7900000 | 74$\times 10^{8}$ | 4
3C 445 S | VLA | 2000000 | 330$\times 10^{8}$ | 4
3C 445 S | VLA | 520000 | 1.4$\times 10^{9}$ | 1
3C 445 S | VLA | 135000 | 4.8$\times 10^{9}$ | 1
3C 445 S | VLA | 81000 | 8.4$\times 10^{9}$ | 1
3C 445 S | VLT | $16.5\pm 1.5$ | 1.38$\times 10^{14}$ | 2
3C 445 S | VLT | $15.2\pm 3.0$ | 1.81$\times 10^{14}$ | 2
3C 445 S | VLT | $13.6\pm 1.4$ | 2.47$\times 10^{14}$ | 2
3C 445 S | VLT | $5.65\pm 0.57$ | 3.33$\times 10^{14}$ | 2
3C 445 S | VLT | $4.56\pm 0.90$ | 4.29$\times 10^{14}$ | 2
3C 445 S | VLT | $2.95\pm 0.60$ | 6.82$\times 10^{14}$ | 2
3C 445 S | VLT | $0.95\pm 0.19$ | 8.33$\times 10^{14}$ | 2
3C 445 S | Chandra | $9.38\times 10^{-4}$ | 1.21$\times 10^{18}$ | 3
References. — (1) Brunetti et al. (2003); (2) Mack et al. (2009); (3) This
paper (4) Kassim et al. (2007) and Kassim, private communication
There are clear differences between the morphologies seen in the near-IR,
X-ray and radio (Figure 1). In contrast to the X-ray emission, in the near-IR
and radio we see a concave arc, which appears to contain the X-ray emission in
its hollow part. This can be seen in Figure 1’s left-hand panels, which show
the data at a common pixel scale. There also appear to be offsets between the
position of the X-ray emission component and those seen in the near-IR and
radio. Looking at higher resolution (middle right and bottom right panels),
the near-IR emission breaks up into three discrete regions, which we term NIR
1-3 in east-west order, shown also in Mack et al. (2009). The brightest near-
IR emission is located within region NIR 1. The radio emission peak is also
near this position, and the higher-resolution VLA data of Mack et al. (2009)
show that the radio emission shows a broad plateau that includes the entire
maximum region of NIR 1.
Figure 1.— The southern hotspot of 3C 445, as seen in the X-rays (top left
panel), radio (middle panels), and near IR (bottom panels). The images at left
were resampled to 0.492 ′′/pix, while the middle right and bottom right panels
are at 0.220′′/pix. The contours were taken from the Chandra image, smoothed
with a 1-pixel (FWHM) Gaussian. Contours are plotted at 2, 4, 6… counts/pixel.
The top right panel shows the centroid of the emission components in each
band, plotted with error bars relative to the radio frame. See §3 for
discussion.
We have attempted to quantify the displacements between components in
different bands by fitting elliptical Gaussians to each major component within
AIPS, using the task JMFIT. In order to utilize the full resolution of our
data, we did this measurement at the native resolution in each band, using
small boxes to zero in on the visible maxima. These positions are reported in
Table 2, and are also compared in the top right-hand panel of Figure 1. In the
case of the near-IR emission, we report three centroids, one for each of the
three regions seen in that image (Figure 1); these have been labeled 1-3, in
order from east to west. We report in Table 2 with parentheses the internal
errors from JMFIT (estimated at 0.2 pixels where smaller values were
reported). However, for cross-comparison between bands, we must emphasize that
the errors are dominated by the uncertainty in registration between bands,
i.e., $\sim 0.2-0.4^{\prime\prime}$, as detailed in §2. Note that these are
errors in cross-comparison – i.e., they do not constitute errors on each
individual position but rather get added only once to the internal errors
reported in Table 2. As can be seen, the displacement between the X-ray and
radio peak is significant at $>3\sigma$, while the displacements between the
X-ray peak and those of NIR 1 and NIR 2 are at the 2.4-3 $\sigma$ level.
Table 2Positions of Emission Regions in Southern Hotspot Band | RA (J2000) | Dec | Delta(radio)
---|---|---|---
X-ray | 22 23 52.67 (0.01) | -02 10 43.29 (0.05) | (-1.94,+0.25)
NIR 1 | 22 23 52.77 (0.01) | -02 10 43.67 (0.12) | (-0.45,+0.13)
NIR 2 | 22 23 52.61 (0.01) | -02 10 44.37 (0.12) | (-2.85,-0.83)
NIR 3 | 22 23 52.48 (0.01) | -02 10 42.97 (0.22) | (-4.80,+0.57)
Radio | 22 23 52.80 (0.01) | -02 10 43.54 (0.02) | (0,0)
Our Chandra data include the position of the northern hotspot, which was
detected for the first time in the near-IR by Mack et al. (2009), who describe
its radio and optical morphology. We do not detect it in the Chandra image,
and the flux quoted in Table 1 reflects a $3\sigma$ upper limit. Note that due
to the northern hotspot’s off-center location, the sensitivity of Chandra was
reduced by about 60% at this position.
We extracted an X-ray spectrum for the southern hotspot of 3C 445, using
_specextract_ in CIAO. In order to create the spectra of the AGN, we defined
two regions in ds9: the hot spot and a background region that was free from
other sources. This allowed us to create a series of files, including source
and background PI spectra, weighted ARF, and RMF files, FEF weight files and a
grouped spectrum. The energy range was left unrestricted for both source
spectra. Following this step, XSPEC was used to fit both the spectra of the
hot spot and the AGN. Here we discuss the spectral fits and broadband spectrum
of the southern hotspot only, as the AGN was off-center for these observations
and our results for it are fully consistent with those of Sambruna et al.
(2007) and Grandi et al. (2007).
The southern hot spot’s X-ray spectrum was easily modeled by a basic power law
with fixed galactic absorption N${}_{H}^{Gal}=5.33\times 10^{20}~{}{\rm
cm^{-2}}$ , a photon index $\Gamma=1.95^{+0.38}_{-0.34}$ and a $\chi^{2}$
value of 0.85, and is shown in Figure 2.
Figure 2.— The X-ray spectrum of the southern hot spot with the best fit power
law overlayed. Below it are the residuals.
## 4\. Discussion
One of the key results of our Chandra observations is that the X-ray emission
of the southern hotspot has a very different morphology than that seen in the
radio and optical and also shows a likely displacement. This displacement
rules out all forms of one-zone models. Before discussing the possible
interpretations for the X-ray emission, we turn to the radio and optical
emission that are approximately cospatial. The radio-optical SED of the
southern and northern hotspots are plotted in Figure 3, along with the X-ray
points. The southern hotspot is brighter in the radio and optical by a factor
of $\approx 3$. If we attribute the brightness difference to beaming, we are
forced to conclude that the beaming of the radio-optical emitting plasma is
mild. Below, we discuss the multiwavelength emissions of both hotspots and
model possible X-ray emission mechanisms.
### 4.1. A high value of $\gamma_{min}$ in the radio-optical hotspot?
The southern hotspot’s radio–optical SED can be modeled as synchrotron
emission from a population of relativistic electrons in energy equipartition
with the hotspot magnetic field. Assuming that beaming is not important for
the radio–optical emission, the equipartition magnetic field is
$\displaystyle
B_{eq}=\left[{96\pi^{2}m_{e}cL_{r}\nu_{r}^{(s-1)/2}(\gamma_{min}^{2-s}-\gamma_{max}^{2-s})\over
c_{1}^{(s-3)/2}\sigma_{\tau}V(s-2)}\right]^{2/(s+5)},$ (1)
where $L_{r}$ is the radio luminosity at frequency $\nu_{r}$, $m_{e}$ is the
electron mass, $c$ is the speed of light, $\sigma_{\tau}$ is the Thomson cross
section, and $V$ is the volume of the emitting region. The injected electron
distribution is a power law of index $s=2\alpha_{r}+1=2.6$ from Lorentz factor
$\gamma_{min}$ to $\gamma_{max}$, with $\alpha_{ro}=0.9=\alpha_{r}$ being the
radio-optical spectral index (Mack et al., 2009). To derive equation (1), we
assumed that an electron of Lorentz factor $\gamma$ in a magnetic field $B$
radiates most of its energy at the characteristic frequency
$\nu=c_{1}B\gamma^{2}$, with $c_{1}=e/(2\pi m_{e}c)$. In equation (1),
$\gamma_{max}$ can in many cases be determined observationally from the
maximum observed synchrotron frequency. However, $\gamma_{min}$ is customarily
set to a value chosen by hand. As we discuss now, there is a way to determine
observationally, or at least constrain the value of $\gamma_{min}$, and
through this get a more appropriate value for $B_{eq}$. This in turn affects
significantly the level of both the SSC and the EC/CMB emission.
Because in this case $s>2$ and the synchrotron emission extends for at least
six decades in frequency, $\gamma_{max}/\gamma_{min}\gg 1$, to a very good
approximation
$B_{eq}\propto\gamma_{min}^{-2(s-2)/(s+5)}=\gamma_{min}^{-1.2/7.6}$: an
increase in $\gamma_{min}$ results in a mild decrease of $B_{eq}$. There are
two observational constraints on $\gamma_{min}$. An upper limit on
$\gamma_{min}$ is derived from the fact that it has to be low enough to
produce the lowest observed radio frequency from the hotspot
$\nu_{r,min}>c_{1}B_{eq}\gamma_{min}^{2}$. A lower limit comes from the fact
that there is no sign of radiative cooling in the radio-optical SED, because
the optical flux level is found practically on the extrapolation of the radio
spectrum. This sets an upper limit on the magnetic field in the radio-optical
hotspot (regardless of equipartition arguments), which in turn sets a lower
limit on $\gamma_{min}$.
To demonstrate these considerations, in Figure 3 we plot with a thin solid
line the synchrotron emission in the case of $\gamma_{min}=1$. This
corresponds to an equipartition magnetic field, $B_{eq}=70.4\;\mu$G. As can be
seen, while the synchrotron SED clearly extends below the lowest radio
frequency safely associated with the hotspot [this is the 4.8 GHz point,
because the 1.4 GHz point may be contaminated with non-hotspot emission, (see
Mack et al. (2009); Prieto et al. (2002)) as is also true for the lower
frequency points (Kassim et al. 2007).], the synchrotron spectrum breaks at
$\sim 10^{12-13}$ Hz and by doing so, underproduces the optical emission of
the hotspot. This is because the high value of $B_{eq}$ causes a break in the
electron energy distribution due to radiative cooling (a cooling break is
expected at $\gamma_{b}=3m_{e}c^{2}/[4\sigma_{\tau}(B^{2}/8\pi+U_{CMB})R]$,
where $U_{CMB}$ is the energy density of the cosmic microwave background and
$R$ is the size of the hotspot that determines the electron escape time
$R/c$). To fit the observed SED, we need significantly higher values of
$\gamma_{min}$. A value of $\gamma_{min}\approx 1840$ similar to the proton to
electron mass ratio $m_{p}/m_{e}$ is required to ensure that there is no
cooling break signature at frequencies lower than optical. The value of the
corresponding equipartition magnetic field is $B_{eq}=21.5\;\mu$G. We plot the
resulting synchrotron SED in Figure 3 with a thick solid line. Note that at
low frequencies the model with $\gamma_{min}\approx 1840$ (thick solid line)
exhibits a break due to the high value of $\gamma_{min}$ (the slope below the
break is due to the $\nu^{1/3}$ synchrotron emissivity of electrons with
Lorentz factor $\gamma_{min}$). Note also that this model manages to reproduce
the optical emission of the hotspot.
Figure 3.— The SED of the southern hotspot of 3C 445 is shown with diamonds
for the radio and optical and bow-tie for the X-rays. The SED of the northern
hotspot is also plotted with asterisks, including the upper limit for the
X-ray flux. The data used are taken from Table 1. Due to angular resolution
constraints, the three lowest radio frequencies may include lobe emission and
should be considered upper limits for the hotspot fluxes. The thin (thick)
lines represent emission in equipartition conditions, assuming
$\gamma_{min}=1$ ($\gamma_{min}=1840$). Solid lines represent the synchrotron,
short dash lines the SSC and long dash lines the EC/CBM emission.
There is practically little freedom for $\gamma_{min}$ around $m_{p}/m_{e}$ if
we want to model the radio to optical SED with synchrotron in equipartition.
Let us mention that observationally driven arguments for similarly high values
of $\gamma_{min}$ in hotspots of other radio galaxies have been presented by
other astronomers (e.g. Blundell et al. 2006, Stawarz et al. 2007, Godfrey et
al. 2009). However, some jet sources require much lower values, e.g., PKS
0637–752 (Mehta et al. 2009). Values of $\gamma_{min}\approx m_{p}/m_{e}$ are
particularly interesting, because this is the minimum energy that electrons
crossing a shock must have to be picked up efficiently by Fermi acceleration
(e.g. Spitkovsky 2008). The level of both the EC and SSC emission depend on
the value of $B_{eq}$, which in turns depends on $\gamma_{min}$: because
$L_{SSC,EC}/L_{S}\propto U_{B}^{-1}\propto B^{-2}$, and
$B_{eq}\propto\gamma_{min}^{-2(s-2)/(s+5)}$, for $B=B_{eq}$,
$L_{SSC,EC}\propto\gamma_{min}^{4(s-2)/(s+5)}=\gamma_{min}^{2.4/7.6}$.
Therefore, an increase from $\gamma_{min}=1$ to $\gamma_{min}=1840$, should
increase the EC and SSC level by a factor of $\approx 10.7$, as seen in Figure
3. Even for $\gamma_{min}=1840$, the X-ray emission of the radio-optical
hotspot is much weaker than the upstream detected X-ray emission.
If analyses like the above point toward a high value of $\gamma_{min}$, then
future low frequency-high angular resolution observations (e.g., with the LWA)
should detect the SED break at low frequencies. Existing low fequency radio
observations suggest such a break for the eastern hotspot of Cygnus A (Lazio
et al., 2006). For 3C 445, VLA observations at 74 MHz and 330 MHz (Kassim et
al. 2007), provide us with upper limits for the emission of the hotspots (due
to lobe contamination). The low frequency data plotted in figure 3 may be
contaminated by lobe emissions and hence are not sufficient to constrain the
spectral shape below 4.8 GHz. If a low frequency break is found by future low
frequency-high resolution observations, they will strengthen the above
picture. If on the other hand the low frequency radio spectrum exhibits no
such break, we will have to search for an alternative reason for the lack of
cooling break (as we discussed above, a low value for $\gamma_{min}$,
manifested through a lack of low frequency break, requires a higher value for
$B_{eq}$, which in turns produces a cooling break below the optical, as can be
seen from the thin solid line in Figure 3). Distributed reacceleration is a
very plausible candidate and it has been claimed to explain the optical
emission of 3C 445 (e.g. Prieto et al. 2002).
### 4.2. What is the X-ray emission mechanism?
Any interpretation of the X-rays must take into account ($i$) the level and
spectrum of the X-ray emission, ($ii$) the apparent upstream ‘nesting’ of the
X-ray emission into the hollow part of the east-west arc of radio–optical
emission, and ($iii$) the mostly westward displacement of the X-ray emission
relative to the peak of the radio - NIR 1 hotspot. Also, because of the
moderate hotspot to counterhotspot flux ratio in the radio and optical, it
should not invoke significant beaming for the radio-optical emitting plasma.
These considerations automatically exclude the possibilities of one zone
EC/CMB (Tavecchio et al., 2005) and SSC in equipartition, because in these
models both the SSC and EC/CMB emission is cospatial with the radio–optical
hotspot emission and (see Fig. 3) is much lower than the X-ray detected flux).
A displacement between the X-ray and radio emission is predicted in the case
of UC emission from a decelerating flow, in which freshly accelerated
relativistic electrons from the fast base of the flow upscatter to high
energies the radio photons produced in the downstream slow part of the flow by
electrons that have cooled radiatively (Georganopoulos & Kazanas, 2003). This
version of the UC emission from a decelerating flow is not favored, however,
as it predicts that the optical and the X-rays are co-spatial and that the
radio emission is shifted downstream. This is because optical emission can
only be produced by the fast, upstream part of the flow, where the freshly
accelerated electrons are found, the same place from which the UC X-rays are
produced. Also, the model predicts the optical to be more beamed than the
radio, something not supported by the very similar hotspot to counter-hotspot
flux ratio in radio and optical. Finally, the model uses as seed photons the
synchrotron photons produced in the source, making the implicit assumption
that these photons dominate the local photon energy density. This is not the
case, as can be seen in Figure 4.
The fact that the X-ray emission is displaced from the radio-optical hotspot
requires the X-ray emitting electrons to also be displaced from the radio-
optical emitting electrons. If the X-ray emission is of IC nature, these
electrons will experience the seed photon field of Figure 4, provided they are
located at a distance from the radio-optical hostspot not much greater than
the radio-optical hotspot size. This immediately excludes the IR-optical
photons as seed photons for the X-ray emission, because this would require the
presence of a powerful component of IC emission due to CMB photons upscattered
in $\sim$ the optical band, co-located with the X-ray component. This is not
observed. We therefore reach the conclusion that if the X-ray emission is due
to a particle population located upstream of those in other bands, then the
seed photons that these electrons upscatter must be the CMB.
Figure 4.— The energy density at the location of the southern hotspot. The
straight line represents the photon energy density due to the radio optical-
emission of the hotspot, while the blackbody is the CMB.
A model that produces co-spatial radio-optical synchrotron emission and EC/CMB
X-ray emission shifted upstream, has been proposed by Georganopoulos & Kazanas
(2004) to address such displacements observed in the large scale jets of
quasars. The model assumes that distributed particle acceleration offsets
radiative losses. In this model, a relativistic decelerating flow results in
an increase of the magnetic field and electron density at the slow downstream
part of the flow, increasing the synchrotron emissivity. At the same time, the
faster upstream part of the flow experiences a higher CMB comoving energy
density $U_{CMB}\propto\Gamma^{2}$, where $\Gamma$ is the bulk Lorentz factor
of the flow (see Figure 3 of Georganopoulos & Kazanas 2004). In this scenario
the only displacement observed is along the flow axis. In our case, however,
besides the general upstream shift of the X-rays relative to the radio–NIR
emission, we note that the peak of the radio–NIR 1 emission is shifted by
$\sim 2"$ to the east relative to the center of the X-rays ($1"$ corresponds
to $1.07$ Kpc). Therefore, for this model to still be viable, the flow needs
to bend to the east after producing the X-ray emission.
To demonstrate how this could reproduce the observed upstream X-ray emission,
we plot in Figure 5 the emission that would result from plasma in
equipartition moving with a bulk Lorentz factor $\Gamma=4$ at an angle to the
line of sight $\theta=14^{\circ}$ and carrying the same power as that injected
in the radio-optical hotspot ($L_{kin}=1.3\times 10^{44}$ erg s-1). Although
such a small angle to the line of sight, bordering angles typical of blazars
is admittedly uncomfortable, it cannot be excluded for this broad line FR II
radio galaxy. To reproduce the shift of the radio-optical emission in NIR 1
relative to location of the X-ray emission, the flow must bend to the east,
forming an angle of $30^{\circ}$ to the line of sight. The projected physical
distance between the X-ray and radio-optical components is 4.6 kpc. Bends in
the flow and velocity gradients at the hotspots, such as the one suggested
here, are seen in numerical simulations of relativistic flows (e.g. Aloy et
al. 1999). Deceleration from $\Gamma=4$ to the subrelativistic speed of the
radio-optical hotspot could be achieved by a series of oblique shocks that
could also aid electron re-acceleration. This latter possibility could be
tested by future, high-resolution radio imaging of the hotspot. If indeed this
is the X-ray emission mechanism, we expect that due to relativistic dimming,
the X-ray emission from the counter hotspot would be undetectable, even for
very deep Chandra exposures. On the assumption of hotspot-counter hotspot
symmetry, X-ray detection of the counter hotspot would exclude this last
alternative for an inverse Compton interpretation of the X-rays.
Figure 5.— X-rays due to EC/CMB (long dash line). The emission is assumed to
come from plasma moving relativistically (bulk Lorentz factor $\Gamma=4$) at
an angle $\theta=14^{\circ}$ to the line of sight. To reproduce the observed
displacements we assume that the flow decelerates and bends to
$\sim\theta=30^{\circ}$ and terminates $\sim 4.6$ Kpc downstream, producing
the radio optical emission (solid line). The radio-optical emission coming
from the X-ray spot is much weaker (short dash line). The data points are the
same plotted in Figure 3.
An alternative model, which avoids the uncomfortably small jet angle to the
line of sight and the relatively high Lorentz factor of the X-ray emitting
plasma needed for the above interpretation, together with the arguments
against strong beaming in other sources (see §1), is to generate the X-ray
emission via synchrotron radiation from a high energy population ($\gamma\sim
10^{8}$) of electrons accelerated upstream of the radio-optical hotspot. A
possible way to produce this electron population upstream the radio-optical
emitting region is through acceleration in the reverse shock of a reverse-
forward shock structure (in this picture the radio-optical comes from electron
acceleration in the forward shock; Kataoka et al. (2008) proposed it,
motivated by the upstream, relative to the radio, X-ray emission seen in both
hotspots and jets of 3C 353). This scheme, however, as Kataoka et al. (2008)
note, does not address the reason the two shocks accelerate electrons at
different energies, with the reverse shock consistently reaching higher
electron energies. If the X-ray emission of the southern hotspot is
synchrotron, then the observed emission in the Chandra band must lie close to
the peak of its $\nu f_{\nu}$ emission. This is because the X-ray photon index
is $\sim 2$ and the luminosity of this component at optical energies must be
below the level of the optical emission detected downstream. The emission
could in principle extend to energies higher than the Chandra band but this
would require the presence of unrealistically energetic electrons.) Because
cooling of the X-ray emitting electrons is severe, even if we only consider
the CMB photon energy density, these electrons are suffering strong cooling,
which must be balanced by continuous reacceleration.
Finally, an account of most of the phenomenology may rest with the possibility
that the entire radio-NIR-X-ray emission is synchrotron, and that the jet beam
moves laterally with time to the west, implying that the maximum X-ray
emission is the most recent and naturally further displaced from the AGN core.
For this same reason (age) there is less IR and almost no radio emission in
along this direction (the corresponding electron radiative times are longer).
The fact that the radio emitting electrons have the longest radiative lifetime
would then explain the displacement of the maximum emission at this frequency
(and of the IR) to the East and the absence of X-rays in the same region (the
X-ray emitting electrons have all cooled to lower energies). There is some
indication of such a motion from the fact that the AGN core and the two lobes
do not all line on a straight line.
To conclude, both the EC/CMB from a decelerating flow and synchrotron
interpretation of the X-rays face important problems, although the EC/CMB
model is more constrained and, therefore, easier to falsify, while the
synchrotron interpretation can reproduce any observed emission by introducing
additional electron populations as needed. A purely spectral discrimination
between the synchrotron and inverse-Compton models is not possible, as both
can be made compatible with various X-ray slopes as well as the forms for the
’valley’ in between the X-ray and radio-IR components (see e.g., Uchiyama et
al. 2006, 2007; Jester et al. 2007; and Hardcastle et al. 2004). The
synchrotron interpretation, however, does not require beaming and is the only
alternative among those examined that could produce detectable X-ray emission
from the counter-hotspot. Therefore, detection of bright X-ray emission from
the counter hotspot in future X-ray observations would rule out any reasonable
IC models for the hotspots of 3C 445. An additional test of the nature of the
X-ray emission could come from future X-ray polarimeters like GEMS: while the
synchrotron emission is expected to be highly polarized, the EC/CMB should
produce negligible polarization (Begelman & Sikora 1987, McNamara et al. 2009,
Uchiyama & Coppi in prep.). An approach that can yield results with our
current observational capabilities is based on HST UV polarimetry: if the far
UV emission is shown to be the low energy tail of the X-ray component as is
the case with 3C 273 (e.g. Jester et al. 2007), then UV HST imaging
polarimetry of the hotspots will distinguish between the EC/CMB and
synchrotron mechanisms (e.g., McNamara et al. 2009, Uchiyama & Coppi, in
prep.).
We thank an anonymous referee for comments that significantly strengthened
this paper. This work was supported at FIT and UMBC by the Chandra grant
G07-8113A and the NASA LTSA grant NNX07AM17G. This project was also partially
funded by a partnership between the National Science Foundation (NSF
AST-0552798), Research Experiences for Undergraduates (REU), and the
Department of Defense (DoD) ASSURE (Awards to Stimulate and Support
Undergraduate Research Experiences) programs.
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|
arxiv-papers
| 2009-10-16T02:13:45 |
2024-09-04T02:49:05.847596
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Eric S. Perlman (FIT), Markos Georganopoulos (UMBC and Goddard), Emily\n M. May (U. Wyoming), Demosthenes Kazanas (Goddard)",
"submitter": "Eric S. Perlman",
"url": "https://arxiv.org/abs/0910.3021"
}
|
0910.3103
|
# Submanifolds with harmonic mean curvature vector field in contact
$3$-manifolds 111Colloquium Mathematicum 100 (2004), no. 2, 163–179. Minor
misprints are corrected.
Jun-ichi Inoguchi
###### Abstract
Biharmonic or polyharmonic curves and surfaces in $3$-dimensional contact
manifolds are investigated.
AMS Mathematics Subject Classification: 2000 53C42 53D10
Keywords and Phrases: Biharmonicity, polyharmonicity, Sasaki manifolds
## Introduction
This paper concerns curves and surfaces in 3-dimensional contact manifolds,
whose mean curvature vector field is in the kernel of certain elliptic
differential operators.
First we study submanifolds whose mean curvature vector field is in the kernel
of Laplacian (submanifolds with harmonic mean curvature vector fields).
The study of such submanifolds is inspired by a conjecture of Bang-yen Chen
[14]:
Harmonicity of the mean curvature vector field implies harmonicity of the
immersion ?
The harmonicity equation $\Delta\mathbb{H}=0$ for the mean curvature vector
field $\mathbb{H}$ of an immersed submanifold
$\mathbf{x}:M^{m}\to\mathbf{E}^{n}$ in Euclidean $n$-space is equivalent to
the biharmonicity of the immersion: $\Delta\Delta\mathbf{x}=0$, since
$\Delta\mathbf{x}=-m\mathbb{H}$.
A submanifold $\mathbf{x}:M\to\mathbf{E}^{n}$ is said to be a biharmonic
submanifold if $\Delta\mathbb{H}=0$.
In 1985, Chen proved the nonexistence of proper biharmonic surfaces in
Euclidean 3-space. The conjecture by Chen is still open.
Some partial and positive answers have been obtained by several authors
[16]-[19], [25]-[27].
The biharmonicity equation is regarded as a special case of the following
condition:
$\Delta\mathbb{H}=\lambda\>\mathbb{H},\ \lambda\in\mathbf{R}.$
Namely the mean curvature vector field is an eigenfunction of the Laplacian.
The study of Euclidean submanifolds with $\Delta\mathbb{H}=\lambda\mathbb{H}$
was initiated by Chen in 1988 (See [14]).
It is known that submanifolds in $\mathbf{E}^{n}$ satisfying
$\Delta\mathbb{H}=\lambda\mathbb{H}$ are either biharmonic ($\lambda=0$), of
$1$-type or null $2$-type. In particular all surfaces in $\mathbf{E}^{3}$ with
$\Delta\mathbb{H}=\lambda\mathbb{H}$ are of constant mean curvature. Moreover
a surface in $\mathbf{E}^{3}$ satisfies $\Delta\mathbb{H}=\lambda\mathbb{H}$
if and only if it is minimal, an open portion of a totally umbilical sphere or
an open portion of a circular cylinder.
F. Defever [17] showed that hypersurfaces satisfying
$\Delta\mathbb{H}=\lambda\mathbb{H}$ are of constant mean curvature. Note that
Chen [12], [13] studied spacelike submanifolds with
$\Delta\mathbb{H}=\lambda\mathbb{H}$ in Minkowski space, hyperbolic space or
de Sitter space. M. Barros and O. J. Garay showed that Hopf cylinders in
$S^{3}$ with $\Delta\mathbb{H}=\lambda\mathbb{H}$ are Hopf cylinders over
circles in the $2$-sphere $S^{2}$. A. Ferrández, P. Lucas and M. A. Meroño
[24] studied such submanifolds in anti de Sitter $3$-space $H^{3}_{1}$.
In non-constant curvature ambient spaces, results on biharmonic submanifolds
are very few.
Recently, T. Sasahara [37]–[38] studied Legendre surfaces in the Sasakian
space form $\mathbf{R}^{5}(-3)$ satisfying
$\Delta\mathbb{H}=\lambda\mathbb{H}$. Moreover Sasahara introduced the notion
of “$\varphi$-position vector field” and “$\varphi$-mean curvature vector
field” for submanifolds in Sasakian space form $\mathbf{R}^{2n+1}(-3)$.
Sasahara investigated submanifolds in $\mathbf{R}^{2n+1}(-3)$ whose
$\varphi$-mean curvature vector field $\mathbb{H}_{\varphi}$ satisfies
$\Delta\mathbb{H}_{\varphi}=\lambda\mathbb{H}_{\varphi}$. In particular he
classified curves and surfaces in $\mathbf{R}^{3}(-3)$ with
$\Delta\mathbb{H}_{\varphi}=\lambda\mathbb{H}_{\varphi}$. Since both
$\mathbf{R}^{2n+1}(-3)$ and $S^{2n+1}$ are typical examples of Sasakian space
form, it seems to be interesting to study biharmonic submanifolds in general
Sasakian space forms.
Based on these observations, in the first part of this paper, we shall study
harmonicity of mean curvature vector fields of curves and surfaces in
3-dimensional Sasakian space forms. Several results for 3-dimensional sphere
$S^{3}$ due to Spanish research group (Barros, Garay Ferrández, Lucas and
Meroño) will be generalised to $3$-dimensional Sasakian space forms.
Next, in the second part, we shall study another “biharmonicity” suggested by
J. Eells and J. H. Sampson [23]. A smooth map $\phi:M\to N$ between Riemannian
manifolds is said to be a biharmonic map (or polyharmonic map of order $2$) if
its bitension field $\mathscr{T}_{2}(\phi)$ vanishes. In [9], “biharmonic”
curves and surfaces in $S^{3}$ are classified. We shall classify Legendre
curves and Hopf cylinders in $3$-dimensional Sasakian space forms, which are
biharmonic in this sense.
In particular we shall show the existence of non-minimal biharmonic Hopf
cylinders in Sasakian space forms of holomorphic sectional curvature greater
than $1$ (Berger spheres).
The author would like to thank Dr. Cezar Dumitru Oniciuc (University “AL. I.
Cuza ”) and Dr. Tooru Sasahara (Hokkaido University) for their useful
comments.
## Part I
## 1 Preliminaries
### 1.1 Contact manifolds
We begin by recalling fundamental ingredients of contact Riemannian geometry
from [7].
Let $M$ be a $(2n+1)$-manifold. A one form $\eta$ is called a contact form on
$M$ if $(d\eta)^{n}\wedge\eta\not=0$. A $(2n+1)$-manifold $M$ together with a
contact form is called a contact manifold. The contact distribution $D$ of
$(M,\eta)$ is defined by
$D=\left\\{X\in TM\ |\ \eta(X)=0\right\\}.$
On a contact manifold $(M,\eta)$, there exists a unique vector field $\xi$
such that
$\eta(\xi)=1,\ \ d\eta(\xi,\cdot)=0.$
This vector field $\xi$ is called the Reeb vector field or characteristic
vector field of $(M,\eta)$.
Moreover there exists an endomorphism field $\varphi$ and a Riemannian metric
$g$ on $M$ such that
(1) $\varphi^{2}=-I+\eta\otimes\xi,\ \eta(\xi)=1,$ (2) $g(\varphi X,\varphi
Y)=g(X,Y)-\eta(X)\eta(Y),\ \ g(\xi,\cdot)=\eta,$ (3) $d\eta(X,Y)=2g(X,\varphi
Y)$
for all vector fields $X,\ Y$ on $M$. On an almost contact manifold
$(M,\eta;\xi,\varphi)$, there exists a Riemannian metric $g$ satisfying (2).
Such a metric $g$ is called an compatible metric of $M$. A contact manifold
$(M,\eta)$ together with structure tensors $(\xi,\varphi,g)$ is called a
contact Riemannian manifold.
###### Proposition 1.1
Let $(M,\eta,\xi,\varphi,g)$ be a contact Riemannian manifold.
Then $M$ $\xi$ is a Killing vector field if and only if
(4) $\nabla_{X}\xi=-\varphi X,\ \ X\in\mathfrak{X}(M).$
Here $\nabla$ is the Levi-Civita connection of $(M,g)$.
###### Definition 1.1
A contact Riemannian manifold $(M,\eta,\xi,\varphi,g)$ is said to be a Sasaki
manifold if
(5) $(\nabla_{X}\varphi)Y=g(X,Y)\xi-\eta(Y)X,\ \ X,Y\in\mathfrak{X}(M).$
Note that on a Sasaki manifold, $\xi$ is a Killing vector field.
Let $(M,\eta;\xi,\varphi,g)$ be a contact Riemannian manifold. A tangent plane
at a point of $M$ is said to be a holomorphic plane if it is invariant under
$\varphi$. The sectional curvature of a holomorphic plane is called
holomorphic sectional curvature. If the sectional curvature function of $M$ is
constant on all holomorphic planes in $TM$, then $M$ is said to be of constant
holomorphic sectional curvature. Complete and connected Sasaki manifolds of
constant holomorphic sectional curvature are called Sasakian space forms. Let
us denote by $R$ the Riemannian curvature tensor field of the metric $g$ which
is defined by
$R(X,Y):=\nabla_{X}\nabla_{Y}-\nabla_{Y}\nabla_{X}-\nabla_{[X,Y]},\ \
X,Y\in\mathfrak{X}(M).$
When $(M,\eta;\xi,\varphi,g)$ is a Sasakian space form of constant holomorphic
sectional curvature $c$, then $R$ is described by the following formula:
$\displaystyle R(X,Y)Z$ $\displaystyle=$
$\displaystyle\frac{c+3}{4}\\{g(Y,Z)X-g(Z,X)Y\\}$
$\displaystyle+\frac{c-1}{4}\>\\{\eta(Z)\eta(X)Y-\eta(Y)\eta(Z)X$
$\displaystyle+g(Z,X)\eta(Y)\xi-g(Y,Z)\eta(X)\xi$ $\displaystyle-g(Y,\varphi
Z)\varphi X-g(Z,\varphi X)\varphi Y+2g(X,\varphi Y)\varphi Z\ \>\\}.$
Note that even if the holomorphic sectional curvature is negative, a Sasakian
space form is not negatively curved. In fact, the sectional curvature of plane
sections containing $\xi$ is $1$ on any Sasaki manifold.
It is known that every $3$-dimensional Sasakian space form is realised as a
Lie group together with a left invariant Sasaki structure. More precisely the
following is known (cf. [6]):
###### Proposition 1.2
Simply connected $3$-dimensional Sasakian space form of constant holomorphic
sectional curvature is isomorphic to
1. (1)
special unitary group $\mathrm{SU}(2);$
2. (2)
Heisenberg group $\mathbf{R}^{3}(-3);$
3. (3)
the universal covering group of the special linear group
$\mathrm{SL}_{2}\mathbf{R}$
together with canonical left invariant Sasaki structure. In particular simply
connected Sasakian space form of constant holomorphic sectional curvature $1$
is the $\mathrm{SU}(2)$ with biinvariant metric of constant curvature $1$
(hence isometric to the unit $3$-sphere $S^{3}$).
### 1.2 Boothby-Wang fibration
Let $(M^{2n+1},\eta;\xi,\varphi,g)$ be a contact Riemannian manifold. Then $M$
is said to be regular if $\xi$ generates a one-parameter group $K$ of
isometries on $M$, such that the action of $K$ on $M$ is simply transitive.
Note that if $M$ is regular, then both $\varphi$ and $\eta$ are automatically
$K$-invariant, i.e, $\pounds_{\xi}\varphi=0$ and $\pounds_{\xi}\eta=0$. The
Killing vector field $\xi$ induces a regular one-dimensional Riemannian
foliation on $M$. We denote by ${\overline{M}}:=M/K$ the orbit space (the
space of all leaves) of a regular contact Riemannian manifold $M$ under the
$K$-action.
Let $\bar{X}_{\bar{p}}$ be a tangent vector of the orbit space $\overline{M}$
at ${\bar{p}}=\pi(p)$. Then there exists a tangent vector ${\bar{X}}^{*}_{p}$
of $M$ at $p$ which is orthogonal to $\xi$ such that
$\pi_{*p}{\bar{X}}^{*}_{p}=\bar{X}_{\bar{p}}$. The tangent vector
${\bar{X}}^{*}_{p}$ is called the horizontal lift of $\bar{X}_{\bar{p}}$ to
$M$ at $p$. The horizontal lift operation
$*:{\bar{X}}_{\bar{p}}\mapsto{\bar{X}}^{*}_{p}$ is naturally extended to
vector fields.
The contact structure on $M$ induces an almost Hermitian structure on the
orbit space ${\overline{M}}$:
(6) $J{\bar{X}}=\pi_{*}(\varphi{\bar{X}}^{*}),\
\bar{X}\in\mathfrak{X}(\bar{M}).$
Let us denote by $\bar{\nabla}$ the Levi-Civita connection of $\bar{M}$. Then,
by using the fundamental equations for Riemannian submersions due to O’Neill
[33], we have the following results.
###### Proposition 1.3
([32]) Let $M$ be a regular contact Riemannian manifold. Then for any
$\bar{X},\bar{Y}\in\mathfrak{X}(\bar{M}):$
(7)
$\nabla_{{\bar{X}}^{*}}\bar{Y}^{*}=(\bar{\nabla}_{\bar{X}}{\bar{Y}})^{*}-g(\bar{X}^{*},\varphi\bar{Y}^{*})\xi.$
###### Proposition 1.4
([32]) Sasakian space forms are regular Sasaki manifolds. The orbit space of a
Sasakian space form of constant holomorphic sectional curvature $c$ is a
complex space form of constant holomorphic sectional curvature $c+3$.
W. M. Boothby and H. C. Wang [8] proved that if $M$ is a compact regular
contact manifold, then the natural projection $\pi:M\rightarrow{\bar{M}}$
defines a principal circle bundle over a symplectic manifold ${\bar{M}}$ and
the symplectic form $\Omega$ of ${\overline{M}}$ determines an integral
cocycle. Furthermore the contact form $\eta$ gives a connection form of this
circle bundle and satisfies $\pi^{*}\Omega=d\eta$. The fibering
$\pi:M\rightarrow{\bar{M}}$ is called the Boothby-Wang fibering of a regular
compact contact manifold $M$. Based on this result, we call the fibering
$\pi:M\to\bar{M}$ of a regular contact Riemannian manifold $M$, the “Boothby-
Wang fibering” of $M$ even if $M$ is noncompact.
The unit sphere $S^{2n+1}$ is a typical example of regular compact Sasaki
manifold. For $S^{2n+1}$, the Boothby-Wang fibering coincides with the Hopf
fibering $S^{2n+1}\rightarrow\mathbb{C}P^{n}$.
In $3$-dimensional case, the Boothby-Wang fibering of Sasakian space forms
have the following matrix group models [6]:
$\displaystyle\pi:\mathrm{SU}(2)\to S^{2}(c)$ $\displaystyle=$
$\displaystyle\mathrm{SU}(2)/\mathrm{U}(1),$
$\displaystyle\pi:\mathbf{R}^{3}(-3)\to\mathbf{C}$ $\displaystyle=$
$\displaystyle\mathbf{R}^{3}(-3)/\mathbf{R},$
$\displaystyle\pi:\mathrm{SL}_{2}\mathbf{R}\to H^{2}(c)$ $\displaystyle=$
$\displaystyle\mathrm{SL}_{2}\mathbf{R}/\mathrm{SO}(2).$
Here $S^{2}(c)$ and $H^{2}(c)$ are sphere and hyperbolic space of curvature
$c$, respectively.
### 1.3 Hopf cylinders
Now we shall restrict our attention to $3$-dimensional regular contact
Riemannian manifold $M$.
Let ${\bar{\gamma}}$ be a curve parameterized by arc length in
${\overline{M}}$ with curvature ${\bar{\kappa}}$. Taking the inverse image
$S_{\bar{\gamma}}:=\pi^{-1}\\{{\bar{\gamma}}\\}$ of ${\bar{\gamma}}$ in
$M^{3}$.
Here we compute the fundamental quantities of $S_{\bar{\gamma}}$.
Let us denote by $\bar{P}=({\bar{\mathbf{p}}}_{1},{\bar{\mathbf{p}}}_{2})$ the
Frenet frame field of $\bar{\gamma}$. By using the complex structure $J$ of
${\overline{M}}^{2}$, ${\bar{\mathbf{p}}}_{2}$ is given by
${\bar{\mathbf{p}}}_{2}=J{\bar{\mathbf{p}}}_{1}$
Then the Frenet-Serret formula of $\bar{\gamma}$ is given by
$\bar{\nabla}_{\bar{\gamma}^{\prime}}P=P\left(\begin{array}[]{cc}0&-\bar{\kappa}\\\
\bar{\kappa}&0\end{array}\right).$
Here the function $\bar{\kappa}$ is the (signed) curvature of $\bar{\gamma}$.
Let $\mathbf{t}=(\bar{\mathbf{p}}_{1})^{*}$ the horizontal lift of
$\bar{\mathbf{p}}_{1}$ with respect to the Boothby-Wang fibering. Then
$(\mathbf{t},\xi)$ gives an orthonormal frame field of $S$. We choose a unit
normal vector field $\mathbf{n}$ by $\mathbf{n}=(\bar{\mathbf{p}}_{2})^{*}$.
Since ${\bar{\mathbf{p}}}_{2}$ is defined by
${\bar{\mathbf{p}}}_{2}=J{\bar{\mathbf{p}}}_{1}$,
$\mathbf{n}=\varphi\>\mathbf{t}$. In fact,
$(\bar{\mathbf{p}}_{2})^{*}=(J\bar{\mathbf{p}}_{1})^{*}=\varphi(\bar{\mathbf{p}}_{1})^{*}=\varphi\>\mathbf{t}.$
Let us denote by $\nabla^{S}$ the Levi-Civita connection of $S$. The second
fundamental form $I\\!I$ derived from $\mathbf{n}$ is defined by the Gauß
formula:
(8) $\nabla_{X}Y=\nabla^{S}_{X}Y+I\\!I(X,Y)\mathbf{n},\ \
X,Y\in\mathfrak{X}(S).$
By using (7),
$\nabla_{\mathbf{t}}\>\mathbf{t}=(\bar{\nabla}_{\bar{\mathbf{p}}_{1}}\bar{\mathbf{p}}_{1})^{*}-g(\mathbf{t},\varphi\>\mathbf{t})\xi=(\bar{\kappa}\circ\pi)\mathbf{n}.$
Hence $\nabla^{S}_{\mathbf{t}}{\mathbf{t}}=0$. Since $\xi$ is Killing, we have
$\nabla^{S}_{\mathbf{t}}\xi=\nabla^{S}_{\xi}\xi=0$. Thus $S_{\bar{\gamma}}$ is
flat. The second fundamental form $I\\!I$ is described as
$I\\!I(\mathbf{t},\mathbf{t})=\bar{\kappa}\circ\pi,\ \
I\\!I(\mathbf{t},\xi)=-1,\ \ I\\!I(\xi,\xi)=0.$
The mean curvature is $H=(\bar{\kappa}\circ\pi)/2$ and the mean curvature
vector field $\mathbb{H}$ is $\mathbb{H}=H\>\mathbf{n}$.
In case $M=S^{3}$, $S_{\bar{\gamma}}$ is called the Hopf cylinder. In
particular if ${\bar{\gamma}}$ is closed, then $S_{\bar{\gamma}}$ is a flat
torus in $S^{3}$ and called the Hopf torus over ${\bar{\gamma}}$ (H. B.
Lawson, cf. [31], [35]). The Hopf torus over a geodesic in $S^{2}(4)$
coincides with the Clifford minimal torus. We call the flat surface
$S_{\bar{\gamma}}$ in a regular contact Riemannian manifold $M$ a Hopf
cylinder over the curve $\bar{\gamma}$ in $\overline{M}$.
### 1.4 Curves in Riemannian $3$-manifolds
Let $(M,g)$ be a Riemannian manifold and $\gamma=\gamma(s):I\to M$ a curve
parametrised by the arclength parameter in $M$. We regard $\gamma$ as a
1-dimensional Riemannian manifold with respect to the metric induced by $g$.
We recall the following definition (cf. [2]).
###### Definition 1.2
If $\gamma(s)$ is a unit speed curve in a Riemannian $3$-manifold $(M^{3},g)$,
we say that $\gamma$ is a Frenet curve if there exists an orthonomal frame
field $P=(\mathbf{p}_{1},\mathbf{p}_{2},\mathbf{p}_{3})$ along $\gamma$ and
two nonnegative functions $\kappa$ and $\tau$ such that $P$ satisfies the
following Frenet-Serret formula:
$\nabla_{\gamma^{\prime}}P=P\left(\begin{array}[]{ccc}0&-\kappa&0\\\
\kappa&0&-\tau\\\ 0&\tau&0\end{array}\right),\ \
\mathbf{p}_{1}=\gamma^{\prime}(s).$
The functions $\kappa$ and $\tau$ are called the curvature and torsion of
$\gamma$ respectively.
Geodesics can be regarded as Frenet curves with $\kappa=0$. A curve with
constant curvature and zero torsion is called a (Riemannian) circle. A helix
is a curve whose curvature and torsion are constants. Riemannian circles are
regarded as degenerate helices. Helices, which are not circles, are frequently
called proper helices.
Note that, in general ambient space $(M^{3},g)$, geodesics may have non-
vanishing torsion. In fact, as we shall see later, Legendre geodesics in a
Sasakian 3-manifold have constant torsion $1$.
The Frenet-Serret formula of $\gamma$ implies that the mean curvature vector
field $\mathbb{H}$ of a Frenet curve $\gamma$ is given by
$\mathbb{H}=\nabla_{\gamma^{\prime}}\gamma^{\prime}=\kappa\mathbf{p}_{2}.$
Let us denote by $\Delta$ the Laplace operator acting on the space
$\Gamma(\gamma^{*}TM)$ of all smooth sections of the vector bundle:
$\gamma^{*}TM:=\bigcup_{s\in I}T_{\gamma(s)}M$
over $I$. Then $\Delta$ is given explicitly by
$\Delta=-\nabla_{\gamma^{\prime}}\nabla_{\gamma^{\prime}}.$
###### Lemma 1.1
The mean curvature vector field $\mathbb{H}$ of a Frenet curve $\gamma$ is
harmonic in $\gamma^{*}TM$ ($\Delta\mathbb{H}=0$) if and only if
$\nabla_{\gamma^{\prime}}\nabla_{\gamma^{\prime}}\nabla_{\gamma^{\prime}}\gamma^{\prime}=0.$
When $M$ is the Euclidean space $\mathbf{E}^{m}$, a curve $\gamma$ satisfies
$\Delta\mathbb{H}=0$ if and only if $\gamma$ is biharmonic, i.e.,
$\Delta\Delta\gamma=0$ since $\Delta\gamma=-\mathbb{H}$.
The following general result is essentially obtained in [24].
###### Theorem 1.1
Let $\gamma$ be a Frenet curve in a Riemannian $3$-manifold $(M,g)$. Then
$\gamma$ satisfies $\Delta\mathbb{H}=\lambda\mathbb{H}$ in $\gamma^{*}TM$ if
and only if $\gamma$ is a geodesic $(\lambda=0)$ or a helix satisfying
$\lambda=\kappa^{2}+\tau^{2}$.
Proof. Let $I$ be an open interval and $\gamma=\gamma(s):I\to M$ be a curve
parametrised by the arclength parameter $s$ with Frenet frame field
$P=(\mathbf{p}_{1},\mathbf{p}_{2},\mathbf{p}_{3})$. Direct computation shows
that
(9)
$\nabla_{\gamma^{\prime}}\mathbb{H}=-\kappa^{2}\mathbf{p}_{1}+\kappa^{\prime}\>\mathbf{p}_{2}+\kappa\tau\mathbf{p}_{3}.$
Let us compute the Laplacian of $\mathbb{H}$:
$-\Delta\mathbb{H}=\nabla_{\gamma^{\prime}}\nabla_{\gamma^{\prime}}\mathbb{H}=-3\kappa\kappa^{\prime}\>\mathbf{p}_{1}+(\kappa^{\prime\prime}-\kappa^{3}-\kappa\tau^{2})\mathbf{p}_{2}+(2\kappa^{\prime}\tau+\kappa\tau^{\prime})\mathbf{p}_{3}.$
Hence $\Delta\mathbb{H}=\lambda\mathbb{H}$ if and only if
$\kappa\>\tau^{\prime}=0,\ \ \kappa^{3}+\kappa\>\tau^{2}=\lambda\kappa.$
These formulae imply that $\gamma$ is a geodesic or a helix satisfying
$\lambda=\kappa^{2}+\tau^{2}$.
Conversely every geodesic satisfies $\Delta\mathbb{H}=0$. Helices satisfy
$\Delta\mathbb{H}=\lambda\mathbb{H}$ with $\lambda=\kappa^{2}+\tau^{2}$.
$\Box$
###### Corollary 1.1
([19]) Let $\gamma$ be a curve in Euclidean $3$-space $\mathbf{E}^{3}$. Then
$\gamma$ is biharmonic if and only if $\gamma$ is a straight line.
On the contrary, in indefinite semi-Euclidean space, there exist nongeodesic
biharmonic curves. Chen and Ishikawa [15] classified biharmonic spacelike
curves in $\mathbf{E}^{m}_{\nu}$. (See also [28]).
### 1.5 Curves with normal-harmonic mean curvature
The results in the preceding subsection say that to characterise curves which
are non geodesics we need to use another differential operator for our
purpose.
In this subsection we use the normal Laplacian.
Let $\gamma:I\to M$ be a Frenet curve in an oriented Riemannian $3$-manifold
$M$ parametrised by the arclength. Denote by
$P=(\mathbf{p}_{1},\mathbf{p}_{2},\mathbf{p}_{3})$ the Frenet frame field of
$\gamma$ as before. Then the normal bundle $T^{\perp}\gamma$ of the curve
$\gamma$ is given by
$T^{\perp}\gamma=\bigcup_{s\in I}T^{\perp}_{s}\gamma,\
T^{\perp}_{s}\gamma=\mathbf{R}\>\mathbf{p}_{2}(s)\oplus\mathbf{R}\>\mathbf{p}_{3}(s).$
The normal connection $\nabla^{\perp}$ is a connection of $T^{\perp}\gamma$
defined by
$\nabla_{\gamma^{\prime}}^{\perp}X=\mathrm{normal}\ \mathrm{component}\
\mathrm{of}\ \nabla_{\gamma^{\prime}}X$
for any section $X$ of the normal bundle $T^{\perp}\gamma$.
By using the Frenet frame field, $\nabla^{\perp}$ can be represented as
$\nabla^{\perp}_{\gamma^{\prime}}X=\nabla_{\gamma^{\prime}}X-g(\nabla_{\gamma^{\prime}}X,\mathbf{p}_{1})\mathbf{p}_{1}.$
Let us denote by $\Delta^{\perp}$ the Laplace operator acting on the space
$\Gamma(T^{\perp}\gamma)$ of all smooth sections of the normal bundle
$T^{\perp}\gamma$. The operator $\Delta^{\perp}$ is called the normal
Laplacian of $\gamma$ in $M$. The normal Laplacian $\Delta^{\perp}$ is given
by
$\Delta^{\perp}X=-\nabla^{\perp}_{\gamma^{\prime}}\nabla^{\perp}_{\gamma^{\prime}}X,\
\ X\in\Gamma(T^{\perp}\gamma).$
Now we compute $\Delta^{\perp}\mathbb{H}$. From (9), we have
$\nabla^{\perp}_{\gamma^{\prime}}\mathbb{H}=\kappa^{\prime}\>\mathbf{p}_{2}+\kappa\tau\mathbf{p}_{3}.$
From this equation, we get
$-\Delta^{\perp}\mathbb{H}=(\kappa^{\prime\prime}-\kappa\tau^{2})\mathbf{p}_{2}+(2\kappa^{\prime}\tau+\kappa\tau^{\prime})\mathbf{p}_{3}.$
###### Theorem 1.2
(cf. [24]) A curve $\gamma$ satisfies
$\Delta^{\perp}\mathbb{H}=\lambda\mathbb{H}$ if and only if
$\kappa^{\prime\prime}-\kappa\tau^{2}=-\lambda\kappa,\ \
2\kappa^{\prime}\tau+\kappa\tau^{\prime}=0.$
###### Corollary 1.2
A curve $\gamma$ satisfies $\Delta^{\perp}\mathbb{H}=0$ if and only if
$\kappa^{\prime\prime}-\kappa\tau^{2}=0,\ \
2\kappa^{\prime}\tau+\kappa\tau^{\prime}=0.$
We shall apply these general results for curves in Sasakian $3$-manifolds in
the next section. Note that Barros and Garay classified curves, which satisfy
$\Delta^{\perp}\mathbb{H}=\lambda\mathbb{H}$ in space forms [4],[5].
## 2 Curves and surfaces in $3$-dimensional
Sasaki manifolds
### 2.1 Curves in $3$-dimensional Sasaki manifolds
Now let $M^{3}=(M,\eta,\xi,\varphi,g)$ be a contact Riemannian $3$-manifold
with an associated metric $g$. A curve $\gamma=\gamma(s):I\to M$ parametrised
by the arclength parameter is said to be a Legendre curve if $\gamma$ is
tangent to the contact distribution $D$ of $M$. It is obvious that $\gamma$ is
Legendre if and only if $\eta(\gamma^{\prime})=0$.
Let $\gamma$ be a Legendre curve in $M^{3}$. Then we can take a Frenet frame
field $P=(\mathbf{p}_{1},\mathbf{p}_{2},\mathbf{p}_{3})$ so that
$\mathbf{p}_{1}=\gamma^{\prime}$ and $\mathbf{p}_{3}=\xi$. (See [2]).
Now we assume that $M$ is a Sasaki manifold. Then by definition, the Frenet-
Serret formula of $\gamma$ is given explicitly by
$\nabla_{\gamma^{\prime}}P=P\left(\begin{array}[]{ccc}0&-\kappa&0\\\
\kappa&0&-1\\\ 0&1&0\end{array}\right).$
Namely every Legendre curve has constant torsion $1$ [2].
Now we investigate curves with harmonic or normal-harmonic mean curvature
vector field in Sasakian 3-manifolds.
The following two results are direct consequence of Theorem 1.1 and Theorem
1.2, respectively.
###### Corollary 2.1
Let $\gamma$ be a Legendre curve in $3$-dimensional Sasaki manifold. Then
$\gamma$ satisfies $\Delta\mathbb{H}=\lambda\mathbb{H}$ in $\gamma^{*}TM$ if
and only if $\gamma$ is a Legendre geodesic $(\lambda=0)$ or a Ledendre helix
satisfying $\lambda=\kappa^{2}+1$ $(\lambda\not=0)$.
###### Remark 2.1
Sasaki manifolds together with compatible Lorentz metric are called Sasakian
spacetimes ([20],[40]). On Sasakian spacetimes, the Reeb vector fields are
timelike. Every $3$-dimensional Sasakian spacetime contains proper biharmonic
Legendre curves. In fact, in a $3$-dimensional Sasakian spacetime biharmonic
Legendre curves are Legendre geodesics or Legendre helices with curvature $1$.
(cf. [28]).
###### Proposition 2.1
Let $\gamma$ be a Legendre curve in a Sasakian $3$-manifold. Then
$\Delta^{\perp}\mathbb{H}=\lambda\mathbb{H}$ if and only if $\gamma$ is a
Legendre geodesic $(\lambda=0)$ or a Legendre helix with constant nonzero
curvature $(\lambda\not=0)$. In the latter case, $\lambda=1$.
### 2.2 Biharmonic Hopf cylinders
In this section we study harmonicity and normal-harmonicity of the mean
curvature of Hopf cylinders.
Let $M^{3}$ be a regular Sasaki manifold with Boothby-Wang fibration
$\pi:M\to\bar{M}$.
Take a curve $\bar{\gamma}=\bar{\gamma}(s)$ parametrised by the arclength $s$
in the base space form $\bar{M}$. Let us denote by $S=S_{\bar{\gamma}}$ the
Hopf cylinder of $\bar{\gamma}$. (See Section 1.3)
Let $\mathbf{t}=(\bar{\mathbf{p}}_{1})^{*}$ be the horizontal lift of
$\bar{\mathbf{p}}_{1}$ with respect to the Boothby-Wang fibering. Then
$(\mathbf{t},\xi)$ gives an orthonormal frame field of $M$. The unit normal
vector field $\mathbf{n}$ is the horizontal lift of $\bar{\mathbf{p}}_{2}$.
Note that $\mathbf{n}=\varphi\mathbf{t}$.
The mean curvature vector field $\mathbb{H}$ of $S$ is
$\mathbb{H}=H\>\mathbf{n}=(\bar{\kappa}\circ\pi)\mathbf{n}/2$.
Now we study harmonicity and normal-harmonicity of $\mathbb{H}$. Denote by
$\iota$ the inclusion map of $S$ into $M$. Then the Laplace operator $\Delta$
acting on the space $\Gamma(\iota^{*}TM)$ and the normal Laplacian
$\Delta^{\perp}$ of $S$ are given by
$\Delta=-\left(\nabla_{\mathbf{t}}\nabla_{\mathbf{t}}+\nabla_{\xi}\nabla_{\xi}\right),\
\
\Delta^{\perp}=-\left(\nabla^{\perp}_{\mathbf{t}}\nabla^{\perp}_{\mathbf{t}}+\nabla^{\perp}_{\xi}\nabla^{\perp}_{\xi}\right),$
respectively. Direct computation shows that
$\nabla_{\mathbf{t}}\mathbb{H}=-2H^{2}\mathbf{t}+H^{\prime}\mathbf{n}+H\xi,\ \
\nabla^{\perp}_{\mathbf{t}}\mathbb{H}=H^{\prime}\mathbf{n},\ \
\nabla_{\xi}\>\mathbb{H}=H\>\mathbf{t},\ \
\nabla^{\perp}_{\xi}\>\mathbb{H}=0,$
$\nabla_{\xi}\nabla_{\xi}\mathbb{H}=-H\mathbf{n}.$
Thus we get
$-\Delta\mathbb{H}=-6HH^{\prime}\mathbf{t}+(H^{\prime\prime}-4H^{3}-2H)\mathbf{n}+2H^{\prime}\xi,$
$-\Delta^{\perp}\mathbb{H}=H^{\prime\prime}\mathbf{n}.$
###### Theorem 2.1
A Hopf cylinder $S_{\bar{\gamma}}$ in a $3$-dimensional regular Sasaki
manifold satisfies $\Delta\mathbb{H}=\lambda\mathbb{H}$ in $\iota^{*}TM$ if
and only if $\bar{\gamma}$ is a geodesic $(\lambda=0)$ or a Riemannian circle
$(\lambda\not=0)$. In case that $\lambda\not=0$, the eigenvalue $\lambda$ is
$\lambda=4H^{2}+2>2$.
###### Remark 2.2
Every Hopf cylinder in a $3$-dimensional regular Sasaki manifold is anti
invariant. Sasahara showed that an anti invariant surface in
$\mathbf{R}^{3}(-3)$ satisfies $\Delta\mathbb{H}=\lambda\mathbb{H},\
\lambda\not=0$ if and only if it is a Hopf cylinder over a circle with
$\lambda>2$. See Proposition 11 in [37].
###### Lemma 2.1
A Hopf cylinder $S_{\bar{\gamma}}$ satisfies
$\Delta^{\perp}\mathbb{H}=\lambda\mathbb{H}$ if and only if $\gamma$ is
defined by one of the following natural equations:
1. (1)
$\bar{\kappa}(s)=as+b,\ a,b\in\mathbf{R},\ \lambda=0;$
2. (2)
$\bar{\kappa}(s)=a\cos(\sqrt{\lambda}s)+b\sin(\sqrt{\lambda}s),\ \lambda>0;$
3. (3)
$\bar{\kappa}(s)=a\exp(\sqrt{-\lambda}s)+b\exp(-\sqrt{-\lambda}s),\
\lambda<0.$
Proof. The Hopf cylinder $S_{\bar{\gamma}}$ satisfies
$\Delta^{\perp}\mathbb{H}=\lambda\mathbb{H}$ if and only if $\bar{\gamma}$
satisfies
${\bar{\kappa}}^{\prime\prime}+\lambda{\bar{\kappa}}=0.$
Thus the result follows. $\Box$
###### Theorem 2.2
A Hopf cylinder $S_{\bar{\gamma}}$ satisfies $\Delta^{\perp}\mathbb{H}=0$ if
and only if $\bar{\gamma}$ is one of the following:
1. (1)
a geodesic;
2. (2)
a Riemannian circle or;
3. (3)
a Riemannian clothoid ( Cornu spiral ).
Here a Riemannian clothoid is a curve in $\bar{M}^{2}$ whose curvature is a
linear function of the arclength.
###### Remark 2.3
On curves in Riemannian 2-space forms, the following result is obtained [24]:
###### Theorem 2.3
Let $\bar{\gamma}$ be a curve in Riemannian $2$-manifold $\bar{{M}}^{2}$. To
avoid the confusion, let us denote by $\Delta^{\perp}_{\bar{\gamma}}$ and
$\mathbb{H}_{\bar{\gamma}}$ the normal Laplacian of $\bar{\gamma}$ and the
mean curvature vector in $\bar{{M}}^{2}$ respectively. Then
$\Delta_{\bar{\gamma}}^{\perp}\mathbb{H}_{\bar{\gamma}}=\lambda\mathbb{H}_{\bar{\gamma}}$
if and only if
1. (1)
$\bar{\gamma}$ is a geodesic, Riemannian circle or a Riemannian clothoid;
2. (2)
$\bar{\kappa}(s)=a\cos(\sqrt{\lambda}s)+b\sin(\sqrt{\lambda}s),\ \lambda>0$;
3. (3)
$\bar{\kappa}(s)=a\exp(\sqrt{-\lambda}s)+b\exp(-\sqrt{-\lambda}s),\
\lambda<0$.
###### Corollary 2.2
Let $M$ be a $3$-dimensional regular Sasaki manifold
with Boothby-Wang fibering $\pi:M\to\bar{M}$. Let $\bar{\gamma}$ be a curve in
$\bar{M}$. Then the Hopf cylinder $S=S_{\bar{\gamma}}$ satisfies
$\Delta^{\perp}\mathbb{H}=\lambda\mathbb{H}$ if and only if $\bar{\gamma}$
satisfies
$\Delta_{\bar{\gamma}}^{\perp}\mathbb{H}_{\bar{\gamma}}=\lambda\mathbb{H}_{\bar{\gamma}}$.
Theorem 2.2 is a generalisation of a result obtained by Barros and Garay [3].
In fact, if we choose $M^{3}=S^{3}$ then we obtain the following.
###### Theorem 2.4
([3]) A Hopf cylinder $S_{\bar{\gamma}}$ in the unit $3$-sphere $S^{3}$
satisfies $\Delta^{\perp}\mathbb{H}=0$ if and only if $\gamma$ is one of the
following:
1. (1)
a geodesic,
2. (2)
a Riemannian circle or
3. (3)
a Riemannian clothoid.
Here a Riemannian clothoid is a curve in the $2$-sphere $S^{2}(1/2)$ of radius
$1/2$ whose curvature is a linear function of the arclength.
Riemannian clothoids are called “Cornu spirals” in [3].
## Part II
## 3 Polyharmonic maps
Let $(M^{m},g)$ and $(N^{n},h)$ be Riemannian manifolds and $\phi:M\to N$ a
smooth map. The tension field $\mathscr{T}(\phi)$ is a section of the vector
bundle $\phi^{*}(TN)$ defined by
$\mathscr{T}(\phi):=\mathrm{tr}(\nabla d\phi).$
A smooth map $\phi$ is said to be a harmonic map if its tension field
vanishes. It is well known that $\phi$ is harmonic if and only if $\phi$ is a
critical point of the energy:
$E(\phi)=\int\frac{1}{2}|d\phi|^{2}dv_{g}$
over every compact supported region of $M$.
Now let $\phi$ be a harmonic map. Then the Hessian $\mathcal{H}_{\phi}$ of the
energy is given by the following second variation formula:
$\mathcal{H}_{\phi}(V,W)=\int h(\mathcal{J}_{\phi}(V),W)dv_{g},\ \
V,W\in\Gamma(\phi^{*}TN).$
Here the operator $\mathcal{J}_{\phi}$ is the Jacobi operator of the harmonic
map $\phi$ defined by
$\mathcal{J}_{\phi}(V):=\bar{\Delta}_{\phi}V-\mathcal{R}_{\phi}(V),\ \
V\in\Gamma(\phi^{*}TN),$
$\bar{\Delta}_{\phi}:=-\\{\sum_{i=1}^{m}(\nabla^{\phi}_{e_{i}}\nabla^{\phi}_{e_{i}}-\nabla^{\phi}_{\nabla_{e_{i}}e_{i}}\\},\
\ \mathcal{R}_{\phi}(V)=\sum_{i=1}^{m}R^{N}(V,d\phi(e_{i}))d\phi(e_{i}).$
Here $\nabla^{\phi},\ R^{N}$ and $\\{e_{i}\\}$ denote the induced connection
of $\phi^{*}TN$, curvature tensor of $N$ and a local orthonormal frame field
of $M$, respectively.
For general theory of harmonic maps and their Jacobi operators, we refer to
[21] and [42].
J. Eells and J. H. Sampson suggested to study polyharmonic maps (See [23] and
[21], p. 77 (8.7)). Let $\phi:M\to N$ be a smooth map as before. Then $\phi$
is said to be a polyharmonic map of order $k$ if it is an extremal of the
functional:
$E_{k}(\phi)=\int|(d+d^{*})^{k}\phi|^{2}dv_{g}.$
Here $d^{*}$ is the codifferential operator. In particular, if $k=2$, we have
$E_{2}(\phi)=\int|\mathscr{T}(\phi)|^{2}dv_{g}.$
The Euler-Lagrange equation of the functional $E_{2}$ was computed by Caddeo
and Oproiu (See [9], p. 867) and G. Y. Jiang [29]–[30], independently. The
Euler-Lagrange equation of $E_{2}$ is
$\mathscr{T}_{2}(\phi):=-\mathcal{J}_{\phi}(\mathscr{T}(\phi))=0.$
###### Remark 3.1
Let $\phi:M\to N$ be an isometric immersion. Then its tension field is
$m\mathbb{H}$. Thus the functional $E_{2}$ is given by
$E_{2}(\phi)=m^{2}\int|\mathbb{H}|^{2}dv_{g}.$
In case that $M$ is $2$-dimensional, $E_{2}(\phi)$ the total mean curvature of
$M$ up to constant multiple. See [11], Section 5.3.
In particular, if $N=\mathbf{E}^{n}$ and $\phi$ an isometric immersion, then
$\mathscr{T}_{2}(\phi)=-\Delta_{M}\Delta_{M}\phi,$
since $\Delta_{M}\phi=m\mathbb{H}$. Here $\Delta_{M}$ is the Laplacian of
$(M,g)$. Thus the polyharmonicity (of order $2$) for an isometric immersion
into Euclidean space is equivalent to the biharmonicity in the sense of Chen.
On this reason, polyharmonic maps of order $2$ are frequently called
biharmonic maps (or $2$-harmonic maps) [9], [29], [30], [34].
Obviously, the notion of $p$-harmonic map in the sense of [22], p. 397 is
different from that of polyharmonic map of order $p$.
Hereafter we call polyharmonic maps of order $2$ by the name “polyharmonic
maps” in short.
Caddeo, Montaldo and Oniciuc classified polyharmonic curves in
$3$-dimensional Riemannian space forms. More precisely they showed the
following two results.
###### Theorem 3.1
([9]) Let $N$ be a $3$-dimensional Riemannian space form of nonpositive
curvature. Then all the polyharmonic curves are geodesics.
Next for the study of polyharmonic curves in positively curved space forms, we
may assume that $N^{3}$ is the unit $3$-sphere.
###### Theorem 3.2
([9]) Let $\gamma:I\to S^{3}$ be a polyharmonic curve parametrised by the
arclength. Then $0\leq\kappa\leq 1$ and $\gamma$ is one of the following:
1. (1)
$\kappa$ is a geodesic ($\kappa=0$) .
2. (2)
If $k=1$ then $\gamma$ is a Riemannian circle of curvature $1;$
3. (3)
If $0<\kappa<1$ then $\gamma$ is a geodesic of the Clifford minimal torus of
$S^{3}$.
The preceding theorem implies the following result:
###### Corollary 3.1
Let $\gamma:I\to S^{3}$ be a Legendre curve parametrised by the arclength.
Then $\gamma$ is polyharmonic if and only if $\gamma$ is a Legendre geodesic.
In fact, curves in the the latter two classes can not be Legendre. (Recall
that every Legendre curve has constant torsion $1$).
Now we study polyharmonic Legendre curves in contact Riemannian $3$-manifolds.
Let $M^{3}$ be a contact Riemannian $3$-manifold and $\gamma:I\to M$ a Frenet
curve framed by $(\mathbf{p}_{1},\mathbf{p}_{2},\mathbf{p}_{3})$. Then direct
computation shows that
$\mathscr{T}_{2}(\gamma)=-3\kappa\kappa^{\prime}\mathbf{p}_{1}+(\kappa^{\prime\prime}-\kappa^{3}-\kappa\tau^{2})\mathbf{p}_{2}+(2\kappa^{\prime}\tau+\kappa\tau^{\prime})\mathbf{p}_{3}+\kappa
R(\mathbf{p}_{2},\mathbf{p}_{1})\mathbf{p}_{1}.$
Now assume that $M$ is a Sasakian space form of constant holomorphic sectional
curvature $c$ then
$\displaystyle R(\mathbf{p}_{2},\mathbf{p}_{1})\mathbf{p}_{1}$
$\displaystyle=$ $\displaystyle\frac{c+3}{4}\mathbf{p}_{2}$ $\displaystyle+$
$\displaystyle\frac{c-1}{4}\\{\eta(\mathbf{p}_{2})\eta(\mathbf{p}_{1})\mathbf{p}_{1}$
$\displaystyle-$
$\displaystyle\eta(\mathbf{p}_{1})^{2}\mathbf{p}_{2}-\eta(\mathbf{p}_{2})\xi+3g(\mathbf{p}_{2},\varphi\mathbf{p}_{1})\varphi\mathbf{p}_{1}\\}.$
In particular, if $\gamma$ is Legendre, then
$R({\mathbf{p}}_{2},{\mathbf{p}}_{1}){\mathbf{p}}_{1}=c\>\mathbf{p}_{2}$. Thus
a Legendre curve $\gamma$ in $M$ is polyharmonic if and only if
$\kappa=\mathrm{constant},\ \kappa^{3}-(c-1)\kappa=0,\ \tau=1.$
If we look for nongeodesic polyharmonic Legendre curves, we obtain
$\kappa=\mathrm{constant},\ \kappa^{2}=c-1,\ \tau=1.$
Thus we obtain the following result which is a generalisation of Corollary
3.1.
###### Theorem 3.3
Let $M^{3}(c)$ be a Sasakian space form of constant holomorphic sectional
curvature $c$ and $\gamma:I\to M$ a polyharmonic Legendre curve parametrised
by the arclength.
1. (1)
If $c\leq 1$, then $\gamma$ is a Legendre geodesic;
2. (2)
If $c>1$, then $\gamma$ is a Legendre geodesic or a Legendre helix of
curvature $\sqrt{c-1}$.
Let $\phi:M\to N$ be an isometric immersion. Then $\phi$ is a critical point
of the volume functional if and only if $\phi$ is minimal. The Jacobi operator
$\mathscr{J}$ of a minimal immersion $\phi$ (with respect to the volume
functional) is appeared in the second variation formula of the volume and
given by [39]
$\mathscr{J}V=\Delta^{\perp}V-\mathscr{S}V+\mathscr{R}(V),\ \
V\in\Gamma(T^{\perp}M).$
Here the operators $\mathscr{S}$ and $\mathscr{R}$ are defined by
$h(\mathscr{S}V,W)=\mathrm{tr}(\mathcal{A}_{V}\circ\mathcal{A}_{W}),\ \
\mathscr{R}(V)=\sum_{i=1}^{m}(R^{N}(d\phi(e_{i}),V)d\phi(e_{i}))^{\perp}.$
Here $\mathcal{A}_{V}$ denotes the Weingarten operator with respect to $V$.
Arroyo, Barros and Garay studied submanifolds in $S^{3}$ whose mean curvature
vector fields are eigen-section of the Jacobi operator with respect to the
volume functional [1], [4], [5]. Such study for surfaces in $5$-dimensional
Sasakian space forms can be found in [37].
It seems to be interesting to study similar problems for submanifolds in space
forms or Sasakian space forms with respect to the energy functional.
In [9], all the polyharmonic surfaces in $S^{3}$ are classified. More
precisely, the only non-minimal polyharmonic surfaces are totally umbilical
$2$-spheres.
Based on this result, we would like to propose the following problem:
Are there non-minimal and non totally umbilical polyharmonic submanifolds in
homogeneous Riemannian manifolds ?
To close this paper, we study polyharmonic Hopf cylinders in
$3$-dimensional Sasakian space forms. Moreover we show the existence of non-
minimal and non totally umbilical polyharmonic surfaces in Sasakian space
forms.
First we recall the following result which is a consequence of the main result
in [9]:
###### Proposition 3.1
There are no non minimal polyharmonic Hopf cylinders in the unit $3$-sphere
$S^{3}$.
Now we generalise this result to Sasakian space forms.
Let $S=S_{\bar{\gamma}}$ be a Hopf cylinder and $\iota:S\subset M^{3}(c)$ its
inclusion map into a Sasakian space form $M^{3}(c)$. Then the bitension field
$\mathscr{T}_{2}(\iota)$ is given by
$\mathscr{T}_{2}(\iota)=-\mathcal{J}_{\iota}(\mathscr{T}(\iota))=-2\mathcal{J}_{\iota}(\mathbb{H}).$
We use the orthonormal frame field $\\{{\mathbf{t}},\xi\\}$ as before. Then
since $S$ is flat, we have
$\bar{\Delta}_{\iota}\mathbb{H}=\Delta\mathbb{H},\ \
\mathcal{R}(\mathbb{H})=H(R(\mathbf{n},\mathbf{t})\mathbf{t}+R(\mathbf{n},\xi)\xi).$
Using the curvature formula of Sasakian space form, we get
$\mathcal{R}(\mathbb{H})=(c+1)H\>\mathbf{n}.$
Hence
$\mathcal{J}_{\iota}(\mathbb{H})=6HH^{\prime}\mathbf{t}-(H^{\prime\prime}-4H^{3}+(c-1)H)\mathbf{n}-2H^{\prime}\xi.$
Thus $\mathcal{J}_{\iota}(\mathbb{H})=\lambda\>\mathbb{H}$ if and only if
$H^{\prime}=0,\ 4H^{3}=(c-1+\lambda)H$
and hence $H=0$ or $\lambda=4H^{2}+1-c,\ H\not=0$.
###### Theorem 3.4
Let $S$ be a Hopf cylinder in a Sasakian space form $M^{3}(c)$. Then $S$
satisfies $\mathcal{J}_{\iota}(\mathbb{H})=\lambda\>\mathbb{H}$ if and only if
the base curve of $S$ is a Riemannian circle or a geodesic. In case that the
base curve is not a geodesic, then $\lambda=4H^{2}+1-c$.
###### Corollary 3.2
Let $\iota:S_{\bar{\gamma}}\to M^{3}(c)$ be a polyharmonic Hopf cylinder in a
Sasakian space form.
1. (1)
If $c\leq 1$ then $\bar{\gamma}$ is a geodesic;
2. (2)
If $c>1$ then $\bar{\gamma}$ is a geodesic or a Riemannian circle of curvature
$\bar{\kappa}=\sqrt{c-1}$.
In particular, there exist nonminimal polyharmonic Hopf cylinders in
Sasakian space forms of holomorphic sectional curvature greater than $1$.
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Department of Mathematics Education, Faculty of Education,
Utsunomiya University, Minemachi 350, Utsunomiya, 321-8505, Japan
E-mail address: inoguchi@cc.utsunomiya-u.ac.jp
Current Address:
Department of Mathematical Sciences, Faculty of Science,
Yamagata University, Kojirakawa 1-4-12, Yamagata, 990-8560, Japan
E-mail address: inoguchi@sci.kj.yamagata-u.ac.jp
|
arxiv-papers
| 2009-10-16T12:37:38 |
2024-09-04T02:49:05.856479
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jun-ichi Inoguchi",
"submitter": "Jun-ichi Inoguchi",
"url": "https://arxiv.org/abs/0910.3103"
}
|
0910.3104
|
# Invariant minimal surfaces in the real special linear group of degree 2
††thanks: 2000 Mathematics Subject Classification. Primary 53A99; Secondly
53C42, 53D15
Key words and phrases. Real special linear group, minimal surfaces, constant
mean curvature surfaces, Hopf cylinders, tangential Gauß maps, contact
structures
Jun-ichi Inoguchi 111partially supported by Grand-in-Aid for Encouragement of
Young Scientists 12740051, 14740053, Japan Society for Promotion of Science
(Dedicated to professor Koichi Ogiue on his 60th birthday )
###### Abstract
Invariant minimal surfaces in the real special linear group
$\mathrm{SL}_{2}\mathbf{R}$ with canonical Riemannian and Lorentzian metrics
are studied.
Constant mean curvature surfaces with vertically harmonic Gauß map are
classified.
## Introduction
In our previous works [15]–[16], we have investigated fundamental properties
of the real special linear group $\mathrm{SL}_{2}\mathbf{R}$ furnished with
canonical left invariant Riemannian metric. It is known that
$\mathrm{SL}_{2}\mathbf{R}$ with canonical Riemannian metric admits a
structure of naturally reductive homogeneous space and left invariant Sasaki
structure. The isometry group of the canonical left invariant metric is
$4$-dimensional.
On the other hand, it is well known that the Killing form of
$\mathrm{SL}_{2}\mathbf{R}$ induces a biinvariant Lorentz metric of constant
curvature on $\mathrm{SL}_{2}\mathbf{R}$.
Thus $\mathrm{SL}_{2}\mathbf{R}$ with biinvariant metric is identified with
anti de Sitter $3$-space $H^{3}_{1}$.
As we will see in Section 1, the canonical left invariant Riemannian metric
and biinvariant Lorentzian metric (of constant curvature $-1$) belong to same
one-parameter family of left invariant semi-Riemannian metrics. Based on this
fact, in this paper, we shall give a unified approach to geometry of
$H^{3}_{1}$ and $\mathrm{SL}_{2}\mathbf{R}$ with canonical metric.
Since the canonical left invariant metric is of non-constant curvature,
geometry of surfaces in $\mathrm{SL}_{2}\mathbf{R}$ is complicated.
In fact, we have shown in [6], there are no extrinsic spheres (totally
umbilical surfaces with constant mean curvature), especially no totally
geodesic surfaces in $\mathrm{SL}_{2}\mathbf{R}$.
In [18], Kokubu introduced the notions of rotational surface and conoid in
$\mathrm{SL}_{2}\mathbf{R}$ with canonical left invariant Riemannian metric.
Further he classified constant mean curvature rotational surfaces and minimal
conoids.
Gorodski [12] independently investigated constant mean curvature rotational
surfaces.
In [6], Belkhelfa, Dillen and the author gave a characterisation of rotational
surfaces with constant mean curvature. More precisely a surface in
$\mathrm{SL}_{2}\mathbf{R}$ is congruent to a rotational surface of constant
mean curvature if and only if its second fundamental form is parallel.
In this paper we give some other characterisations of rotational surfaces (of
constant mean curvature).
First we show that rotational surfaces in the sense of Kokubu coincide with
Hopf cylinders (over curves in the hyperbolic $2$-space $H^{2}$) in the sense
of Pinkall [22] and Barros–Ferrández–Lucas–Meroño [4]. Based on this fact, we
give a unified viewpoint for [4] and [18].
Similarly we shall show that conoids in the sense of Kokubu coincide with Hopf
cylinders over curves in Lorentz $2$-sphere $S^{2}_{1}$.
When we identify the Lie algebra $\mathfrak{g}$ of $\mathrm{SL}_{2}\mathbf{R}$
with (semi) Euclidean $3$-space, both $H^{2}$ and $S^{2}_{1}$ are given by
adjoint orbits in $\mathfrak{g}$. The adjoint orbits of
$\mathrm{SL}_{2}\mathbf{R}$ in $\mathfrak{g}$ are $H^{2}$, $S^{2}_{1}$ and
lightcone $\Lambda$. Based on this fact, we shall introduce a new class of
surfaces in $\mathrm{SL}_{2}\mathbf{R}$. More precisely, in section 4, we
shall investigate surfaces in $\mathrm{SL}_{2}\mathbf{R}$ derived from curves
in $\Lambda$.
For every surface in $\mathrm{SL}_{2}{\mathbf{R}}$, we associate a map into
the Grassmannian bundle $Gr_{2}(T\>\mathrm{SL}_{2}\mathbf{R})$ of
$2$-planes—called the Gauß map of the surface. We shall give a
characterisation of constant mean curvature rotational surfaces in terms of
harmonicity for Gauß maps.
More precisely, in the final section, we shall prove that a constant mean
curvature surface in $\mathrm{SL}_{2}\mathbf{R}$ is congruent to a rotational
surface with constant mean curvature if and only if its Gauß map is vertically
harmonic.
The author would like to thank professor Luis Jose Alías (Universidad de
Murcia, Spain) for his careful reading of the manuscript and invaluable
suggestions.
## 1 The special linear group
1.1 Let $G=\mathrm{SL}_{2}{\mathbf{R}}$ be the real special linear group of
degree $2$:
$\mathrm{SL}_{2}\mathbf{R}=\left\\{\left(\begin{array}[]{cc}a&b\\\
c&d\end{array}\right)\ \biggr{|}\ a,b,c,d\in\mathbf{R},\ ad-bc=1\right\\}.$
By using the Iwasawa decomposition $G=NAK$ of $G$;
$None$ $N=\left\\{\left(\begin{array}[]{cc}1&x\\\ 0&1\end{array}\right)\
\biggr{|}\ x\in\mathbf{R}\right\\},$ $None$
$A=\left\\{\left(\begin{array}[]{cc}\sqrt{y}&0\\\
0&1/\sqrt{y}\end{array}\right)\ \biggr{|}\ y>0\right\\},$ $None$
$K=\left\\{\left(\begin{array}[]{cc}\cos\theta&\sin\theta\\\
-\sin\theta&\cos\theta\end{array}\right)\ \biggr{|}\ 0\leq\theta\leq
2\pi\right\\},$
we can introduce the following global coordinate system $(x,y,\theta)$ of $G$:
(1) $(x,y,\theta)\longmapsto\left(\begin{array}[]{cc}1&x\\\
0&1\end{array}\right)\left(\begin{array}[]{cc}\sqrt{y}&0\\\
0&1/\sqrt{y}\end{array}\right)\left(\begin{array}[]{cc}\cos\theta&\sin\theta\\\
-\sin\theta&\cos\theta\end{array}\right).$
We equip on $G$ the following one-parameter family $\\{g[\nu]\\}$ of semi-
Riemannian metrics:
$g[\nu]=\frac{dx^{2}+dy^{2}}{4y^{2}}+\nu\left(d\theta+\frac{dx}{2y}\right)^{2},\
\ \nu\in\mathbf{R}^{*}.$
Every metric $g[\nu]$ is left invariant. Clearly $g[\nu]$ is Riemannian for
$\nu>0$ and Lorentzian for $\nu<0$.
Throughout this paper we restrict our attention to $\nu=\pm 1$ for simplicity.
One can see that $g[1]$ is only left invariant but $g[-1]$ is a biinvariant
Lorentz metric on $G$.
We take the following orthonormal coframe field of $(G,g[\nu])$:
(2) $\omega^{1}=\frac{dx}{2y},\ \ \omega^{2}=\frac{dy}{2y},\ \
\omega^{3}=d\theta+\frac{dx}{2y}.$
The dual frame field of $\\{\omega^{1},\omega^{2},\omega^{3}\\}$ is given by
$\epsilon_{1}=2y\frac{\partial}{\partial x}-\frac{\partial}{\partial\theta},\
\epsilon_{2}=2y\frac{\partial}{\partial y},\
\epsilon_{3}=\frac{\partial}{\partial\theta}.$
Note that this orthonormal frame field is not left invariant.
The Levi-Civita connection $\nabla$ of $g[\nu]$ is given by the following
formulae:
$\nabla_{\epsilon_{1}}\epsilon_{1}=2\epsilon_{2},\ \
\nabla_{\epsilon_{1}}\epsilon_{2}=-2\epsilon_{1}-\epsilon_{3},\ \
\nabla_{\epsilon_{1}}\epsilon_{3}=\nu\epsilon_{2},$ (3)
$\nabla_{\epsilon_{2}}\epsilon_{1}=\epsilon_{3},\ \
\nabla_{\epsilon_{2}}\epsilon_{2}=0,\ \
\nabla_{\epsilon_{2}}\epsilon_{3}=-\nu\epsilon_{1},$
$\nabla_{\epsilon_{3}}\epsilon_{1}=\nu\epsilon_{2},\ \
\nabla_{\epsilon_{3}}\epsilon_{2}=-\nu\epsilon_{1},\ \
\nabla_{\epsilon_{3}}\epsilon_{3}=0.$
The commutation relations of the basis are given by
(4) $[\epsilon_{1},\epsilon_{2}]=-2\epsilon_{1}-2\epsilon_{3},\ \
[\epsilon_{1},\epsilon_{3}]=0,\ \ [\epsilon_{2},\epsilon_{3}]=0.$
The Riemannian curvature tensor $R$ of the metric $g$ defined by
$R(X,Y)Z:=\nabla_{X}\nabla_{Y}Z-\nabla_{Y}\nabla_{X}Z-\nabla_{[X,Y]}Z,\ \
X,Y,Z\in\mathfrak{X}(G)$
is described by the following formulae:
(5)
$\begin{array}[]{cc}{R}(\epsilon_{1},\epsilon_{2})\epsilon_{1}=(3\nu+4)\epsilon_{2},&{R}(\epsilon_{1},\epsilon_{2})\epsilon_{2}=-(3\nu+4)\epsilon_{1},\\\
{R}(\epsilon_{1},\epsilon_{3})\epsilon_{1}=-\nu\epsilon_{3},&{R}(\epsilon_{1},\epsilon_{3})\epsilon_{3}=\nu\>\epsilon_{1},\\\
{R}(\epsilon_{2},\epsilon_{3})\epsilon_{2}=-\nu\>\epsilon_{3},&{R}(\epsilon_{2},\epsilon_{3})\epsilon_{3}=\nu\>\epsilon_{2}.\end{array}$
1.2 The one-form $\eta=-d\theta-dx/(2y)$ is a contact form on $G$, i.e.,
$d\eta\wedge\eta\not=0$.
Let us define an endmorphism field $F$ by
$F\>\epsilon_{1}=\epsilon_{2},\ F\>\epsilon_{2}=-\epsilon_{1},\
F\>\epsilon_{3}=0.$
And put $\xi=-\epsilon_{3}$. Then $(\eta,\xi,F,g[\nu])$ satisfies the
following relations:
$F^{2}=-I+\eta\otimes\xi,\ \ d\eta(X,Y)=2g(X,FY),$
$g(FX,FY)=g(X,Y)-\nu\eta(X)\eta(Y),$ $\nabla_{X}\xi=-\nu FX,$
$(\nabla_{X}F)Y=g(X,Y)\xi-\nu\eta(Y)X$
for all $X,Y\in\mathfrak{X}(G)$.
These formulae say that the structure $(\xi,F,g[\nu])$ is the associated
almost contact structure of the contact manifold $(G,\eta)$ [15]. The
resulting almost contact manifold $(G;\eta,\xi,F,g[\nu])$ is a homogeneous
Sasaki manifold [24]. The structure $(\eta,\xi,F,g[\nu])$ is called the
canonical Sasaki structure of $G$. With respect to the canonical Sasaki
structure, $(G,g[\nu])$ is a Sasaki manifold of constant holomorphic sectional
curvature $-(3\nu+4)$. The vector field $\xi$ is called the Reeb vector field
of $G$ associated to $\eta$. In Lorentzian case, since $\xi$ is a globally
defined unit timelike vector field on $G$, $\xi$ time-orients $G$.
###### Remark 1.1
The Riemannian curvature tensor $R$ of $(G,g[\nu])$ is given explicitly by
$\displaystyle R(X,Y)Z$ $\displaystyle=$ $\displaystyle-g(Y,Z)X+g(Z,X)Y$
$\displaystyle-(1+\nu)\>\\{\eta(Z)\eta(X)Y-\eta(Y)\eta(Z)X$
$\displaystyle+g(Z,X)\eta(Y)\xi-g(Y,Z)\eta(X)\xi$
$\displaystyle-g(Y,FZ)FX-g(Z,FX)FY+2g(X,FY)FZ\>\\}$
in terms of the canonical Sasaki structure. In particular, this explicit
formula says $g[-1]$ is a Lorentz metric of constant curvature $-1$. As we
will see later $(G,g[-1])$ is identified with the anti de Sitter space
$H^{3}_{1}$.
For more informations on the canonical Sasaki structure of $G$, we refer to
[15].
1.3 The special linear group $G$ acts transitively and isometrically on the
upper half plane:
$H^{2}(1/2)=\left(\\{(x,y)\in{\mathbf{R}}^{2}\ |\
y>0\\},\frac{dx^{2}+dy^{2}}{4y^{2}}\right)$
of constant curvature $-4$. The isotropy subgroup of $G$ at $(0,1)$ is the
rotation group $K=\mathrm{SO}(2)$. The natural projection $\pi:(G,g[\nu])\to
G/K=H^{2}(1/2)$ is a semi-Riemannian submersion with totally geodesic fibres.
Moreover $\pi$ is given explicitly by
$\pi(x,y,\theta)=(x,y)\in H^{2}(1/2)$
in terms of the global coordinate system (1).
The horizontal distribution of this semi-Riemannian submersion coincides with
the contact distribution determined by $\eta$. The submersion
$\pi:(G,g[-1])\to H^{2}(1/2)$ is traditionally called the Hopf fibering of
$H^{2}(1/2)$. The Sasaki manifold $(G,\eta;\xi,F,g[-1])$ is an example of
regular contact spacetime which is not globally hyperbolic.
1.4 Let us denote by $\mathfrak{g}$ the Lie algebra of $G$, i.e., the tangent
space of $G$ at the identity matrix $\mathbf{1}$:
$\mathfrak{g}=\left\\{\left(\begin{array}[]{cc}a&b\\\ c&d\end{array}\right)\
\biggr{|}\ a,b,c,d\in\mathbf{R},\ a+d=0\right\\}.$
We take the following (split-quaternion) basis of $\mathfrak{g}$:
$\mathbf{i}=\left(\begin{array}[]{cc}0&-1\\\ 1&0\end{array}\right),\ \
\mathbf{j}^{\prime}=\left(\begin{array}[]{cc}0&1\\\ 1&0\end{array}\right),\ \
\mathbf{k}^{\prime}=\left(\begin{array}[]{cc}-1&0\\\ 0&1\end{array}\right).$
Hereafter we identify $\mathfrak{g}$ with Cartesian $3$-space $\mathbf{R}^{3}$
via the linear isomorphism:
$X=x_{1}\,{\mathbf{i}}+x_{2}\,{\mathbf{j}}^{\prime}+x_{3}\,{\mathbf{k}}^{\prime}\longmapsto(x_{1},x_{2},x_{3}).$
Equivalently,
$X=\left(\begin{array}[]{cc}-x_{3}&-x_{1}+x_{2}\\\
x_{1}+x_{2}&x_{3}\end{array}\right)\longmapsto(x_{1},x_{2},x_{3}).$
We denote the scalar product on ${\mathfrak{g}}$ induced by $g[1]$ and $g[-1]$
by $\langle\cdot,\cdot\rangle^{(+)}$ and $\langle\cdot,\cdot\rangle^{(-)}$
respectively.
The scalar products $\langle\cdot,\cdot\rangle^{(\pm)}$ are given explicitly
by the following formulae:
$\langle X,Y\rangle^{(+)}=\frac{1}{2}\mathrm{tr}({}^{t}XY),\
X,Y\in\mathfrak{g},$ $\langle X,Y\rangle^{(-)}=\frac{1}{2}\mathrm{tr}(XY),\
X,Y\in\mathfrak{g}.$
For $X\in\mathfrak{g}$,
$\langle X,X\rangle^{(\pm)}=\pm x^{2}_{1}+x^{2}_{2}+x^{2}_{3}.$
Thus we identify $(\mathfrak{g},\langle\cdot,\cdot\rangle^{(+)})$ with
Euclidean $3$-space:
$\mathbf{E}^{3}=(\mathbf{R}^{3}(x_{1},x_{2},x_{3}),dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}).$
And $(\mathfrak{g},\langle\cdot,\cdot\rangle^{(-)})$ is identified with
Minkowski $3$-space:
$\mathbf{E}^{3}_{1}=(\mathbf{R}^{3}(x_{1},x_{2},x_{3}),-dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2})$
respectively.
Moreover the semi-Euclidean $4$-space
$\mathbf{E}^{4}_{2}=(\mathbf{R}^{4}(x_{0},x_{1},x_{2},x_{3})\ ,\
-dx_{0}^{2}-dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}\ )$
is identified with the space $\mathrm{M}_{2}\mathbf{R}$ of all real $2$ by $2$
matrices:
$\mathrm{M}_{2}\mathbf{R}=\left\\{x_{0}\mathbf{1}+x_{1}\mathbf{i}+x_{2}\mathbf{j}^{\prime}+x_{3}\mathbf{k}^{\prime}\right\\}.$
The semi-Euclidean metric of $\mathbf{E}^{4}_{2}$ corresponds to the scalar
product
$\langle
X,Y\rangle=\frac{1}{2}\left\\{\mathrm{tr}(XY)-\mathrm{tr}(X)\mathrm{tr}(Y)\right\\},\
X,Y\in\mathrm{M}_{2}\mathbf{R}.$
Since $\langle X,X\rangle=-\det X$ for all $X\in\mathrm{M}_{2}\mathbf{R}$, the
special linear group $G$ with biinvariant Lorentz metric $g[-1]$ is identified
with anti de Sitter $3$-space:
$H^{3}_{1}=\\{(x_{0},x_{1},x_{2},x_{3})\in\mathbf{E}^{4}_{2}\ |\
-x_{0}^{2}-x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=-1\ \\}.$
1.5 The Lie group $G$ acts on $\mathfrak{g}$ by the Ad-action:
$\mathrm{Ad}:G\times\mathfrak{g}\to\mathfrak{g};\
\mathrm{Ad}(a)X=a\>X\>a^{-1},\ a\in G,\ X\in\mathfrak{g}.$
Since the determinant function $\det$ is Ad-invariant, the Ad-orbits in
$\mathfrak{g}$ are parametrised in the following way:
$\mathcal{O}_{c}=\left\\{X\in\mathfrak{g}\ |\ \det X=c\ \right\\},\
c\in\mathbf{R}.$
For $c\geq 0$, put
$\mathcal{O}^{\pm}_{c}=\\{(x_{1},x_{2},x_{3})\in\mathcal{O}_{c}|\ \pm
x_{1}>0\\}.$
Then
$\displaystyle\mathcal{O}_{c}$ $\displaystyle=$
$\displaystyle\mathcal{O}_{c}^{+}\cup\mathcal{O}_{c}^{-},\ c>0,$
$\displaystyle\mathcal{O}_{0}$ $\displaystyle=$
$\displaystyle\mathcal{O}_{0}^{+}\cup\\{0\\}\cup\mathcal{O}_{0}^{-},\ c=0.$
###### Proposition 1.1
The $\mathrm{Ad}$-orbits of $G$ are
$\mathcal{O}^{\pm}_{c},\ (c>0),\ \ \mathcal{O}_{0}^{\pm},\ (c=0),\ \\{0\\},\ \
\mathrm{or}\ \ \mathcal{O}_{c},\ (c<0).$
With respect to the Lorentz scalar product $\langle\cdot,\cdot\rangle^{(-)}$,
the non-trivial $\mathrm{Ad}$-orbit $\mathcal{O}_{c}$ are classified as
follows:
(1) $c<0$: The $\mathrm{Ad}$-orbit $\mathcal{O}_{c}$ is the pseudo-$2$-sphere
$S^{2}_{1}(\sqrt{-c})$ of radius $\sqrt{-c}$.
In this case $\mathcal{O}_{c}=G/A\mathbf{Z}^{2}$.
(2) $c>0$: The $\mathrm{Ad}$-orbit $\mathcal{O}^{\pm}_{c}$ is the upper or
lower imbedding of hyperbolic
$2$-space $H^{2}(\sqrt{c})$ with radius $\sqrt{c}$ in $\mathbf{E}^{3}_{1}$. In
this case $\mathcal{O}^{\pm}_{c}=G/K$.
(3) $c=0$: The $\mathrm{Ad}$-orbit $\mathcal{O}^{\pm}_{0}$ is the future or
past lightcone:
$\Lambda_{\pm}=\\{(x_{1},x_{2},x_{3})\not=0\ |\
-x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=0,\ \pm x_{1}>0\\}.$
The future lightcone $\Lambda_{+}$ is represented as
$\Lambda_{+}=G/N\mathbf{Z}_{2}$.
1.6 The Riemannian metric $g[1]$ is not only $G$-left invariant but also right
$K$-invariant. Thus the product group $G\times K$ acts isometrically on
$(G,g[1])$. Note that $(G,g[1])$ is represented by $(G\times K/K,g[1])$ as a
naturally reductive (Riemannian) homogeneous space (See [25]).
On the other hand, since $g[-1]$ is biinvariant, $G\times G$ acts
isometrically on $(G,g[1])$. Moreover $(G,g[-1])$ is represented by $(G\times
G/G,g[-1])$ as a Lorentzian symmetric space.
Hence every subgroup of $G\times K$ acts isometrically on both $(G,g[\nu])$.
Kokubu introduced the notion of helicoidal motion for $(G,g[1])$. This notion
can be naturally extended for $(G,g[\nu])$.
###### Definition 1.1
Let $\\{\sigma^{\mu}_{t}\\}_{t\in{\mathbf{R}}}$ be a one parameter subgroup of
$G\times K$ defined by
(6) $\sigma^{\mu}_{t}(X)=\left(\begin{array}[]{cc}1&\mu t\\\
0&1\end{array}\right)X\left(\begin{array}[]{cc}\cos t&\sin t\\\ -\sin t&\cos
t\end{array}\right),\ \ \mu\in\mathbf{R}.$
An element of $\\{\sigma^{\mu}_{t}\\}_{t\in{\mathbf{R}}}$ is called a
helicoidal motion with pitch $\mu$.
Kokubu called surfaces in $(G,g[1])$ which are invariant under some helicoidal
motion group $\\{\sigma^{\mu}_{t}\\}$ helicoidal surfaces.
## 2 Hopf cylinders
3.1 We recall two classes of surfaces in $(G,g[1])$ studied by Kokubu.
###### Definition 2.1
([18]) An immersed surface in $G$ is said to be a rotational surface if it is
invariant under the right $K$-action.
A rotational surface can be parametrised as
(7) $\varphi(u,v)=\left(\begin{array}[]{cc}1&x(v)\\\
0&1\end{array}\right)\left(\begin{array}[]{cc}\sqrt{y(v)}&0\\\
0&1/\sqrt{y(v)}\end{array}\right)\left(\begin{array}[]{cc}\cos u&\sin u\\\
-\sin u&\cos u\end{array}\right).$
Obviously this definition is also valid for $H^{3}_{1}$.
Next we recall the notion of Hopf cylinder introduced by Pinkall [22].
Let $\pi:S^{3}\to S^{2}(1/2)$ be the Hopf fibering of $S^{2}(1/2)$. Take a
curve ${\bar{\gamma}}$ in the base space $S^{2}(1/2)$. Then the inverse image
$M:=\pi^{-1}\\{{\bar{\gamma}}\\}$ is a flat surface in $S^{3}$ which is called
a Hopf cylinder over ${\bar{\gamma}}$ [22]. This construction is valid for
other Hopf fiberings: $H^{3}_{1}\to H^{2}(1/2)$ and $H^{3}_{1}\to
S^{2}_{1}(1/2).$
In particular Hopf cylinders in $H^{3}_{1}$ over curves in $H^{2}(1/2)$ are
timelike. Barros, Ferrández, Lucas and Meroño [4], [5], [10] developed
detailed studies on Hopf cylinders in $H^{3}_{1}$. It is easy to see that the
notion of Hopf cylinder can be extended naturally to the fibering:
$\pi:(G,g[\nu])\to H^{2}(1/2).$
By using $\mathrm{SL}_{2}{\mathbf{R}}$-model of $H^{3}_{1}$ and the coordinate
system (1), we can see that Hopf cylinders over curves in $H^{2}$ are nothing
but surfaces in $G$ invariant under the right action of $K$.
###### Proposition 2.1
Let $M$ be a surface in $(G,g[\nu])$. Then $M$ is a Hopf cylinder over a curve
in $H^{2}(1/2)$ if and only if it is a rotational surface.
Thus we can unify two theories of “Hopf cylinders in $H^{3}_{1}$” and of
“rotational surfaces in $(G,g[1])$”.
###### Proposition 2.2
Let $\varphi:I\times S^{1}\rightarrow(G,g[\nu])$ be a Hopf cylinder over a
curve $(x(v),y(v))$ in $H^{2}(1/2)$ parametrised by arclength parameter $v$.
Then the induced metric of $\varphi$ is
(8)
$\mathrm{I}[\nu]=\nu\left(du+\frac{x^{\prime}(v)}{2y}dv\right)^{2}+dv^{2}.$
Hence the Hopf cylinder $(I\times S^{1},\varphi)$ is flat.
Hopf cylinders of constant mean curvature are classified as follows (Compare
[4]–[5] and Proposition 4.3 in [18]):
###### Proposition 2.3
(Classification of CMC Hopf cylinders)
Let $c$ be a unit speed curve in $H^{2}(1/2)$ with curvature $\kappa$ and
$M_{c}$ the Hopf cylinder over $c$ in $(G,g[\nu])$. Then $M_{c}$ is of
constant mean curvature if and only if $c$ is a Riemannian circle in
$H^{2}(1/2)$. The mean curvature of $M_{c}$ is $H=\kappa/2$.
The Hopf cylinder $M_{c}$ is classified in the following way:
(1) $M_{c}$ is a minimal complex circle if $\kappa=0$,
(2) $M_{c}$ is a non-minimal complex circle or a Hopf cylinder over a line
segment $y=\pm(\sqrt{1-4\kappa^{2}}/(2\kappa))x$ if $0<\kappa^{2}<4$,
(3) $M_{c}$ is a Hopf cylinder over a horocycle or $y=$constant if
$\kappa^{2}=4$,
(4) $M_{c}$ is an imbedded torus if $\kappa^{2}>4$.
Note that, in $H^{3}_{1}$ case, $M_{c}$ is a $B$-scroll of the horizontal lift
${\hat{c}}$ of $c$. ([8], [4]. Compare with Theorem 3.2).
###### Remark 2.1
The notion of complex circle is introduced by Magid. (See [19], Example 1.12.)
The non-minimal complex circle is an isometric immersion
$\varphi:{\mathbf{E}}_{1}^{2}(u,v)\to H^{3}_{1}$ of Minkowski plane into
$H^{3}_{1}$ defined by
$\varphi(u,v)=\left(\begin{array}[]{c}b\cosh v\cos u-a\sinh v\sin u\\\ a\sinh
v\cos u+b\cosh v\sin u\\\ a\cosh v\cos u+b\sinh v\sin u\\\ a\cosh v\sin
u-b\sinh v\cos u\end{array}\right),$
where $a^{2}-b^{2}=-1,\ ab\not=0$. The non-minimal complex circle $\varphi$ is
a non-minimal flat timelike surface in $H^{3}_{1}$. (cf. Alías, Ferrández and
Lucas [1], Example 3.3.)
If we interchange $+$ and $-$ in the third and fourth components of $\varphi$,
then we obtain a timelike minimal surface in $H^{3}_{1}$. This timelike
minimal surface has the following expression $\exp(u{\mathbf{i}})\
\exp(v{\mathbf{k}}^{\prime})\ \exp(t{\mathbf{j}}^{\prime})$. Here we put
$b=\cosh t$ and $a=\sinh t$.
###### Remark 2.2
It is straightforward to check that every rotational surface of constant mean
curvature in $(G,g[1])$ has parallel second fundamental form (especially
constant principal curvatures). Conversely, one can see that surfaces with
parallel second fundamental form in $(G,g[1])$ are congruent to rotational
surfaces of constant mean curvature. See [6]. Since rotational surfaces of
constant mean curvature are not totally umbilical, there are no extrinsic
spheres (totally umbilical surfaces with constant mean curvature) in
$(G,g[1])$.
On the other hand, timelike isometric immersion of $\mathbf{E}^{2}_{1}$ into
$H^{3}_{1}$ with parallel second fundamental form are classified in p. 93,
Corollary in [8]. See also [19].
###### Remark 2.3
Let $c(t)=(x(t),y(t))$ be a curve in $H^{2}(1/2)$ parametrised by the
arclength parameter $t$ and $M$ the Hopf cylinder over $c$. Then it is easy to
see that $\xi$ is tangent to $M$. Moreover the horizontal lift
$c^{\prime}(t)^{*}$ of the tangent vector field $c^{\prime}(t)$ of $c$ to $G$
also tangents to $M$. The tangent space of $M$ at $(x(t),y(t),\theta)$ is
spanned by $c^{\prime}(t)^{*}$ and $\xi$. Denote by $\mathscr{D}^{\perp}$ the
distribution spanned by $c^{*}(t)$ and put $\mathscr{D}=\\{0\\}$. Then we have
$TM=\mathscr{D}\oplus\mathscr{D}^{\perp}\oplus\langle\xi\rangle,\ \
F(\mathscr{D})\subset\mathscr{D},\ \ F(\mathscr{D}^{\perp})=T^{\perp}M.$
Here $\langle\xi\rangle$ is the distribution spanned by $\xi$. Thus the Hopf
cylinder $M$ is an anti invariant submanifold of $G$ in the sense of [27].
3.2 Next we shall recall the notion of conoid introduced by Kokubu.
###### Definition 2.2
([18]) An immersed surface in $(G,g[1])$ of the form:
(9) $\varphi(u,v)=\left(\begin{array}[]{cc}1&x(u)\\\
0&1\end{array}\right)\left(\begin{array}[]{cc}\sqrt{v}&0\\\
0&1/\sqrt{v}\end{array}\right)\left(\begin{array}[]{cc}\cos u&\sin u\\\ -\sin
u&\cos u\end{array}\right)$
is called a conoid in $G$.
If we use the metric $g[-1]$, then $(x,\theta)=(x(u),u)$ is a curve in the
double covering manifold ${\widetilde{S}}^{2}_{1}$ of $S^{2}_{1}$. Hence
conoids in $(G,g[-1])$ may be regarded as Hopf cylinders over curves in
$S^{2}_{1}$.
Constant mean curvature Hopf cylinders in $H^{3}_{1}$ over curves in
$S^{2}_{1}$ are classified by Barros, Ferrández, Lucas and Merõno.
###### Proposition 2.4
(Classification of CMC Hopf cylinders [5])
Let $c$ be a unit speed curve in $S^{2}_{1}(1/2)$ with curvature $\kappa$ and
$M_{c}$ the Hopf cylinder over $c$ in $H^{3}_{1}$. Then $M_{c}$ is of constant
mean curvature if and only if $c$ is a semi-Riemannian circle in
$S^{2}_{1}(1/2)$. The mean curvature of $M_{c}$ is $H=\kappa/2$.
(1) $M_{c}$ is a minimal complex circle if $\kappa=0$,
(2) $M_{c}$ is a non-minimal complex circle if $0<\kappa^{2}<4$,
(3) $M_{c}$ is a Hopf cylinder over a pseudo-horocycle if $\kappa^{2}=4$,
(4) $M_{c}$ is the semi-Riemannian product $H^{1}_{1}(-r^{2})\times
S^{1}_{1}(r^{2}-1)$ if $\kappa^{2}>4$,
(5) $M_{c}$ is the Riemannian product $H^{1}(-r^{2})\times H^{1}(r^{2}-1)$
with $r$ such that
$\frac{1-2r^{2}}{r\sqrt{1-r^{2}}}=\kappa.$
On the other hand in $(G,g[1])$, Kokubu obtained the following
###### Proposition 2.5
([18]) The only (complete) minimal conoids in $(G,g[1])$ are helicoidal
surfaces:
$\varphi(u,v)=\left(\begin{array}[]{cc}1&\mu u+a\\\
0&1\end{array}\right)\left(\begin{array}[]{cc}\sqrt{v}&0\\\
0&1/\sqrt{v}\end{array}\right)\left(\begin{array}[]{cc}\cos u&\sin u\\\ -\sin
u&\cos u\end{array}\right)$ $=\sigma^{\mu}_{u}\left(\
\left(\begin{array}[]{cc}1&a\\\
0&1\end{array}\right)\left(\begin{array}[]{cc}\sqrt{v}&0\\\
0&1/\sqrt{v}\end{array}\right)\ \right).$
Namely these minimal conoids are $\\{\sigma^{\mu}_{t}\\}$-orbits of a line
$\\{(a,y,0)\in H^{2}\times S^{1}\>|\>y>0\\}$. In particular $\varphi$ is an
imbedding.
The results in this section motivate us to study the class of surfaces which
will be introduced in the next section.
## 3 Surfaces derived from curves in the lightcone
In this section, we shall introduce a new class of surfaces in $G$. As we saw
before, $\mathrm{Ad}$-orbits of vectors in
${\mathfrak{g}}={\mathfrak{s}}{\mathfrak{l}}_{2}{\mathbf{R}}$ are classified
in three types. The $\mathrm{Ad}$-orbit of a spacelike [resp. timelike] vector
is a hyperbolic $2$-space [resp. Lorentz sphere]. The $\mathrm{Ad}$-orbit of a
null vector is the lightcone. In the preceding section, we saw that two kinds
of surfaces, “rotational surfaces” and “conoids” coincide Hopf cylinders over
curves in hyperbolic $2$-space or Lorentz sphere. It seems to be interesting
to study surfaces obtained by curves in $\mathrm{Ad}$-orbit of a null vector,
i.e., the lightcone. This section is devoted to study such surfaces.
Let $c$ be a curve in lightcone $\Lambda$. Then its inverse image $M$ in
$H^{3}_{1}=(G,g[-1])$ is given by
(10) $\varphi(u,v)=\left(\begin{array}[]{cc}1&v\\\
0&1\end{array}\right)\left(\begin{array}[]{cc}\sqrt{y(u)}&0\\\
0&1/\sqrt{y(u)}\end{array}\right)\left(\begin{array}[]{cc}\cos u&\sin u\\\
-\sin u&\cos u\end{array}\right).$
The partial derivatives of $\varphi$ are
$\varphi_{*}\frac{\partial}{\partial
u}=\frac{y^{\prime}}{2y}\epsilon_{2}+\epsilon_{3},\ \
\varphi_{*}\frac{\partial}{\partial
v}=\frac{1}{2y}\left(\epsilon_{1}+\epsilon_{3}\right).$
The induced metric $\mathrm{I}[\nu]$ of $M$ is
$\mathrm{I}[\nu]=\left\\{\nu+\left(\frac{y^{\prime}(u)}{2y(u)}\right)^{2}\right\\}du^{2}+\frac{\nu}{y(u)}dudv+\frac{1+\nu}{4y(u)^{2}}dv^{2}.$
The determinant of $\mathrm{I}[\nu]$ is
$\det\mathrm{I}[\nu]=\frac{1}{16y(u)^{4}}\left\\{(1+\nu)y^{\prime}(u)^{2}+4\nu
y(u)^{2}\right\\}.$
In particular $\det\mathrm{I}[-1]=-1/(4y^{2})$, hence $(M,\varphi)$ is
timelike in $H^{3}_{1}$. Direct computations using (3) show that
$\nabla_{\partial_{u}}\varphi_{*}\frac{\partial}{\partial
u}=-\nu\left(\frac{y^{\prime}}{y}\right)\epsilon_{1}+\left(\frac{y^{\prime}}{2y}\right)^{\prime}\epsilon_{2},$
$\nabla_{\partial_{u}}\varphi_{*}\frac{\partial}{\partial
v}=\frac{1}{4y^{2}}\left\\{-y^{\prime}(\nu+2)\epsilon_{1}+2\nu
y\epsilon_{2}-y^{\prime}\epsilon_{3}\right\\},$
$\nabla_{\partial_{v}}\varphi_{*}\frac{\partial}{\partial
v}=\frac{\nu+1}{2y^{2}}\epsilon_{2}.$
The unit normal vector field ${\mathbf{n}}[\nu]$ is
$\mathbf{n}[\nu]=\frac{1}{\sqrt{1+(1+\nu)(\frac{y^{\prime}}{2y})^{2}}}\left(\frac{y^{\prime}}{2y}\epsilon_{1}+\epsilon_{2}-\frac{\nu
y^{\prime}}{2y}\epsilon_{3}\right).$
Let us denote by
${\mathrm{I}}\\!{\mathrm{I}}={\mathrm{I}}\\!{\mathrm{I}}[\nu]$ the second
fundamental form derived form $\mathbf{n}[\nu]$. The second fundamental form
${\mathrm{I}}\\!{\mathrm{I}}$ is defined by the Gauß formula:
$None$
$\nabla_{X}\varphi_{*}Y=\varphi_{*}(\nabla^{M}_{X}Y)+{\mathrm{I}}\\!{\mathrm{I}}(X,Y)\mathbf{n},\
\ X,Y\in\mathfrak{X}(M).$
Here $\nabla^{M}$ is the Levi-Civita connection of $(M,\mathrm{I}[\nu])$.
Put $\alpha=\sqrt{1+(1+\nu)\\{y^{\prime}/(2y)\\}^{2}}$. Then
$\det\mathrm{I}[\nu]=\nu\alpha^{2}/(4y^{2})$.
The second fundamental form ${\mathrm{I}}\\!{\mathrm{I}}$ is described by the
following formulae:
${\mathrm{I}}\\!{\mathrm{I}}(\frac{\partial}{\partial
u},\frac{\partial}{\partial
u})=\frac{-(1+\nu)y^{\prime}(u)^{2}+y^{\prime\prime}(u)y(u)}{2\alpha
y(u)^{2}},$ ${\mathrm{I}}\\!{\mathrm{I}}(\frac{\partial}{\partial
u},\frac{\partial}{\partial v})=\frac{-(1+\nu)y^{\prime}(u)^{2}+4\nu
y(u)^{2}}{8\alpha y(u)^{3}},$
${\mathrm{I}}\\!{\mathrm{I}}(\frac{\partial}{\partial
v},\frac{\partial}{\partial v})=\frac{(1+\nu)}{2\alpha y(u)^{2}}.$
The mean curvature $H[\nu]$ of $\varphi$ is
(11)
$H[\nu]=\frac{1}{4\alpha^{3}y(u)^{2}}\left\\{(1+\nu)y^{\prime\prime}(u)y(u)+4y(u)^{2}\right\\}.$
Here we used the formula:
$H[\nu]=\frac{1}{2}\>\mathrm{tr}\\{{\mathrm{I}}\\!{\mathrm{I}}[\nu]\cdot\mathrm{I}[\nu]^{-1}\\}.$
Case 1: $\nu=1$
From (11), we have $\varphi$ is minimal if and only if
$y^{\prime\prime}=-2y.$
###### Theorem 3.1
Let $\varphi(u,v)$ an immersed surface in $(G,g[1])$ obtained by taking
inverse image of a curve in $\Lambda$ which is parametrised as (10). Then
$\varphi$ is minimal if and only if $\varphi$ is the inverse image of
$(A\cos(\sqrt{2}u)+B\sin(\sqrt{2}u),u)\in\mathbf{R}^{+}\times S^{1}.$
Case 2: $\nu=-1$
On the other hand, in $(G,g[-1])$, $\varphi$ has constant mean curvature $1$
and Gaußian curvature $0$. Denote by $\mathcal{D}$ the discriminant of the
characteristic equation:
$\det(tI-S)=0$
for the shape operartor $S={\mathrm{I}}\\!{\mathrm{I}}\cdot\mathrm{I}^{-1}$.
Then $\mathcal{D}$ is given by the following formula:
$\mathcal{D}=H^{2}-K-1.$
Thus $(M,\varphi)$ has real and repeated principal curvatures in $H^{3}_{1}$.
Hence $\varphi$ is a $B$-scroll of a null Frenet curve with constant torsion
$1$ in $H^{3}_{1}$. In particular $M$ is flat totally umbilical timelike
surface if and only if it is a $B$-scroll of a null geodesic with constant
torsion $1$. (See Theorem 3 in [8]. )
Comparing the first and second fundamental forms we have the following
###### Proposition 3.1
Let $\varphi(u,v)$ an immersed surface in $H^{3}_{1}$ obtained by taking
inverse image of a curve in $\Lambda$ which is parametrised as (10). Then
$\varphi$ is totally umbilical if and only if $y$ is a solution to
(12) $y^{\prime\prime}-\frac{(y^{\prime})^{2}}{2y}+2y=0.$
The ordinary differential equation (12) with $y>0$ can be solved explicitly.
In fact let us introduce an auxiliary function $\mathscr{T}$ by
$\mathscr{T}(u):=\frac{d}{du}\log y(u).$
Then (12) is rewritten as
$\mathscr{T}^{\prime}+\frac{1}{2}\mathscr{T}^{2}+2=0.$
The general solutions of this ordinary equation are given explicitly by
$\mathscr{T}(u)=-2\tan(u+u_{0}),\ u_{0}\in\mathbf{R}.$
Thus the solutions $y$ to (12) are given by
$y(u)=A\>\cos^{2}(u+u_{0}),\ A>0.$
###### Theorem 3.2
Let $\varphi(u,v)$ an immersed surface in $H^{3}_{1}$ obtained by taking
inverse image of a curve in $\Lambda$. Then $\varphi$ is a $B$-scroll of a
null Frenet curve with constant torsion $1$ in $H^{3}_{1}$. In particular
$\varphi$ is totally umbilical if and only if $\varphi$ is the inverse image
of the curve:
$(A\>\cos^{2}(u+u_{0}),u)\in\mathbf{R}^{+}\times S^{1},\ \ A>0.$
###### Remark 3.1
(Weierstraß-type representations for surfaces in $H^{3}_{1}$)
1. (1)
Hong [13] obtained a Bryant-type representation formula
for timelike constant mean curvature $1$ surfaces in $H^{3}_{1}$.
2. (2)
Balan and Dorfmeister [3] established a loop group theoretic
Weierstraß-type representation (so-called DPW representation) for harmonic
maps of Riemann surface into general Lie group with biinvariant
semi-Riemannian metric. Their general scheme is applicable to maximal
(spacelike) surfaces in $H^{3}_{1}=(\mathrm{SL}_{2}{\mathbf{R}},g[-1])$.
###### Remark 3.2
Hopf cylinders over curves in $H^{2}$ [resp. $S^{2}_{1}$] are surfaces in $G$
which are invariant under $K$-action [resp. $A\mathbf{Z}_{2}$-action].
Surfaces considered in this section are invariant under $N$-action. Thus all
the surfaces investigated in preceding section and present section are
invariant under $1$-dimensional closed subgroup of the isometry group $G\times
K$. In [11], Figueroa, Mercuri and Pedrosa classified all constant mean
curvature surfaces in the Heisenberg group which are invariant under
$1$-dimensional closed subgroups of the isometry group. Some results in [11]
are independently obtained in [14]. Recently S. D. Pauls studied minimal
surfaces in the Heisenberg group with Carnot-Carathéodory metric [21]
## 4 Tangential Gauß maps
5.1 Let $(N^{n},g_{N})$ be a Riemannian $n$-manifold and $O(N)$ the
orthonormal frame bundle of $N$. As is well known, $O(N)$ is a principal
$\mathrm{O}(n)$-bundle over $N$.
Denote by $Gr_{\ell}(T_{p}N)$ be the Grassmannian manifold of $\ell$-planes in
the tangent space $T_{p}N$ of $N$ at $p\in N$. The set
$Gr_{\ell}(TN):=\cup_{p\in N}Gr_{\ell}(T_{p}N)$ of all $\ell$-planes in the
tangent bundle $TN$ admits a structure of fibre bundle over $N$. In fact,
$Gr_{\ell}(TN)$ is a fibre bundle associated to $O(N)$:
$Gr_{\ell}(TN)=O(N)\times_{{\mathrm{O}}(n)}Gr_{\ell}(\mathbf{E}^{n})$
whose standard fibre is the Grassmannian manifold $Gr_{\ell}(\mathbf{E}^{n})$
of $\ell$-planes in Euclidean $n$-space. This fibre bundle $Gr_{\ell}(TN)$ is
called the Grassmannian bundle of $\ell$-planes over $N$.
The canonical 1-form of $O(N)$ and the Levi-Civita connection 1-forms of
$g_{N}$ naturally induces an invariant Riemannian metric
$\langle\cdot,\cdot\rangle$ on $Gr_{\ell}(TN)$. with respect to this metric
the projection $pr:Gr_{\ell}(TN)\to N$ becomes a Riemannian submersion with
totally geodesic fibres. For more details about the metric, see Jensen and
Rigoli [17] and Sanini [23].
###### Definition 4.1
Let $\varphi:M^{m}\to N^{n}$ be an immersed submanifold. Then the (tangential)
Gauß map $\psi:M\to Gr_{m}(TN)$ is defined by
$\psi(p):=\varphi_{*p}(T_{p}M)\in Gr_{m}(T_{p}N),\ \ p\in M.$
###### Remark 4.1
In case the ambient Riemannian $n$-manifold $N$ is a Lie group with left
invariant metric and $M$ is a hypersurface, we can introduce another kind of
Gauß map.
Let $G$ be an $n$-dimensional Lie group with left invariant metric. For an
immersed hypersurface $\varphi:M\to G$ with unit normal ${\mathbf{n}}$, the
normal Gauß map $\Upsilon$ of $M$ is a smooth map into the unit $(n-1)$-sphere
in the Lie algebra ${\mathfrak{g}}$ of $G$ defined by
$\Upsilon(p):=L_{\varphi(p)*}^{-1}{\mathbf{n}}_{p}\in
S^{n-1}\subset\mathfrak{g}.$
In our study for surfaces in $(G,g[1])$, to distinguish the Gauß maps into the
$Gr_{2}(TG)$ from the normal Gauß maps, we use the name “tangential Gauß maps”
for the Gauß maps defined in Definition 4.1.
5.2 Here we recall and colllect fundamental ingredients in the theory of
harmonic maps from the lecture note [9] by Eells and Lemaire.
Let $(M,g_{M})$ and $(P,g_{P})$ be Riemannian manifolds. And let $f:M\to P$ be
a smooth map of a manifold $M$ into $P$. The energy density $e(f)$ of $f$ is a
smooth function on $M$ defined by $e(f):=|df|^{2}/2$. It is obvious that
$e(f)=0$ if and only if $f$ is constant.
The energy $E(f)$ of $f$ is
$E(f):=\int_{M}e(f)\>dV_{M}.$
Here $dV_{M}$ is the volume element of $(M,g_{M})$.
The tension field $\tau(f)$ of $f$ is a smooth section of $f^{*}(TP)$ defined
by
$\tau(f):=\mathrm{tr}\>\nabla df.$
It is known that $f$ is a critical point of the energy if and only if
$\tau(f)=0$.
A map $f$ is said to be a harmonic map if $\tau(f)=0$.
Baird and Eells introduced the notion of stress-energy tensor in [2]. The
stress-energy tensor $\mathcal{S}(f)$ of a map $f$ is a symmetric (0,2)-tensor
field on $M$ defined by
$\mathcal{S}(f):=e(f)g_{M}-f^{*}g_{P}.$
In particular in case $\dim M=2$ and $f$ is nonconstant, $f$ is conformal if
and only if $\mathcal{S}(f)=0$.
Since $\mathcal{S}(f)$ is symmetric $(0,2)$-tensor field, the divergence
$\mathrm{div}\>\mathcal{S}(f)$ of $\mathcal{S}(f)$ can be defined by the
formula:
$\mathrm{div}\>\mathcal{S}(f):=\mathbb{C}_{13}(\nabla\mathcal{S}(f)\>).$
(See p. 86 in [20])
Here $\mathbb{C}_{13}$ is the metric contraction operator in the 1st and 3rd
entries. See p. 83 in [20]. The divergence of $\mathcal{S}(f)$ is given
explicitly by [2]:
$\mathrm{div}\>\mathcal{S}(f)=-g_{P}(\tau(f),df).$
Thus if $f$ is a harmonic map then its stress-energy tensor is conservative.
5.3 Next we recall the notion of vertically harmonic map [26].
Let $(P,g_{P})$ be a Riemannian manifold and $pr:(P,g_{P})\to(N,g_{N})$ a
Riemannian submersion. With respect to the metric $g_{P}$, the tangent bundle
$TP$ of $P$ is decomposed as:
$T_{u}P=\mathcal{H}_{u}\oplus\mathcal{V}_{u},\ u\in P.$
Here $\mathcal{V}_{u}:=\mathrm{Ker}\ (pr_{*})_{u}$ and
$\mathcal{H}_{u}=\mathcal{V}_{u}^{\perp}$ are called the vertical subspace and
horizontal subspace of $T_{u}P$ at $u$ respectively.
Now let $f:(M,g_{M})\to(P,g_{P})$ be a smooth map. With respect to the
Riemannian submersion $pr$, $\tau(f)$ is decomposed into its horizontal and
vertical components:
$\tau(f)=\tau^{\mathcal{H}}(f)+\tau^{\mathcal{V}}(f).$
The map $f$ is said to be a vertically harmonic map if the vertical component
$\tau^{\mathcal{V}}(f)$ vanishes.
In case $f:M=N\to P$ is a section of $P$, i.e., a smooth map satisfying
$pr\circ f=\mathrm{identity}$, C. M. Wood [26] showed that the vertical
harmonicity for maps is equivalent to the criticality for the vertical energy
under the vertical variations.
5.4 Now we investigate harmonicity of tangential Gauß maps for surfaces in
$(G,g[1])$.
The following fundamental result is due to Sanini (See (3.2)–(3.3) in [23]).
###### Lemma 4.1
Let $N$ be a Riemannian $3$-manifold and $\varphi:M\to N$ an immersed surface
with unit normal vector field ${\mathbf{n}}$. Take a principal frame field
$\\{e_{1},e_{2},e_{3}=\mathbf{n}\\}$, i.e., an orthonormal frame field such
that $\\{e_{1},e_{2}\\}$ diagonalise the shape operator. Put
$R_{ijkl}=g_{N}(R(e_{i},e_{j})e_{k},e_{l})$
and denote by $\psi$ the tangential Gauß map of $(M,\varphi)$. Then the
following holds.
(1) The tangential Gauß map $\psi$ is conformal if and only if $(M,\varphi)$
is totally umbilical or minimal.
(2) Assume that $(M,\varphi)$ has constant mean curvature. Then $\psi$ is
vertically harmonic if and only if $R_{1213}=R_{2123}=0$. Moreover when
$(M,\varphi)$ is minimal, $\psi$ is harmonic if and only if, in addition,
$R_{3113}=R_{3223}=0$.
(3) Assume that the mean curvature is nonzero constant. Then the tangential
Gauß map is vertically harmonic if and only if the stress energy tensor
$\mathcal{S}(\psi)$ of the tangential Gauß map $\psi$ is conservative
(divergence free).
Sanini applied this Lemma to surfaces in 3-dimensional Heisenberg group with
canonical left invariant metric [23].
Lemma 4.1 together with the nonexistence of extrinsic spheres (See Remark 2.2
and [6]) implies the following.
###### Corollary 4.1
Let $M$ be a constant mean curvature surface in $(G,g[1])$. Then $M$ is
minimal if and only if its tangential Gauß map is conformal.
The following is the main result of this section222This result is generalised
to $3$-dimensional Sasakian space forms by M. Tamura (Comment. Math. Univ. St.
Pauli 52 (2003), no. 2, 117–123..
###### Theorem 4.1
Let $M$ be a surface in $(G,g[1])$ with constant mean curvature. Then the
tangential Gauß map of $M$ is vertically harmonic if and only if $M$ is a Hopf
cylinder ( rotational surface) of constant mean curvature. Hopf cylinders with
nonzero constant mean curavture are (only) constant mean curvature surfaces
whose tangential Gauß map are vertically harmonic but nonharmonic and have
conservative stress-energies.
In particular the only minimal surface in $(G,g[1])$ with vertically harmonic
tangential Gauß map is a Hopf cylinder over a geodesic. In this case the
tangential Gauß map is a harmonic map.
Proof. Let $\varphi:M\to(G,g[1])$ be a surface with constant mean curvature
and unit normal vector field ${\mathbf{n}}$. Denote by $\theta^{3}$ the dual
one-form of ${\mathbf{n}}$. Express $\theta^{3}$ by
$\theta^{3}=a\>\omega^{1}+b\>\omega^{2}+c\>\omega^{3},\ \ a^{2}+b^{2}+c^{2}=1$
in terms of the coframe field (2).
1. (1)
Case 1 $c\neq 0$: In this case,
$v_{1}=-c\epsilon_{2}+b\epsilon_{3},\
v_{2}=(b^{2}+c^{2})\epsilon_{1}-ab\epsilon_{2}-ac\epsilon_{3}$
gives a orthogonal frame field of $M$.
Direct computations show the following formulae:
$g[1](R(v_{1},v_{2})v_{1},\mathbf{n})=8ac^{2}(b^{2}+c^{2}),\ \
g[1](R(v_{1},v_{2})v_{2},\mathbf{n})=8bc(b^{2}+c^{2}).$
Take a principal frame $\\{e_{1},e_{2}\\}$. Then $\\{e_{1},e_{2}\\}$ is
expressed as
(13) $e_{1}=\cos\mu\frac{v_{1}}{|v_{1}|}+\sin\mu\frac{v_{2}}{|v_{2}|},\ \
e_{2}=-\sin\mu\frac{v_{1}}{|v_{1}|}+\cos\mu\frac{v_{2}}{|v_{2}|}.$
Then we have
$R_{1213}=\frac{8c(b^{2}+c^{2})}{|v_{1}||v_{2}|}\left(\frac{ac}{|v_{1}|}\cos\mu+\frac{b}{|v_{2}|}\sin\mu\right),$
$R_{2123}=\frac{8c(b^{2}+c^{2})}{|v_{1}||v_{2}|}\left(\frac{ac}{|v_{1}|}\sin\mu-\frac{b}{|v_{2}|}\cos\mu\right).$
From these we have $\tau^{\mathcal{V}}(\psi)=0$ if and only if $a=b=0$. Hence
$\theta^{3}=-\eta$. Namley $M$ is an integral surface of the distribution
$\eta=0$, but this is impossible, since $\eta$ is contact. (See p. 36, Theorem
in [7]).
2. (2)
Case 2 $c=0$: Since $a^{2}+b^{2}=1$, we may write $a=\cos\phi,\ b=\sin\phi$.
In this case $u_{1}=\sin\phi\epsilon_{1}-\cos\phi\epsilon_{2},\
u_{2}=\epsilon_{3}$ are orthonormal and tangent to $M$. The unit normal
${\mathbf{n}}$ is given by
${\mathbf{n}}=\cos\phi\epsilon_{1}+\sin\phi\epsilon_{2}$. Then we have
(14) $R(u_{1},u_{2})u_{1}=-\sin^{2}\phi\epsilon_{3},\ \
R(u_{2},u_{1})u_{2}=-\sin\phi\epsilon_{1}+\cos\phi\epsilon_{2}.$
Let us denote by $\mu$ the angle between the principal frame
$\\{e_{1},e_{2}\\}$ and $\\{u_{1},u_{2}\\}$, i.e.,
(15) $e_{1}=\cos\mu\>u_{1}+\sin\mu\>u_{2},\ \
e_{2}=-\sin\mu\>u_{1}+\cos\mu\>u_{2}.$
Using (14) and (15), we have $R_{1213}=R_{2123}=0$. Thus
$\tau^{\mathcal{V}}(\psi)=0$ is fulfilled automatically for $M$ with $c=0$.
We have shown in [6] that constant mean curvature surfaces with $c=0$ are Hopf
cylinder of constant mean curvature. See the proof of Thereom in [6].
Furthermore, the second fundamental form ${\mathrm{I}}\\!{\mathrm{I}}$ of $M$
relative to ${\mathbf{n}}$ is given by (cf. (5) and (8) in [6])
(16) ${\mathrm{I}}\\!{\mathrm{I}}(u_{1},u_{1})=2H,\
{\mathrm{I}}\\!{\mathrm{I}}(u_{1},u_{2})=1,\ \
{\mathrm{I}}\\!{\mathrm{I}}(u_{2},u_{2})=0.$
Next we see the case $\psi$ is harmonic. Using (14) and (15) again, we have
$R_{3113}=-7\cos^{2}\mu+\sin^{2}\mu,\ \ R_{3223}=-7\sin^{2}\mu+\cos^{2}\mu.$
Thus $R_{3113}=R_{3223}$ if only if $\mu=\pm\pi/4$. Without loss of
generality, we may assume $\mu=\pi/4$. In this case the principal frame
$\\{e_{1},e_{2}\\}$ is given by
$e_{1}=\frac{1}{\sqrt{2}}(u_{1}+u_{2}),\ \
e_{2}=\frac{1}{\sqrt{2}}(-u_{1}+u_{2}).$
By definition, ${\mathrm{I}}\\!{\mathrm{I}}(e_{1},e_{2})=0$. On the other
hand, direct computation using (16) shows
${\mathrm{I}}\\!{\mathrm{I}}(e_{1},e_{2})=-H$. Thus a constant mean curvature
surface $M$ with $c=0$ satisfying $R_{3113}=R_{3223}$ is minimal.
Conversely one can check that every rotational surface of constant mean
curvature has vertically harmonic tangential Gauß map and when $H\not=0$, the
tension field does not vanish by direct computations. It is also
straightforward to check that every minimal Hopf cylinder has harmonic
tangentail Gauß map. $\Box$
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* [15] J. Inoguchi, T. Kumamoto, N. Ohsugi and Y. Suyama, Differential geometry of curves and surfaces in $3$-dimensional homogeneous spaces $\mathrm{I}\\!\mathrm{I}\\!\mathrm{I}$, Fukuoka Univ. Sci. Rep. 30 (2000), 131–160.
* [16] J. Inoguchi, T. Kumamoto, N. Ohsugi and Y. Suyama, Differential geometry of curves and surfaces in $3$-dimensional homogeneous spaces $\mathrm{I}\\!\mathrm{V}$, Fukuoka Univ. Sci. Rep. 30 (2000), 161–168.
* [17] G. Jensen and M. Rigoli, Harmonic Gauss maps, Pacific J. Math. 136 (1989), 261–282.
* [18] M. Kokubu, On minimal surfaces in the real special linear group $SL(2,{\mathbf{R}})$, Tokyo J. Math. 20 (1997), 287–297.
* [19] M. Magid, Isometric immersions of Lorentz space with parallel second fundamental forms, Tsukuba J. Math. 8 (1984), 31–54.
* [20] B. O’Neill, Semi-Riemannian Geometry with Application to Relativity, Academic Press, 1983.
* [21] S. D. Pauls, Minimal surfaces in the Heisenberg group, Geom. Dedicata 104 (2004), 201–231, math.DG/0108048.
* [22] U. Pinkall, Hopf tori in $S^{3}$, Invent. Math. 81 (1985), 379–386.
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* [24] T. Takahashi, Sasakian manifold with pseudo-Riemannian metric, Tôhoku Math. J. (2) 21 (1969), 271–290.
* [25] F. Tricerri and L. Vanhecke, Homogeneous Strctures on Riemannian manifolds, London Math. Soc. Lecture Note Series 83, Cambridge University Press, Cambridge, 1983\.
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J. Inoguchi
Department of Applied Mathematics
Fukuoka University
Nanakuma, Fukuoka, 814-0180
Japan
inoguchi@bach.sm.fukuoka-u.ac.jp
Current address
Department of Mathematical Sciences
Yamagata University
Yamagata 990-8560
Japan
inoguchi@sci.kj.yamagata-u.ac.jp
|
arxiv-papers
| 2009-10-16T12:43:32 |
2024-09-04T02:49:05.863159
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jun-ichi Inoguchi",
"submitter": "Jun-ichi Inoguchi",
"url": "https://arxiv.org/abs/0910.3104"
}
|
0910.3158
|
1–30
# Eta Carinae in the Context of the Most Massive Stars
Theodore R. Gull1 Augusto Damineli2 1Laboratory for Extraterrestial Planets
and Stellar Astrophysics, Code 667, NASA/GSFC, Greenbelt, MD, 20771, USA,
email: Theodore.R.Gull@nasa.gov
2IAGUSP, Universidade de Sao Paulo, Rua do Matao 1226, Sao Paulo, 05508-900,
Brazil, email: damineli@astro.iag.usp.br
(2009; Sep 30, 2009 and in revised form ??)
††volume: Volume 15††journal: Highlights of Astronomy, Volume 14††editors: Ian
F Corbett, ed.
Eta Car, with its historical outbursts, visible ejecta and massive, variable
winds, continues to challenge both observers and modelers. In just the past
five years over 100 papers have been published on this fascinating object. We
now know it to be a massive binary system with a 5.54-year period. In January
2009, Eta Car underwent one of its periodic low-states, associated with
periastron passage of the two massive stars. This event was monitored by an
intensive multi-wavelength campaign ranging from $\gamma$-rays to radio. A
large amount of data was collected to test a number of evolving models
including 3-D models of the massive interacting winds. August 2009 was an
excellent time for observers and theorists to come together and review the
accumulated studies, as have occurred in four meetings since 1998 devoted to
Eta Car. Indeed, Eta Car behaved both predictably and unpredictably during
this most recent periastron, spurring timely discussions.
Coincidently, WR140 also passed through periastron in early 2009. It, too, is
a intensively studied massive interacting binary. Comparison of its
properties, as well as the properties of other massive stars, with those of
Eta Car is very instructive. These well-known examples of evolved massive
binary systems provide many clues as to the fate of the most massive stars.
What are the effects of the interacting winds, of individual stellar rotation,
and of the circumstellar material on what we see as hypernovae/supernovae? We
hope to learn.
Topics discussed in this 1.5 day Joint Discussion were:
Eta Car: the 2009.0 event: Monitoring campaigns in X-rays, optical, radio,
interferometry
WR140 and HD5980: similarities and differences to Eta Car
LBVs and Eta Carinae: What is the relationship?
Massive binary systems, wind interactions and 3-D modeling
Shapes of the Homunculus & Little Homunculus: what do we learn about mass
ejection?
Massive stars: the connection to supernovae, hypernovae and gamma ray bursters
Where do we go from here? (future directions)
The Science Organizing Committee:
Co-chairs: Augusto Damineli (Brazil) & Theodore R. Gull (USA). Members: D.
John Hillier (USA), Gloria Koenigsberger (Mexico), Georges Meynet
(Switzerland), Nidia I. Morrell (Chile), Atsuo T. Okazaki (Japan), Stanley P.
Owocki (USA), Andy M.T. Pollock (Spain), Nathan Smith (USA), Christiaan L.
Sterken (Belgium), Nicole St Louis (Canada), Karel A. van der Hucht
(Netherlands), Roberto Viotti (Italy) and Gerd Weigelt (Germany)
Website for talks and posters:
http://astrophysics.gsfc.nasa.gov/research/etacar/IAUJD.html
## 1 Oral Presentations
### 1.1 Dedication to Prof. Sveneric Johansson (Henrik Hartman)
Professor Sveneric Johansson is remembered for his important contributions to
the know- ledge on atomic data, focusing on the iron group elements in general
and singly ionized iron, Fe ii, in particular. His work includes term analysis
of several important ions, and measurements of atomic parameters for
astrophysicaly important elements. His thorough knowledge of atomic structure
also allowed major contributions to the analysis of complex astronomical
spectra and atomic photo processes. Sveneric is greatly missed as an ingenious
scientist, positive colleague and a great friend.
Figure 1: Professors Sveneric Johansson and Vladelin Letokhov discussing the
stimulated emission properties of the ionized ejecta surrounding Eta Car. Both
researchers passed away this past year. Their interest in atomic spectroscopy
and enthusiasm was infectious to all.
Sveneric received his PhD from Lund University in 1978 under the supervision
of Professor Bengt Edlén, on the subject of term analysis of Fe ii (the
spectrum of Fe+). This work continued to be his main research topic for more
than 35 years. Sveneric led classical atomic spectroscopy into a new era of
measurements with crucial astronomical applications. He spent a sabbatical
year at NASA’s Goddard Space Flight Center with Dave Leckrone during
1987-1988, starting up a collaboration for the upcoming Hubble Space Telescope
(HST) mission and the $\chi$ Lupi pathfinder project. The high resolution
spectrographs onboard HST challenged existing laboratory atomic data bases.
Sveneric foresaw the need of high-accuracy ultraviolet data and directed,
together with Ulf Litzén, the Lund University spectroscopy laboratory to
measure wavelengths, isotopic shifts and line structures needed to interpret
astronomical observations. Spectroscopic investigations included iron,
yttrium, mercury, boron, gold, ruthenium, nickel, thallium, platinum, and
zirconium.
The high cosmic abundance of iron makes Fe ii lines abundant in a variety of
astronomical objects. For quantitative analyses the intrinsic strength of the
spectral lines need to be known. In 2001 Sveneric founded the Atomic
Astrophysics group at Lund University and organized the FERRUM project, an
international collaboration on oscillator strengths for iron group elements.
The aim of this project is to present a fully evaluated and consistent set of
values, experimental and theoretical, that can be used for astronomical
analyses.
Throughout his career Sveneric also analyzed complex astronomical emission
line spectra, and was especially interested in atomic photo processes.
Together with Professor Vladilen Letokhov he identified and developed the idea
of stimulated emission (LASER) in gas condensations close to the massive star
Eta Carinae. From the strange behavior observed and ionization structure of
the high ionization lines, they derived the concept of resonance-enhanced two-
photon ionization (RETPI) of Ne and Ar atoms as an explanation for the
production of these ions.
In addition, it is with great sadness, that we learnt of Dr. Vladelen
Letokhov’s passing during 2009. He is greatly missed by colleagues and friends
all over the world. During his productive career he published nearly 900
articles and 16 monographs. Sveneric’s and Vladilen’s work on photo processes
culminated in their book ’Astrophysical Lasers’ (Oxford Press, 2009) published
earlier this year.
### 1.2 The historical background on Eta Car (D. John Hillier)
Eta Carinae, a spectacular object, is one of the most luminous stars in the
galaxy, and exhibits a wide range of interesting phenomena with implications
for many areas of astrophysics. In this presentation we provide a brief
summary of key discoveries and an introduction to some jargon associated with
Eta Car.
Figure 2: An example of what is so intriguing about Eta Car: the extended wind
and ejecta. A 0.1”-wide slit of the Hubble Space Telescope Imaging
Spectrograph samples the extended structure surrounding Eta Car as imaged by
Hubble Space Telescope. Continuum and broad line emission at the center of the
spectrum originate from the extended interacting winds. Narrow forbidden
emission lines shifted with velocities up to 500 km s-1 come from the interior
of the Homunculus, thrown out in the 1840s. An estimated 10 to 20 $M_{\odot}$
was ejected during the Great Eruption as well as up to 0.5 $M_{\odot}$ in the
lesser eruption of the 1890s. How did the ejecting star survive and what clues
does this provide us on the late stages of massive stellar evolution? (Image
courtesy of NASA and STScI)
Eta Carinae, a spectacular object, is one of the most luminous stars in the
galaxy, and exhibits a wide range of interesting phenomena with implications
for many areas of astrophysics. In this presentation we provide a brief
summary of key discoveries and an introduction to some jargon associated with
Eta Carinae.
In the 1840’s Eta Carinae underwent a giant outburst and ejected a nebula
which we call the Homunculus. The event was so impressive that Eta Carinae was
classified as a peculiar SN. With the onset of dust formation, it suffered a
dramatic drop in brightness by $\sim 6$ magnitudes (e.g., van Genderen et al.
1984, Space Sci. Rev., 39, 317). In the early 1890’s Eta Carinae underwent a
smaller outburst ejecting the Little Homunculus nebula (discovered with the
HST; Ishibashi et al. 2003, AJ, 125, 3222).
The Homunculus is a bipolar nebula whose axis is tilted at about 41∘ to our
line of sight. H2 emission and dust is confined to a thin outer layer, while
[Fe ii] & [Ni ii] emission lines originate inside this shell (Smith et al.
2006, ApJ, 644, 1151). From infrared observations the mass of the Homunculus
is inferred to exceed 10$M_{\odot}$ (Smith et al, 2003, AJ, 125, 1458), and is
possibly as large as 20$M_{\odot}$ (Smith et al. 2007, ApJ, 655, 911). In
contrast, the mass of the Little Homunculus is $\sim 0.1M_{\odot}$ (Smith
2005, MNRAS, 357, 1330).
S-condensation ejecta (a condensation to the south of the Homunculus) are N
enhanced and CO depleted, consistent with the influence of CNO processing
(Davidson et al. 1982, ApJ, 254, L47). A similar abundance pattern is seen in
the star (Hillier et al. 2001, ApJ, 553, 837). As Eta Carinae is located in a
region of massive star formation (Walborn et al. 1977, ApJ, 211, 181), it is
inferred that it is a young, but evolved, massive star.
Speckle observations showed that Eta Carinae is composed of 4 ‘star-like’
objects (Weigelt et al. 1986, A&A, 163, L5). Subsequent HST observations
revealed that the brightest of these is truly star-like, while the remaining 3
are small nebula (the Weigelt blobs) which emit the narrow permitted and
forbidden lines that are prominent in ground-based spectra (Davidson et al.
1995, AJ, 109, 1784); they are prominent because the primary star suffers
additional extinction ($\sim$ 5 magnitudes in 1997; Hillier et al. 2001, ApJ,
553, 837).
The discovery of a 5.5 year variability cycle (Damineli 1996, ApJ, 460, L49)
led to the realization that Eta Carinae is a binary system (Damineli et al.
1997, New Astr., 2, 107). A wide range of phenomena, including infrared
(Whitelock et al. 2004, MNRAS, 352, 447), X-ray (Ishibashi et al. 1999, ApJ,
524, 983; Corcoran 2005, AJ, 129, 2018), radio (Duncan et al. 1999, ASP Conf.
Ser. 179, 54), and line variability (Damineli et al. 2008, MNRAS, 386, 2330)
indicate that we are dealing with a binary system with a large orbital
eccentricity ($\epsilon\sim 0.9$).
The spectrum of the primary is similar to the P Cygni star HDE 316285.
Modeling places a lower limit of 60$R_{\odot}$ on the radius of the central
star, although with a re-interpretation of the He i emission lines a larger
radius ($\sim 240R_{\odot}$) is now preferred. Because of the very dense wind
we observe the wind — not the “normal” photosphere of the star ($\dot{M}\sim
10^{-3}$ $M_{\odot}$/yr; Hillier et al. 2001, ApJ, 553, 837).
UV spectra reveal multiple systems of narrow absorption lines arising from
neutral and singly ionized metals, and from H2 (Gull et al. 2005, ApJ, 620,
442). The two dominant systems are associated with the Little Homunculus and
the Homunculus, with other systems thought to be related to structures arising
from the periodic interaction between the winds of the primary and secondary
stars.
HST observations show that the central star has brightened – by over a factor
of 3 since the first HST observations (Martin et al. 2004, AJ, 127, 2352).
This is presumably due to a reduction in extinction, since spectra of the
star, and the Weigelt blobs, have not shown dramatic changes. Variability
observations show that spectral changes occur throughout the 5.5 year cycle.
This provides additional evidence for binarity; the variability most likely
arises from illumination effects of the Weigelt blobs as the secondary star
(believed to be responsible for ionizing the Weigelt blobs) moves in its
orbit.
HST observations show that the broad He i emission lines most likely originate
in the neighborhood of the wind-wind interface, and are not excited by the
primary star. They exhibit complex radial velocity and profile variations
which are broadly consistent with those expected in a binary system (Nielsen
et al. 2007, ApJ, 660, 669).
### 1.3 The 2009 monitoring campaign
#### 1.3.1 The X-ray light curve (Michael F. Corcoran & Kenji Hamaguchi)
Figure 3: Left: Overplot of Eta Car’s 3 X-ray minima observed by RXTE (in the
2–10 keV band). The 2009 minimum showed an abrupt recovery compared to the two
earlier minima. Right: RXTE hardness ratio compared to the PCU2 net rate. All
three minima show a marked increase in hardness towards the end of the X-ray
minima through flux recovery.
X-ray photometry in the 2–10 keV band of the the supermassive binary star Eta
Car has been measured with the Rossi X-ray Timing Explorer from 1996–2009 (see
Fig. 1). The ingress to X-ray minimum is consistent with a period of 2024
days. The 2009 X-ray minimum began on January 16 2009 and showed an
unexpectedly abrupt recovery starting after 12 Feb 2009. This is about one
month earlier than the flux recovery in the two earlier minima (in 2003.5 and
1998). This recovery roughly corresponds in phase to the “shallow minimum” of
Hamaguchi et al (2007 ApJ 663, 522), and suggests that for the most recent
cycle the “shallow minimum” was very shallow indeed, or did not occur at all.
Figure 1 also shows the hardness ratio measured by RXTE compared to the RXTE
fluxes. The X-ray colors become harder about half-way through all three minima
and continue until flux recovery. The behavior of the fluxes and X-ray colors
for the most recent X-ray minimum (which corresponds to the time of periastron
passage of an unseen companion star) suggests a significant change in the
inner wind of Eta Car and might suggest that the star is entering a new
unstable phase of variable mass loss.
#### 1.3.2 Optical photometry of the 2009.0 event of Eta Car (Eduardo
Fernandez-Lajus, Cecilia Farina, Juan P. Calderon, Martan A. Schwartz, Nicolas
E. Salerno, Carolina von Essen, Andrea F. Torres, Federico N. Giudici,
Federico A. Bareilles, M. Cecilia Scalia & Cintia S. Peri)
Figure 4: Optical BVRI light curves from monitoring Eta Car by the La Plata
Observatory. While the fluxes are trending brighter, most noticeable are the
broad bump associable with apastron and the two narrow drops associated with
the 2003.5 and 2009.0 periastron events.
During the last “event” that ocurred in 2009.0, Eta Car was the target of
several observing programs. Through our optical photometric monitoring
campaign, we recorded in detail the behavior of the associated “eclipse-like”
event, which happened fairly on schedule. In this work we present the
resulting $UBVRI$ and H$\alpha$ light curves, and a new determination of the
present period length.
Our ground-based photometry was performed from the beginning of the 2009
observing season of Eta Car, using two telescopes at La Plata Observatory and
Complejo Astronómico El Leoncito, both located in Argentina. The $UBVRI$ and
H$\alpha$ light curves obtained display once more an “eclipse-like”
appearance. This feature is preceded by an ascending branch which peaks a
maximum one month later. A sudden drop of 0.15 - 0.26 mag (depending on the
band) reaches a minimum nearly simultaneously in the six bands. Then, the
recovery phase starts and the brightness increases steeply up to the end of
the season. The color indices show some particularities during the event,
specially a blueing peak in $V-R$. Although the general trend of this event is
quite similar to that of the 2003.5, there are some differences, specially the
deeper dips of the minima and the high increasing rate after the “eclipse-
like” feature. Our long term photometry shows some evidence of systematic
brightenings of the central region (r $<$ 3”) relative to the complete
“Homunculus” (r $<$ 12”) occurring just after each of these last two events.
Our results provided more observational evidence on the periodic origin of the
events occurring at Eta Car, in accordance with the proposed binary nature of
this object.
#### 1.3.3 VLTI/AMBER interferometry and VLT/CRIRES spectroscopy of Eta Car
across the 2009.0 spectroscopic event (Gerd Weigelt, José H. Groh, Thomas
Driebe, Karl-Heinz Hofmann, Stefan Kraus, Dieter Schertl, P. Bristol, Augusto
Damineli, Theodore Gull, Henrik Hartman, Florian Kerber, Florentin Millour,
Koji Murakawa & Krister E. Nielsen)
Eta Car’s 2009.0 spectroscopic event provided a unique opportunity to study
the changes of Eta Car’s primary wind and wind-wind interaction region. The
goals of VLTI/AMBER observations in 2008 and 2009 were to study the
wavelength-dependent shape of Eta Car’s aspherical stellar wind and wind-wind
interaction region across the 2009.0 spectroscopic event. We carried out a
large number of VLTI/AMBER observations with spectral resolution of 12000 in
April 2008, January 2009, March 2009, and April 2009. We measured that the
size of the wind did not significantly change at the wavelength of the
Br$\gamma$ 2.16 $\mu$m line during our event observations from Jan 1 to 8.
However, during the event, the size of the He I 2.06 $\mu$m emitting region
collapsed from 17 mas (continuum-subtracted 50% encircled energy diameter
before the event) to only 6 mas during the event. Therefore, we found strong
evidence for the collapse of the wind-wind interaction zone during periastron
passage.
In addition, we obtained near-IR long-slit spectroscopy of Eta Car with very
high spatial ($0.2^{\prime\prime}$) and spectral ($R$ = 100 000) resolution
using VLT/CRIRES. These unique data provided definitive evidence that high-
velocity material, up to $\sim-1900~{}{\rm km\,s^{-1}}$, was present in the
wind region of Eta Car during the 2009.0 periastron passage. A broad, high-
velocity absorption is seen in He I $\lambda$10833 only in the spectrum of
2008 Dec 26 – 2009 January 07, which strongly suggests a connection with the
periastron passage, since a brief appearance of high-velocity material was
also detected during previous periastron passages. We suggest that the high-
velocity absorption is either formed directly in the wind of the companion
star or, most likely, is due to shocked, high-velocity material from the wind-
wind collision zone.
#### 1.3.4 HeII 4686A in Eta Car: The Data and Modeling (Augusto Damineli,
Mairan Teodoro, Joao E. Steiner, Nidia I. Morrell, Rodolfo H. Barba, Gladys
Sollivela, Roberto C. Gamen, Eduardo Fernandez-Lajus, Federico Gonzalez,
Carlos A. O. Torres, Jose Groh, Luciano Fraga, Claudio B. Pereira , Marcelo
Borges Fernandes, Maria I. Zevallos & Peter McGregor)
Figure 5: Line flux (photons per second) in the He ii $\lambda$4686 spectral
line along cycles #11 (2003.5) and #12 (2009.0).
The intrinsic emission of He ii is quite repeatable from cycle to cycle. The
He ii $\lambda$4686 line flux rises by a factor of $\approx$10 in the 2 months
preceding phase zero. There are two local maxima in the month preceding the
minimum and a secondary maximum $\approx$50 days after phase zero. The rising
before phase zero resembles that seen in X-rays, but with remarkable
differences. The He ii line flux increases by a factor of $\approx$10 as
compared to only a few times in X-ray emission. Both light curves collapse
before phase zero, but the collapse of He ii is shifted by 16.5 days relative
to the X-ray collapse. The minimum in He ii is reached a week after phase
zero. Since the X-ray variability is measured in the range 2–10 keV, and comes
mainly from the vertex of the wind-wind shock cone, it is probably not common
to the He ii emitting region, which comes from gas at lower temperature. The
He ii line indicates a high luminosity source in the system, but it is not
clear where it comes from. One possible source is the collision of the
secondary star wind, since the SED derived from Parkin et al. 2009 (MNRAS 394,
1758) models indicates the presence of 10 times more He+ ionizing photons than
those passing through this atomic transition. Recombination of the shocked
secondary wind is not the only source for the He+ ionizing photons. As the
shock cone migrates deep in the wind of the primary star, a huge amount of
hard photons are free to escape and ionize the inner walls of the wind-wind
collision zone.
### 1.4 3-D Modeling and Application
#### 1.4.1 3-D models of the colliding winds in Eta Car (Julian M. Pittard,
E. Ross Parkin, Michael F. Corcoran, Kenji Hamaguchi & Ian R. Stevens)
A 5.5 yr periodicity is now firmly established for Eta Car, with variations
seen at radio, sub-mm, infrared, optical, and X-ray energies (Duncan & White
2003 (MNRAS 338, 425), Abraham et al. 2005 (MNRAS 364, 922), Corcoran 2005 (AJ
129, 2018), Damineli et al. 2008 (MNRAS 384, 1649)). The overwhelming
consensus is that this emission is regulated by the presence of an (unseen)
companion, with the emission either originating in the wind-wind collision
region between the stars (e.g., as for the X-rays, see Pittard & Corcoran 2002
(A&A 383, 636)), or being influenced by its presence and the low-density
cavity which the wind of the companion star bores into the dense wind of the
LBV primary (e.g., as for the radio emission).
The X-ray emission from Eta Car is believed to originate in the hot plasma
created by the high-speed wind of the companion star shocking against the
denser LBV wind (e.g., Pittard et al. 1998 (MNRAS 299, L5),Pittard & Corcoran
2002 (A&A 383, 636)). We present a recent analysis of the RXTE X-ray
lightcurve, using a 3-D model with spatially and energy dependent X-ray
emission (Parkin et al. 2009 (MNRAS 394, 1758)). The model fails to obtain a
good match to the data through the minimum and overpredicts the hardness of
XMM-Newton spectra (Hamaguchi et al. 2007 (ApJ 663, 522)). We find that the
pre-shock speed of the companion wind must substantially decrease around
periastron passage, and that this reduction lasts for longer than expected
post-periastron. This implies that the companion wind no longer shocks at high
speed against the LBV wind at this time. We speculate that this is either
because the wind-wind collision region deforms into a multitude of oblique,
radiative shocks, or the LBV wind completely overwhelms it and accretes onto
the companion star (Soker 2005 (ApJ 635, 540)). We conclude by presenting 3-D
hydrodynamical models of the colliding winds, noting several interesting
features as the stars swing through periastron passage.
#### 1.4.2 3-D Numerical Simulations of Colliding Winds in Eta Car & WR140
(Atsuo T. Okazaki, Stanley P. Owocki, Christopher M. P. Russell, Thomas I.
Madura & Michael F. Corcoran)
We report on the results from 3-D SPH simulations of colliding winds in the
supermassive binary, Eta Car, and the proto-typical Wolf-Rayet binary, WR 140.
For simplicity, both winds are assumed to be either isothermal or adiabatic.
Our simulations show that in Eta Car the lower-density, faster wind from the
secondary carves out a spiral cavity in the higher-density, slower wind from
the primary, whereas in WR 140 it is the lower-density, primary (O4-5V) wind
that carves out a spiral cavity in the denser wind from the secondary (WC7).
Because of their very-high orbital eccentricities, both systems show a
similar, strongly asymmetric interaction surface: the cavities are very thin
on the periastron side, whereas on the apastron side they occupy a large
volume separated by thin dense shells. A closer look, however, reveals
differences caused by the differences in the wind momentum ratio and the speed
of the slower wind: the shock opening angle is wider and the spiral structure
is more tightly wound in Eta Car than in WR 140. These differences are likely
to affect the observational appearance of these systems.
#### 1.4.3 Precession and Nutation in Eta Car (Zulema Abraham & Diego
Falceta-Goncalves)
Although the overall shape of the X-ray light curve of Eta Car can be
explained by the high eccentricity of the binary orbit, other features, like
the asymmetry near periastron passage and the short quasi-periodic
oscillations seen at those epochs, have not yet been accounted for. We explain
these features assuming that the rotation axis of Eta Car is not perpendicular
to the orbital plane of the binary system. As a consequence, the companion
star will face Eta Car on the orbital plane at different latitudes for
different orbital phases and, since both the mass loss rate and the wind
velocity are latitude dependent, they would produce the observed asymmetries
in the X-ray flux. We were able to reproduce the main features of the X-ray
light curve assuming that the rotation axis of Eta Car forms an angle of 29
degrees with the axis of the binary orbit. We also explained the short quasi-
periodic oscillations by assuming nutation of the rotation axis, with
amplitude of about 5 degrees and period of about 22 days. The nutation
parameters, as well as the precession of the apsis, with a period of about 274
years, are consistent with what is expected from the torques induced by the
companion star.
#### 1.4.4 Accretion onto the Companion of Eta Car (Amit Kashi & Noam Soker)
The Accretion Model was introduced to explain observations along the entire
orbit, mainly those close around the spectroscopic event. We use the standard
parameters of the system and show that near periastron the secondary is very
likely to accrete mass from the slow dense wind blown by the primary. The
condition for accretion (that the accretion radius is large) lasts for several
weeks. The exact duration of the accretion phase is sensitive to the winds’
properties that can vary from cycle to cycle.
We find that: (1) The secondary accretes $\sim 2\times 10^{-6}\rm
M_{\odot}yr^{-1}$ close to periastron. (2) This mass possesses enough angular
momentum to form a geometrically thick accretion belt, around the secondary.
(3) The viscous time is too long for the establishment of equilibrium, and the
belt must dissipate as its mass is blown in the re-established secondary wind.
This processe requires about half a year, which we identify with the recovery
phase of Eta Car from the spectroscopic event.
We attribute the early exit in the 2009 event to the primary wind that we
assume was somewhat faster and of lower mass loss rate than during the two
previous X-ray minima. This results in a much lower mass accretion rate during
the X-ray minimum, and consequently faster recovery of the secondary wind and
the conical shell.
Mass transfer is an important process in the evolution of close massive star
binaries. The high luminosity and ejected mass of many eruptive events can be
explained by mass transfer, e.g., the Great Eruption of Eta Car.
#### 1.4.5 The outer interacting winds of Eta Car revealed by HST/STIS
(Theodore R. Gull – presented by Michael F. Corcoran)
High spatial resolution (0.1”) with moderate spectral resolution has been
applied to mapping the extended wind structure of Eta Car. Emission lines of
[Ne iii], [Ar iii]. [Fe iii], [S iii] and [N ii] show an extended outer
structure associable with the extended wind interaction regions. [Fe ii]
reveals the structure of the primary wind. We followed the spectro-images of
these lines from the 1998.0 through the 2003.5 minima, finding changes in
structure and velocity as the two massive winds, originating from a highly
eccentric massive binary, interact.
Comparison of the forbidden line emission spatial structures to 3-D models
(see Gull et al., 2009, MNRAS 396, 1308) shows 1) that the He i and H i,
consistent with the observations of Weigelt et al (2007, A&A 474, 87),
originate deep within the 0.1” limit of HST angular resolution, 2) that the
broad [Ne iii], [Fe iii], [Ar iii], [S iii] and [N ii] profiles are blue-
shifted relative to the broad H i, Fe ii and [Fe ii] profiles. Moreover, the
spatial distributions of the high excitation, forbidden emissions are oriented
in a NE to SW distribution in the form of arcuate velocity loops that evolve
in strength and spatial location across the broad high state of the binary
system.
Based upon the 3-D SPH models of Okazaki (see above), the forbidden high
excitation emissions originate from compressed structures in the outer regions
of the interacting winds that flowed out in the previous cycle. FUV radiation
is channeled by the spiral cavity carved out by the lesser wind of Eta Car B,
the less massive, but hotter companion, with a spectral distribution of a mid
O-star. As Eta Car B, in the highly eccentric orbit, spends the majority of
the orbit near apastron, the blue-shifted, spatial distributions of the high
excitation, forbidden emission, and the excitation of the blue-shifted Weigelt
condensations, demonstrate that apastron is on the near side of Eta Car A with
periastron passing on the far side. Moreover, because of the high eccentricity
of the binary system, the outer, hot, low density cavity is spirally-shifted
in the orbital plane by about 45 to 60o relative to the orbital major axis,
known from the X-ray curve to be tilted at 45o from the sky. Combining this
information leads to placement of the orbital plane close to, if not in, the
plane defined by the skirt of the Homunculus, whose planar axis is aligned to
the axis of symmetry of the bipolar Homunculus and Little Homunculus.
Continued mapping of the spatial distribution provides the potential to map
portions of the interacting winds as they distort throughout the 5.5 year
period.
### 1.5 Mass loss in single and binary massive stars
#### 1.5.1 What causes the X-ray flares in Eta Carinae? (Anthony F. J. Moffat
& Michael F. Corcoran)
We examine the rapid variations in X-ray brightness (“flares”), plausibly
assumed to arise in the hard X-ray emitting wind-wind collision zone (WWCZ)
between the two stars in eta Car, as seen during the past three orbital cycles
by RXTE. The observed flares tend to be shorter in duration and more frequent
as periastron is approached (see the figure), although the largest flares tend
to be roughly constant in strength at all phases. Among the plausible
scenarios (1. the largest of multi-scale stochastic wind clumps from the LBV
component entering and compressing the hard X-ray emitting WWCZ, 2. large-
scale corotating interacting regions (CIR) in the LBV wind sweeping across the
WWCZ, or 3. instabilities intrinsic to the WWCZ), the first one appears to be
most consistent with the observations. This requires homologously expanding
clumps as they propagate outward in the LBV wind and a turbulence-like power-
law distribution of clumps, decreasing in number towards larger sizes, as seen
in Wolf-Rayet winds.
Figure 6: Full width half maximum (in days) for the identified flares vs.
orbital phase. Green symbols are from cycle 1, blue symbols cycle 2, and red
symbols cycle 3. The smooth curves are the best-fit models: clump model, long-
dashed line; CIR model, short dashed line.
#### 1.5.2 Revealing the mechanism of the Deep X-ray Minimum of Eta Car
(Kenji Hamaguchi, Michael F. Corcoran & the Eta Car 2009 Campaign
Observational Team)
The multi-wavelength observing campaign of the colliding wind binary system,
Eta Car, targeted at its periastron passage in 2003 presented a detailed view
of the flux and spectral variations of the X-ray minimum phase. The X-ray
spectra showed a strange Fe K line profile, without significantly varying the
hard band slope above 7 keV. The result, combined with 3-D modeling studies,
suggests that the X-ray minimum originates from either an eclipse of most of
the emission by a porous absorber or a large change of the plasma emissivity.
The key to solve this problem would be in the deep X-ray minimum phase when
X-ray emission from the central point source plunges. We therefore launched
another focussed observing campaign of Eta Car with the Chandra, XMM-Newton
and Suzaku observatories during the periastron passage in early 2009. Five
Chandra spectra taken during the deep minimum revealed an underlying non-
variable X-ray component from the central point source. With similar X-ray
characteristics, it would be the Central Constant Emission (CCE) component
discovered in 2003. Instead, the 2009 data showed it has a very hot plasma of
kT $\sim$4$-$6 keV. The other, variable component, probably originating in the
wind-wind collision (WWC), decreased from the hard energy band above $\sim$4
keV around the onset of the deep minimum and recovered only in the hard band
at the end. These phenomena are consistent with a picture that the hottest
plasma at the WWC convex was hidden behind an optically thick absorber first
and cooler plasmas in the WWC tail followed: i.e., the deep minimum would be
driven by an X-ray eclipse. On the other hand, Suzaku did not find any
extremely embedded X-ray source ($N_{\rm H}\lesssim$ 1025 $cm^{-2}$) in
spectra above 10 keV during the X-ray minimum; XMM-Newton spectra showed
strong deformation in the iron K line as in the last cycle; the X-ray minimum
recovered earlier in 2009 without significant $N_{\rm H}$ change from the 2003
cycle. These results suggest that the WWC plasma activity significantly
changed during the X-ray minimum.
### 1.6 LBVs, Massive Binaries and SNs: Is there a Connection?
#### 1.6.1 Connections between LBVs and Supernovae (Nathan Smith)
I will discuss the properties of LBV eruptions inferred from their
circumstellar nebulae and from their light curves in historical examples and
extragalactic Eta Carinae analogs. Recent observations of supernovae,
especially those of the Type IIn class, suggest that these supernovae undergo
precursor outbursts with masses, velocities, kinetic energies, and composition
similar to the 1843 giant eruption of Eta Carinae and non-terminal giant
eruptions of other LBVs. This possible connection offers valuable clues to the
final pre-SN evolution of massive stars that contradict current paradigms, and
it emphasizes that giant LBV eruptions (or events like them) represent a key
long-standing mystery in astrophysics that begs for our attention.
#### 1.6.2 The S-Dor phenomenon in Luminous Blue Variables (Jose H. Groh)
While Luminous Blue Variables (LBVs) have been classically thought to be
rapidly evolving massive stars in the transitory phase from O-type to Wolf-
Rayet stars, recent works have suggested that LBVs might surprisingly explode
as a core-collapse supernova. Such a striking result highlights that the
evolution of massive stars through the LBV phase is far from being understood.
LBVs exhibit photometric, spectroscopic, and polarimetric variability on
timescales from days to decades, probably caused by different physical
mechanisms.
I presented the latest results on the long-term S Dor-type variability of
LBVs, in particular regarding changes in bolometric luminosity, the Humphreys-
Davidson limit, and the role of rotation. The S Dor-type variability
characterized by irregular visual magnitude changes on timescales of decades,
with a typical amplitude of $\Delta V\simeq 1-2$ mag, and corresponding
changes in effective temperature and hydrostatic radius. During visual
minimum, the star is typically hot, while at visual maximum, a cooler
effective temperature is obtained. How the S Dor-type variability relates to
the powerful giant eruptions is not clear, although it could be possible that
a relatively large amount of stellar mass, which is not ejected from the star,
is taking part in the S Dor-type variability. This would suggest that the S
Dor-type variability is a failed giant eruption.
At least for AG Car, a significant reduction ($\sim 50\%$) in the inferred
bolometric luminosity from visual minimum to maximum has been determined, and
a high rotational velocity has been obtained during minimum. I will present
evidence that fast rotation is typical in Galactic LBVs that show S-Dor type
variability, and will discuss how these recent results put strong constraints
on the progenitor, current evolutionary stage, and fate of LBVs.
#### 1.6.3 Pulsational instability in massive stars: implications for SN and
LBV progenitors (Matteo Cantiello & Sung-Chul Yoon)
Most massive stars experience a pulsational instability induced by
$\ukappa-$mechanism, when the surface temperature sufficiently decreases. The
amplitude of pulsations grows very fast, and may result in very high mass loss
rates. We propose a new scenario for mas- sive star evolution based on our new
calculations of this pulsational instability, where the initial mass of SNe
progenitors increases according to the order: SN IIp$-->$ SN IIn$-->$SN
IIL$-->$SN IIb$-->$SN Ib/c. Moreover, the pulsation appears strong in the
early core He-burning stage for M $\geq$40Mo, and may lead to the formation of
LBVs. We also argue that stellar eruptions like SN 2008S may be related to
this instability.
#### 1.6.4 Hydrodynamical Models of Type II-P SN Light Curves (Melina C.
Bersten, Omar Benvenuto, & Mario Hamuy)
Figure 7: Hydrodynamical models for TypeII-P SN Light Curves. Left: Bolometric
correction versus B-V. Right: Light curve for SNeII-P
We present computations of bolometric light curves (LC) of type II plateau
supernovae (SNe II-P) obtained using a newly developed, one-dimensional
Lagrangian hydrodynamic code with flux-limited radiation diffusion. We derive
a calibration for bolometric corrections (BC) from $BVI$ photometry (see
figure 7, left) with the goal of comparing our models with a large database of
high-quality $BVI$ light curves of SNe II-P. The typical scatter of our
calibration is 0.1 mag. As a first step, in our comparison we have determined
the physical parameters (mass, radius and energy) of two very well observed
supernovae, SN 1999em (see figure 7, right) and SN 1987A. Despite the
simplifications used in our code we obtain a remarkably good agreement with
the observations and the parameters derived are in excellent concordance with
previous studies of these objects.
### 1.7 Massive Binaries and Eta Car: What is the Relationship?
#### 1.7.1 WR140 & WR25 in X-ray relation to Eta Car (Andrew M. Pollock &
Michael F. Corcoran)
WR 25 (WN6ha+O) and WR 140 (WC7+O5) are both X-ray bright binaries of long
period and high eccentricity, whose individual stellar and wind and collective
binary parameters are much better known than those of Eta Car. Observations at
different orbital phases thus show how X-rays are produced by colliding winds
under physical and geometrical conditions that are quite well defined at any
one time but which vary considerably around the orbit. As WR 25 is 7’ from Eta
Car, there are more observations than would otherwise be the case, a few of
which during the 2003 XMM–Newton campaign led to the recognition of brightness
and absorption variations that were soon shown to coincide with a periastron
passage of the 208-day $e\approx 0.6$ optical radial velocity orbit discovered
by Gamen et al. 2006 (A&A 460, 777). Their orbit was used in early 2008 to
plan a month-long daily ToO campaign with the soft X-ray XRT instrument aboard
the Swift GRB Observatory. As well as the relatively shallow eclipse by the
extended Wolf-Rayet wind, a sudden overall decrease between quadrature and
conjunction is most obviously interpreted as a stellar eclipse by the WN6ha
primary, thought to be one of the most massive stars in the Galaxy.
Repeatability is good within the relatively modest statistical limits of the
few dozen measurements available, spread unevenly over several cycles. The
luminosity increases monotonically between apastron and periastron from the
surface that provides the backdrop for the eclipses.
Observing conditions for WR 140 are more favourable. It has an orbit well-
established by Marchenko et al. 2003 (ApJ, 596, 1295) of longer 7.94-year
period and higher $e\approx 0.881$ eccentricity. It is also a brighter X-ray
source. As a result, measurements are more precise and the phase density much
higher. Weekly hard X-ray monitoring with RXTE started just before the 2001
periastron passage, increasing to daily measurements in the approach to the
2009 periastron with recent measurements also made with Swift, Suzaku and
XMM–Newton. Preliminary analysis of the RXTE data show the same general type
of eclipse events seen in WR 25 but in greater detail and with significant
differences. For example, the luminosity maximum apparently occurs a few weeks
before periastron and even before conjunction. with asymmetries before and
after periastron. The adiabatic $1/D$ luminosity law gives a poor description
throughout the orbit and there were no obvious flares like those seen in Eta
Car. High resolution Chandra data obtained at 4 phases show very small changes
in shape between apastron and O-star conjunction in a spectrum dominated,
perhaps surprisingly given the expected collisionless nature of the shocks
concerned, by a smooth continuum probably from hot electrons. The lines imply
complete mixing of shocked material from both winds. Details of the velocity
profiles are more difficult to understand, especially the absence of the
highest velocity blue-shifted material near periastron.
#### 1.7.2 The Erupting Wolf-Rayet System HD 5980 in the SMC: A (Missing)
Link in Massive Stellar Evolution or a Freak? (Rodolfo H. Barba)
The Wolf-Rayet eclipsing binary system HD 5980 in the Small Magellanic Cloud
has shown a peculiar behaviour along the past years. In 1994 the star
developed an un- predicted eruption and changed its spectrum from WN-type to
one resembling those of Luminous Blue variables (LBV). In this presentation, I
will review observational aspects of this unique system, emphasizing those
similarities and differences with extreme LBV objects like Eta Car. I will
briefly describe a century of photometric and spectroscopic records of the
star, and depict a new analysis of the spectroscopic data obtained during the
outburst phase, and the present WN-E stage. Also, I will discuss the different
scenarios proposed to explain the LBV-like behaviour (rapid rotators, tidal
interactions, single star evolution).
#### 1.7.3 The Extragalactic Eta Car Analogs (Schuyler D. Van Dyk)
Powerful eruptions of massive stars, such as Eta Car are often referred to as
“supernova (SN) impostors,” because some observational aspects can mimic the
appearance of a true SN. During the Great Eruption during the 1800’s of Eta
Car, the star greatly exceeded the Eddington limit, with its bolometric
luminosity increasing by $\sim$2 mag. The total luminous output of such an
eruption ($\sim 10^{49.7}$ erg) can rival that of a SN, to such a degree that
some impostors initially are assigned designations as SNe, even in modern
extragalactic SN searches. A number of extragalactic SN impostors are known,
such as SNe 1954J, 1961V, 1997bs, 1999bw, 2000ch, 2001ac, 2002kg, 2003gm, NGC
2363-V1, etc. I will present here the latest results for those that can be
considered Eta Car analogs. Not all impostors are as powerful as Eta Car, and
are therefore not considered true analogs to Eta Car; some cases are more like
the “classical” LBVs (e.g., S Dor), where the bolometric luminosity remains
constant during an eruption, as the star’s envelope expands or its wind
becomes optically thick, and the apparent temperature cools to $\sim$8000 K.
Like Eta Car, the precursor star for each analog is expected to survive the
eruption and return to relative quiescence. Some have had eruption survivors
identified (SNe 1954J, 1961V), using the Hubble Space Telescope, some have
seemingly ”vanished” after outburst (SNe 1997bs, 1999bw), and one (SN 2000ch)
continues in outburst after almost a decade. Only one (SN 1999bw) has shown
evidence for dust emission, based on Spitzer Space Telescope observations, and
the emission has apparently faded from detection. Studying the characteristics
of the analogs provides us with a greater understanding of Eta Car itself and
of the evolution of very massive stars.
### 1.8 Summary and Discussion (Nidia I. Morrell, Michael F. Corcoran,
Anthony F.Moffat & Julian Pittard)
After a brief brainstorm session, Mike Corcoran, Tony Moffat, Nidia Morrell
and Julian Pittard came up with the following list of questions and
highlights, which served as a basis for a half-hour open discussion on future
studies of Eta Car:
How to better constrain the orbital and wind parameters of both stars in Eta
Car?
What is its future evolution?
What caused the Great Eruption? Which star erupted?
What is the nature of the companion star? (Very urgent!)
What s the connection between WRs, LBVs and supernovae?
How to explain the strictly cyclic, bizarre behavior of the He II 4686
emission, which emerges only within several months of periastron passage?
What is the role of a companion star in driving the formation, evolution and
instabilities of Eta Car and other binary LBVs?
Does dust form in Eta Car?
Does Eta Car pulsate?
## 2 Posters
### 2.1 A full cycle 7 mm light-curve of Eta Car (Zulema Abraham, Pedro P.
Beaklini & Carlo Miceli)
It is now well established that the light curve of Eta Carinae has a periodic
behavior at all wavelengths, from mm waves to X-rays. These light curves are
characterized by the presence of a sharp dip, with duration that depends on
wavelength, being longer at X-rays. At mm wavelengths, the dip was detected
during the last four cycles, but only during the 2003.5 minimum the light
curve was obtained with daily resolution. At that epoch, the 7 mm light curve,
obtained with the Itapetinga radiotelescope, in Atibaia, Brazil, followed the
X-ray decaying behavior but showed a strong peak, not seen at other
wavelengths, before reaching the minimum. This peak was attributed to free-
free emission of the 107 K optically thick gas located at the wind-wind
collision contact surface. Here, we report the 7 mm light curve of the
complete 2003-2009 cycle, including the 2003.5 and 2009.0 minima, both
obtained with daily resolution. We show for the rst time that: (a) the
duration of the minima are the same at 7 mm and at X-rays; (b) The peak at 7
mm seen after the minimum is 2003.5 appeared again in 2009.0, with the same
phase, duration and shape; (c) two other strong peaks were observed before the
2009.0 minimum, coincident with the peaks observed at X-rays, which supports
the previous assumption that they are formed at the wind-wind shock interface.
### 2.2 The multiple zero-age main-sequence O star Herschel 36 (Julia I.
Arias, Rodolfo H. Barba, Roberto C. Gamen, Nidia I. Morrell, Jesus Maiz
Apellaniz, Emilio J. Alfaro, Nolan R. Walborn, Alfredo Sota, Christian M.
Bidin)
We present a study of the zero-age main-sequence O star Herschel 36 in M8,
based on high-resolution optical spectroscopic observations spanning six
years. This object is de nitely a multiple system. We propose a picture of a
close massive binary and a companion of spectral type O, most probably in wide
orbit about each other. The components of the close pair are identi ed as O9 V
and B0.5 V. The orbital solution for this binary is characterized by a period
of 1.5415$\pm$0.00001 days. With a spectral type O7.5 V, the third body is the
most luminous component of the system. It also presents radial velocity
variations with short (a few days) and long (hundreds of days) timescales,
although no accurate temporal pattern can be discerned from the available
data. Some possible hypotheses to explain the variability are brie y addressed
and further observations are suggested.
### 2.3 Spatially extended wind emission in the massive binary systems VV Cep
& KQ Pup (Wendy Hagen Bauer, Theodore R. Gull, Philip Bennett & Jahanara
Ahmad)
VV Cep and KQ Pup are binary systems consisting of M supergiant primaries with
B main-sequence companions which orbit within the extensive M supergiant
winds. VV Cep undergoes total eclipses and was observed with the HST/STIS
Spectrograph at several epochs which spanned total eclipse through
”chromospheric eclipse” as lines from ions like Fe i weakened and disappeared
through first quadrature. KQ Pup comes close to eclipsing its hot companion
and was observed to be in chromospheric eclipse (showing weak absorption from
Fe i in the M supergiant s chromosphere) by STIS in October 1999. Two-
dimensional reprocessing of the STIS echelle spectra has revealed spatially
extended emission in all observations of these two systems. Emission arising
from gas thought to be associated with the hot component shows spatial
extension consistent with the STIS spatial point spread function. The
spatially extended flux seen outside total eclipse arises from emission in
transitions expected to be observed from the winds of cool supergiants. VV Cep
was observed at enough epochs to map out radial velocity structure within the
wind. It is consistent with model predictions for wind flow in a binary system
in which the wind outflow is comparable with the M supergiant s orbital
velocity. Spatially resolved wind and wind interaction structures of these two
stars and of Eta Car reinforce the need for imaging spectroscopy and added
capabilities of integral field units for mapping these complex interacting
binary systems.
### 2.4 Abundances and depletion of iron-peak elements in the Strontium
filament of Eta Car (Manuel A. Bautista, Henrik Hartman, Marcio Melendez,
Theodore R. Gull, Katharina Lodders & Mariela Martinez)
We carried out a systematic study of elemental abundances in the Strontium
Filament, a peculiar metal-ionized structure located in the skirt plane of the
Homunculus, ejecta surrounding Eta Car. To this end we interpret the emission
spectrum of neutral C and singly ionized Al, Sc, Ti, Cr, Mn, Fe, Ni, and Sr
using multilevel non-LTE models for each ion. The atomic data for most of
these ions is limited and of varying quality, so we carried out ab initio
calculations of radiative transition rates and electron impact excitation rate
coef cients for each of these ions. The observed spectrum is consistent with
an electron density $\approx 10^{7}$cm-3 and a temperature between 6000 and
7000 K. The observed spectra are consistent with large enhancements in the gas
phase Sr/Ni, Sc/Ni, and Ti/Ni abundance ratios relative to solar values. Yet,
the abundance ratios Cr/Ni, Mn/Ni, and Fe/Ni are roughly solar. We explore
various scenarios of elemental depletion in the context of nitrogen-rich
chemistry, given that the stellar ejecta has enriched nitrogen at the expense
of greatly depleted oxygen and carbon due to mixing in the $>$60 $M_{\odot}$
star. Finally, we discuss the implications of these findings for the
generation of dust during the evolution of supermassive stars from main
sequence to pre-supernova stage.
### 2.5 A fast ray tracing disk model for 10$\mu$ interferometric data
fitting: First application on the B[e] star CPD57 2874 (Philippe Bendjoya,
Giles Niccolini, & Amando D. de Souza)
We present here a parametric dust disk model (P2DM) developed to fit
interferometric observations in a much faster computing time than the
classical Monte Carlo Modeling Approach. P2DM combined with a Levenberg-
Markward minimisation algorithm allows us to derive both crucial physical and
geometrical parameters. This model is restricted to wavelengths around and
above 10 microns (no gas, no scattering) making it useful for VLTI-MIDI (and
future MATISSE) observations and implies that more elaborate modelling is
necessary to get a deeper understanding of the physical processes responsible
of the observed disks. Neverthelss, this fast and physical model is useful for
exploring the physical parameter phase space and to provide starting values
for more powerful models. We present the model and its applica- tion to the
supergiant B[e] CPD -57 2874 star observed with VLTI-MIDI.
### 2.6 A search for relics of interstellar bubbles originated by LBV
progenitors (Cristina E. Cappa, Silvina Cichowolski, Javier Vasquez & J. R.
Rizzo)
The strong stellar winds of massive O stars sweep up and compress the
surrounding gas creating interstellar bubbles in their environs. In this
modified environment, massive stars evolve into Luminous Blue Variables
(LBVs), which are the immediate progenitors of WR stars. Using the Canadian
Galactic Plane Survey (CGPS) and Southern Galactic Plane Survey (SGPS) we
searched for Hi interstellar bubbles associable with O-type progenitors of a
number of galactic LBVs and LBV candidates. We found Hi cavities and shells
that probably originated from the massive progenitors of P Cygni, G79.29+0.46,
AG Carinae, and He3-519.
### 2.7 Massive binaries and rotational mixing (Selma E. de Mink, Matteo
Cantiello, Norbert Langer & Onno R. Pols)
In massive stars fast rotation is the cause of efficient internal mixing,
which leads to the transport of hydrogen burning products from the core to the
stellar envelope. This results in hot and overluminous stars, which stay
compact as they gradually evolve into massive helium stars (e.g. Yoon &
Langer, 2005). While non-rotating stars in close binaries experience severe
mass loss as soon as their radius exceeds the Roche lobe radius, fast-rotating
stars, which are efficiently mixed, stay compact and can avoid the onset of
mass transfer.
This can occur in wide binaries (orbital periods much larger than about 10
days) where the rotation rate of the stars is not affected by tides during the
main sequence evolution. Alternatively, this can occur in massive binaries
with orbital periods smaller than 3 days. Tides force the stars to rotation
rates high enough to trigger efficient mixing (De Mink et al. 2008, 2009).
This type of evolution leads naturally to the formation of compact Wolf-Rayet
binaries and is potentially interesting as an explanation for the formation of
massive black hole binaries such as M33 X-7 and IC10 X-1.
### 2.8 MHD numerical simulations of wind-wind collisions in massive binary
systems (Diego Falceta-Goncalves & Zulema Abraham)
In past years, several massive binary systems have been studied in details at
both radio and X-rays wavelengths, revealing a whole new physics present in
such systems. Large emission intensities from thermal and non-thermal sources
showed us that most of the radiation in these wavelengths originates at the
wind-wind collision region. OB and WR stars present supersonic and massive
winds that, when under collision, emit largely in X-rays and radio due to the
free-free radiation, as well as in radio due to synchrotron emission. However,
in the latter case, magnetic fields play an important role on the emission
distribution. Astrophysicists have been modeling free-free and synchrotron
emission from massive binary systems based on purely hydrodynamical
simulations and ad hoc assumptions regarding the distribution of magnetic
energy and the field geometry in order to study the non-thermal source. In
this work we provide a number of the first MHD numerical simulations of wind-
wind collision in massive binary systems. We study the free-free emission,
characterizing its dependence on the stellar and orbital parameters. We also
study, self consistently, the evolution of the magnetic field at the shock
interfaces, obtaining also the synchrotron energy distribution integrated
along different lines of sight.
### 2.9 On the peculiar variations of two southern B[e] stars (Marcelo Borges
Fernandes, Michaela Kraus, Olivier Chesneau, Jiri Kubat, Armando Domiciano de
Souza, Francisco X. de Araujo, Philippe Stee & Anthony Meilland)
In this work, we present the peculiar variations shown by two B[e] stars,
namely the SMC supergiant LHA115-S23 and the galactic unclassified object
HD50138, mainly based on high resolution optical spectroscopic data. The
spectra of LHA115-S23 revealed the disappearance of photospheric He i
absorption lines in a period of only 11 years. Due to this, the star has
changed its MK classi cation from B8I to A1Ib, becoming the first A[e] star
identified. Concerning HD50138, the brightest known B[e] star, based on our
data, taken with a difference of 8 years, it is possible to see the presence
of strong spectral variations, probably associated with a new outburst, which
took place prior to 2007. A detailed spectroscopic description, the projected
rotational velocities, the modeling of their spectral energy distributions,
and the discussion about the possible nature and circumstellar scenarios for
these two curious B[e] stars are provided.
### 2.10 Interferometric analysis of peculiar stars with the B[e] phenomenon
(Marcelo Borges Fernandes, Olivier Chesneau, Denis Mourard, Michaela Kraus,
Philippe Stee, Armando Domiciano de Souza, Alex Carciofi, Florentin Millour,
Anthony Meilland, Philippe Bendjoya, Samer Kanaan, Giles Niccolini & Olga
Suarez)
Stars that present the B[e] phenomenon are known to form a very heterogeneous
group. This group is composed by objects in different evolutionary stages,
like high- and low- mass evolved stars, intermediate-mass pre-main sequence
stars and symbiotic objects. However, more than 50% of the confirmed B[e]
stars have unknown evolutionary stages, being called as unclassified B[e]
stars. The main problem is the absence of reliable physical parameters and of
knowledge of their circumstellar geometries. Based on this, high-angular
resolution interferometry is certainly an important tool to answer several
questions concerning the nature of these stars, including a possible
evolutionary link between B[e] supergiants and LBV stars, like Eta Car. In
this work, we present the results related to a sample of objects, namely
HD50138, HD45677, HD62623 and MWC361 based on observations using VLTI/MIDI,
VLTI/AMBER and CHARA/VEGA.
### 2.11 Numerical models for 19th century outbursts of Eta Car (Ricardo F.
Gonzalez Dominguez)
We present new results of two-dimensional hydrodynamical simulations of the
eruptive events of the 1840s (the great) and the 1890s (the minor) eruptions
suffered by the massive star, Eta Car. The two bipolar nebulae commonly known
as the Homunculus (H) and the Little Homunculus (LH) were formed from the
interaction of these eruptive events with the underlying stellar wind. We
assume a colliding wind scenario to explain the shape and the kinematics of
both Homunculi. Adopting a more realistic parametrization of the phases of the
wind, we show that the LH is formed at the end of the 1890s eruption when the
post-outburst Eta Car wind collides with the eruptive flow, rather than at the
beginning (as claimed in previous works; González et al. 2004a, 2004b). The
regions at the edge of the LH become Rayleigh-Taylor unstable and develop
filamentary structuring that shows some resemblance with the observed spatial
structures in the polar caps of the inner Homunculus (Smith 2005). We also
find the formation of some tenuous equatorial, high-speed features.
### 2.12 Discovery of a new WNL star in Cygnus with Spitzer (Vasilii
Gvaramadze, Sergei Fabrika, Wolf-Rainer Hamann, Olga Sholukhova, Azamat F.
Valeev, Vitaly P. Goranskij, Anatol M. Cherepashchuk, Dominik J. Bomans &
Lidia M. Oskinova)
We report the serendipitous discovery of an infrared ring nebula in Cygnus
using the archival data from the Cygnus-X Spitzer Legacy Survey and present
the results of study of its central point source. The optical counterpart to
this source was identi ed by Dolidze (1971) as a possible Wolf-Rayet star. Our
follow-up spectoscopic observations with the Russian 6-m telescope confirmed
the Wolf-Rayet nature of this object and showed that it belongs to the WN8-9h
subtype.
### 2.13 VLT-CRIRES observations of Eta Car’s Weigelt blobs & Strontium
Filament (Henrik Hartman, José Groh, Thedore R. Gull, Hans U. Kaufl, Florian
Kerber, Vladilen Letokhov & Krister E. Nielsen)
We have obtained Very Large Telescope-CRIRES observations of Eta Car, focused
on the Weigelt condensations (WC) and the Strontium Filament (SrF). These are
nebular regions, in the close vicinity to Eta Car, with complex emission line
spectra. The two regions show, however, strikingly different physical
conditions and abundances. The WC are driven by far-UV radiation from the hot
companion (Eta Car B). The radiation is internally redistributed to hydrogen
emission which enables exotic atomic photo processes, such as Resonance
Enhanced Two-Photon Ionization (RETPI) and stimulated emission (LASER). The
lines proposed for the stimulated emission are the 1.68 and 1.74 mm
transitions from the c4F7/2 level in Fe ii (i.e. the spectrum of Fe+).
The Strontium Filament received its name from the initial discovery of [Sr
ii], lines from singly-ionized strontium. Modeling of the emission spectrum
has revealed strange abundances (see separate poster by Bautista et al. at
this meeting), and spectral lines with complex line profiles. The main
emission component is consistent with a creation of the ejecta in the 1890s.
We present a preliminary analysis of the ejecta in the NIR, using high
spectral (R= 90,000) and spatial resolution ($\approx$0.3”) spectra obtained
with CRIRES in April 2007. The data allow us to study the individual ejecta in
detail, at a spectroscopic phase where the effects due to Eta Car B’s
periastron passage is negligible.
We all acknowledge the tremendous contributions by Sveneric Johansson and
Vladilen Letokhov to the field of plasma physics, the understanding of the
physical processes in the WC, and the final contribution with their book
Astrophysical Lasers (Oxford, 2009).
### 2.14 Radiative transfer Modeling of rotational modulations in the B
supergiant HD 64760 (Alex Lobel & Ronny Blomme)
We develop parameterized models for the large-scale structured wind of the
blue supergiant, HD 64760 (B0.5 Ib), based on best fits to Rotational
Modulations and Discrete Absorption Components (DACs) observed with IUE in Si
iv $\lambda$1400\. The fit procedure employs the Wind3D code with non-LTE
radiative transfer (RT) in 3-D. We parameterize the density structure of the
input models in wind regions (we term ”Rotational Modulation Regions” or RMRs)
that produce Rotational Modulations, and calculate the corresponding radial
velocity field from CAK-theory for radiatively-driven rotating winds. We find
that the Rotational Modulations are caused by a regular pattern of radial
density enhancements that are almost linearly shaped across the equatorial
wind of HD 64760. Unlike the Co-rotating Interaction Regions (CIRs) that warp
around the star and cause DACs, the RMRs do not spread out with increasing
distance from the star. The detailed RT fits show that the RMRs in HD 64760
have maximum density enhancements of $\sim$17 % above the surrounding smooth
wind density, about two times smaller than hydrodynamic models for CIRs.
Parameterized modelling of Rotational Modulations reveals that nearly linear-
shaped (or ‘spoke-like’) wind regions co-exist with more curved CIRs in the
equatorial plane of this fast rotating B-supergiant. We present a preliminary
hydrodynamic model computed with Zeus3D for the RMRs, based on mechanical wave
excitation at the stellar surface of HD 64760.
### 2.15 Parameterized structured wind modelling of massive hot stars with
Wind3D (Alex Lobel & Jesus A. Toala)
We develop a new and advanced computer code for modelling the physical
conditions and detailed spatial structure of the extended winds of massive
stars with three-dimensional (3-D) non-LTE radiation transport calculations of
important diagnostic spectral lines. The Wind3D radiative transfer code is
optimized for parallel processing of advanced input models that adequately
parameterize large-scale wind structures observed in these stars.
Parameterized 3-D input models for Wind3D offer crucial advantages for high-
performance transfer computations over ab-initio hydrodynamic input models.
The acceleration of the input model calculations permits us to investigate and
model a much broader range of physical (3-D) wind conditions with Wind3D. We
apply the new parameterization procedure to the equatorial wind-density
structure of Co-rotating Interaction Regions (CIRs) and calculate the wind
velocity-structure from CAK-theory for radiatively-driven rotating winds. We
use the parameterized CIR models in Wind3D to compute the detailed evolution
of Discrete Absorption Components (DACs) in Si iv UV resonance lines. The new
method is very flexible and efficient for constraining physical properties of
extended 3-D CIR wind structures (observed at various inclination angles) from
best fits to DACs in massive hot stars. We compare the results with an
accurate hydrodynamical model for the DACs of B0.5 Ib-supergiant HD 64760, and
apply it to best fit the detailed DAC evolution observed with $IUE$ in B0
Iab/Ib-supergiant HD 164402.
### 2.16 3D modeling of eclipse-like events in Eta Car (Thomas I. Madura,
Theodore R. Gull, Atsuo Okazaki & Stanley Owocki)
We discuss recent efforts to apply 3D Smoothed Particle Hydrodynamics (SPH)
simulations to model the binary wind collision in Eta Car, with emphasis on
reproducing BVRI photometric variations observed from La Plata Observatory.
Photometric dips occurring concurrently with X-ray minima seen with RXTE
provide further evidence for binarity in the system. We investigate the role
of the unseen secondary star, focusing on two effects: 1) an occultation of
the secondary by the slower, extended optically thick primary wind; and 2)a
Bore-Hole effect, wherein the fast wind from the secondary carves a cavity in
the dense primary wind, allowing increased escape of radiation from the
hotter/deeper layers of the pri mary s extended photosphere. Such models may
provide clues on how/where light is escaping the system, the directional
illumination of distant material (e.g., the Homunculus, the Little Homunculus,
the purple haze , Weigelt blobs, etc.) and the parameters/orientation of the
binary orbit.
### 2.17 The Other Very Massive Stars in the Carina Nebula as observed with
HST (Jesus Maiz Apellaniz, Nolan R. Walborn, Nidia I. Morrell, Ed P. Nelan &
Virpi S. Niemela)
We have used HST/ACS+FGS and ground-based data to study 10 WNha, O2-4
supergiant, and O3.5 main-sequence stars in the Carina Nebula. HD 93129 Aa+Ab
is the most massive known astrometric binary. Its motion is currently being
followed with STIS spectroscopic observations planned for the fall of 2009.
Previously unknown resolved components are detected: an $\sim$8 M⊙ star for HD
93162 (=WR 25) and two $\sim$1 M⊙ stars for Tr 16-244. Overall, at least 8 of
the 11 most massive stars in the Carina Nebula are members of multiple
systems. The NUV-to-NIR photometry has been processed with the new version
(v3.1) of the CHORIZOS code using Geneva isochrones with ages of 1.0 Ma and
1.8 Ma. Most stars in our sample are found to have visual total extinctions
between 1.0 and 2.2 mag but HD 93162 and Tr 16-244 are more extinguished. The
ratio of total to selective extinction $R_{5495}$ is found to vary between 3.0
and 4.5 and is positively correlated with the total extinction. For a fixed
age for the full sample, the Trumpler 14 stars are underluminous for their
spectral types, hence implying a small age ($\lesssim$1 Ma) for the cluster.
HD 93250 is overluminous for its spectral type, a possible indication of an
undetected (by spectroscopic, interferometric, or imaging methods) massive
companion. The three WRs (22, 24, and 25) and HD 93129 Aa have evolutionary
(initial) masses above 90 M⊙, i.e. values comparable to that of Eta Car.
### 2.18 The High Angular Resolution Multiplicity of Massive Stars (Brian D.
Mason, William I. Hartkopf, Douglas R. Gies, Theo A. ten Brummelaar, Nils H.
Turner, Chris D. Farrington & Todd J. Henry)
Conducted on NOAO 4-m telescopes in 1994, the first speckle survey of O stars
(Mason et al. 1998) had success far in excess of our expectations. In addition
to the frequently cited multiplicity analysis, many of the new systems which
were first resolved in this paper are of significant astrophysical importance.
Now, some ten years after the original survey, we have re-examined all systems
analyzed before. Improvements in detector technology allowed for detection of
companions missed before as well as systems which may have been closer than
the resolution limit in 1994. Also, we made a first high-resolution inspection
of the additional O stars in the recent Galactic O Star Catalog of Maíz-
Apellániz & Walborn (2004). In these analyses we resolved four binaries not
detected in 1994 due to the enhanced detection capability of our current
system or kinematic changes in their relative separation. We also recovered
four pairs, confirming their original detection. In the new sample, stars are
generally more distant and fainter, decreasing the chance of detection.
Despite this, eight pairs were detected for the first time.
In addition to many known pairs observed for testing, evaluation and detection
characterization, we also investigated several additional samples of
interesting objects, including accessible Galactic WR stars from the
contemporaneous speckle survey of Hartkopf et al. (1999), massive, hot stars
with separations which would indicate their applicability for mass
determinations (for fully detached O stars masses are presently known for only
twelve pairs), and additional datasets of nearby red, white, sub and G dwarf
stars to investigate other astrophysical phenomena. In these observations, in
addition to those enumerated above we resolved seventeen pairs for the first
time.
Massive stars have also been a important observing program for the CHARA
Array. Preliminary results from Separated Fringe Packet solutions of
interferometric binaries are also presented.
### 2.19 Far-IR Spectroscopic Imaging of the ISM around Eta Car (Hiroshi
Matsuo, Takaaki Arai, Tom Nitta & Aya Kosaka)
To study interstellar material around Eta Car, we have performed far-infrared
imaging spectroscopic observations using a Fourier transform spectrometer
onboard the Japanese infrared satellite AKARI. We have obtained images of C
ii, N ii, and O iii covering the 15 arcmin $\times$10 arcmin area centered at
Eta Car. The O iii and C ii lines were found wide-spread, but peaked toward
Carinae nebulae, which gives an indication of interaction of ejecta and
molecular clouds. The N ii line is weak and only partially observed around Eta
Car. Comparison with ionized hydrogen and non-thermal emission at millimeter-
wave O iii emission is coincident with ionized region while C ii emission is
peaked at different positions but similar to the position angle of the
Homunculus nebulae, which may indicate that we are observing interactions of
old ejecta with molecular clouds.
### 2.20 Stellar forensics with SNe & GRBs: Deciphering the size &
metallicity of their massive progenitors (Maryam Modjaz)
Massive stars die violently. Their explosive demise gives rise to brilliant
fireworks that constitute supernovae and long GRBs, and that are seen over
cosmological distances. By interpreting their emission and probing their
environment, we get insights into the size, make-up, mass loss history and
metallicity of their massive progenitor stars that are situated at
extragalactic distances.
I will present extensive X-ray, optical and NIR data on SN 2008D which was
dis- covered serendipitously with the NASA Swift satellite via its X-ray
emission from shock breakout. It is a supernova of Type Ib, that is, a core-
collapse supernova whose massive stellar progenitor had been been stripped of
most, if not all, of its outermost hydrogen layer, but had retained its next-
inner helium layer, before explosion. I will discuss the signi cance of this
supernova, the derived size of its Wolf-Rayet progenitor, what it tells us
about the explosive demise of massive stars, and its implications for the
supernova-GRB connection. Furthermore, I will present observational results
that confirm low metallicity as a key player in determining whether some
massive stars die as GRB-SN or as an ordinary SN without a GRB. I show that
the oxygen abundances at the SN-GRB sites are systematically lower than those
found near ordinary broad-lined SN Ic, at a cut-off value of 0.3$-$0.5 Zsolar.
### 2.21 Rapid Spectrophotometric Changes in R127 and Reversal of the Decline
(Nidia I. Morrell, Roberto C. Gamen, Nolan R. Walborn, Rodolfo H. Barba,
Katrien Uytterhoeven, Artemio Herrero, Christiopher Evans, Ian Howarth &
Nathan Smith)
R127, the famous Luminous Blue Variable in the Large Magellanic Cloud, was
found in the peculiar early-B state and fainter in January 2008, suggesting
that the major outburst which started sometime between 1978 and 1980 was
drawing to a close, and that the star would presumably continue to fade and
move to earlier spectral types until reaching its quiescent Ofpe/WN9 state.
Archival data showed that the main spectral transformation from the peculiar
A-type state at maximum started between 2005 and 2007, and that it was in
close concordance with features in the light curve. However, subsequent
observations during 2008 and early 2009 have shown that the spectrum of R127
is now returning to a cooler, lower excitation state, while the photometry
shows a new brightening of the star. A speculative 7-year cycle during the
decline bears further investigation. The curious behavior of R127 provides an
opportunity to gain further insight into the rapid transitional stages in the
late evolution of very massive stars.
### 2.22 The Luminous Blue Variable Stars in M33: the Extended Hot Phase of
Romano’s Star (GR 290) (Corinne Rossi, Vito Francesco Polcaro, Silvia Galleti,
Roberto Gualandi, Laura Norci & Roberto F. Viotti)
Romano’s Star (GR290) is an LBV in M33. Recently, the star underwent a
dramatic decrease in the visual, that was accompanied by a marked increase of
the spectral line excitation. Presently, GR290 appears to be in the hottest
phase ever observed in an LBV. More than 100 emission lines have been
identified in the 3100$-$10000Å range covered by the WHT spectra, including
the hydrogen Balmer and Paschen series, Hei̇ and He ii, C iii, N ii-iii, Si
iii-iv, and many forbidden lines of [O iii], [N ii], [S iii], [Ar iii] and [Fe
iii]. Many lines, especially the He i triplets, show a P Cygni profile with an
E-A radial velocity difference of about 400 km/s. The 2008 spectrum appears
quite similar to that of a typical WN8-9 star. During 2003$-$2009 GR290 varied
between the WN11$-$WN8 spectral types, with the hottest spectrum corresponding
to a fainter visual magnitude. This temperature-visual luminosity
anticorrelation suggests variation at constant Mbol. GR290 might just present
the key evidence that will help to bridge the LBV and WNL evolutionary phases.
### 2.23 X-Ray modeling of Eta Car and WR140 from hydrodynamic simulations
(Christopher Russell, Michael F. Corcoran, Atsuo Okazaki, Thomas I.Madura &
Stanley Owocki)
The colliding wind binary (CWB) systems Eta Car and WR140 provide unique
laboratories for X-ray astrophysics. Their wind-wind collisions produce hard
X-rays, which have been monitored extensively by several X-ray telescopes,
such as RXTE and Chandra. To interpret these X-ray light curves and spectra,
we apply 3D hydrodynamic simulations of the wind-wind collision using both
smoothed particle hydrodynamics (SPH) and nite difference methods. We nd
isothermal simulations that account for the absorption of X-rays from an
assumed point source of X-ray emission at the apex of the wind-collision shock
cone can closely match the RXTE light curves of both Eta Car and WR140. We are
now applying simulations with self-consistent energy balance and extended
X-ray emission to model the observed X-ray spectra. We present these results
and discuss efforts to understand the earlier recovery of Eta Car’s RXTE light
curve from the 2009 minimum.
### 2.24 Accretion onto the secondary of Eta Car during the spectroscopic
event (Noam Soker & Amit Kashi)
We show that near periastron passage the shocked primary wind becomes
gravitationally bound to the secondary star. This results in accretion flow
onto the secondary star that almost shuts down the secondary wind. The
accretion process is the mechanism of the deep X-ray minimum. Not only in the
present Eta Car, but also during the great eruption, accretion played a key
role.
### 2.25 New massive, eclipsing, double-lined spectroscopic binaries: Cyg
OB2-17 & NGC 346-13 (V. E. Stroud, J. S. Clark, I. Negueruela, D. J. Lennon &
C. J. Evans)
Massive, eclipsing, double-lined spectroscopic binaries are not common but
necessary to understand the evolution of massive stars as they are the only
direct way to determine the masses of OB stars and therefore obtain mass-
luminosity functions. They are also the progenitors of energetic phenomena
such as X-ray binaries and $\gamma$-ray bursts. We discuss results from
photometric and spectroscopic studies of two binary systems: Cyg OB2-B17 which
is a semidetached binary located in the Cyg OB2 association and comprised of 2
O supergiants; and NGC 346-13 which is a system located in the Small
Magellanic Cloud and comprised of a semi-evolved B1 star and a hotter,
optically fainter secondary, suggesting mass transfer in the system.
### 2.26 Monte Carlo radiative transfer in stellar wind (Brankica Surlan &
Jiri Kubat)
As a first step towards solution of the radiative transfer equation in clumped
stellar wind we started to develop a code for the formal solution of the
radiative transfer equation for given velocity, temperature, and density
stratification. Wind structure was taken from a model calculated using a NLTE
code by Krtička & Kubát (2004, A&A 417, 1003). Wind opacity consists of line
scattering under Sobolev approximation and of the electron scattering. As our
first preliminary results we plot the P Cygni profile of the line obtained
from our calculation. This work has been supported by grants 205/08/0003 and
205/08/H005 (GA ČR).
### 2.27 Gamma-ray observations of the Eta Car region (Marco Tavani, Sabina
Sabatini, Roberto Viotti, Michael F. Corcoran, Elena Pian & the AGILE Team
We present the results of extensive observations by the gamma-ray AGILE
satellite of the Galactic region hosting the Carina nebula and the colliding
wind binary Eta Car. The AGILE gamma-ray satellite monitored the Eta Car
region in several occasions during the period 2007 July to 2009 January. AGILE
detects a gamma-ray source consistent with the position of Eta Car. The
average gamma-ray flux above 100 MeV integrated over the pre-periastron period
2007 July - 2008 October is F = (37 +/- 5)$\times$10-8 ph/cm2/sec
corresponding to an average gamma-ray luminosity of L = 3.4$\times$1034
erg/sec for a distance of 2.3 kpc. AGILE also detected a remarkable 2-day
gamma-ray flaring episode of on 11-13 October 2008, most likely caused by a
colliding wind transient particle acceleration episode. The pre-periastron
gamma-ray emission appears to be erratic, and is possibly related to transient
acceleration and radiation episodes in the strongly variable colliding wind
shocks in the system. Our results provide the long sought first detection
above 100 MeV of a colliding wind binary, and have important theoretical
implications.
### 2.28 Long-term variability of Eta Car (Mairan Teodoro)
During the last 50 years, Eta Car has increased its brigthness at variable
rates. For instance, the central source presented V=8 from 1910 to 1940, when
it suddenly increased its brightness by 1 magnitude in a few years. Since
then, the brightness has increased almost linearly with time at a rate of
approximately 0.03 mag per year. However, after the spectroscopic event of
1997.9, the rate increased to 0.2 mag per year and remained so until mid-2006,
when a drop in the brightness of the central source was observed (almost 30
per cent in less than one year!). In this work we present the results of our
study on the long-term variability of the central source of Eta Car, showing
that, while the central source is getting brighter, the equivalent width of
the lines are getting weaker from cycle to cycle. Besides, our results
indicate that at least in the last 4 events, the behaviour of the high- and
intermediary-excitation lines near the spectroscopic event have not changed
signi cantly.
### 2.29 Eta Car around the 2009 periastron - a new view with X-shooter
(Christina Thone, Theodore R. Gull, Guido Chincarini, Elena Pian, Henrik
Hartman, Sandro D’Odorico & Lex Kapor)
We observed the Eta Car binary system with the X-shooter spectrograph at the
VLT during commissioning phase that spanned the latest periastron event of the
system on Jan. 11 2009. X-shooter covers the whole spectral range from the UV
(3000Å to the IR (2.5 $\mu$m) simultaneously with medium resolution
($R=\lambda/\delta\lambda=4000-9000$). Two long slits were placed on the
Homunculus skirt radially extending out from the star in opposite directions
at three different epochs in January (5 10 d after periastron), March and
June. At visible wavelengths, the Strontium Filament was sampled with three
sub-slits of the 1.8” $\times$4” Integral Field Unit (IFU) in January. The
shape of the Balmer lines in the opposite slit positions can give us
information about the orientation of the orbit of the secondary star. The
absence of PCygni absorption on the south-west slit indicates that the
secondary enters from the south-western side ionizing the wind material
causing the absorption in the north-east slit. The X-ray emission, which
disappears during periastron due to the collapse of the shock front of the
winds, recovered surpisingly early in 2009. High ionization lines were still
not visible again in the data of the March run while they are still visible in
the outer regions of the radial slits in January since those regions had not
yet seen the shut off of the FUV radiation due to the light travel time.
### 2.30 INTEGRAL observations of Eta Car (Roland Walter & Jean-Christophe
Leyder)
If relativistic particle acceleration takes place in colliding-wind binaries,
then hard X-rays and $\gamma$-rays are expected through inverse Compton
scattering of the copious UV radiation field. The INTEGRAL satellite provided
hard X-ray images of the Carina region with a much higher spatial resolution
than previously available. Based on observations taken far from periastron, a
bright source was detected at the position of Eta Car up to 100 keV. Two
additional nearby hard X-ray sources could also be resolved. This is the first
unambiguous detection of Eta Car at hard X-rays. There is no other X-ray
source in the hard X-ray error circle, bright enough to match the hard X-ray
flux.
The average hard X-ray emission of Eta Car in the 22-100 keV energy range is
very hard (with a photon index $\Gamma\approx 1$) and its luminosity ($7\times
10^{33}$erg/s) is in agreement with the predictions of inverse Compton models
and corresponds to about 0.1% of the energy available in the wind collision.
New INTEGRAL observations were taken during the 2009 periastron passage, and
the first results are presented. Only a 5-$\sigma$ upper-limit could be
derived. This is consistent with a lower fraction of very energetic particles
during periastron than outside. This could perhaps be linked with electron
cooling by the extreme radiation field.
### 2.31 BRITE-Constellation (Werner W. Weiss, Anthony F. Moffat & the
BRITE-Constellation Team)
BRITE-Constellation, a project developed since 2003 by researchers at Canadian
and Austrian Universities presently consists of UniBRITE and BRITE-
Austria/TUG-SAT1, which are two 20 cm cube nanosatellites. Each will host a 30
mm aperture telescope with a CCD camera equipped with either a red (550 to 700
nm) or a blue (390 to 460 nm) lter, to perform high-precision two-color
photometry of the brightest stars in the sky for up to several years.
Depending on the orbit and the position of the BRITE targets the photometry
can be obtained contiguously during many orbits for many months, with gaps
during individual orbits, or only for certain periods of the year.
The primary science goals are studies of luminous stars in our neighbourhood,
representing objects which dominate the ecology of our Universe, and of
evolved stars to probe the future development of our Sun.
A launch of UniBRITE and BRITE-Austri in 2009 is envisioned and an expansion
proposal of the BRITE-Constellation by two additional spacecraft of the same
construction is currently under review in Canada.
|
arxiv-papers
| 2009-10-16T16:53:19 |
2024-09-04T02:49:05.872701
|
{
"license": "Public Domain",
"authors": "Theodore R. Gull and Augusto Damineli",
"submitter": "Theodore Gull",
"url": "https://arxiv.org/abs/0910.3158"
}
|
0910.3201
|
# Instability of liquid jet
penetrated into stream in channel
Naoto Oka1, Ichiro Ueno2
1Graduate School, Tokyo University of Science,
2641 Yamazaki, Noda, Chiba 278-8510, JAPAN
2Tokyo University of Science, Noda, JAPAN
###### Abstract
Penetration process and an instability on a liquid jet impinging to a stream
of the same fluid in a channel is focused. The jet penetrated into the stream
is wrapped by entrained air, and coalesces with the stream when the air sheath
around the jet collapses. We introduce instability arisen on the jet and the
vigorous effect of the entrained-air sheath on the dynamic behavior of the jet
in this fluid dynamics video.
## 1 Introduction
In the present fluid dynamics video, we focus on instability of the penetrated
jet of 100-cSt silicone oil to the same liquid flowing in the channel. The
penetrated jet exhibits a Rayleigh-Plateau-like instability to break up into
droplets. We are interested in a unique behavior of the jet affected by the
entrained air; once the broken tip of the jet is completely capped by chance
by the air film around the jet, the disturbance is unexpectedly destabilized
to form the stable jet penetrating further without breakdown. This instability
might be essential to realize the jet bouncing off from the fluid flowing in
the channel (1).
REFERENCES
(1) Thrasher, M. et al., Phys. Rev. E 76,056319, 2007.
|
arxiv-papers
| 2009-10-16T19:50:21 |
2024-09-04T02:49:05.883480
|
{
"license": "Public Domain",
"authors": "Naoto Oka, Ichiro Ueno",
"submitter": "Naoto Oka",
"url": "https://arxiv.org/abs/0910.3201"
}
|
0910.3241
|
# Interpolation and Iteration for Nonlinear Filters
Alexandre J. Chorin and Xuemin Tu
Department of Mathematics,
University of California at Berkeley and Lawrence Berkeley National
Laboratory,
Berkeley, CA, 94720
###### Abstract
We present a general form of the iteration and interpolation process used in
implicit particle filters. Implicit filters are based on a pseudo-Gaussian
representation of posterior densities, and are designed to focus the particle
paths so as to reduce the number of particles needed in nonlinear data
assimilation. Examples are given.
Keywords: Implicit sampling, filter, pseudo-Gaussian, Jacobian, chainless
sampling, particles
## 1 Introduction
There are many problems in science in which the state of a system must be
identified from an uncertain equation supplemented by a stream of noisy data
(see e.g. [7]). A natural model of this situation consists of an Ito
stochastic differential equation (SDE):
$dx=f(x,t)\,dt+g(x,t)\,dw,$ (1)
where $x=(x_{1},x_{2},\dots,x_{m})$ is an $m$-dimensional vector, $w$ is
$m$-dimensional Brownian motion, $f$ is an $m$-dimensional vector function,
and $g(x,t)$ is an $m$ by $m$ diagonal matrix. The initial state $x^{0}$ is
assumed given and may be random as well.
As the solution of the SDE unfolds, it is observed, and the values $b^{n}$ of
a measurement process are recorded at times $t^{n},n=1,2,...$ For simplicity
assume $t^{n}=n\delta$, where $\delta$ is a fixed time interval. The
measurements are related to the evolving state $x(t)$ by
$b^{n}={h}(x^{n})+QW^{n},$ (2)
where $h$ is a $k$-dimensional, generally nonlinear, vector function with
$k\leq m$, $Q$ is a $k$ by $k$ diagonal matrix, $x^{n}=x(n\delta)$, and
$W^{n}$ is a vector whose components are $k$ independent Gaussian variables of
mean zero and variance one, independent also of the Brownian motion in
equation (1). The task is to estimate $x$ on the basis of equation (1) and the
observations (2).
If the system (1) and equation (2) are linear and the data are Gaussian, the
solution can be found via the Kalman-Bucy filter (see e.g. [3]). In the
general case, it is natural to try to estimate $x$ via its evolving
probability density. The initial state $x^{0}$ is known and so is its
probability density; all one has to do is evaluate sequentially the density
$P_{n+1}$ of $x^{n+1}$ given the probability densities $P_{k}$ of $x^{k}$ for
$k\leq n$ and the data $b^{n+1}$. This can be done by following “particles”
(replicas of the system) whose empirical distribution approximates $P_{n}$. A
standard construction (see e.g [13, 12, 8, 1, 11, 5, 10, 9]) uses the
probability density function (pdf) $P_{n}$ and equation (1) to generate a
prior density, and then uses the new data $b^{n+1}$ to generate a posterior
density $P_{n+1}$ through weighting and resampling. In addition, one has to
sample backward to take into account the information each measurement provides
about the past, as well as avoid having too many identical particles after
resampling. This can be very expensive, in particular because the number of
particles needed can grow catastrophically (see e.g. [14, 2] and also Example
2 below). Sophisticated methods for generating efficient priors can be found
e.g. in [8, 1]. The challenge is to generate high probability samples so as to
minimize the effort of computing particle paths whose weight is very low.
In [6] we introduced an alternative to the standard approach. In our method
the posterior density is sampled directly by iteration and interpolation, as
suggested by our earlier work on chainless sampling [4], and by the
observation in [15] connecting interpolation and the marginalization process
used in chainless sampling. The new filter aims the particle trajectories as
accurately as possible in the direction of the observations so that fewer
particles are needed. In that earlier paper our approach was presented by
means of simple examples. In the present paper we present a general, more
abstract, formulation, introduce an extension to the case of sparse
observations, and discuss additional examples.
## 2 Forward step
To begin, assume that at time $t^{n}=n\delta$, where $\delta>0$ is fixed, we
have a collection of $M$ particles $X_{i}^{n}$, $1\leq i\leq M$,
$n=0,1,\dots$, whose empirical density approximates $P_{n}$, the probability
density at time $n\delta$ of the particles that obey the evolution equation
(1) subject to the observations (2) at times $t=k\delta$ for $k\leq n$. In the
present section we explain how to find positions for the same particles at
time $(n+1)\delta$ given only the positions at time $n\delta$ and the pdf
$P_{n}$, taking into account the next observation and the equation of motion.
Let $N(a,v)$ denote a Gaussian variable of mean $a$ and variance $v$. First,
approximate the SDE (1) by a difference scheme of the form
$X^{n+1}=X^{n}+F(X^{n},t^{n})\delta+G(X^{n},t^{n})V^{n+1},$ (3)
where we assume temporarily that $\delta$ equals the interval between
observations, i.e., we assume that there is an observation at every time step.
$X^{n}$ stands for $X(n\delta)$, $G$ is assumed to be diagonal, and
$X^{n},X^{n+1}$ are $m$ dimensional vectors. $F,G$ determine the scheme used
to solve the SDE, see for example [6]. $V^{n+1}$ is a vector of $N(0,\delta)$
Gaussian variables, independent of each other for each $n$, with the vectors
$V^{n+1}$ independent of each other for differing $n$, independent also of the
$W^{k},k=1,...,$ in the observation equation (2). The sequence of
$X^{n},n=0,1,\dots$ approximates a sample solution of the SDE, $X^{0}$ is
assumed given and may be random. The function $G$ in (3) does not depend on
$X^{n+1}$ for an Ito equation, and we assume for simplicity that $F$ does not
depend on $X^{n+1}$ either, because this was the case in all the examples we
have worked on so far. The analysis below can be easily repeated for the case
where $F$ does depend on $X^{n+1}$, at the cost of slightly more complicated
formulas. Equation (3) states that $X^{n+1}-X^{n}$ is an
$N(F(X^{n},t^{n})\delta,\delta G(X^{n},t^{n})^{*}G(X^{n},t^{n}))$ vector,
where the star * denotes a transpose.
We have one sample solution $X_{i}^{n}$ of the SDE for each particle. Our task
is to sample, for each particle, the vector $X^{n+1}_{i}$ whose probability
density is determined by the approximation of the SDE as well as by the next
observation for each of the $M$ particles. We keep the notation $X^{n+1}_{i}$
for the positions of the particles even though once the observation is taken
into account these positions no longer coincide with the positions of sample
solutions of equation (3).
Consider the $i$-th particle. We are going to work particle by particle, so
that the particle index $i$ will be temporarily suppressed. Suppose we already
know the posterior vector $X^{n+1}$. Its probability density $P_{n+1}$ of
$X^{n+1}$ given $X^{n}$ is
$\displaystyle P_{n+1}(X^{n+1})$ $\displaystyle=$ $\displaystyle
Z^{-1}\exp\left(-\left(X^{n+1}-X^{n}-F_{n}\right)^{*}(G_{n}^{*}G_{n})^{-1}\left(X^{n+1}-X^{n}-F_{n}\right)/2\right.$
$\displaystyle\left.-\left(h(X^{n+1})-b^{n+1}\right)^{*}(Q^{*}Q)^{-1}\left(h(X^{n+1})-b^{n+1}\right)/2\right),$
where the functions $F_{n}=F(X^{n},t^{n})\delta$, and
$G_{n}=\sqrt{\delta}G(X^{n},t^{n})$ can be read from the approximation of the
SDE, and $Z$ is a normalization constant, the integral of the numerator over
all $X^{n+1}$ with $X^{n}$ fixed. The value of this $Z$ is not available. Our
goal is to find samples $X^{n+1}$ whose probability is high, and which are
well distributed with respect to $P_{n+1}$. We do that by picking the
probability in advance: we first pick samples of $m$ $N(0,1)$ variables
$(\xi_{1},\xi_{2},\dots,\xi_{m})=\xi$, whose joint pdf (probability density
function) is $\exp(-\xi^{*}\xi/2))/(2\pi)^{m/2}$, and require that each
$X^{n+1}$ be a function of a sample $\xi$ with the same probability as $\xi$,
up to the Jacobian of the transformation. This should produce likely and well-
distributed samples.
A little thought shows that this can be done, not by equating $P_{n+1}$ to
$\exp(-\xi^{*}\xi/2)/(2\pi)^{m/2}$, but by equating the arguments of the two
exponentials. For example, if one wants to represent a $N(0,v)$ random
variable $x$ with pdf $\exp(-\frac{x^{2}}{2v})/\sqrt{2\pi v}$ as a function of
a $N(0,1)$ variable $\xi$ with pdf $\exp(-\xi^{2}/2)/\sqrt{2\pi}$, equating
the arguments yields $x=\sqrt{v}\,\xi$, clearly a good choice. Thus, we wish
to solve the equation
$\displaystyle{\xi}^{*}\xi/2=$ $\displaystyle=$
$\displaystyle\left(X^{n+1}-X^{n}-F_{n}\right)^{*}(G_{n}^{*}G_{n})^{-1}\left(X^{n+1}-X^{n}-F_{n}\right)/2+\left(h(X^{n+1})-b^{n+1}\right)^{*}(Q^{*}Q)^{-1}\left(h(X^{n+1})-b^{n+1}\right)/2$
and obtain $X^{n+1}$ as a function of $\xi$.
We proceed point by point— given a vector $\xi$, we find the corresponding
$X^{n+1}$ rather than look for an expression for the function $X^{n+1}(\xi)$
as a whole—and by iteration: we find a sequence of approximations
$X^{n+1}_{j}$ ($=X_{j}$ for brevity) which converges to $X^{n+1}$; we set
$X_{0}=0$, and now explain how to find $X_{j+1}$ given $X_{j}$. First, expand
the function $h$ in the observation equation (2) in Taylor series around
$X_{j}$:
$h(X_{j+1})=h(X_{j})+H_{j}\cdot(X_{j+1}-X_{j}),$ (6)
where $H_{j}$ is a Jacobian matrix evaluated at $X_{j}$. The observation
equation (2) can be approximated as:
$z_{j}=H_{j}X_{j+1}+QW^{n+1},$ (7)
where $z_{j}=b^{n+1}-h(X_{j})+H_{j}X_{j}$.
The left side of equation (LABEL:args) can be approximated as:
$\displaystyle\left(X_{j+1}-X^{n}-F_{n}\right)^{*}(G_{n}^{*}G_{n})^{-1}\left(X_{j+1}-X^{n}-F_{n}\right)/2+\left(H_{j}X_{j+1}-z_{j}\right)^{*}(Q^{*}Q)^{-1}\left(H_{j}X_{j+1}-z_{j}\right)/2$
$\displaystyle=$
$\displaystyle\left(X_{j+1}-\bar{m}_{j}\right)^{*}\Sigma_{j}^{-1}\left(X_{j+1}-\bar{m}_{j}\right)/2+\Phi_{j},$
(8)
where
$\Sigma_{j}^{-1}=(G_{n}^{*}G_{n})^{-1}+H_{j}^{*}(Q^{*}Q)^{-1}H_{j},\quad\bar{m}_{j}=\Sigma_{j}\left((G_{n}^{*}G_{n})^{-1}(X^{n}+F_{n})+H_{j}^{*}(Q^{*}Q)^{-1}z_{j}\right),$
and
$K_{j}=H_{j}G_{n}^{*}G_{n}H_{j}^{*}+Q^{*}Q,\quad\Phi_{j}=\left(z_{j}-H_{j}(X^{n}+F_{n})\right)^{*}K_{j}^{-1}\left(z_{j}-H_{j}(X^{n}+F_{n})\right)/2.$
We now solve for $X_{j+1}$ as a function of $\xi$. To make the computation
tractable, in this step we ignore the remainder $\Phi_{j}$; this is a key
step. We thus solve the simpler equation
$(X_{j+1}-\bar{m}_{j})^{*}\Sigma_{j}^{-1}(X_{j+1}-\bar{m}_{j})/2=\xi^{*}\xi/2.$
(9)
This can be done in any of a number of ways; for example, one can write
$\Sigma_{j}=L_{j}L_{j}^{*}$, where $L_{j}$ is a lower triangular matrix and
$L_{j}^{*}$ is its transpose, and then set $X_{j+1}=\bar{m}_{j}+L_{j}\xi$ (a
different algorithm was suggested in [6]). The iteration is done.
If the sequence $X_{j}$ converges to a limit, call the limit $X^{n+1}$. One
can readily check that the approximate equation (7) converges to the full
observation equation (2). The remainders $\Phi_{j}$ also converge to a limit
$\Phi^{n+1}$. Equation (LABEL:args) becomes:
$\displaystyle\xi^{*}\xi/2+\Phi^{n+1}=$ $\displaystyle=$
$\displaystyle\left(X^{n+1}-X^{n}-F_{n}\right)^{*}(G_{n}^{*}G_{n})^{-1}\left(X^{n+1}-X^{n}-F_{n}\right)/2+(h(X^{n+1})-b^{n+1})(Q^{*}Q)^{-1}(h(X^{n+1})-b^{n+1})/2.$
Multiply this equation by $-1$ and exponentiate both sides:
$\displaystyle\exp(-\xi^{*}\xi/2)\exp(-\Phi^{n+1})=$ $\displaystyle=$
$\displaystyle\exp\left(-\left(X^{n+1}-X^{n}-F_{n}\right)^{*}(G_{n}^{*}G_{n})^{-1}\left(X^{n+1}-X^{n}-F_{n}\right)/2-\left(h(X^{n+1})-b^{n+1})^{*}(Q^{*}Q)^{-1}(h(X^{n+1})-b^{n+1}\right)/2\right).$
This differs from what we set out to do in equation (LABEL:args) by the factor
$\exp(-\Phi^{n+1})$ on the right hand side.
Let $P(\alpha|\beta)$ be the probability of $\alpha$ given $\beta$. The factor
$\exp(-\Phi^{n+1})$ is proportional to $P(b^{n+1}|X^{n})$, and equation
(LABEL:main) is the statement
$P(X^{n+1}|X^{n},b^{n+1})P(b^{n+1}|X^{n})=P(X^{n+1}|X^{n})P(b^{n+1}|X^{n+1}),$
(12)
i.e., this is Bayes’ theorem. Note also that equation (9) is a pseudo-Gaussian
representation of $X^{n+1}$, not a Gaussian representation; the matrix
$\Sigma_{j}$ is a function of the sample.
We next compute the Jacobian determinant
$J=\det({\partial}X^{n+1}/{\partial\xi})$. This can be often done
analytically. Equation (9) relates $X^{n+1}$ to $\xi$ implicitly. We have
values of $\xi$ and the corresponding values of $X^{n+1}$; to find $J$ there
is no need to solve again for $X^{n+1}$; an implicit differentiation is all
that is needed. Alternately, $J$ can be found numerically, by taking nearby
values of $\xi$, redoing the iteration (which should converge in one step,
because one can start from the known value of $X^{n+1}$), and differencing.
The expression on the right-hand side of equation (LABEL:main) is proportional
to $P(b^{n+1}|X^{n+1})P(X^{n+1}|X^{n})$, with a proportionality constant
independent of $X^{n}$. When $X^{n+1}$ is sampled as just described, each
value of $X^{n+1}=X^{n+1}(\xi)$ appears with probability
$\frac{1}{(2\pi)^{m/2}}\exp(-\xi^{*}\xi/2)/|J|$, and then the value of this
expression is $\exp(-\xi^{*}\xi/2)\exp(-\Phi^{n+1})$. To get the right value
of the expression on the average, one has to give each proposed $X^{n+1}$ the
sampling weight $W=\frac{1}{(2\pi)^{m/2}}\exp(-\Phi^{n+1})|J|$, (with another
factor $P(X^{n})$ if such factors are not all equal). Since
$\frac{1}{(2\pi)^{m/2}}$ is a constant and the same to every particle, we will
drop it from now on. Here we see an advantage of starting from a prechosen
reference variable $\xi$: the factor $\exp(-\xi^{*}\xi/2)$, which varies from
sample to sample, has been discounted in advance and does not contribute to
the non-uniformity of the weights. We shall see that the other factors can be
expected to vary little.
Do this for all the particles and obtain new positions with weights
$W_{i}=\exp(-\Phi^{n+1}_{i})|J_{i}|$, where $\Phi^{n+1}_{i},J_{i}$ are the
values of these quantities for the $i$-th particle. One can get rid of the
weights after the fact by resampling, i.e., for each of $M$ random numbers
$\theta_{k},k=1,\dots,M$ drawn from the uniform distribution on $[0,1]$,
choose a new ${\widehat{X}}^{n+1}_{k}=X^{n+1}_{i}$ such that
$A^{-1}\sum_{j=1}^{i-1}W_{j}<\theta_{k}\leq A^{-1}\sum_{j=1}^{i}W_{j}$ (where
$A=\sum_{j=1}^{M}W_{j}$), and then suppress the hat.
Note also that the resampling does not have to be done at every step- for
example, one can add up the phases for a given particle and resample only when
the ratio of the largest cumulative weight $\exp(-\sum(\phi_{i}-\log|J_{i}|))$
to the smallest such weight exceeds some limit $L$ (the summation is over the
weights accrued to a particular particle $i$ since the last resampling). If
one is worried by too many particles being close to each other (”depletion” in
the usual Bayesian terminology), one can divide the set of particles into
subsets of small size and resample only inside those subsets, creating a
greater diversity. As will be seen in the numerical results section, none of
these strategies is used here and we resample fully at every step.
The computational complexity of this construction depends on the sparseness of
the matrix $\Sigma_{j}$, which depends on the sparseness of $H_{j}$ in the
expression (8), which depends on the structure of the function $h$ in equation
(2). In the frequently encountered situation where $h$ is diagonal, in the
sense that each quantity measured is a function of a single component of the
vector whose dynamics are given by equation (1), one finds that $\Sigma_{j}$
and $H_{j}$ are diagonal, and the computations, including the computation of
the Jacobian $J$, are easy, whether $h$ is linear or not. The more arguments
in each of the components of the function $h$, the more labor is required.
If both equations (1) and (2) are linear and the initial data are Gaussian,
then the pdfs $P_{n}$ are Gaussian. We only need to find the mean and the
variance of the pdf, which can be found as above by considering a single
particle; the iterations converge in one step. The resulting means and
variances are identical to those produced by the Kalman filter. If one had
needed multiple particles, their weights would have been all equal. If
equation (1) is nonlinear but equation (2) is linear (or can be well
approximated by a linear function in each interval $(n\delta,({n+1})\delta)$),
then the $P_{n+1}$ are in general not Gaussian and one needs multiple
particles. The iterations still converge in one step, and what one obtains is
a version of the forward step in a filter with an optimal importance function
(as described e.g in [6]).
The convergence of the iteration will be very briefly discussed further below.
We have chosen the variables $\xi$ to be independent $N(0,1)$ variables, but
there is nothing sacred about this choice. The goal is to pick samples whose
probability is high, and in some contexts other choices may be better. We will
discuss those other choices when they are made in further work.
## 3 Backward sampling
In the previous section we described how to sample the pdf at time
$(n+1)\delta$ given the pdf at time $n\delta$. In general, this is not
sufficient. Every observation provides information not only about the future
but also about the past- it may, for example, tag as improbable earlier states
that had seemed probable before the observation was made. Furthermore, in non-
Gaussian settings, the pdf one obtains by going directly from time
$(n-1)\delta$ to step $(n+1)\delta$ by a step of duration $2\delta$ may be
different from the pdf one obtains after two steps that include an
intermediate step. After one has sampled at time $(n+1)\delta$, one has to go
back, correct the past, and resample (this backward sampling is often
misleadingly explained in the literature solely by the need to create greater
diversity among the particles). We resample by interpolation, which we present
explicitly for one backward step. It is quite obvious one can do that for as
many backward steps as are needed.
Given a set of particles at time $(n+1)\delta$, after a forward step and maybe
a subsequent resampling, one can figure out where each particle $i$ was in the
previous two steps, and have a partial history for each particle $i$:
$X_{i}^{n-1},X_{i}^{n},X_{i}^{n+1}$ (if resamplings had occurred, some parts
of that history may be shared among several current particles). Knowing the
first and the last members of this sequence, we recompute $X^{n}$ by
interpolation, thus projecting information backward one step.
The probability of the $X^{\text{new}}$ that will replace $X^{n}$ is the
product of the three probabilities (properly normalized): the probability of
the new leg from $X^{n-1}$ to $X^{n}$, the probability of the resulting leg
from $X^{n}$ to $X^{n+1}$ (the end result being known), and the probability of
the resulting observation at time $n\delta$, i.e.:
$\displaystyle\exp\left(-\left(X^{\text{new}}-X^{n-1}-F_{n-1}\right)^{*}(G_{n-1}^{*}G_{n-1})^{-1}\left(X^{\text{new}}-X^{n-1}-F_{n-1}\right)/2\right.$
$\displaystyle\left.-\left(X^{n+1}-X^{\text{new}}-F_{n}\right)^{*}(G_{n}^{*}G_{n})^{-1}\left(X^{n+1}-X^{n}-F_{n}\right)/2-\left(h(X^{\text{new}})-b^{n}\right)^{*}(Q^{*}Q)^{-1}\left(h(X^{\text{new}})-b^{n}\right)/2\right).$
Here we recall that $F_{n-1}=F(X^{n-1},t^{n-1})\delta$ and
$G_{n-1}=\sqrt{\delta}G(X^{n-1},t^{n-1})$ are known from the approximation of
the SDE, $F_{n}$ and $G_{n}$ are functions of $X^{\text{new}}$, and the
subscript $i$ referring to the particle has been omitted. This expression
differs from equation (LABEL:Pn+1) by having an additional exponential factor.
Once again, we set up an iteration, with iterates $X_{j}$, that converges to
$X^{\text{new}}$, and start with $X_{0}=0$. We expand $h(X_{j+1})$ in a Taylor
series around $X_{j}$, so that the last factor in the expression (LABEL:new)
becomes a quadratic in $X_{j+1}$. We complete squares so that the argument of
the exponential in (LABEL:new) can be written as
$(X_{j+1}-\bar{m}_{j})\Sigma_{j}^{-1}((X_{j+1}-\bar{m}_{j})/2+\Phi_{j}$;
equate $(X_{j+1}-\bar{m}_{j})\Sigma_{j}^{-1}((X_{j+1}-\bar{m}_{j})/2$ to
$\xi^{*}\xi/2$, solve to get $X_{j+1}$ as a function of $\xi$, calculate the
Jacobian, and find the weight. We do this for all the particles, and resample
as needed. This concludes the backward sampling step. Note that as a result of
the backward step and the subsequent forward step, $P_{n+1}$ depends, not only
on the positions of the particles at time $n\delta$, but also on the earlier
history of the system.
## 4 Sparse observations
Consider now a situation where we do not have observations at every time step.
First, assume that one has observation at time $(n+1)\delta$ but not at time
$n\delta$. We try to sample $X^{n}$ and $X^{n+1}$ together given the
observation information at time step $(n+1)\delta$. Consider the $i$-th
particle. Suppose we are given the vector $X^{n-1}_{i}$ for that particle.
Suppress again the particle index $i$. The joint probability density
$P_{n,n+1}$ of $X^{n}$ and $X^{n+1}$ given $X^{n-1}$ is
$\displaystyle P_{n,n+1}(X^{n},X^{n+1})$ $\displaystyle=$ $\displaystyle
Z^{-1}\exp\left(-\left(X^{n}-X^{n-1}-F_{n-1}\right)^{*}(G_{n-1}^{*}G_{n-1})^{-1}\left(X^{n}-X^{n-1}-F_{n-1}\right)/2\right.$
$\displaystyle\left.-\left(X^{n+1}-X^{n}-F_{n}\right)^{*}(G_{n}^{*}G_{n})^{-1}\left(X^{n+1}-X^{n}-F_{n}\right)/2-\left(h(X^{n+1})-b^{n+1}\right)^{*}(Q^{*}Q)^{-1}\left(h(X^{n+1})-b^{n+1}\right)/2\right),$
where $Z$ is the normalization constant. We recall that
$F_{n-1}=F(X^{n-1},t^{n-1})\delta$, $G_{n-1}=\sqrt{\delta}G(X^{n-1},t^{n-1})$
are known from the approximation of the SDE, $F_{n}$ and $G_{n}$ depend on
$X^{n}$.
In the now familiar sequence of steps, we pick two independent samples
$\xi_{n}$ and $\xi_{n+1}$, each with probability density
$\exp(-\xi^{*}\xi/2)/(2\pi)^{m/2}$, and try to solve the equation
$\displaystyle{\xi_{n}}^{*}\xi_{n}/2+\xi_{n+1}^{*}\xi_{n+1}/2$
$\displaystyle=$
$\displaystyle\left(X^{n}-X^{n-1}-F_{n-1}\right)^{*}(G_{n-1}^{*}G_{n-1})^{-1}\left(X^{n}-X^{n-1}-F_{n-1}\right)/2$
$\displaystyle+\left(X^{n+1}-X^{n}-F_{n}\right)^{*}(G_{n}^{*}G_{n})^{-1}\left(X^{n+1}-X^{n}-F_{n}\right)/2+\left(h(X^{n+1})-b^{n+1}\right)^{*}(Q^{*}Q)^{-1}\left(h(X^{n+1})-b^{n+1}\right)/2,$
(15)
to obtain $X^{n}$ and $X^{n+1}$ as functions of $\xi_{n}$ and $\xi_{n+1}$.
We define a sequence of approximations $X^{n}_{j}$ and $X^{n+1}_{j}$ which
converge to $X^{n}$ and $X^{n+1}$, respectively; set $X^{n}_{0}=0$ and
$X_{0}^{n+1}=0$, and at each iteration find $X^{n}_{j+1}$ and $X^{n+1}_{j+1}$
given $X^{n}_{j}$ and $X^{n+1}_{j}$. First, expand the function $h$ in the
observation equation (2) in Taylor series around $X^{n+1}_{j}$:
$h(X^{n+1}_{j+1})=h(X^{n+1}_{j})+H^{n+1}_{j}\cdot(X^{n+1}_{j+1}-X^{n+1}_{j}),$
(16)
where $H^{n+1}_{j}$ is a Jacobian matrix evaluated at $X^{n+1}_{j}$. The
observation equation (2) is approximated as:
$z_{j}^{n+1}=H^{n+1}_{j}X^{n+1}_{j+1}+QW^{n+1},$ (17)
where $z^{n+1}_{j}=b^{n+1}-h(X^{n+1}_{j})+H^{n+1}_{j}X^{n+1}_{j}$.
Let $F_{n,j}=F(X^{n}_{j},t^{n})\delta$ and
$G_{n,j}=\sqrt{\delta}G(X^{n}_{j},t^{n})$. The right side of equation (15) can
be approximated as:
$\displaystyle\left(X_{j+1}^{n}-X^{n-1}-F_{n-1}\right)^{*}(G_{n-1}^{*}G_{n-1})^{-1}\left(X_{j+1}^{n}-X^{n-1}-F_{n-1}\right)/2$
$\displaystyle+$
$\displaystyle\left(X_{j+1}^{n+1}-X_{j+1}^{n}-F_{n,j}\right)^{*}(G_{n,j}^{*}G_{n,j})^{-1}\left(X_{j+1}^{n+1}-X_{j+1}^{n}-F_{n,j}\right)/2$
$\displaystyle+$
$\displaystyle\left(H_{j}^{n+1}X_{j+1}^{n+1}-z_{j}^{n+1}\right)^{*}(Q^{*}Q)^{-1}\left(H_{j}^{n+1}X_{j+1}^{n+1}-z_{j}^{n+1}\right)/2.$
We first combine the last two terms in (LABEL:bH121) and obtain
$\displaystyle\left(X_{j+1}^{n+1}-X_{j+1}^{n}-F_{n,j}\right)^{*}(G_{n,j}^{*}G_{n,j})^{-1}\left(X_{j+1}^{n+1}-X_{j+1}^{n}-F_{n,j}\right)/2+\left(H_{n+1}X_{j+1}^{n+1}-z^{n+1}\right)^{*}(Q^{*}Q)^{-1}\left(H_{n+1}X_{j+1}^{n+1}-z^{n+1}\right)/2$
$\displaystyle=$
$\displaystyle\left(X^{n+1}_{j+1}-\bar{m}_{j}^{n+1}\right)^{*}(\Sigma_{j}^{n+1})^{-1}\left(X^{n+1}_{j+1}-\bar{m}_{j}^{n+1}\right)/2+\Phi^{n+1}_{j},$
(19)
where
$(\Sigma_{j}^{n+1})^{-1}=(G_{n,j}^{*}G_{n,j})^{-1}+(H_{j}^{n+1})^{*}(Q^{*}Q)^{-1}H_{j}^{n+1},$
$\bar{m}_{j}^{n+1}=\Sigma_{j}^{n+1}\left((G_{n,j}^{*}G_{n,j})^{-1}(X^{n}_{j+1}+F_{n,j})+(H_{j}^{n+1})^{*}(Q^{*}Q)^{-1}z_{j}^{n+1}\right),$
$K_{j}^{n+1}=H_{j}^{n+1}G_{n,j}^{*}G_{n,j}(H_{j}^{n+1})^{*}+Q^{*}Q,$
and
$\Phi^{n+1}_{j}=\left(z_{j}^{n+1}-H_{j}^{n+1}(X^{n}_{j+1}+F_{n,j})\right)^{*}(K_{j}^{n+1})^{-1}\left(z_{j}^{n+1}-H_{j}^{n+1}(X^{n}_{j+1}+F_{n,j})\right)/2.$
We combine the first term in (LABEL:bH121) and the second term in (19) and
obtain
$\displaystyle\left(X_{j+1}^{n}-X^{n-1}-F_{n-1}\right)^{*}(G_{n-1}^{*}G_{n-1})^{-1}\left(X_{j+1}^{n}-X^{n-1}-F_{n-1}\right)/2+\Phi^{n+1}_{j}$
(20) $\displaystyle=$
$\displaystyle\left(X_{j+1}^{n}-X^{n-1}-F_{n-1}\right)^{*}(G_{n-1}^{*}G_{n-1})^{-1}\left(X_{j+1}^{n}-X^{n-1}-F_{n-1}\right)/2$
$\displaystyle+\left(z_{j}^{n+1}-H_{j}^{n+1}(X^{n}_{j+1}+F_{n,j})\right)^{*}(K_{j}^{n+1})^{-1}\left(z_{j}^{n+1}-H_{j}^{n+1}(X^{n}_{j+1}+F_{n,j})\right)/2$
$\displaystyle=$
$\displaystyle\left(X^{n}_{j+1}-\bar{m}_{j}^{n}\right)^{*}(\Sigma_{j}^{n})^{-1}\left(X^{n}_{j+1}-\bar{m}_{j}^{n}\right)/2+\Phi^{n}_{j},$
where
$(\Sigma_{j}^{n})^{-1}=(G_{n-1}^{*}G_{n-1})^{-1}+(H_{j}^{n+1})^{*}(K_{j}j^{n+1})^{-1}H_{j}^{n+1},$
$\bar{m}_{j}^{n}=\Sigma_{j}^{n}\left((G_{n-1}^{*}G_{n-1})^{-1}(X^{n-1}+F_{n-1})+(H_{j}^{n+1})^{*}(K_{j}^{n+1})^{-1}(z_{j}^{n+1}-H_{j}^{n+1}F_{n,j})\right),$
$K_{j}^{n}=H_{j}^{n+1}G_{n-1}^{*}G_{n-1}(H_{j}^{n+1})^{*}+K_{j}^{n+1},$
and
$\displaystyle\Phi_{j}^{n}=\left(z_{j}^{n+1}-H_{j}^{n+1}(F_{n,j}+X^{n-1}+F_{n-1})\right)^{*}(K_{j}^{n})^{-1}\left(z_{j}^{n+1}-H_{j}^{n+1}(F_{n,j}+X^{n-1}+F_{n-1})\right)/2.$
Combining (15), (16), (LABEL:bH121), (19), and (20), we try to solve
$\displaystyle{\xi_{n}}^{*}\xi_{n}/2+\xi_{n+1}^{*}\xi_{n+1}/2$
$\displaystyle=$
$\displaystyle\left(X^{n+1}_{j+1}-\bar{m}_{j}^{n+1}\right)^{*}(\Sigma_{j}^{n+1})^{-1}\left(X^{n+1}_{j+1}-\bar{m}_{j}^{n+1}\right)/2+\left(X^{n}_{j+1}-\bar{m}_{j}^{n}\right)^{*}(\Sigma_{j}^{n})^{-1}\left(X^{n}_{j+1}-\bar{m}_{j}^{n}\right)/2+\Phi^{n}_{j}.$
(21)
We now solve for $X^{n}_{j+1}$ and $X^{n+1}_{j+1}$ as functions of $\xi_{n}$
and $\xi_{n+1}$, ignoring the remainders $\Phi^{n}_{j}$, i.e. we solve the
simpler equations
$(X^{k}_{j+1}-\bar{m}_{j}^{k})^{*}(\Sigma_{j}^{k})^{-1}(X^{k}_{j+1}-\bar{m}_{j}^{k})/2=\xi_{k}^{*}\xi_{k}/2,\quad
k=n,n+1$ (22)
If the sequences $X^{n}_{j}$ and $X^{n+1}_{j}$ converge to limits, call the
limits $X^{n}$ and $X^{n+1}$. In the limit, the approximate equation (17)
converges to the full observation equation (2). The remainders $\Phi^{n}_{j}$
and $\Phi^{n+1}_{j}$ also converge to limits $\Phi^{n}$ and $\Phi^{n+1}$.
Equation (15) becomes:
$\displaystyle\xi_{n}^{*}\xi_{n}/2+\xi_{n+1}^{*}\xi_{n+1}/2+\Phi^{n}$
$\displaystyle=$
$\displaystyle\left(X^{n}-X^{n-1}-F_{n-1}\right)^{*}(G_{n-1}^{*}G_{n-1})^{-1}\left(X^{n}-X^{n-1}-F_{n-1}\right)/2$
$\displaystyle+\left(X^{n+1}-X^{n}-F_{n}\right)^{*}(G_{n}^{*}G_{n})^{-1}\left(X^{n+1}-X^{n}-F_{n}\right)/2+(h(X^{n+1})-b^{n+1})(Q^{*}Q)^{-1}(h(X^{n+1})-b^{n+1})/2.$
Multiply by $-1$ and exponentiate:
$\displaystyle\exp(-\xi_{n}^{*}\xi_{n}/2)\exp(-\xi_{n+1}^{*}\xi_{n+1}/2)\exp(-\Phi^{n})$
$\displaystyle=$
$\displaystyle\exp\left(\left(X^{n}-X^{n-1}-F_{n-1}\right)^{*}(G_{n-1}^{*}G_{n-1})^{-1}\left(X^{n}-X^{n-1}-F_{n-1}\right)/2\right.$
$\displaystyle+$
$\displaystyle\left.\left(X^{n+1}-X^{n}-F_{n}\right)^{*}(G_{n}^{*}G_{n})^{-1}\left(X^{n+1}-X^{n}-F_{n}\right)/2+\left(h(X^{n+1})-b^{n+1})^{*}(Q^{*}Q)^{-1}(h(X^{n+1})-b^{n+1}\right)/2\right).$
As before, one has to give each proposed $X^{n}$ and $X^{n+1}$ the sampling
weight $W=\exp(-\Phi^{n})|J|$, where $J$ is the Jacobian
$J=\det(\partial(X^{n},X^{n+1})/\partial(\xi_{n},\xi_{n+1}))$ which must be
computed. One does this for all particles and resamples as needed. This
process can be generalized if one wishes to sample at more times between
observations. One should also note that the procedure just described may make
the evaluation of Jacobians significantly more onerous, but still often
tractable.
The construction of this paragraph is important because many data sets one
tries to assimilate are indeed sparse, and also for the following reason. We
have not provided in this present paper a discussion of the convergence of the
iterations we use. This convergence depends on the structure of the underlying
SDE, on the scheme used to approximate it, and on the specific ways one solves
for the new increments in terms of the reference variables $\xi$, and cannot
be analyzed without considering these specifics. In our previous paper [6] we
analyzed a special case and found that there the convergence depended on the
size of the time step. We conjecture that this happens frequently. The present
section provides a way to decrease the time step as a device for repairing
diverging iterations without much additional thought.
## 5 Example 1
We apply our filter to a prototypical marine ecosystem model studied in [10].
We set the main parameters equal to the ones in [10]; however, we will also
present some results with a range of noise variances to make a particular
point. We did the data assimilation with the filter described above, without
back sampling, and also by the a standard particle filter SIR (Sampling
importance resampling), see [1].
The model involves four state variables: phytoplankton P (microscopic plants),
zooplankton Z (microscopic animals), nutrients N (dissolved inorganics), and
detritus D (particulate organic non-living matter). At the initial time $t=0$
we have $P(0)=0.125$, $Z(0)=0.00708$, $N(0)=0.764$, and $D(0)=0.136$. The
system is described by the following nonlinear ordinary differential
equations, explained in [10]:
$\displaystyle\frac{dP}{dt}$ $\displaystyle=$
$\displaystyle\frac{N}{0.2+N}\gamma
P-0.1P-0.6\frac{P}{0.1+P}Z+N(0,\sigma^{2}_{P})$ $\displaystyle\frac{dZ}{dt}$
$\displaystyle=$ $\displaystyle 0.18\frac{P}{0.1+P}Z-0.1Z+N(0,\sigma^{2}_{Z})$
$\displaystyle\frac{dN}{dt}$ $\displaystyle=$ $\displaystyle
0.1D+0.24\frac{P}{0.1+P}Z-\gamma P\frac{N}{0.2+N}+0.05Z+N(0,\sigma^{2}_{N})$
$\displaystyle\frac{dD}{dt}$ $\displaystyle=$
$\displaystyle-0.1D+0.1P+0.18\frac{P}{0.1+P}Z+0.05Z+N(0,\sigma^{2}_{D}),$ (25)
where the parameter $\gamma$, the “ growth rate”, is determined by the
equations given by
$\gamma_{t}=0.14+3\Delta\gamma_{t},\quad\Delta\gamma_{t}=0.9\Delta\gamma_{t-1}+N(0,\sigma^{2}_{\gamma}).$
The variances of the noise terms are: $\sigma_{P}^{2}=(0.01P(0))^{2}$,
$\sigma_{Z}^{2}=(0.01Z(0))^{2}$, $\sigma_{N}^{2}=(0.01N(0))^{2}$,
$\sigma_{D}^{2}=(0.01D(0))^{2}$, and $\sigma_{\gamma}^{2}=(0.01)^{2}$.
The observations were obtained from NASA’s SeaWiFS satellite ocean color
images. These observations provide a time series for phytoplankton; the
relation between the observations $P(t)_{\mbox{obs}}$ (corresponding to the
vector $b^{n}$ in the earlier discussion) and the solution $P(t)$ of the
equation of the first equation in (25) is assumed to be:
$\log P(t)_{\mbox{obs}}=\log P(t)+N(0,\sigma^{2}_{\mbox{obs}}),$
where $\sigma^{2}_{\mbox{obs}}=0.3^{2}$. Note that this observation equation
is not linear. There are 190 data points distributed from late 1997 to mid
2002. The sample intervals ranged from a week to a month or more, for details
see [10]. As in [10], we discretize the system (25) by an Euler method with
$\Delta t=1$ day and prohibit the state variables from dropping below 1
percent of their initial values.
We have compared our filter and SIR in three sets of numerical experiments,
all with the same initial values as listed above. In each case we attempted to
find a trajectory of the system consistent with the fixed data, and observed
how well we succeeded. In the first set of the experiments, we used 100
particles and take $\sigma_{P}^{2}=(0.01P(0))^{2}$ as in [10]. In this case,
the (assumed) variance of the system is much smaller than the (assumed)
variance of the observations; the particle paths are bunched close together,
and the results from our filter and from SIR are quite close, see Figure 1,
where we plotted the $P$ component of the reconstructed solution as well as
the corresponding data.
In the second set of the experiments, we still used 100 particle but assumed
$\sigma_{p}^{2}=(P(0))^{2}$. The variance of the system is now comparable to
the variance of the observation. For SIR, after resampling, the number of the
distinct particles is smaller than in the first case, as a result of the loss
of diversity after resampling when the weights are very different from each
other, see Table 1, where we exhibit the average number of distinct particles
left after each resample; there is a resample after each step. Remember that
there is some loss of diversity in resampling even if all the weights are
equal. With 100 particles, the filtered results with SIR are still comparable
to those with our filter. See Figure 2.
In the third set of the experiments, we used only 10 particles and kept
$\sigma_{p}^{2}=(P(0))^{2}$. As one could have foreseen, our filter does
better than SIR, see Figure 3. One should remember however that we are working
with a low dimensional problem where the differences between filters are not
expected to be very significant; the cost if 100 particles is not prohibitive.
Table 1: The number of distinct particles after resampling with different system variances and different numbers of particles $\sigma_{p}$ | # particle | average # particles left after resampling
---|---|---
| | SIR | Our filter
$0.01P(0)$ | 100 | 61 | 61
$P(0)$ | 100 | 19 | 63
$P(0)$ | 10 | 2.2 | 6.3
Figure 1: Results with $\sigma_{P}^{2}=(0.01P(0))^{2}$ and 100 particles
Figure 2: Results with $\sigma_{P}^{2}=P(0)^{2}$ and 100 particles Figure 3:
Results with $\sigma_{P}^{2}=P(0)^{2}$ and 10 particles
## 6 Example 2
We consider next a simple high dimensional example, used in [14] to show how
particle filters fail when the number of dimensions is large. We assume that
each component of $X^{n}$ is an independent Gaussian with zero mean and unit
variance. This is equivalent to taking $\delta=1$, $F(X^{n},\delta)=0$,
$G(X^{n},t^{n})=I$ in equation (3), and eliminating the $X^{n}$ term. We have
$X^{n}=V^{n}.$
Each component of $X^{n}$ is observed individually, so that
$b^{n}=X^{n}+W^{n}.$
We implement our filter with these particular choices. At the $j$-th
iteration, $H_{j}=I$ in equation (6) and $z_{j}=b^{n+1}$ in equation (7).
Therefore, we have $\Sigma_{j}^{-1}=2I$, $\bar{m}_{j}=b^{n+1}/2$, and
$\Phi_{j}=(b^{n+1})^{*}b^{n+1}/4$, in equation (8). The iterations converge in
one step and all the particles have the same weights.
However, with SIR the weights are uneven. We ran the SIR filter 1000 times,
with a 1000 particles each time; in each run we normalized the weights so that
add up to one, and we recorded the maximum weight. In Figures 4 we display a
histogram of these recorded maximum weights. As one can observe, when the
number of dimensions is large, most of time, a single particle in each run
hogs all the probability, and this version of SIR fails.
Figure 4: Histogram of the SIR normalized maximum particle weights with 1000
runs for $100$ dimensions
## 7 Conclusions
We have presented a general form of the iteration and interpolation process
used in our new implicit nonlinear particle filter. The goal is to aim
particle paths sharply so that fewer are needed. We conjecture that there is
no general way to reduce the variability of the weights in particle sampling
further than we have. We also presented additional simple examples that
illustrate the potential of this new sampling. These examples are simple in
that one is low-dimensional, while the second is linear so that other
effective ways of sampling it do exist. High-dimensional nonlinear problems
where our filter may be indispensable will be presented elsewhere, in the
context of specific applications.
## 8 Acknowledgments
We would like to thank Prof. J. Goodman, who urged us to write a more general
version of our previous work and suggested some notations and nomenclature,
Prof. R. Miller, who suggested that we try Dowd’s model plankton problem as a
first step toward an ambitious joint effort and helped us set it up, and Prof.
M. Dowd, who kindly made the data available. This work was supported in part
by the Director, Office of Science, Computational and Technology Research,
U.S. Department of Energy under Contract No. DE-AC02-05CH11231, and by the
National Science Foundation under grant DMS-0705910.
## References
* [1] M. Arulampalam, S. Maskell, N. Gordon, and T. Clapp. A tutorial on particle filters for online nonlinear/nongaussian Bayesia tracking. IEEE Trans. Sig. Proc., 50:174–188, 2002.
* [2] P. Bickel, B. Li, and T. Bengtsson. Sharp failure rates for the bootstrap particle filter in high dimensions. IMS Collections: Pushing the Limits of Contemporary Statistics: Contributions in Honor of Jayanta K. Ghosh, 3:318–329, 2008.
* [3] S. Bozic. Digital and Kalman Filtering. Butterworth-Heinemann, Oxford, 1994.
* [4] A. J. Chorin. Monte Carlo without chains. Comm. Appl. Math. Comp. Sc., 3:77–93, 2008.
* [5] A.J. Chorin and P. Krause. Dimensional reduction for a Bayesian filter. Proc. Nat. Acad. Sci. USA, 101:15013–15017, 2004.
* [6] A.J. Chorin and X. Tu. Implicit sampling for particle filters. Proc. Nat. Acad. Sc. USA, 2009. to appear.
* [7] A. Doucet, N. de Freitas, and N. Gordon. Sequential Monte Carlo Methods in Practice. Springer, New York, 2001.
* [8] A. Doucet, S. Godsill, and C. Andrieu. On sequential Monte Carlo sampling methods for Bayesian filtering. Stat. Comp., 10:197–208, 2000.
* [9] A. Doucet and A. Johansen. Particle filtering and smoothing: Fifteen years later. Handbook of Nonlinear Filtering (eds. D. Crisan et B. Rozovsky), to appear.
* [10] M. Dowd. A sequential Monte Carlo approach for marine ecological prediction. Environmetrics, 17:435–455, 2006.
* [11] W. Gilks and C. Berzuini. Following a moving target -Monte Carlo inference for dynamic Bayesian models. J. Roy. Statist. Soc. B, 63:127–146, 2001.
* [12] J. Liu and C. Sabatti. Generalized Gibbs sampler and multigrid Monte Carlo for Bayesian computation. Biometrika, 87:353–369, 2000.
* [13] S. Maceachern, M. Clyde, and J. Liu. Sequential importance sampling for nonparametric Bayes models: the next generation. Can. J. Stat., 27:251–267, 1999.
* [14] C. Snyder, T. Bengtsson, P. Bickel, and J. Anderson. Obstacles to high-dimensional particle filtering. Mon. Wea. Rev., 136:4629–4640, 2008.
* [15] J. Weare. Efficient Monte Carlo sampling by parallel marginalization. Proc. Nat. Acad. Sc. USA, 104:12657–12662, 2007.
|
arxiv-papers
| 2009-10-16T22:18:16 |
2024-09-04T02:49:05.887469
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Alexandre J. Chorin and Xuemin Tu",
"submitter": "Xuemin Tu",
"url": "https://arxiv.org/abs/0910.3241"
}
|
0910.3280
|
# Slow dynamics of phospholipid monolayers
at the air/water interface
Siyoung Q. Choi and Todd M. Squires
Department of Chemical Engineering ,
University of California, Santa Barbara, CA 93106, USA
###### Abstract
Phospholipid monolayers at the air-water interface serve as model systems for
various biological interfaces, e.g. lung surfactant layers and outer leaflets
of cell membranes. Although the dynamical (viscoelastic) properties of these
interfaces may play a key role in stability, dynamics and function, the
relatively weak rheological properties of most such monolayers have rendered
their study difficult or impossible. A novel technique to measure the
dynamical properties of fluid-fluid interfaces have developed accordingly. We
microfabricate micron-scale ferromagnetic disks, place them on fluid-fluid
interfaces, and use external electromagnets to exert torques upon them. By
measuring the rotation that results from a known external torque, we compute
the rotational drag, from which we deduce the rheological properties of the
interface. Notably, our apparatus enable direct interfacial visualization
while the probes are torqued.
In this fluid dynamics video, we directly visualize
dipalmitoylphosphatidylcholine(DPPC) monolayers at the air-water interface
while shearing. At about 9 mN/m, DPPC exhibits a liquid condensed(LC) phase
where liquid crystalline domains are compressed each other, and separated by
grain boundaries. Under weak oscillatory torque, the grain boundaries slip
past each other while larger shear strain forms a yield surface by deforming
and fracturing the domains. Shear banding, which is a clear evidence of yield
stress, is visualized during steady rotation. Remarkably slow relaxation time
was also found due to slow unwinding of the stretched domains.
Two videos are (low quality, high quaility)
|
arxiv-papers
| 2009-10-17T06:55:10 |
2024-09-04T02:49:05.894550
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Siyoung Q. Choi, Todd M. Squires",
"submitter": "Siyoung Choi Q.",
"url": "https://arxiv.org/abs/0910.3280"
}
|
0910.3545
|
# Random walks on networks: cumulative distribution of cover time
Nikola Zlatanov Macedonian Academy for Sciences and Arts, Skopje, Macedonia
Ljupco Kocarev Macedonian Academy for Sciences and Arts, Skopje, Macedonia
Institute for Nonlinear Science, University of California, San Diego
9500 Gilman Drive, La Jolla, CA 92093-0402
###### Abstract
We derive an exact closed-form analytical expression for the distribution of
the cover time for a random walk over an arbitrary graph. In special case, we
derive simplified, exact expressions for the distributions of cover time for a
complete graph, a cycle graph, and a path graph. An accurate approximation for
the cover time distribution, with computational complexity of $O(2n)$, is also
presented. The approximation is numerically tested only for graphs with $n\leq
1000$ nodes.
††preprint: APS/123-QED
## I Introduction
The random walk is a fundamental dynamic process which can be used to model
random processes inherent to many important applications, such as transport in
disordered media dis , neuron firing dynamics neuron , spreading of diseases
spred or transport and search processes search-1 ; search-2 ; search-3 ;
search-4 ; search-5 .
In this paper, we investigate random walks on graphs laslo and derive exact
expressions for the cumulative distribution functions for three quantities of
a random walk that play the most important role in the theory of random walks:
(1) hitting time $h_{ij}$ (or first-passage time), which is the number of
steps before node $j$ is visited starting from node $i$; (2) commute time
$\kappa_{ij}=h_{ij}+h_{ji}$; and cover time, that is the number of steps to
reach every node.
Average hitting time, average commute time, and average cover time have been
recently studied in several papers. In noh the authors investigate random
walks on complex networks and derive an exact expression for the mean first-
passage time between two nodes. For each node the random walk centrality is
introduced, which determines the relative speed by which a node can receive
and spread information over the network in a random process. Using both
numerical simulations and scaling arguments, the behavior of a random walker
on a one-dimensional small-world network is studied in almas . The average
number of distinct sites visited by the random walker, the mean-square
displacement of the walker, and the distribution of first-return times obey a
characteristic scaling form. The expected time for a random walk to traverse
between two arbitrary sites of the Erdos-Renyi random graph is studied in sood
. The properties of random walks on complex trees are studied in pastor . Both
the vertex discovery rate and the mean topological displacement from the
origin present a considerable slowing down in the tree case. Moreover, the
mean first passage time displays a logarithmic degree dependence, in contrast
to the inverse degree shape exhibited in looped networks pastor . The random
walk on networks has also much relevance to algorithmic applications. The
expected time taken to visit every vertex of connected graphs has recently
been extensively studied. In a series of papers, Cooper and Frieze have
studied the average cover time of various models of a random graph, see for
example cover .
This is an outline of the paper. In section II we derive closed formulas of
the cumulative distribution function for hitting time, commute time, and cover
time; we also present a simple example of a graph with four nodes, and derive
closed formulas of the cumulative distribution function for cover time of
complete graphs, cycle and path graphs. An approximation of the cumulative
distribution function for cover time is proposed in the section III; we also
present some numerical results of the cumulative distribution function for
cover time of different graphs IV. We finish the paper with conclusions.
## II Exact Random Walk Distributions for hitting time, commute time, and
cover time
Let $G=(V,E)$ be a connected graph with $n$ nodes and $m$ edges. Consider a
random walk on $G$: we start at a node $v_{0}$; if at the $t$-th step we are
at a node $v_{t}$, we move to neighbor of $v_{t}$ with probability
$1/d(v_{t})$, if an edge exists between node $v_{t}$ and it’s neighbor, where
$d(v_{t})$ is the degree of the node $v_{t}$. Clearly, the sequence of random
nodes $(v_{t}:t=0,1,\ldots)$ is a Markov chain. We denote by
$M=(m_{ij})_{i,j\in V}$ the matrix of transition probabilities of this Markov
chain:
$m_{ij}=\left\\{\begin{array}[]{cccc}1/d(i),&\mbox{if }ij\in E\\\
0,&\mbox{otherwise,}\end{array}\right.$ (1)
where $d(i)$ is the degree of the node $i$. Recall that the probability
$m^{t}_{ij}$ of the event that starting at $i$, the random walk will be at
node $j$ after $t$ steps, is an entry of the matrix $M^{t}$. It is well known
that $m_{ij}^{t}\rightarrow d(j)/2m$ as $t\rightarrow\infty$.
We now introduce three quantities of a random walk that play the most
important role in the theory of random walks: (1) hitting time $h_{ij}$ is the
number of steps before node $j$ is first visited starting from node $i$; (2)
commute time $\kappa_{ij}=h_{ij}+h_{ji}$ is the number of steps in a random
walk starting at $i$ before node $j$ is visited and then node $i$ is reached
again; and (3) cover time is the number of steps to reach every node.
### II.1 Hitting time
We first calculate the probability mass function for the hitting time. To
calculate the hitting time from $i$ to $j$, we replace the node $j$ with an
absorbing node. Let $D_{j}$ be a matrix such that $d_{ik}=m_{ik}$ for all
$k\neq j$, and $d_{ij}=0$ for all $i\neq j$ and $d_{jj}=1$. This means that
the matrix $D_{j}$ is obtained from $M$ by replacing the original row $j$ with
the basis row-vector $e_{j}$ for which the $j$-th element is 1 and all other
elements are 0. Let $d_{ij}^{t}$ be the $ij$ entry of the matrix $D_{j}^{t}$,
denoting the probability that starting from $i$ the walker is in the node $j$
by time $t$. Since $j$ is an absorbing state, $d_{ij}^{t}$ is the probability
of reaching $j$, originating from $i$, in not more then $t$ steps , i.e.
$d_{ij}^{t}$ is the cumulative distribution function (CDF) of hitting time.
Note that the $j-$th column of the matrix $D_{j}^{t}$ approaches the all $1$
vector, as $t\rightarrow\infty$. The probability mass function of the hitting
time $h_{ij}$ to reach $j$ starting from $i$ is, therefore, given by
$p_{h_{ij}}(t)=d_{ij}^{t}-d_{ij}^{t-1},\quad t\geq 1$
Let $E_{x_{j}}^{t}$ be the event of reaching the node $x_{j}$ starting from
the node $i\neq x_{j}$ by time $t$. Consider a sequence of events
$\\{E_{x_{1}}^{t},E_{x_{2}}^{t},\ldots,E_{x_{k}}^{t}\\}$. What is the
probability of the event: starting from node $i$, the walker visits one of the
nodes $\\{x_{1},x_{2},\ldots,x_{k}\\}$ by time $t$? Obviously, it is the
probability of the union $\cup_{j=1}^{k}E_{x_{j}}^{t}$. To calculate this
probability, we replace the nodes $\\{x_{1},x_{2},\ldots,x_{k}\\}$ with
absorbing nodes. Let $D_{\mathbf{x}}$ be a matrix obtained from $M$ by
replacing the rows $\\{x_{1},x_{2},\ldots,x_{k}\\}$ with the basis row-vectors
$e_{x_{1}},e_{x_{2}},\ldots,e_{x_{k}}$, respectively. Let $d_{ix_{j}}^{t}$ be
the $ix_{j}$ entry of the matrix $D_{\mathbf{x}}^{t}$.
$\sum_{j=1}^{k}d_{ix_{j}}^{t}$ is the probability that starting from $i$ we
reach for the first time one of the $\\{x_{1},x_{2},\ldots,x_{k}\\}$ nodes in
$\leq t$ steps. Therefore,
$F_{x_{1},\ldots,x_{k}}(t)=\sum_{j=1}^{k}d_{ix_{j}}^{t}$ (2)
is the cumulative distribution function (CDF) of the hitting time $h_{ix}=t$
of the union of events. The probability of reaching one of the nodes
$\\{x_{1},x_{2},\ldots,x_{n}\\}$, starting from $i$, in the $t$-th step is
given by
$\displaystyle
p_{h_{ix}}(t)=F_{x_{1},\ldots,x_{k}}(t)-F_{x_{1},\ldots,x_{k}}(t-1),\quad
t\geq 1$
which actually gives the probability mass function (PMF) of hitting time
$h_{ix}$ of the union $\cup_{j=1}^{k}E_{x_{j}}^{t}$.
### II.2 Commute time
Probability mass function of the commute time $\kappa_{ij}=h_{ij}+h_{ji}$ is
obtained as the convolution of PMFs of the two random variables $h_{ij}$ and
$h_{ji}$:
$p_{\kappa_{ij}}(t)=p_{h_{ij}}(t)\star
p_{h_{ji}}(t)=\sum_{\tau=1}^{t}p_{h_{ij}}(\tau)p_{h_{ji}}(t-\tau).$
The cumulative distribution function of the commute time can also be derived
as follows: we copy our Markov chain and we modify the original Markov chain
by deleting all outgoing edges of the node $j$, we modify the original Markov
chain by deleting all outgoing edges of the node $j$ and we modify the copied
Markov chain by replacing all outgoing edges of the node $i^{\prime}$ (which
is a copy of the node $i$ of the original Markov chain) with a self-loop. We
then connect the two chains by adding one directed edge from node $j$ to its
copy $j^{\prime}$ of the copied chain. Let $O$ be $n\times n$ matrix of all
0s, $O_{j}=(o_{kl})$ be the $n\times n$ matrix for which all elements are 0
except $o_{jj}=1$, and $D^{*}_{j}$ be the matrix obtained from $M$ by
replacing the $j-$th row with all 0. Define the $2n\times 2n$ matrix $C$ as
$C=\left[\begin{array}[]{cc}D^{*}_{j}&O_{j}\\\ O&D_{i}\end{array}\right].$
The matrix $C$ is a transition matrix of the modified Markov chain with $2n$
elements (original Markov chain and its copy). Let $c_{i,i+n}^{t}$ be the
$(i,i+n)$ element of the matrix $C^{t}$. This element is the cumulative
distribution function for the commute time $\kappa_{ij}-1$.
### II.3 Cover time
Cover time is defined as the number of steps to reach all nodes in the graph.
In order to determine the CDF of the cover time, we consider the event
$\cap_{j=1,j\neq z}^{n}E_{x_{j}}^{t}$, and use the well known equation for the
inclusion-exclusion of multiple events
$\displaystyle P\bigg{(}\bigcap_{k=1,k\neq
z}^{n}E_{x_{k}}^{t}\bigg{)}=\sum_{i=1,i\neq z}^{n}P(E_{x_{i}}^{t})-$
$\displaystyle-\sum_{i=1,i\neq z}^{n}\sum_{j=i+1,j\neq
z}^{n}P(E_{x_{i}}^{t}\cup E_{x_{j}}^{t})+\ldots+$
$\displaystyle+(-1)^{n-1}P(E_{x_{1}}^{t}\cup E_{x_{2}}^{t}\cup\ldots\cup
E_{x_{n}}^{t}),$
From the last equation and equation (2), we determine the cumulative
distribution function of the cover time as
$\displaystyle F_{\mbox{cover}}(t)$ $\displaystyle=$
$\displaystyle\sum_{i=1,i\neq z}^{n}F_{x_{i}}(t)-\sum_{i=1,i\neq
z}^{n}\sum_{j=i+1,j\neq z}^{n}F_{x_{i},x_{j}}(t)+$ (3)
$\displaystyle\ldots+(-1)^{n-1}F_{x_{1},x_{2},\ldots,x_{n}}(t).$
where $z$ is the starting node of the walk.
Equation (3) is the main result of this paper. The probability mass function
of the cover time can be easily computed from the Eq. (3). We note that Eq.
(3) is practically applicable only for small values of $n$; in fact the
computational complexity of Eq. (3) at a single time step is:
$\sum_{j=1}^{n}\frac{n!}{j!(n-j)!}=2^{n-1}-1.$
### II.4 An Example
Figure 1: Random walk on a network with four nodes
We now present a simple example to illustrate our results. Consider a random
walk on a network with four nodes, see Fig. 1, such that the matrix of
transition probabilities of the corresponding Markov chain is given by
$M=\left[\begin{array}[]{cccc}0&1/3&1/3&1/3\\\ 1/2&0&1/2&0\\\ 1/3&1/3&0&1/3\\\
1/2&0&1/2&0\end{array}\right].$
Let $m_{ij}^{t}$ be the $(i,j)$-th element of the matrix $M^{t}$. Since, in
this example, $M$ is a $4\times 4$ matrix, one can compute analytically, using
for example the software package Mathematica, the elements of the matrix
$M^{t}$. Thus, it can be found, for example,
$m_{14}^{t}=\frac{1}{5}\left(1-(-1)^{t}\left(\frac{2}{3}\right)^{t}\right),$
which is the probability that the walker starting from $i=1$ at the time $t$
is in $j=4$. Note that $\lim_{t\to\infty}m_{14}^{t}=1/5$.
Figure 2: Modified random walk for computing the hitting time to reach the
node 4 starting from an arbitrary node
To compute the hitting time to reach the node 4 starting from an arbitrary
node, we modify the existing random walk to the random walk shown on Fig. 2.
The transition matrix of the modified walk is:
$D_{4}=\left[\begin{array}[]{cccc}0&1/3&1/3&1/3\\\ 1/2&0&1/2&0\\\
1/3&1/3&0&1/3\\\ 0&0&0&1\end{array}\right].$
Let $d_{ij}^{t}$ be the elements of the matrix $D^{t}$. Again the elements of
the matrix $D^{t}$ can be computed analytically. For example, the probability
of reaching the node 4 starting form 1 in time steps $\leq t$ is equal to
$\displaystyle d_{14}^{t}=1$ $\displaystyle-$
$\displaystyle\frac{6^{-t}}{2\sqrt{13}}\bigg{[}\left(1-\sqrt{13}\right)^{t}\left(\sqrt{13}-3\right)$
$\displaystyle+$
$\displaystyle\left(1+\sqrt{13}\right)^{t}\left(\sqrt{13}+3\right)\bigg{]}.$
Clearly, as for any cumulative distribution, $\lim_{t\to\infty}d_{14}^{t}=1$.
Let us now compute the probability starting from node 1 to reach for the first
time the node 4 and then to reach 1 for the first time starting from 4 in time
$t$. For this, we consider the modified random walk shown on Fig. 3, with the
transition matrix given by:
$C=\left[\begin{array}[]{cccccccc}0&1/3&1/3&1/3&0&0&0&0\\\
1/2&0&1/2&0&0&0&0&0\\\ 1/3&1/3&0&1/3&0&0&0&0\\\ 0&0&0&0&0&0&0&1\\\
0&0&0&0&1&0&0&0\\\ 0&0&0&0&1/2&0&1/2&0\\\ 0&0&0&0&1/3&1/3&0&1/3\\\
0&0&0&0&1/2&0&1/2&0\end{array}\right].$
The element $c_{15}^{t}$ of the matrix $C^{t}$ is the cumulative distribution
function of the commute time $\kappa_{14}-1$ and it is given by:
$\displaystyle c_{15}^{t}=\frac{2^{-t-2}3^{-t}}{13}$ $\displaystyle\times$
$\displaystyle\bigg{[}13\times 2^{t}3^{t/2}\left(3+2\sqrt{3}+4\times
3^{t/2}\right)$ (4) $\displaystyle-$
$\displaystyle\left(1+\sqrt{13}\right)^{t}\left(65+19\sqrt{13}\right)$
$\displaystyle+$
$\displaystyle\left(1-\sqrt{13}\right)^{t}\left(-65+19\sqrt{13}\right)$
$\displaystyle+$
$\displaystyle(-2)^{t}3^{t/2}\left(39-26\sqrt{3}\right)\bigg{]}.$
Notice that again $\lim_{t\to\infty}c_{15}^{t}=1$.
As the last example, we consider the probability of reaching the node 4 or the
node 2 from the node 1 for the first time in time steps $t$. The modified
random walk is shown on Fig. 4 and the transition probability matrix of the
modified walk is given by the matrix $D_{4;2}$, which has the form:
$D_{4;2}=\left[\begin{array}[]{cccc}0&1/3&1/3&1/3\\\ 0&1&0&0\\\
1/3&1/3&0&1/3\\\ 0&0&0&1\end{array}\right].$
The elements $\tilde{d}_{1,2}^{t}$ and $\tilde{d}_{1,4}^{t}$ of the matrix
$D_{4;2}^{t}$ are
$\tilde{d}_{12}^{t}=\tilde{d}_{14}^{t}=\frac{1}{2}-\frac{3^{-t}}{2}$
The cumulative distribution function of the event: the node 2 or the node 4 is
reached from the node 1 for the first time by the $t$-th step, is given by
$\tilde{d}_{12}^{t}+\tilde{d}_{14}^{t}$.
Figure 3: Modified random walk for computing the commute time starting from
node 1 to reach for the first time the node 4 and then to reach 1 for the
first time starting from 4
Figure 4: Modified random walk for computing the probability of reaching the
node 4 or the node 2 from arbitrary node
### II.5 Cover time for complete, cycle and path graph
In this subsection we derive exact expressions for the CDF of cover time for
three particular graphs: complete, cycle, and path graph.
#### II.5.1 Complete graph
A complete graph is a simple graph in which every pair of distinct vertices is
connected by an edge. The complete graph on $n$ vertices has $n$ vertices and
$n(n-1)/2$ edges, and is denoted by $K_{n}$. We can now easily derive
analytical results for the PMF of a complete graph. It is easy to see that for
the complete graph we have
$\displaystyle P(E_{x_{i}}^{t})$ $\displaystyle=$ $\displaystyle
1-\frac{(n-2)^{t}}{(n-1)^{t}}$ $\displaystyle P(E_{x_{i}}^{t}\cup
E_{x_{j}}^{t})$ $\displaystyle=$ $\displaystyle 1-\frac{(n-3)^{t}}{(n-1)^{t}}$
$\displaystyle P(E_{x_{i}}^{t}\cup E_{x_{j}}^{t}\cup E_{x_{k}}^{t})$
$\displaystyle=$ $\displaystyle 1-\frac{(n-4)^{t}}{(n-1)^{t}}.$
Therefore,
$\displaystyle
P\bigg{(}\bigcup_{i=1}^{k}E_{x_{i}}^{t}\bigg{)}=1-\frac{(n-k-1)^{t}}{(n-1)^{t}}$
Thus, the cumulative distribution function of the cover time for complete
graph with $n$ nodes can be expressed as
$\displaystyle F_{\mbox{cover}}(t)$ $\displaystyle=$
$\displaystyle\sum_{\gamma=1}^{n-1}(-1)^{\gamma-1}\frac{(n-1)!}{\gamma!(n-\gamma-1)!}P\bigg{(}\bigcup_{i=1}^{\gamma}E_{x_{i}}^{t}\bigg{)}$
$\displaystyle=$
$\displaystyle\sum_{\gamma=1}^{n-1}(-1)^{\gamma-1}\frac{\Gamma(n)}{\gamma!\Gamma(n-\gamma)}\bigg{(}1-\frac{(n-\gamma-1)^{t}}{(n-1)^{t}}\bigg{)}$
Therefore, the probability mass function is:
$\displaystyle
f_{c}(t)=\sum_{\gamma=1}^{n-1}(-1)^{\gamma-1}\frac{\Gamma(n-1)}{\Gamma(\gamma)\Gamma(n-\gamma)}\bigg{(}1-\frac{\gamma}{n-1}\bigg{)}^{t}.$
#### II.5.2 Cycle graph
Figure 5: Cycle graph with 12 nodes
A cycle graph is a graph that consists of a single cycle, or in other words,
some number of vertices connected in a closed chain. Let us denote the cycle
graph with $n$ vertices as $C_{n}$. The number of vertices in a $C_{n}$ equals
the number of edges, and every vertex has degree 2; that is, every vertex has
exactly two edges incident with it. An example of a cycle graph with 12 nodes
is given in Fig. 5.
Let us assume that the first node of the cycle graph is the starting node of
the walk. We need to find the intersection of the events of reaching nodes
2,3, $\ldots$ to $n$. These events form a path. A path in a graph is a
sequence of vertices such that, from each of its vertices, there is an edge to
the next vertex in the sequence. A cycle is a path such that the start vertex
and end vertex are the same. Note that the choice of the start vertex in a
cycle is arbitrary. By exploiting the Remark of Corollary 3.1.16 given in
dohmen , and proved in naiman for events that form a path, we find that the
cumulative distribution function of the cover time for a cycle graph is:
$\displaystyle F_{\mbox{cover}}(t)$ $\displaystyle=$
$\displaystyle\sum_{i=2}^{n}P\left(E_{i}^{t}\right)-\sum_{i=2}^{n-1}P\left(E_{i}^{t}\cup
E_{i+1}^{t}\right).\quad$
#### II.5.3 Path graph
Figure 6: Path graph with 7 nodes
A path graph is a particularly simple example of a tree, namely one which is
not branched at all, that is, contains only nodes of degree two and one. In
particular, two of its vertices have degree 1 and all others (if any) have
degree 2\. An example of a path graph with 7 nodes is given in Fig. 6.
To find the cumulative distribution function of the cover time for a path
graph we note that all the nodes will be covered if the first and the last
nodes are reached by the random walker. Therefore, the cumulative distribution
function of the cover time for a path graph is
$\displaystyle F_{\mbox{cover}}(t)$ $\displaystyle=$ $\displaystyle
P\left(E_{1}^{t}\cap E_{n}^{t}\right)$ $\displaystyle=$ $\displaystyle
P\left(E_{1}^{t}\right)+P\left(E_{n}^{t}\right)-P\left(E_{1}^{t}\cup
E_{n}^{t}\right).$
We note that if the first node is the starting node then $P\left(E_{1}^{t}\cap
E_{n}^{t}\right)=P\left(E_{n}^{t}\right)$ and if the last node is the starting
node then $P\left(E_{1}^{t}\cap E_{n}^{t}\right)=P\left(E_{1}^{t}\right)$.
## III Approximation of the CDF of cover time
The cumulative distribution functions for hitting and commute time can be
computed for reasonable large graphs. The complexity of matrix multiplication,
if carried out naively, is $O(n^{3})$, but more efficient algorithms do exist,
which are computationally interesting for matrices with dimensions $n>100$
press .
The inclusion-exclusion formula (3) has little practical value in graphs with
large number of nodes since it then requires extensive computational times. In
the following, we present an accurate and useful approximation of (3) that can
be evaluated in a reasonable time. The first inequality for the inclusion-
exclusion was discovered by Ch. Jordan jordan and from then until now a lot
of work has been done in sharpening the bounds or the approximation. An
excellent survey of the various results for the inclusion-exclusion is given
in dohmen .
We propose the following approximation for inclusion-exclusion formula:
$\displaystyle
P\bigg{(}\bigcap_{k=1}^{n}E_{x_{k}}\bigg{)}\approx\prod_{i=1}^{n}P(E_{x_{i}})$
$\displaystyle\qquad\qquad\times\prod_{i=1}^{n-1}\frac{P(E_{x_{i}}\cap
E_{x_{i+1}})}{P(E_{x_{i}})P(E_{x_{i+1}})}$ (5)
where $P(E_{x_{i}}\cap
E_{x_{i+1}})=P(E_{x_{i}})+P(E_{x_{i+1}})-P(E_{x_{i}}\cup E_{x_{i+1}})$. The
node indexes must be arranged in such way that there exists an edge between
nodes $x_{i}$ and $x_{i+1}$. This condition is not strict, and there can exist
a small number of nodes that do not satisfy this condition. The Appendix
presents the heuristic derivation of (5) by using the method proposed in
kessler .
As can be seen, the single-step computational complexity of Eq. (5) is
$O(2n)$. The proposed approximation is very accurate for strongly connected
graphs like the complete graph and is less accurate for poorly connected
graphs like the path graph, as can be seen from the figures below. We note
that the error of the approximation for the path graph is the upper bound of
the error when the middle node is the starting node. This is due to the fact
that the proposed approximation equation reduces to the exact equation for
independent events, while diminishing as the events become more and more
dependent. When almost all the nodes are subset of the rest of the nodes then
the approximation formula is the least accurate. Thus the formula is the least
accurate when is applied to a path graph with $n$ nodes and the walk starts
from the middle, $n/2$-th node (assuming that $n$ is an even number). In this
case the event of reaching nodes 1 to $n/2-1$ is the event of reaching node
$1$ and the event of reaching nodes $n/2+1$ to $n$ is the event of reaching
node $n$.
Interestingly, if we start changing the starting node and evaluate the error
of the approximation for a path graph, then when the starting node approaches
the first or the $n$-th node the formula becomes more and more accurate and
when the starting node is the first or the last node, then the approximation
formula reduces to the exact formula. To prove this we let the first node be
the starting node, then the event of reaching node $i$ and node $j$, if $j>i$
is $P(E_{i}^{t}\cap E_{j}^{t})=P(E_{j}^{t})$. Thus the approximation formula
is given by
$\displaystyle
P\bigg{(}\bigcap_{k=1}^{n}E_{k}\bigg{)}=\prod_{i=1}^{n}P(E_{i})\prod_{i=1}^{n-1}\frac{1}{P(E_{i})}=P(E_{n})$
(6)
A similar proof is when the $n$-th node is the starting node. An example is
given in Fig.7 where the CDF of a path graph is given by the exact and the
approximate formula for two path graphs with 30 and 40 nodes when starting
node is the 4-th and the 20-th node, respectively. The second worst case error
is when the approximation formula is applied to a cycle graph and in this case
the error is independent of the starting node.
Figure 7: Exact and the approximate formula of the CDF for a path graph with
two different starting nodes
(a)
(b)
Figure 8: Analytical, approximated and simulated (a) cumulative distribution
function and (b) probability mass function of cover time for a random graph
with 20 nodes
(a)
(b)
Figure 9: Exact vs approximated CDF for (a) a path graph with 50 nodes and (b)
a path graph with 200 nodes
(a)
(b)
Figure 10: Exact vs approximated CDF (a) and PMF (b) for complete graph with
50 nodes
## IV Numerical Examples
In this section, several numerical examples are presented. First, we validate
the cover time formula and the approximation by Monte Carlo simulations, Fig.
8, for a Erdos-Renyi random graph with 20 nodes. Fig. 8a illustrates the CDF,
while Fig. 8b the PDF of cover time. We illustrate the accuracy of the
approximation for a path graph and a complete graph, Figures 9 and 10
respectively, where the starting node of the walk for the path graph is the
middle node for both Figures 9a and 9b. We have performed various numerical
simulations of the cumulative distribution functions using exact and
approximate expressions for complete, path and cycle graphs with up to 1000
nodes. For small $n$ ($n\leq 1000$) we found that increasing $n$ to up to
1000, the accuracy of the approximation is maintained. We believe that Eq. (5)
is a good approximation for cumulative distribution of cover time even for
larger graphs, but since at the moment we do not have estimates for accuracy
of our approximation, we leave this as a subject of our next research. More
detailed analysis on how the CDF of cover time depends on graph topology will
be discussed in a forthcoming paper.
## V Conclusions
In this paper we have derived the exact closed-form expressions for the PMF
and CDF of three random walk parameters that play pivotal role in the theory
of random walks: hitting time, commute time, and cover time. We also have
derived simpler closed formulas for the cumulative distribution function of
cover time for complete, cycle and path graphs. An approximation of the
cumulative distribution function for cover time is proposed, and several
numerical results for the CDF of cover time for different graphs are
presented.
## Appendix
If A is the union of the events $A_{1},A_{2},\ldots,A_{n}$ then, writing
$p_{i}$ for the probability of $A_{i}$, $p_{ij}$ for the probability of
$A_{i}\cap A_{j}$ , $p_{ijk}$ for the probability of $A_{i}\cap A_{j}\cap
A_{k}$ etc, the probability of $A$ is given by
$p(A)=\sum_{i}p_{i}-\sum_{i<j}p_{ij}+\sum_{i<j<k}p_{ijk}-\ldots+(-1)^{n-1}p_{12\ldots
n}$
The inclusion – exclusion principle tells us that if we know the
$p_{i},p_{ij},p_{ijk}\ldots$ then we can find $P(A)$. However, in practice we
are unlikely to have full information on the $p_{i},p_{ij},p_{ijk}\ldots$.
Therefore, we are faced with the task of approximating $P(A)$ taking into
account whatever partial information we are given. In certain cases where the
events $A_{i}$ are in some sense close to being independent, then there are a
number of known results approximating $P(A)$. In this paper we use the
following result [kessler , equation (9)]:
$P(A)\approx 1-\prod_{i=1}^{n}(q_{i})\prod_{i=1}^{n}\prod_{j=i+1}^{n}(q_{ij})$
(7)
where
$\displaystyle q_{i}$ $\displaystyle=$ $\displaystyle P(\bar{A}_{i})$ (8)
$\displaystyle q_{ij}$ $\displaystyle=$
$\displaystyle\frac{P(\bar{A}_{i}\cap\bar{A}_{j})}{P(\bar{A}_{i})P(\bar{A}_{j})}$
(9)
Let the event $B_{i}$ be defined as $B_{i}=\bar{A}_{i}$. Then
$B=\bigcup_{i=1}^{n}B_{i}=\bigcup_{i=1}^{n}\bar{A}_{i}$ and the probability of
this event is:
$P(B)=P\left(\bigcup_{i=1}^{n}\bar{A}_{i}\right)=1-P\left(\bigcap_{i=1}^{n}A_{i}\right)$
(10)
The approximated form (7) of the event $B$ is:
$P(B)\approx
1-\prod_{i=1}^{n}P(\bar{B}_{i})\prod_{i=1}^{n}\prod_{j=i+1}^{n}\frac{P(\bar{B}_{i}\cap\bar{B}_{j})}{P(\bar{B}_{i})P(\bar{B}_{j})}$
(11)
Replacing (10) and $\bar{B}_{i}=A_{i}$ into the (11), we get:
$\displaystyle
P\left(\bigcap_{i=1}^{n}A_{i}\right)\approx\prod_{i=1}^{n}P(A_{i})\prod_{i=1}^{n}\prod_{j=i+1}^{n}\frac{P(A_{i}\cap
A_{j})}{P(A_{i})P(A_{j})},$ (12)
where $P(A_{i}\cap A_{j})$ can be expressed as:
$P(A_{i}\cap A_{j})=P(A_{i})+P(A_{j})-P(A_{i}\cup A_{j})$
When the events $A_{i}$ and $A_{j}$ are not close to being independent but on
the contrary, one of the events is a subset of the other, as the case for the
events in the cover time formula, the approximation formula (12) is not
accurate.
The inaccuracy can be seen from the following example: Let the events $A_{j}$
for $j>i$ are all subsets of the event $A_{i}$. Then the probability of the
event $A_{i}\cap A_{j}$ is
$P(A_{i}\cap A_{j})=P(A_{j})$
if we now replace this expression in (12) we get
$P\left(\bigcap_{i=1}^{n}A_{i}\right)\approx\prod_{i=1}^{n}P(A_{i})\prod_{i=1}^{n}\left(\frac{1}{P(A_{i})}\right)^{n-i}$
Then if $n$ is large and the probabilities $P(A_{i})$ are a very small
numbers, this probability expression can be a number much bigger then one.
One way to solve the accuracy problem is not to take the second product over
all node pairs, but just over $n-1$ different neighboring pairs. We suggest
the following approximation for the cumulative distribution of cover time:
$\displaystyle
P\left(\bigcap_{i=1}^{n}A_{i}\right)\approx\prod_{i=1}^{n}P(A_{i})\prod_{i=1}^{n-1}\frac{P(A_{i}\cap
A_{i+1})}{P(A_{i})P(A_{i+1})}.$
We note that this approximate probability expression reduces to the exact
probability expression in the two limiting cases: first, when all events are
mutually independent, and second, when all events are subset of just one
event. The first claim can be proved just by noting that $P(A_{i}\cap
A_{i+1})=P(A_{i})P(A_{i+1})$ and the second claim was previously proved, see
equation (6) when the events $E_{i}$ for $i=1,\ldots,n-1$ are all subsets of
the event $E_{n}$.
## References
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* (11) E. Almaas, R. V. Kulkarni, and D. Stroud, “Scaling properties of random walks on small-world networks”, Phys. Rev. E 68, 056105 (2003)
* (12) V. Sood, S. Redner, and D. ben-Avraham, “First Passage Properties of the Erdos-Renyi Random Graph”, J. Phys. A 38, 109-123 (2005)
* (13) A. Baronchelli, M. Catanzaro, and R. Pastor-Satorras, “Random walks on complex trees”, Phys. Rev. E 78, 011114 (2008)
* (14) C. Cooper and A. Frieze, “The cover time of sparse random graphs”, Random Structures and Algorithms, Vol. 30, 1-16 (2007); C. Cooper and A. Frieze, “The cover time of random geometric graphs”, Proceedings of SODA, p. 48-57 (2009).
* (15) W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (3rd ed.), Cambridge University Press, (2007).
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* (17) K. Dohmen, “Improved Inclusion-Exclusion Identities and Bonferroni Inequalities with Applications to Reliability Analysis of Coherent Systems”, Habilitationsschrift, Math.-Nat. Fak. II, Humboldt-Universitat zu Berlin, (2001) (available online at http://www.htwm.de/mathe/neu/?q=dohmen-publikationen)
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|
arxiv-papers
| 2009-10-19T12:22:05 |
2024-09-04T02:49:05.904301
|
{
"license": "Public Domain",
"authors": "Nikola Zlatanov, Ljupco Kocarev",
"submitter": "Nikola Zlatanov",
"url": "https://arxiv.org/abs/0910.3545"
}
|
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